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 |
|---|---|---|---|---|
129,854 | <p>I have a four points in plane and need to test (based on point coordinates) whether they are able to form a convex quadrilateral:</p>
<p><img src="https://i.stack.imgur.com/qMiCH.png" alt="convex quadrilateral"></p>
<p>Of course, the test should avoid configurations like these:</p>
<p><img src="https://i.stack.im... | Eric Haengel | 1,089 | <p>Maybe a simpler way to solve this problem would be to write down the four vectors which describe the sides of the polygon, and calculate at each vertex the angle between these vectors. If this angle is ever obtuse, then you don't have a convex polygon. This procedure would only require going through 4 calculations.<... |
1,315,970 | <p>I am trying to prove the following using complex analysis:</p>
<p>$$\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}}{a^{2}+n^{2}}=\frac{\pi}{a\sinh(a\pi)}$$</p>
<p>I am told to use the following function:</p>
<p>$$f(z)=\frac{1}{(a^{2}+z^{2})\sin(\pi z)}$$</p>
<p>So we note that $f(z)$ has singularities at $z = \{\pm i a... | Mark Viola | 218,419 | <p>Change the contour to a square box $C_{R}$ of sides $2R$ and centered at the origin. The residues are from singularities at $z=\pm i$ and $z=n$ for all integers $|n|<R$ .</p>
<p>The residues at $z=\pm i$ are both equal to </p>
<p>$$\frac{1}{2ia\sin(\pi ia)}=-\frac{1}{2a \sinh(\pi a)}$$</p>
<p>The residues at ... |
4,425,648 | <p>Consider a discrete dynamical system <span class="math-container">$x_{k+1} = f(x_k)$</span>, where <span class="math-container">$f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$</span>, sufficiently smooth, and let <span class="math-container">$C \subseteq \mathbb{R}^2$</span> be an invariant, closed curve in the phase spac... | S.C. | 544,640 | <p>I will retain the notation found in the original question...starting from:</p>
<p><span class="math-container">$$\left|x+\frac{1}{2y} \right|=\sqrt{1+\frac{1}{4y^2}}$$</span></p>
<p>This expression means that for any <span class="math-container">$y \in \mathbb R \setminus \{0\}$</span>:</p>
<blockquote>
<p><span cla... |
4,646,558 | <p>I´m trying to find this using Lagrange Error bound.
<span class="math-container">$$|R|\leq \frac{f^{n+1}(t) (x-a)^{n+1}}{(n+1)!}$$</span>
Basically I thought of using the function <span class="math-container">$f(x) = \sqrt{x+1}$</span> centered on <span class="math-container">$a= 8$</span> (but when I was calculatin... | orangeskid | 168,051 | <p>The <a href="https://en.wikipedia.org/wiki/Taylor%27s_theorem" rel="nofollow noreferrer">Taylor formula</a> of order <span class="math-container">$n$</span> at point <span class="math-container">$a$</span>, with Lagrange remainder, for the function <span class="math-container">$f(x) = x^N$</span>, (<span class="mat... |
3,263,900 | <p>Request help in understanding the telescoping sum for the given series. <br>
For <span class="math-container">$n\geq1$</span>,
<span class="math-container">\begin{align}
(a-b)\sum_{i=0}^{n-1}a^ib^{n-1-i}&=\sum_{i=0}^{n-1}a^{i+1}b^{n-1-i}-\sum_{i=0}^{n-1}a^ib^{n-i}\\
&=\sum_{i=0}^{n-1}(a^{i+1}b^{n-(i+1)}-a^ib... | trancelocation | 467,003 | <p>If you write <span class="math-container">$c_i = a^ib^{n-i}$</span>, then your sum turns into
<span class="math-container">$$\sum_{i=0}^{n-1}(a^{i+1}b^{n-(i+1)}-a^ib^{n-i}) = \sum_{i=0}^{n-1}(c_{i+1}-c_i) = c_n - c_0 = a^n-b^n$$</span></p>
|
4,281,859 | <p><a href="https://math.stackexchange.com/a/4280920/26327">Proof by another user</a> of the formula for <span class="math-container">$k = 1$</span>, that is the total number of pairs <span class="math-container">$(x, x + 2k)$</span> of odd numbers in in interval <span class="math-container">$[a,b]$</span> such that <s... | D Left Adjoint to U | 26,327 | <p>This is an extension of <strong>mathlove</strong>'s answer. The purpose is to verify & correct their formula if need be.</p>
<pre><code>from math import floor, sqrt, ceil
def hole_actual_count(a, b, n, k):
count = 0
for x in range(a, b-2*k + 1):
if x % 2 == 1:
if x % n == 0 or ... |
907,620 | <p>We will call a matrix <em>positive matrix</em> if all elements in the matrix are positive, and we will denote the <em>largest eigenvalue</em> with <span class="math-container">$\lambda_{\max}$</span>, what is exist because of the <a href="http://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem" rel="nofollow n... | user2097 | 170,091 | <p>For (almost) all positive vectors $x$, the <a href="http://en.wikipedia.org/wiki/Power_method">Power iterations</a> do work, and we have $$\lambda_\max=\lim_{k\rightarrow\infty}\left(\|A^kx\|/\|x\|\right)^{1/k}$$ which shows that $\lambda_\max$ can not decrease as entries of $A$ increase. If $A'$ is obtained from $A... |
907,620 | <p>We will call a matrix <em>positive matrix</em> if all elements in the matrix are positive, and we will denote the <em>largest eigenvalue</em> with <span class="math-container">$\lambda_{\max}$</span>, what is exist because of the <a href="http://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem" rel="nofollow n... | user1551 | 1,551 | <p>This is just a simple consequence of Perron-Frobenius theorem. We can prove something slightly more general:</p>
<blockquote>
<p>Let $A,B$ be two nonnegative square matrices. If one of the following cases holds, then $\rho(A)<\rho(A+B)$:</p>
<ul>
<li>$A$ is irreducible (in particular, when $A>0$ entr... |
1,965,189 | <blockquote>
<p>If $\lim\sup$ of $\{s_n\}$ equals $M$, prove that the $\lim\sup$ of any subsequence $\{s_{n_k}\}$ is $\le M$</p>
</blockquote>
<p><strong>PROOF</strong>: Given that $\{s_n\}$ is a sequence of real numbers such that $\displaystyle \limsup_{n\to\infty}s_n = M$, then by definition we know that $\{s_n\}$... | H. H. Rugh | 355,946 | <p>Your $M_{n_k}$ is not correctly defined (it shouldn't be a subsequence of $M_n$ if the limit should be $\limsup_k s_{n_k}$). A possible path: For $j\geq k\geq 1$
$$ s_{n_j} \leq M_{n_k} $$
so for every $k$
$$ s^*=\limsup_j s_{n_j} \leq M_{n_k}$$
and then
$$ s^* \leq \lim_k M_{n_k} =\lim_n M_n=s$$</p>
|
3,097,542 | <p>So far I have showed that <span class="math-container">$K$</span> is a normal subgroup and that the Operation defined on <span class="math-container">$G/K$</span> is abelian. </p>
<blockquote>
<p>Now I have to show that if <span class="math-container">$\phi:G\rightarrow A$</span> is a group
homomorphism and <sp... | Elias Costa | 19,266 | <p>We can also use the <a href="https://en.wikipedia.org/wiki/Cauchy%27s_convergence_test" rel="nofollow noreferrer">Cauchy's convergence test</a>. This test tells us that a serie <span class="math-container">$\sum_{k=1}^\infty a_k$</span> is convergent if, and only if, for all <span class="math-container">$\epsilon>... |
281,870 | <p>can you help me with this question
find $\sin$ Fourier transform of $\frac{\sin{x}}{x}$ </p>
| bs_math | 801,062 | <p>Here is another perspective on the problem: Using <span class="math-container">$\sin(x) = \frac{1}{2i}(e^{ix}-e^{-ix})$</span> one easily calculates
<span class="math-container">$$ \mathcal{F} \sin = \frac{1}{2} i \left( \delta_{-\frac{1}{2\pi}} - \delta_{\frac{1}{2\pi}} \right), \tag{1} $$</span>
where <span class=... |
936,213 | <p>This is a question from my previous year analysis exam .The question says :</p>
<blockquote>
<p>Can we construct a sequence which converges but never attains its limits,such that its terms consist of a finite number of distinct values ?</p>
<p>If yes,give an example.If not explain ..</p>
</blockquote>
<p>I cannot fr... | drhab | 75,923 | <p>Limits are normally practicized in topologies where the singletons, hence the finite sets are closed. This to ensure their uniqueness. </p>
<p>If $S$ is the set of the distinct values of a convergent sequence $(a_n)_{n\in\mathbb N}$ - i.e. $S=\{a_n\mid n\in\mathbb N\}$ - then the limit must belong to the closure of... |
936,213 | <p>This is a question from my previous year analysis exam .The question says :</p>
<blockquote>
<p>Can we construct a sequence which converges but never attains its limits,such that its terms consist of a finite number of distinct values ?</p>
<p>If yes,give an example.If not explain ..</p>
</blockquote>
<p>I cannot fr... | Karl | 175,709 | <p>Ok, take a sequence $a_n$ whose terms have either the value $b$ or $c$.
Neither of both can be the limit, so the limit must be some number $d$ with
$d \neq b,c$. Now think about the distance of an arbitrary term in the sequence $a_n$ to its supposed limit $d$.</p>
<p>You should get it from here ;-).</p>
|
936,213 | <p>This is a question from my previous year analysis exam .The question says :</p>
<blockquote>
<p>Can we construct a sequence which converges but never attains its limits,such that its terms consist of a finite number of distinct values ?</p>
<p>If yes,give an example.If not explain ..</p>
</blockquote>
<p>I cannot fr... | Lehs | 171,248 | <p>No, there is no such sequence of real numbers.</p>
<p>Suppose $f:\mathbb N\rightarrow \{a_1,\dots,a_m\}\subset \mathbb R$ and that $x_n=f(n)$. If $x_n$ converges to $x$ then given $\epsilon>0$ there is a $N\in\mathbb N$ such that
$|x-x_n|<\epsilon$ for all $n>N$. Select $\epsilon<\frac{1}{2}\min\{|a_i-... |
4,388,965 | <p>Let <span class="math-container">$h(X) = X/2$</span>, and <span class="math-container">$f(X) = (3X + 1)/2$</span>. Then clearly every iteration <span class="math-container">$g^i(X), X \in \Bbb{Z}$</span> the Collatz mapping <span class="math-container">$$g(X) = \begin{cases} X/2, \ X=0\pmod 2\\ \dfrac{3X + 1}{2}, \... | D Left Adjoint to U | 26,327 | <p>It's true by the above base cases for all <span class="math-container">$s = f h^n, n \geq 0$</span>. Now suppose that <span class="math-container">$s$</span> has a count of <span class="math-container">$k$</span> total <span class="math-container">$f$</span>'s in it, and that it begins with an <span class="math-con... |
2,136,757 | <p>Let $H=\{ x \in \Bbb R : x+\frac3x-1 \le 0\}$. Find $\sup(H)$, $\max(H)$, $\inf(H)$, and $\min(H)$. I understand the terminology but I am new to this please just offer any first steps or any help on how to get started.</p>
| Ross Millikan | 1,827 | <p>As the expression on the left is continuous, the values of $x$ for which it is true will form one or more intervals. The transitions will either be where $x+\frac 3x-1=0$ or where it is undefined, so solve the equation, add $0$ to the set of roots because the expression is undefined there, and assess the truth of t... |
114,715 | <p>Is a domain $D$, all of whose localizations $D_P$ for $P \in Spec(D)$ are noetherian, itself noetherian ? </p>
<p>The question is motivated by proposition 11.5 of Neukirch's Algebraic Number Theory: </p>
<blockquote>
<p>Let $\mathfrak{o}$ be a noetherian integral domain. $\mathfrak{o}$ is a Dedekind domain if an... | Ralph | 10,194 | <p>No, this isn't true. In the paper <a href="http://www.ams.org/journals/tran/1971-158-02/S0002-9947-1971-0280472-2/S0002-9947-1971-0280472-2.pdf" rel="noreferrer">[Heinzer, Ohm: Locally Noetherian Commutative Rings]</a> the authors (who also mention Nakano's example from Faisal's answer which they call "quite involve... |
227,794 | <p>I'm a really confused about fields.</p>
<p>I know that it means $x$ is the reciprocal element of itself, and I can easily show that $1^2=1$ (not as trivial for $(-1)^2$ though), but I'm not sure how it helps me.</p>
<p>edit: oh... I can only approve one answer. Well Rankeya was first (by a very short time) so I gu... | Brian M. Scott | 12,042 | <p>HINT: $x^2=1$ if and only if $x^2-1=0$. In any field $x^2-1=(x-1)(x+1)$, so $x^2=1$ if and only if $(x-1)(x+1)=0$. Now prove that for any $a,b\in F$, $ab=0$ if and only if at least one of $a$ and $b$ is $0$.</p>
|
27,439 | <p>A car has to pick up each person and take them to their destination. I am trying to find the shortest tour that will do this. How can I find the shortest path between any s and t.... visiting all vertices, and without coming back to the start?</p>
<p><img src="https://i.stack.imgur.com/LEJqM.png" alt="competition">... | Stefan | 2,448 | <p><strong>EDIT 1</strong></p>
<p>Now that the question is getting more specific, here another approach:</p>
<p>I'd like to say that this is not a solution, but it shows you how to set-up your graph accordingly. If you combine this with the comment of @Rahul Narain you should get your result easily.</p>
<p>1) Create... |
951,664 | <p>How would I plot $r^2 = 36\cos(2\phi)$ in $x$ and $y$ coordinate system?</p>
<p>I know that $r^2 = x^2 + y^2$, and I know that $x = r\cos(\phi)$ and $y = r\sin(\phi)$. However I'm not sure how to proceed.</p>
<p>I get $x^2 + y^2 = 36\cos(2\phi)$. But what next?</p>
<p>I know circle equation $x^2 + y^2 = constant$... | robjohn | 13,854 | <p>The implicit equation is
$$
\begin{align}
r^2
&=36\cos(2\phi)\\
&=36[\cos^2(\phi)-\sin^2(\phi)]\\
r^4
&=36[r^2\cos^2(\phi)-r^2\sin^2(\phi)]\\
x^4+2x^2y^2+y^4
&=36x^2-36y^2
\end{align}
$$
But parametrically, to plot the curve,
$$
\begin{align}
x&=6\cos(\phi)\sqrt{\cos(2\phi)}\\
y&=6\sin(\phi)\... |
951,664 | <p>How would I plot $r^2 = 36\cos(2\phi)$ in $x$ and $y$ coordinate system?</p>
<p>I know that $r^2 = x^2 + y^2$, and I know that $x = r\cos(\phi)$ and $y = r\sin(\phi)$. However I'm not sure how to proceed.</p>
<p>I get $x^2 + y^2 = 36\cos(2\phi)$. But what next?</p>
<p>I know circle equation $x^2 + y^2 = constant$... | Pauly B | 166,413 | <p>Using $r^2=x^2+y^2,x=r\cos\phi,y=r\sin\phi$, and the trig identity $\cos(2\phi)=\cos^2\phi-\sin^2\phi$, we have</p>
<p>$$r^2=36\cos(2\phi)$$ $$\implies r^4=36r^2(\cos^2\phi-\sin^2\phi)=(6r\cos\phi)^2-(6r\sin\phi)^2$$</p>
<p>$$\implies(x^2+y^2)^2=36x^2-36y^2$$
$$\implies {x}^{4}+2\,{x}^{2}{y}^{2}+{y}^{4}-36\,{x}^{2... |
2,203,876 | <p>As part of a proof regarding reduced homology, I need to know that the following fact holds for good pairs $(X,A)$:</p>
<p>If $q:X\rightarrow X/A$ is the quotient map, then $q|_{X\setminus A}:X\setminus A \rightarrow (X/A)\setminus (A/A)$ is a homeomorphism.
I can see why the restriction is injective, surjective (... | Henno Brandsma | 4,280 | <p>Thinking out loud:</p>
<p>So you're identifying $A$ to a point, so define $R_A$ to be the unique equivalence relation with classes $\{\{x\}: x \notin A\} \cup \{A\}$ or $xR_A y $ iff ($x \in A \land y \in A$) or ($x=y$). So $X/A$ (the set of aforementioned classes) gets the quotient topology w.r.t. $q; X \rightarro... |
1,746,213 | <p>I'm trying to prove that $$\lim_{x \rightarrow \infty} x^ne^{-x}=0$$</p>
<p>So far I have said:</p>
<p>Let $\epsilon>0.$ </p>
<p>Then $$|x^ne^{-x}-0|=|x^ne^{-x}| \leq |x^n...|$$</p>
<p>I'm not sure how to bound $e^{-x}$ though. How can I do this?</p>
| daniels_pa | 329,439 | <p>Don't try to prove it by definition if you don't have to.</p>
<p>Since $n$ is a fixed number in this limit we can apply L' Hospital rule n times.</p>
<p>$$
\lim_{x \rightarrow +\infty} x^ne^{-x}=\lim_{x \rightarrow +\infty} \frac{x^n}{e^{x}}=\lim_{x\rightarrow +\infty}\frac{n!}{e^x}=0
$$</p>
<p>Even though here i... |
1,278 | <p>Being quite new to the site, I'm curious if there is any established protocol with regards to action toward answers that are incorrect (e.g., produce incorrect / incomplete / etc. results). Is it appropriate to comment on the answer as such, or to place such a caveat on one's own answer if provided for the question?... | David Z | 79 | <blockquote>
<p>yet it feels... confrontational to think of "calling out" an answer.</p>
</blockquote>
<p>This is one of several ways in which Stack Exchange etiquette differs from the etiquette in some other places on the internet and in real life: when you see an incorrect answer, especially if it is getting upvot... |
1,169,314 | <p>We have that:
$$\begin{align}
\lim_{x \to 2} x^2 &= 4\\
D_{x = 2}\left(x^2\right) &= 4
\end{align}$$</p>
<p>In both cases the result is $4$. So limit and derivative are always the same?<br>
If not, what is the relation (if any exists)?</p>
| rubik | 2,582 | <p>The fact that in your example the limit and the derivative are equal is only a matter of coincidence.</p>
<p>You can understand the relation between a limit and a derivative if you look at their definition. First, the concept of limit is defined:
$$\lim_{x \to \alpha} f(x) = l \quad\iff\quad \forall\, U(l),\ \exist... |
3,239,860 | <p>This question is taken from book: Advanced Calculus: An Introduction to Classical Analysis, by Louis Brand. The book is concerned with introductory real analysis.</p>
<p>I request to help find the solution. </p>
<blockquote>
<p>If <span class="math-container">$n$</span> is a positive integer, find all roots of t... | DINEDINE | 506,164 | <p>Hint:
Put <span class="math-container">$$z=\frac{1+i\frac{x}{n}}{1-i\frac{x}{n}}$$</span> then <span class="math-container">$z$</span> will be a <span class="math-container">$n$</span>-root of unity and solve for <span class="math-container">$x:$</span> <span class="math-container">$$z= \frac{1+i\frac{x}{n}}{1-i\fra... |
2,430,407 | <p>My question comes from watching Khan Academy's introduction to Tangent planes <a href="https://www.youtube.com/watch?time_continue=2&v=H2hOwKszKRo" rel="nofollow noreferrer">here.</a> He extends the concept of a tangent line in two dimensions, a line that shares the same slope of a function at a point, to a plan... | user1551 | 1,551 | <p><em>Hint.</em> Let $B=P^{-1}B_1P$. Then $AB=BA$ if and only if $DB_1=B_1D$. Using the fact that $D$ is a diagonal matrix with distinct diagonal entries, you may try to prove that $B_1$ is a diagonal matrix.</p>
|
2,430,407 | <p>My question comes from watching Khan Academy's introduction to Tangent planes <a href="https://www.youtube.com/watch?time_continue=2&v=H2hOwKszKRo" rel="nofollow noreferrer">here.</a> He extends the concept of a tangent line in two dimensions, a line that shares the same slope of a function at a point, to a plan... | Gabriel Romon | 66,096 | <p>If $AB=BA$, then the eigenspaces of $A$ are stable under $B$. Indeed, if $x$ is such that $Ax=\lambda x$, then $A(Bx)=(AB)x=(BA)x=B(Ax)=\lambda(Bx)$.</p>
<p>$A$ is diagonalizable with $n$ distinct eigenvalues, hence each eigenspace has dimension $1$. Consider $\lambda$ an eigenvalue and $\mathbb Rv$ the associated ... |
788,344 | <p>I have shown that $\Gamma(a+1)=a\Gamma(a)$ for all $a>0$. </p>
<p>But I'd also like to show the following 2 things:</p>
<p>1) Using the previous fact, I'd like to show that $\lim_{a \to 0^{+}}a\Gamma(a) = \Gamma(1)=1$</p>
<p>The second equality is obvious, but how about the limit equality?</p>
<p>2) Using 1) ... | robjohn | 13,854 | <ol>
<li><p>Since $\Gamma(x)$ is continuous at $x=1$, we get $$\lim_{a\to0^+}a\Gamma(a)=\lim_{a\to0^+}\Gamma(a+1)=\Gamma(1)=1$$</p></li>
<li><p>Here is how I usually approach this type of integral, but it does not use the Gamma function, just the fact that for $x\gt0$, $-1\le\frac{e^{-x\vphantom{\Large1}}-1}{x}\le0$:
$... |
123,524 | <p>So today in my final for number theory I had to prove that the Fermat numbers ($F_n=2^{2^n}+1$) are coprime. </p>
<p>I know that the standard proof uses the following: $F_n=F_1\cdots F_{n-1}+2$ and then the $\gcd$ divides $2$, and it cannot be two, and hence the numbers are coprime.</p>
<p>However, he asked us to ... | André Nicolas | 6,312 | <p>If $l$ is a prime that divides $F_n$, then $2^{2^n}\equiv -1 \pmod{l}$, and therefore $2^{2^{n+1}}\equiv 1 \pmod{l}$. So $2$ has order $2^{n+1}$ modulo $l$. Equivalently, but in more group-theoretic language, (the equivalence class of) $2$ has order $2^{n+1}$ in $(\mathbb{Z}/l\mathbb{Z})^\times$. (The order of $2$ ... |
2,394,582 | <blockquote>
<p>Show that $$f(x) = \ln \left(x+\sqrt{x^2+1}\right)$$ is an odd function.</p>
</blockquote>
<p>My attempt:</p>
<p>$$f(-x)=\ln\left(-x+\sqrt{(-x)^2+1}\right)=\ln\left(-x+\sqrt{x^2+1}\right).$$</p>
<p>How should I proceed? I know that if $f(-x)=-f(x)$, the function is odd. </p>
| tattwamasi amrutam | 90,328 | <p>$$\ln(-x+\sqrt{x^2+1})=\ln\left(\frac{1}{\sqrt{x^2+1}+x}\right)=-\ln(\sqrt{x^2+1}+x)$$</p>
|
2,394,582 | <blockquote>
<p>Show that $$f(x) = \ln \left(x+\sqrt{x^2+1}\right)$$ is an odd function.</p>
</blockquote>
<p>My attempt:</p>
<p>$$f(-x)=\ln\left(-x+\sqrt{(-x)^2+1}\right)=\ln\left(-x+\sqrt{x^2+1}\right).$$</p>
<p>How should I proceed? I know that if $f(-x)=-f(x)$, the function is odd. </p>
| Jack D'Aurizio | 44,121 | <p>$$ f(\sinh\theta) = \theta $$
and both $\theta$ and $\sinh\theta$ are odd functions, hence $f$ is an odd function as well ($f(x)=\text{arcsinh}(x)$).</p>
|
431,485 | <p>I'm looking at this link:</p>
<p><a href="http://www.mathsisfun.com/geometry/trapezoid.html" rel="nofollow">http://www.mathsisfun.com/geometry/trapezoid.html</a></p>
<p>..and it says "Called an Isosceles trapezoid when the sides that aren't parallel are equal in length and both angles coming from a parallel side a... | Adam | 53,893 | <p>Let $G=(V,E)$, where $V=\{0,1\}^n$, and $E=\{(a,b)\in V\times V:a\text{ and }b \text{ differ in only one place}\}$. How to construct cycle of length $4$? Pick any vertex $v=v_1$, change any digit, let's say $i$-th, obtaining $v_2$. Then change any other digit ($j$-th) to get $v_3$, and return to $v_1$ using the only... |
1,842,689 | <p>I hold a bachelor's degree in mathematics and I have taken undergraduate real analysis course. But I'm interested to know more about it, especially the stuff that will help me to understand more of topology. Could you please recommend me some books that I can start with?</p>
<p>Your help will be appreciated</p>
<p... | Rubertos | 44,669 | <p>Real analysis : authors: Folland, Rudin, Royden (hist newest edition), Frank jones (especially for gamma functions)</p>
<p>Point set topology: authors: Munkres, Hocking & Young (this is not standard, but I like it personally)</p>
<p>In my very opinion, one should learn multivariable calculus before studying re... |
1,067,382 | <p>Is there any way to find the <code>n-th</code> number in the series, by knowing it's genereting function.</p>
<p>For example, I found that the closed form solution for a generating function $\displaystyle\frac{(1+x)^2}{(1-x)^3}$ can be expressed as $2n^2 + 2n + 1$.</p>
<p>I am interested how can I derive similar f... | Leox | 97,339 | <p>Yes, yours $n$-th numbers are called the Hilbert polynomials or quasi-polynomials. See details in Chapter 4 of the book:</p>
<p>R. Stanley, Enumerative combinatorics. Vol. 1., Cambridge Studies in Advanced Mathematics. 49.
Cambridge: Cambridge University Press. </p>
<p>The following theorem ( in some corrected f... |
703,151 | <p>Let $(X,\leq_X )$ and $(Y,\leq_Y)$ be two well-ordered sets, such that the discrete space of $X$ is the same as the order topology on $(X,\leq_X)$ and the same goes for $Y$. I'd like to prove that $(X,\leq_X)\cong (Y,\leq_Y) $. It seems correct, but I'm having trouble on where to start from.</p>
<p>Another thing is... | Chris | 135,478 | <p>For the first part, you first need that |X| = |Y|. Then there exists f: X -> Y which is bijective. Since the order topologies on each is really just the discrete topology, f is open and continuous. So, f is a homeomorphism</p>
|
390,480 | <p>I need to solve the following simultaneous equations for the constants $A$ and $B$:</p>
<p>$$zA + (1-z)B = 1$$
$$\hspace{1.8cm} z^2A + (1-z)B = z^2 - z + 1$$</p>
<p>where$z$ is just a variable. What I did was this:</p>
<p>$$zA = 1 - (1-z)B \implies z^2A = z - (z - z^2)B$$</p>
<p>and so subbing this into the seco... | Clement C. | 75,808 | <p>You system has infinitely many solutions if $A=B=1$ (for instance, substract the second from the first, and see what happens to $A$). Then, any $z$ satisfies it.</p>
<p>If $A\neq1$, then $z$ must be either $0$ or $1$. But plugging $z=1$ in the equations yields $A=1$, so the only solution is $z=0$.</p>
|
1,027,281 | <p>$1^x = 1^y$, and $x,y \in \mathbb{R}$.</p>
<p>Following the rule, same base has powers equal every $x$ should be equal to every $y$.
$$1^x = 1^y$$
$$x = y$$
What went wrong?</p>
| Martín-Blas Pérez Pinilla | 98,199 | <p>The function $x\to 1^x$ isn't <a href="http://en.wikipedia.org/wiki/Injective_function" rel="nofollow">injective</a>.</p>
|
1,153,054 | <p>So I have </p>
<ol>
<li><p>$i=e^{i\pi/2}=e^{5×i\pi/2}$</p></li>
<li><p>If I raise it to the power of $i$,
$i^i=[e^{i\pi/2}]^i=[e^{5×i\pi/2}]^i$</p></li>
</ol>
<p>Hence I have $e^{-\pi/2}=e^{-5\pi/2}$?</p>
<p>Can someone please tell me what I did wrong in my steps?</p>
| kobe | 190,421 | <p>We have </p>
<p>\begin{align}
&\lim_{x\to 0} \frac{x^5 - \frac{x^{10}}{2} + O(x^{15})}{(x^3 + \frac{x^6}{2!} + O(x^9))(x^2 - \frac{x^6}{3!} + O(x^{10}))}\\
&= \lim_{x\to 0}\frac{1 + O(x^5)}{(1 + O(x^3))(1 + O(x))}\\\\
&= \frac{1}{(1)(1)}\\
&= 1.
\end{align}</p>
|
427,794 | <p>The Euler and Weierstrass forms of the gamma function are :</p>
<p>$$\mathop{\mathrm{\Gamma}}\left(z\right)=\frac{1}{z}\prod^{\infty}_{n=1}\frac{\left(1+\frac{1}{n}\right)^{z}}{\left(1+\frac{z}{n}\right)}=\frac{e^{-\gamma z}}{z}\prod^{\infty}_{n=1}\frac{e^{\frac{z}{n}}}{\left(1+\frac{z}{n}\right)},\quad\mbox{for}\;... | Mariano Suárez-Álvarez | 274 | <p>You can say </p>
<blockquote>
<p>Let $\Omega$ be the set of complex numbers excluding the negative integers. Then the above formulas are valid in $\Omega$.</p>
</blockquote>
|
4,224 | <p>Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their sum? What about the special case when the matrices are Hermitian and positive definite?</p>
<p>I am investigating this with regard to finding the normalized graph cut under general convex constraints. Any pointers wi... | doublefelix | 94,409 | <p>Here is a trivial case with a simple solution. Applicable in Quantum Mechanics, for one.</p>
<p>Given two matrices of the form <span class="math-container">$A \otimes Id$</span>, <span class="math-container">$Id \otimes B$</span>, the eigenvalues of their sum are all combinations <span class="math-container">$a_i+b... |
1,026,647 | <p>I'm trying to integrate the following:</p>
<p>$\int_{0}^{\infty} x^2 e^{-2x}$</p>
<p>I know that $\int_{0}^{\infty} x^n e^{-x} dx = n!$</p>
<p>Feels like a $u$ substitution problem but I'm having trouble making use of the above.</p>
<p>Thanks,
Mariogs</p>
| Dr. Sonnhard Graubner | 175,066 | <p>make the ansatz $\int x^2e^{-2x}\,dx=(Ax^2+Bx+C)e^{-2x}$</p>
|
1,187,894 | <p><img src="https://i.stack.imgur.com/LFtIh.jpg" alt="enter image description here"></p>
<p>I thought I knew how to solve this kind of problems, but I really don't know how to start...</p>
| Gyu Eun Lee | 52,450 | <p>As far as I know, there is no unified subject that specifically deals with integral transforms in general. Different integral transforms come up in different contexts. You are probably better off asking yourself, "What sort of engineering/mathematics do I want to study?" Depending on your answer, this may lead you t... |
4,551,007 | <blockquote>
<p>Let <span class="math-container">$A$</span> be a <span class="math-container">$2 \times 2$</span> matrix with <span class="math-container">$-3$</span> and <span class="math-container">$-1$</span> as eigenvalues. The eigenvectors are <span class="math-container">$v_1=[-1,1]$</span> and <span class="math-... | Klaus | 635,596 | <p>You made a mistake when you solved the linear system. Adding the two equations gives <span class="math-container">$5 = -2C_2$</span>, hence <span class="math-container">$C_2 = -\frac{5}{2}$</span>. Plugging this into one of the equations, we obtain <span class="math-container">$C_1 = -\frac{1}{6}$</span>.</p>
|
3,402,072 | <p>I was watching a youtube video on epsilon-delta proofs when I came across this argument. We wanted to show <span class="math-container">$\lim_\limits{x \to 5} \sqrt{x+4} = 3$</span>. He started the proof of by :
<span class="math-container">$|\sqrt{x+4}-3|<\epsilon \implies -\epsilon <\sqrt{x+4}-3<\epsilo... | David P | 49,975 | <p>You know that <span class="math-container">$\sqrt{x+4} \ge 0$</span>. So, you may assume the left-hand side of your inequality is</p>
<p><span class="math-container">$$\max\{0, 3-\epsilon\}$$</span></p>
|
4,081,780 | <p>I have seen the statement that if <span class="math-container">$R$</span> is a commutative ring and <span class="math-container">$R$</span> is a semisimple ring, then <span class="math-container">$R$</span> is Noetherian several times while reading through some algebra resources; however, I have never seen an explan... | Jbag1212 | 509,460 | <p>How do you define "same object"? Those functions are different functions, they are not the same. In geometry we often like to say that two shapes are the "same object" or congruent if we can transform one into the other solely by (proper) rotations and translations. Sometimes we also allow reflec... |
202,012 | <p>Lets say I have a two dimensional list:</p>
<pre><code>list = {{a, b, c}, {aa, bb, cc}, {x, y, z}, {xx, yy, zz}};
</code></pre>
<p>and two pure functions:</p>
<pre><code>f1 = Total[#]/Length[#] &;
f2 = Total[##]/Length[##] &;
</code></pre>
<p>I want to learn how to apply these functions over rows and col... | ramesh | 18,636 | <p>Look at how <code>Map</code> and <code>MapThread</code> are applied to a two-dimensional list. </p>
<pre><code>Map[f, list]
</code></pre>
<blockquote>
<p>{f[{a, b, c}], f[{aa, bb, cc}], f[{x, y, z}], f[{xx, yy, zz}]}</p>
</blockquote>
<p>and <code>MapThread[f, list]</code></p>
<blockquote>
<p>{f[a, aa, x, xx... |
637,017 | <p>$d_1(x,y)=d(x,y)+|f(x)-f(y)|$</p>
<p>Suppose $f(x)=1$ if $x\in\mathbb{Q}$ and 0 otherwise for a metric space = $\mathbb{R}$.</p>
<p>With this function I am shocked that $d_1$ is a metric. I do not doubt my proof though.</p>
<p>It's still very strange, for example consider $d_1(0,p)\le1$, what p satisfy this? p=1 ... | Daniel Fischer | 83,702 | <p>The metric $d_1$ gives an isometry between $(X,d_1)$ and the graph of $f$,</p>
<p>$$\Gamma(f) = \left\lbrace (x,f(x)) : x \in X\right\rbrace \subset X\times\mathbb{R},$$</p>
<p>when the product $X\times\mathbb{R}$ is endowed with the metric</p>
<p>$$\delta((x,s),(y,t)) = d(x,y) + \lvert s-t\rvert,$$</p>
<p>which... |
622,616 | <p>I have attempted a proof of this, but upon looking at my notes, I feel it might be incorrect: it is noticeably simpler than the one in my notes.</p>
<p><strong>Proposition:</strong> any permutation in $S_n$ can be written as a product of disjoint cycles.</p>
<p><em>Proof (attempt):</em> $s\in S_n$. Let $\langle s\... | benh | 115,596 | <p>The proof is indeed correct. There is a nice intuition behind this: </p>
<p>You can interpret a permutation as a directed graph: The nodes are the numbers $1,\dots,n$ and there is an arrow from $a$ to $b$ if the permutation $\pi$ maps $\pi(a)=b$. This is a functional graph and as the number of nodes is finite and t... |
2,323,605 | <p>I am very confused on this question.</p>
<p>How does $(4^{-3} \cdot 2^{-3})^0 = (8^{-3})^0$? Should it not be $8^{-6}$ since $-3+-3 = -6$?</p>
<p>I just don't get where the other power of $-3$ goes off to?</p>
<p><a href="https://i.stack.imgur.com/xC5pY.jpg" rel="nofollow noreferrer"><img src="https://i.stack.img... | Hans Hüttel | 289,137 | <p>Remember that for every $x \neq 0$ we have that $x^0 = 1$.</p>
|
2,323,605 | <p>I am very confused on this question.</p>
<p>How does $(4^{-3} \cdot 2^{-3})^0 = (8^{-3})^0$? Should it not be $8^{-6}$ since $-3+-3 = -6$?</p>
<p>I just don't get where the other power of $-3$ goes off to?</p>
<p><a href="https://i.stack.imgur.com/xC5pY.jpg" rel="nofollow noreferrer"><img src="https://i.stack.img... | StackTD | 159,845 | <p>You seem to think that $a^xb^x = (ab)^{x+x}=(ab)^{2x} $ or perhaps even $a^xb^y = (ab)^{x+y}$, but the rule you need in that step is:
$$a^xb^x = (ab)^x$$
So: $4^{-3}2^{-3} = \left( 4 \cdot 2 \right)^{-3} = 8^{-3}$.</p>
|
2,793,355 | <p>I am interested in bounding the following integral, where $a>0$ is a constant:</p>
<p>$$\int\limits_0^a \exp\left(\left(x^2 - \frac{2a^4+3}{4a^2}\right)^2\right) dx$$</p>
<p>I first conjectured that </p>
<p>$$\int\limits_0^a \exp\left(\left(x^2 - \frac{ba^4+c}{a^2}\right)^2 \right) dx \le a^{-1}\exp \left( \le... | Pagode | 564,189 | <p>For the infinity: you can write for positive a</p>
<p>$$ \int_0^{a}\exp\left(\left(x^2 - \frac{2a^4+3}{4a^2}\right)^2\right)dx=\int_{0}^{\infty}\exp\left(\left(x^2 - \frac{2a^4+3}{4a^2}\right)^2\right)\chi_{[0 , a]}(x) dx $$ </p>
<p>So you can begin by majorate the integrand only independantly of a </p>
<p>And a... |
2,793,355 | <p>I am interested in bounding the following integral, where $a>0$ is a constant:</p>
<p>$$\int\limits_0^a \exp\left(\left(x^2 - \frac{2a^4+3}{4a^2}\right)^2\right) dx$$</p>
<p>I first conjectured that </p>
<p>$$\int\limits_0^a \exp\left(\left(x^2 - \frac{ba^4+c}{a^2}\right)^2 \right) dx \le a^{-1}\exp \left( \le... | Mefitico | 534,516 | <p>Since <span class="math-container">$\exp(z)>0 \,(\forall z \in \mathbb{R})$</span>, you can use the following bounding method:</p>
<p><span class="math-container">$$
\int_0^a f(x)dx =\int_0^a |f(x)|dx
$$</span></p>
<p>Then:</p>
<p><span class="math-container">$$
(a-0) \min \lbrace f(x) \rbrace \leq\int_0^a |f(x... |
2,138,631 | <p>If you reverse the binary digits of a multiple of $3$, the result is always a multiple of $3$. The same is not true for $7$, but it does appear to happen more often than $\frac{1}{7}$ of the time. For fun, I'll call a number $n$ such that the reverse of the binary digits of $7n$ represents a multiple of $7$ a "seven... | mercio | 17,445 | <p>If the ratio converges to a limit, then it is $1/7$.</p>
<p>Let $n'$ be the reverse of $n$.<br>
If $n$'s binary expansion is $\sum_{k=1}^m d_k 2^k$,
then, modulo $7$, we have<br>
$n' = \sum_{k=1}^m d_k 2^{m-k} \equiv \sum_{k=1}^m d_k 2^{m-k} 8^k \equiv 2^m \sum_{k=1}^m d_k 4^k$,<br>
and so $n'$ is a multiple of $7$... |
1,931,222 | <p>I am solving a test and have trouble solving this particular task: given a differentiable function $f : \mathbb{R} \rightarrow \mathbb{R} $ and $ f(0) = f(1) = 0$, are the following limits finite:</p>
<p>$$
(a) \lim_{n \rightarrow +\infty} nf\left(\frac{1}{n}\right)
$$</p>
<p>$$
(b) \lim_{n \rightarrow +\infty} nf... | Jean-Claude Arbaut | 43,608 | <p><strong>The following is rendered useless after the OP added the information that $f$ is differentiable</strong> However, I leave the answer, as it may show (at least to the OP, who didn't bother to mention it), that without differentiability the answer may be entirely different.</p>
<p>You have not specified wheth... |
243,901 | <p>Can anyone come up with a one line answer why a set $A \subset X$ being compact implies it's bounded. I figure that we could take the closure of $A$'s open subcover and every $x \in A$ would be contained in this set. Is that correct? And is there a 'nicer' way of saying it (in one sentence)?</p>
| Asaf Karagila | 622 | <p>Fix $a\in A$ and consider the open cover defined by $B(a,n)=\{x\in X\mid d(a,x)<n\}$ for $n\in\mathbb N$.</p>
|
243,901 | <p>Can anyone come up with a one line answer why a set $A \subset X$ being compact implies it's bounded. I figure that we could take the closure of $A$'s open subcover and every $x \in A$ would be contained in this set. Is that correct? And is there a 'nicer' way of saying it (in one sentence)?</p>
| Community | -1 | <p>Take $X$-balls of radius $1$ around every point in $A$ and then take a finite subcover of that.</p>
|
989,857 | <p>The limit $$\lim_{n\rightarrow\infty}\dfrac{(4n)^{4n}n^n}{(3n)^{3n}(2n)^{2n}}$$ can be calculated as $\lim_{n\rightarrow\infty}\dfrac{4^{4n}}{3^{3n}2^{2n}}=\lim_{n\rightarrow\infty}\left(\dfrac{4^4}{3^32^2}\right)^n=\infty$.</p>
<p>What about $$\lim_{n\rightarrow\infty}\dfrac{(4n-100)^{4n-100}n^n}{(3n)^{3n}(2n)^{2n... | Paul | 17,980 | <p>Hints:</p>
<p>Can you see that $(4n-100)^{4n-100} \sim (4n)^{4n}$ as $n\to \infty$?</p>
<p>Then $$\lim_{n\rightarrow\infty}\dfrac{(4n-100)^{4n-100}n^n}{(3n)^{3n}(2n)^{2n}}=\lim_{n\rightarrow\infty}\dfrac{(4n)^{4n}n^n}{(3n)^{3n}(2n)^{2n}}$$ </p>
|
1,066,761 | <p>I have this problem I can't figure out. </p>
<p>Suppose R is a relation on A, and let S be the reflexive closure of R. Prove
that if R is symmetric, also is S.</p>
<p>Could you suggest me how to do it?</p>
<p>Thanks</p>
| Henno Brandsma | 4,280 | <p>Taking the reflexive closure, we are only adding points (from the diagonal) that cannot be a counterexample to symmetry..</p>
|
1,490,480 | <p>Please help me with the following question: </p>
<p>Let $A$ be $m \times n$, $B$ be $n \times p$ matrices with $\text{rank}(A)=m$, $\text{rank}(B)=p$, where $p < m < n$.<br>
What are conditions such that $\text{rank}(AB)=\text{rank}(B)$???</p>
<p>Thanks!</p>
| userabc | 204,687 | <p>View $A,B,D$ as linear transformation, let $V$ be the domain of $B$, a vector space of dimension $p$.
Recall that for any linear transormation $T$, rank$(T)$ is the dimension of the image of the linear transformation that is given by multiplication by $T$.</p>
<p>rank$(B)=p$ means dim$(B(V))=p$. $D$ is diagonal po... |
2,978,752 | <p>I am wondering how to use the properties of conditional expectations to show that:</p>
<p><span class="math-container">$$
\mathbb{E}[\mathbb{E}[Y|Z,X,W]|X] = \mathbb{E}[\mathbb{E}[\mathbb{E}[Y|Z,X,W]|Z,X]|X]
$$</span></p>
<p>for random variables <span class="math-container">$Y,Z,X,W$</span>? It appears to me that ... | Kavi Rama Murthy | 142,385 | <p>Both sides are equal to <span class="math-container">$E(Y|X)$</span>. All you have to do is to use repeatedly the property <span class="math-container">$E(E(Y|\{X_i\}|\{Z_j\})=E(E(Y|\{Z_j\})$</span> whenever <span class="math-container">$\{Z_j\}$</span> is a subcollection of <span class="math-container">$\{X_i\}$</s... |
2,410,736 | <p>If we have the product of 3 primes,
A<em>B</em>s
two of them large: A, B
and one small: s
We don t know the product of A<em>B,
(Given that if we knew A</em>B we couldn t factorize it, because of the size and the computational complexity of the problem.)
can we find the small number s through factorization?
Is t... | Marios Gretsas | 359,315 | <p>Take the topology $\mathcal{T}$ on $\mathbb{R}$ where $\mathcal{T}=\{[a,b)|a<b,a,b \in \mathbb{R}\}$</p>
<p>If i remember correctly this is called the lower limit topology.</p>
<p>Every interval of the form $[a,b)$ is clopen in this topology.</p>
<p>Take the collection $\{[\frac{1}{n}, +\infty)|n \in \mathbb{N... |
4,474,923 | <p>I am confused as to the relation between Markov chains and random walks. Some sources I look at claim all random walks are markov chains and then other sources say the opposite. What makes a markov chain a random walk?</p>
| Calvinfwc | 897,319 | <p>If you define random walk as <a href="http://www2.math.uu.se/%7Esea/kurser/stokprocmn1/slumpvandring_eng.pdf" rel="nofollow noreferrer">here</a> , that's <span class="math-container">$(S_n)_{n\ge0}$</span> defined by <span class="math-container">$S_0 = 0 , S_n = \sum_{k=1}^n X_k$</span> for i.i.d <span class="math-... |
321,032 | <p>Take any convex set <span class="math-container">$A\subset\mathbb{R}^n$</span> which contains a neighborhood of the origin and let <span class="math-container">$f_A$</span> be the associated <em>Minkowski functional</em>
<span class="math-container">$$
f_A(x) = \inf\{\lambda>0\mid x\in\lambda A\},
$$</span>
which... | Michael Renardy | 12,120 | <p>Consider <span class="math-container">$n=1$</span>, <span class="math-container">$A=[-1,1]$</span>, <span class="math-container">$\Omega=(0,1)$</span>. You want to minimize <span class="math-container">$\int_0^1 |u'(x)|\,dx$</span> subject to Dirichlet conditions, say <span class="math-container">$u(0)=0$</span> and... |
918,430 | <p>I am reading a paper (in biology) which performs a clustering algorithm.</p>
<p>At one point in the paper, it is stated that:</p>
<p>$$
\arg \max_{\lVert X\rVert=1} X^T S\, X
$$</p>
<p>can be computed as the normalized eigenvector of $S$ with largest eigenvalue...</p>
<p>How is this possible? I am guessing that ... | Ian | 83,396 | <p>Rewriting your expression this way:</p>
<p>$$\arg \max_{\| X \| = 1} \frac{X^T S X}{X^T X}$$</p>
<p>what you have is called a Rayleigh quotient. (I insert the division by $X^T X = 1$ to clarify why it is called a quotient.) It is clear that any eigenvalue of $S$ is a possible Rayleigh quotient: just evaluate the f... |
402,655 | <p>I'm interested in doing PhD in computer algebra or symbolic computation, and was wondering if this is an active area of research? Would this area of research also help me in the transition to industry after my PhD? I'm also based in the USA.</p>
<p>I apologize if this question is kind of "bare", I'm a Ph... | Joshua Grochow | 38,434 | <p>[This is certainly a biased view, but too long for a comment, and hopefully these are some helpful starting points. ]</p>
<p>I'd say it's active but in the US not huge (more elsewhere in the world). A few things to look up:</p>
<ul>
<li><p>Computational number theory. Of computer algebra/symbolic computation, this m... |
402,655 | <p>I'm interested in doing PhD in computer algebra or symbolic computation, and was wondering if this is an active area of research? Would this area of research also help me in the transition to industry after my PhD? I'm also based in the USA.</p>
<p>I apologize if this question is kind of "bare", I'm a Ph... | Zach Teitler | 88,133 | <p>Two books and a journal:</p>
<ol>
<li><p><a href="http://www.math.uni-konstanz.de/%7Emichalek/NonLinearAlgebra.pdf" rel="nofollow noreferrer">Invitation to Nonlinear Algebra</a> by Michałek and Sturmfels. Computational nonlinear algebra and a variety of applications (“variety”).</p>
</li>
<li><p><a href="https://boo... |
1,566,275 | <p>I'm familiar with the "construction" of non-measurable sets on $\mathbb{R}$. But of interest to me is if there is a way to construct a countably additive probability measure $\mu$ on $\mathbb{N}$ such that we can't extend $\mu$ to $2^{\mathbb{N}}$. Apparently there's no way to do it when $\mu$ is defined on all sing... | Patrick Da Silva | 10,704 | <p>Let $(\mathbb N, \mathcal F, \mu)$ be a measure space. Consider for each $x \in \mathcal F$ the set
$$
\mathcal F_x = \underset{x \in S}{\bigcap_{S \in \mathcal F}} S.
$$
We can take this intersection to be countable, since for each $y \in \mathcal F_x$, we can pick one $S_y \in \mathcal F$ with $x,y \in \mathcal F... |
150,259 | <p>Fix Grothendieck universes $\mathcal{U} \in \mathcal{V}$ and suppose that $C$ is a locally $\mathcal{U}$-small category which is $\mathcal{V}$-small. Denote by $Set$ the category of $\mathcal{U}$-small sets (which is also $\mathcal{V}$-small). Suppose that $\mathcal{E}$ is a category which can obtained as a left exa... | Zhen Lin | 11,640 | <p>By definition, a <strong>$\mathbf{U}$-pretopos</strong> is a category $\mathcal{C}$ that satisfies Giraud's axioms <em>except</em> for the existence of topological generators, i.e.</p>
<ul>
<li>$\mathcal{C}$ has finite limits,</li>
<li>$\mathcal{C}$ is a $\mathbf{U}$-extensive category, i.e. $\mathcal{C}$ has copro... |
930,121 | <p>Alice rolls a 12 sided die (the faces labeled 1 through 12) and Bob rolls a 20 sided die (the faces labeled 1 through 20). After seeing their roll (but not the other person's roll), each person can choose to roll again. Bob wins if his final roll is strictly greater than Alice's final roll. What is the probability t... | Xoff | 36,246 | <p>What I will do to solve that question :</p>
<p>Consider the 13 cases for Alice and 21 cases for Bob (from reroll always, to never reroll, through reroll on 1, reroll on 2 or less, reroll on 3 or less...) and compute the probability of winning for each cases (use a computer program for the $21\times13$ cases).</p>
... |
3,353,608 | <p>Prove by induction that:</p>
<p><span class="math-container">$64 \mid 3^{2n + 2} - 8n - 9$</span>, <span class="math-container">$n > 0$</span></p>
<p>I've tried to manipulate <span class="math-container">$P(k + 1)$</span> and haven't come up with anything like <span class="math-container">$P(k)$</span>.</p>
<p... | lab bhattacharjee | 33,337 | <p>If <span class="math-container">$f(m)=3^{2n+2}-8n-9,$</span></p>
<p>Eliminate <span class="math-container">$3^{2n}$</span> to establish <span class="math-container">$$f(m+1)-3^2f(m)$$</span> is actually divisible by <span class="math-container">$64$</span></p>
<p><span class="math-container">$$\implies64\mid f(m+1... |
59,888 | <p>Show that the quadratic form of a matrix is $0$ if and only if the matrix is skew–
symmetric, i.e., show that $q_A(x) = 0$ for all $x$ iff $A^t = −A$.</p>
<p>Thanks a lot!</p>
| Community | -1 | <p>Here is the if part,
$$x^TA x = x^TA^Tx= - x^TA x \in \mathbb{R}$$
Only if part does not hold, counter-example
$$
\begin{pmatrix} 1\\ 0\end{pmatrix}^T\begin{pmatrix} 0&1\\1& 0\end{pmatrix}\begin{pmatrix} 1\\ 0\end{pmatrix}=0
$$
unless you are missing "for all $x$" before the "iff"</p>
|
59,888 | <p>Show that the quadratic form of a matrix is $0$ if and only if the matrix is skew–
symmetric, i.e., show that $q_A(x) = 0$ for all $x$ iff $A^t = −A$.</p>
<p>Thanks a lot!</p>
| Davide Giraudo | 9,849 | <p>If $A^t=-A$ then for $x\in\mathbb R^n$ we have $-x^tAx=x^tA^tx=(x^tAx)^t=x^tAx$ since $x^TAx$ is a real number hence $x^tAx=0$.</p>
<p>For the controverse, let $S:=A^t+A$. Since for all $x\in\mathbb R^n$ we have $q_A(x)=x^tAx=0=-x^tAx$, we get $x^tSx=0$ for all $x$. We denote by $e_j$ the vector whose entries are $... |
1,901,722 | <p>I just stared a new math course, and we began by reviewing metric spaces. The professor gave us the set $X = (-\infty,\infty) $ , so all real numbers, and asked us to come up with any metric for it. When we suggested $|x-y|$, the simplest we could think of, he said that we were wrong because this failed to work fo... | Alexis Olson | 11,246 | <p>On $\mathbb{R} = (-\infty, \infty)$, the function defined by $d(x,y) = |x - y|$ is certainly a metric. It's called the <a href="https://en.wikipedia.org/wiki/Euclidean_distance" rel="nofollow">Euclidean metric</a>.</p>
<p>For the extended reals $\overline{\mathbb{R}} = [-\infty, \infty]$, the same definition does n... |
18,771 | <p>Show that if a manifold $M$ is contractible, then every vector bundle over $M$ is equivalent to the trivial bundle.</p>
<p><em>Got this as homework but I'm kind of lost in the hole vector bundle subject, so if anyone could give some hints to this one as well as to understanding the subject, it would be highly appre... | Sean Tilson | 627 | <p>Here are a couple of things that I found helpful when thinking about vector bundles:</p>
<ol>
<li><p>Over nice spaces they they can be classified up to isomorphism. So by nice here we mean compact manifold, or CW complex. The idea is that every finite dimensional vector bundle $\xi: E \to X$ is isomorphic to the pu... |
384,913 | <p><img src="https://i.stack.imgur.com/hgqYR.jpg" alt="enter image description here"></p>
<p>Why is the above statement true? What if $fg$ is a constant function?? (for example, if $f(z)=e^z$ and $g(z)=\frac 1{e^z}$)</p>
<p>(I <em>think</em> that the above is true if $fg$ is definitely non-constant.)</p>
| Dimitris | 37,229 | <p>On the contrary, suppose that there exists $z_0\in G$ such that $\forall R>0$ , with $D'(z_0,R)\subset G$ there exists $z_R\in D'(z_0,R)$ such that:
$$f(z_R)g(z_R)=-f(z_R)-g(z_R)-1$$
Now take a (decreasing) sequence of radii $R_n\rightarrow 0$ such that $D'(z_0,R_n)\subset G$. For each $n\in\mathbb N$ there exist... |
918,624 | <p>I am trying to refresh my memory with linear algebra. I am confused with these set of questions:</p>
<blockquote>
<ol>
<li><p>Show that $a=\begin{pmatrix}
1 \\
i \\
-1 \\
-i
\end{pmatrix},$ $b=\begin{pmatrix}
1 \\
-1 \\
1 \\
-1
\end{pmatrix}, $$\ c=\begin{pmatrix}
1 \\
-i \\
-1 \\... | BigM | 90,395 | <p>$\langle f,g\rangle=\int_{-\pi}^{\pi} f(x)\overline{g(x)}\,dx=\int_{-\pi}^\pi e^{2\pi x}dx=0.$</p>
<p>$\|f\|_2^2=\int_{-1}^1f^2(x)\,dx=\int_{-1}^1 \,dx=2.$</p>
|
1,267,956 | <p>I have the $$\int_{16}^{500} \frac{1}{x^{0.25} - 2} dx,$$ and am trying to find whether it converges or diverges. </p>
<p>I have sketched the graph and noticed that their is an asymptote at $x=16$ (hence why the integral is improper for these boundaries).</p>
<p>I am now trying to evaluate the limits to see if it ... | Clement C. | 75,808 | <p>Do the change of variable $t = x^{\frac{1}{4}}-2$, i.e. $x = (t+2)^4$. Then
$$
\int_{16}^{500} \frac{dx}{x^{\frac{1}{4}}-2} = \int_{0}^{500^{\frac{1}{4}}-2} \frac{4(t+2)^3}{t}dt
$$
and around $0$ the integrand is equivalent to $\frac{32}{t}$ -- is it integrable?</p>
|
627,394 | <p>Let $R$ be a commutative ring and consider the ideal $I = (r_{1}, r_{2})$ where $r_{1}, r_{2}$ are 2 linear independent generators of $R$.</p>
<p>Now consider the $R-$modules. Here $I$ is a $R-$submodule of the $R-$module $R$. Can I then conclude that $I$ is free? </p>
<hr>
<p><strong>Argument</strong></p>
<p>I ... | user119107 | 119,107 | <p>The elements $x = r_1$ and $y = r_2$ are <em>not</em> linearly independent in $R$, since $r_2 x - r_1 y = 0$.</p>
<p>The theorem you mentioned is not applicable. For example, if $B = \{4,6\}$, the free $\mathbf{Z}$-module with basis $B$ would be the set $\{x \cdot 4 + y \cdot 6 \, | \, x, y \in \mathbf{Z}\}$, wh... |
1,794,039 | <p>This is the Guess-The-Number game with a twist!</p>
<h2>Variant 1</h2>
<blockquote>
<p>Take any positive integer <span class="math-container">$n$</span>.</p>
<p>The game-master chooses an <span class="math-container">$n$</span>-bit integer <span class="math-container">$x$</span>.</p>
<p>The player makes queries one ... | ervx | 325,617 | <p>I tried to write a proof in the style you're going for. </p>
<p>Suppose $x\in Z(G)$. Then, $xa=ax$ for all $a\in G$, and hence, $x\in C(a)$ for all $a\in G$. Thus, $x\in\bigcap_{a\in G}C(a)$. Conversely, suppose $x\in\bigcap_{a\in G}C(a)$ for all $a\in G$. Then, $xa=ax$ for all $a\in G$, so that $x\in Z(G)$. </p>
... |
717,679 | <blockquote>
<p>In how many ways $n$ balls can be divided into groups where each group may contain only one ball or two. </p>
</blockquote>
<p>Each ball can be a singleton, or coupled with one of the $n-1$ options. Hence,<br>
$a_0 = 1$<br>
$a_1 = 1$<br>
$a_n = a_{n-1} + (n-1)a_{n-2}$ </p>
<p>The correspond gener... | ShreevatsaR | 205 | <p>Though you did not state it in the question, from your recurrence I guess you are dealing with distinguishable (labelled) balls. Thus for instance, $\{\{1, 2\}, 3\}$ and $\{\{1, 3\}, 2\}$ are distinct groupings. So your recurrence $a_n = a_{n-1} + (n-1)a_{n-2}$ is correct: given $n$ balls labelled $1$ to $n$, you pu... |
1,275,779 | <p>I am taking Abstract Algebra now and it seems most of its examples come from elsewhere. For example, I know what permutations are in counting, but the kind Algebra studies seem to be somewhat different. And to me these concepts Algebra comes bundled with is the most difficult part of studying it. It's like studying ... | J W | 118,539 | <p>If you're studying abstract algebra starting with group theory, Nathan Carter's <a href="http://web.bentley.edu/empl/c/ncarter/vgt/" rel="nofollow">Visual Group Theory</a> is a great way to get used to the concepts before diving in deeper. On the other hand, if what you're having trouble with is proofs, sets, functi... |
3,248,162 | <p>I have an exercise in my Exam that says: Prove that the set V=R (doesn't specify real positive numbers, only Real) is a vector space in the Real Field given the operations:</p>
<p><span class="math-container">$\ x+y = \sqrt[3]{x^3+y^3}$</span></p>
<p><span class="math-container">$ c * x = \sqrt[3]{c*x}$</span></p>... | Ross Millikan | 1,827 | <p>Everything in the problem is about the reals, so you should use the real cube root for the multiplication. Every real number has a unique real cube root. Closure is easy because it is inherited from the reals. Commutativity is also easy because of the symmetry of the definitions. Inverses, associativitiy and dis... |
1,514,981 | <p>I have one question as following:
Can we move f(x) out of $\int_{-\infty}^{\infty}f(x)sinc(x)dx$?</p>
<p>If $f(x')\int_{-\infty}^{\infty}sinc(x)dx$=$\int_{-\infty}^{\infty}f(x)sinc(x)dx$?</p>
| Harish Chandra Rajpoot | 210,295 | <p>Notice, the property of the definite integral $\int_{0}^{2a}f(x)\ dx=2\int_{0}^{a}f(x)\ dx\ \ \ \ \ \ \forall \ \ \ f(2a-x)=f(x)$</p>
<p>$$\int_{0}^{2\pi} \frac{1}{a^2\cos^2 x+b^2\sin^2 x}\ dx$$$$=2\int_{0}^{\pi} \frac{1}{a^2\cos^2 x+b^2\sin^2 x}\ dx$$ $$=4\int_{0}^{\pi/2} \frac{1}{a^2\cos^2 x+b^2\sin^2 x}\ dx$$
... |
173,508 | <p>I am having an issue with the following complex analysis problem. I am suppose to
find the coefficients of $z^{-1}$, $z^{-2}$ and $z^{-3}$ in the Laurent series for
$\displaystyle \frac{1}{\sin z}$ around $z_0 = 0$ which is valid for $2\pi < |z| < 3\pi$.</p>
<p>One way I thought of doing it was to say
$$ ... | Arturo Magidin | 742 | <p>What you are doing wrong is that you need to do the inner sum <em>three times</em>, once for each value of $i$. And when $i$ changes, the value of the inner sum changes by more than just $2i$: you add $2i$ <strong>four times</strong> (once for each of $j=0$, $1$, $2$, and $3$), so you end up adding $8i$, not just $2... |
521,739 | <p>Consider the r-subsets of {1,2,...,n} in lexicographic order. What are the first (n-r+1) r-subsets?</p>
<p>The answer is 12...r, 12...(r-1)(r+1),...,12...(r-1)n </p>
<p>I don't understand where the 12 is coming from.
In my book it says that 12...r is the first r-subset in the lexicographic ordering and (n-r+1)(n-... | vadim123 | 73,324 | <p>Consider the concrete example where $n=7, r=4$. We want subsets of size 4. Since a set can be written in any order, we are specifying "lexicographic" or dictionary order for each set, and then ordering them the same way.</p>
<p>The first set will be $\{1,2,3,4\}$. Four terms is pretty few, but we could abbreviat... |
130,345 | <p>I am trying to implement this function and solve for parameters: <a href="https://i.stack.imgur.com/VLfdJ.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/VLfdJ.jpg" alt="enter image description here"></a></p>
<p>Now as many can see it is a maximum likelihood estimation:
I have this data: </p>
<... | JimB | 19,758 | <p>Following @gwr 's suggestion to lose the subscripts, here is what you might consider:</p>
<pre><code>SeedRandom[123]
n = 120
x = RandomFunction[OrnsteinUhlenbeckProcess[0, 4, 5, 13], {0, n, 1}][[2]][[1, 1]];
t = Range[0, n];
(* Log likelihood *)
logL = Simplify[-(n/2) Log[σ^2/(2 η)] -
1/2 Sum[Log[1 - E^(-2 η (... |
4,044,564 | <p>calculation<span class="math-container">$$\lim\limits_{n\to\infty}\sum\limits_{k=1}^{n}\frac{1}{k}\sin\frac{k\pi}{n+1}.$$</span>
I try to use <span class="math-container">$|\sin x-x|\leqslant\frac{x^2}{2}$</span>, but I don't feel like it's going to work. Is there any other way?</p>
| CHAMSI | 758,100 | <p>Let <span class="math-container">$ n\in\mathbb{N}^{*} $</span>, we have the following : <span class="math-container">\begin{aligned}\sum_{k=1}^{n}{\frac{1}{k}\sin{\left(\frac{k\pi}{n+1}\right)}}&=\frac{2}{n+1}\int_{0}^{\frac{\pi}{2}}{\sum_{k=1}^{n}{\cos{\left(\frac{2kx}{n+1}\right)}}\,\mathrm{d}x}\\ &=\frac{... |
463,500 | <p><a href="http://en.wikipedia.org/wiki/Theory_%28mathematical_logic%29#Theories_associated_with_a_structure" rel="nofollow">According</a> to Wikipedia, every $\sigma$-structure $A$ is associated with a collection of atomic and negated atomic sentences called its <em>diagram</em>. Here's the precise definition.</p>
<... | William | 13,579 | <p>The diagram does not determine the structure. Even stronger, the full theory of a structure does not determine a structure. </p>
<p>There are elementarily equivalent structures that are not isomorphic. In fact, if you take any structure of uncountable cardinality in a countable language, by the downward Lowenheim S... |
7,367 | <p>This question and the described solution are copied from a test-paper :</p>
<p>For the equation $x^2$ + |x| - 6 = 0 analyze the four statements below for correctness.</p>
<ol>
<li>there is only one root</li>
<li>sum of the roots is + 1</li>
<li>sum of the roots is zero</li>
<li>the product of the roots is +4</li>
... | jake | 2,029 | <p>If x<0, then -x>0, which means that the modulus is indeed positive.</p>
|
7,367 | <p>This question and the described solution are copied from a test-paper :</p>
<p>For the equation $x^2$ + |x| - 6 = 0 analyze the four statements below for correctness.</p>
<ol>
<li>there is only one root</li>
<li>sum of the roots is + 1</li>
<li>sum of the roots is zero</li>
<li>the product of the roots is +4</li>
... | Isaac | 72 | <p>$f(x)=x^2+|x|-6$ is an even function—that is, $f(x)=f(-x)$ for all $x$, or the graph of $y=f(x)$ is symmetric over the y-axis—so if $f(c)=0$ then $f(-c)=0$, so the sum of the zeros of $f$ must be 0.</p>
|
521,412 | <p>So I tried this out and got stuck with this:</p>
<p>$$0 = 3x^{(7/6)} + 2x - 10$$</p>
<p>I didn't think I could use a quadratic for this since its to the power of $7/6$</p>
<p>Here is the working I did:</p>
<p>We know its a critical point when f'(a) = 0</p>
<p>So I found the derivative of f(x) which is $$2*(5-x)... | Cameron Buie | 28,900 | <p>I'm not sure what happened when you were finding your derivative, but assuming that $*$ indicates multiplication, then it's wrong. $$x^{2/3}(5-x)=5x^{2/3}-x^{5/3}$$ Can you take it from there? (As a side note, critical points can also occur where the function is defined but the derivative is not.)</p>
|
186,621 | <p>Let $f$ be a monic univariate polynomial with real coefficients:</p>
<p>$$f_A(x) = x^n + a_{n-1}x^{n-1} + ... + a_{0}$$</p>
<p>The values of $A=(a_{n-1},...,a_0)$ are unknown, but are estimated as $B=(b_{n-1},...,b_0)$ with error $\epsilon$. Therefore, $b_i - \epsilon \leq a_i \leq b_i + \epsilon$. Furthermore, ... | Pushpendre | 29,887 | <p>You might find the idea of <a href="http://en.wikipedia.org/wiki/Companion_matrix" rel="nofollow">Companion matrix</a> to be useful. The eigenvalues $\lambda$ of the companion matrix $C$ are equal to the roots of a polynomial and its eigenvectors $v$ are functions of the eigen-values and $v$ are linearly independent... |
94,481 | <p>I have some plots in an old Mathematica notebook that I'd like to re-create in Matplotlib (for stylistic consistency with the rest of the plots in my thesis).</p>
<p>Unfortunately, I've lost the data that I used to generate these plots. Since the plots are still intact, is it possible to extract the coordinates of... | anderstood | 18,767 | <ul>
<li>First step for illustration purpose : generate an image</li>
</ul>
<p><code>MoneyPlot[{{{1, 2, 6}, {6, 3, 3}}}, 0, 0]</code></p>
<p><a href="https://i.stack.imgur.com/8ugZg.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/8ugZg.png" alt="enter code here"></a></p>
<ul>
<li><p>Copy the plot ... |
362,252 | <p>Consider the advection equation $u_t + c\, u_x =0$ with initial condition
$u(x,0)=f(x)$, so that $u(x,t) = f(x-c\,t)$ is the solution.</p>
<p>It is asked to show that
\begin{equation}
\int_{-\infty}^{\infty} u^2(x,t)\, dx = \int_{-\infty}^{\infty} u^2(x,0)\, dx =\int_{-\infty}^{\infty} f^2(x)\, dx
\end{equation}
fo... | Ron Gordon | 53,268 | <p>Multiply the advection equation by the complex conjugate of $u$ and integrate with respect to $x$:</p>
<p>$$\int_{-\infty}^{\infty} dx \: u^* u_t + c \int_{-\infty}^{\infty} dx \: u^* u_x = 0$$</p>
<p>Note that </p>
<p>$$[u^* u]_{t,x} = 2 \Re{[u^* u_{t,x}]}$$</p>
<p>so that, when we take the real part of the abo... |
255,807 | <p>With <code>x</code>, <code>y</code>, <code>z</code> all positive, how can one simplify the following expression:</p>
<pre><code> f[x_,y_,z_]=(Sqrt[1 + x - y -
Sqrt[(-1 + x + y)^2 + 4 z^2]] (-1 + x + y -
Sqrt[(-1 + x + y)^2 + 4 z^2]))/(2 Sqrt[2] z Sqrt[
2 - (2 (-1 + x + y))/(-1 + x + y + Sqrt[(-1 + x + y)^... | zeros | 26,849 | <p>try to include the one that x, y, z are greater than zero inside the expression and to finish write at the end // FullSimplify</p>
<p>Edit
that was my original intention, but it did not produce the desired results, it was out of MMA's reach at the time I responded.</p>
<pre><code> Assuming[x > 0 && y >... |
2,384,939 | <p>There are two boxes, one white, one black. The white box contains 3 white balls and 2 black balls. The black box contains four white balls and six black balls. This particular example 60%/40% and 40%/60% distribution split but in other versions of this problem the distribution could be uneven. I'm hoping for a gener... | Graham Kemp | 135,106 | <p>When we have a two state Markov chain with a symmetric transition matrix, then the steady state will be $50\%$ probability for drawing from the white box. After 50 transitions we will be approximately at steady state. </p>
<p>$${\begin{bmatrix}1 & 0 \end{bmatrix}}\lower{1.5ex}{\begin{bmatrix}0.60 & 0... |
738,940 | <p>Asked a question to evaluate the Gaussian Integral,</p>
<p>$$\dfrac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty x^2 \exp(-x^2/2) dx $$</p>
<p>using the the following approximation,</p>
<p>$J=\Bbb E[X^2] \sim J_N = 1/N \sum_1^NX_n^2 $</p>
<p>where $x_n \sim N(0, 1)$</p>
<p>Examin $J_N$ for different values of $N$ and... | Lucian | 93,448 | <p>$$I(a)=\int_{-\infty}^\infty e^{-ax^2}dx\iff-I'(a)=\int_{-\infty}^\infty x^2\cdot e^{-ax^2}dx$$ Now, since we already know that $I(1)=\sqrt\pi$, it is trivial to show, by a simple change in variable, that $I(a)=\sqrt{\dfrac\pi a}$ , in which case our integral is simply $-\dfrac{I'\Big(\tfrac12\Big)}{\sqrt{2\pi}}$ . ... |
4,243,754 | <p><span class="math-container">$f(x)=x^3-x+1$</span></p>
<p><span class="math-container">$f'(x)=3x^2-1$</span>, It will be zero at <span class="math-container">$x=1/3,-1/3$</span></p>
<p><span class="math-container">$f"(x)=6x,$</span> so <span class="math-container">$1/3$</span> is a local minima and <span class="math... | Mike | 544,150 | <p>Well, that the first derivative <span class="math-container">$f'(x)=3x^2-1$</span> does not imply that the function is always increasing; this is <span class="math-container">$0$</span> at <span class="math-container">$\pm \sqrt{\frac{1}{3}}$</span>.</p>
<p>However, at <span class="math-container">$x^+=\sqrt{\frac{1... |
1,302,459 | <p>I think I understand the proof for square matrices, such that $(AB)^{-1}=B^{-1}A^{-1}$, but I'm not sure if I can just say the same for the bounded linear operators A and B.</p>
<p>Does the existence of $(AB)^{-1}:X\rightarrow X$ imply the existence of $B^{-1}$ and $A^{-1}$?</p>
| Ben Grossmann | 81,360 | <p>Not necessarily. For example, define on $\ell^p$
$$
L(x_1,x_2,\dots) = (x_2,x_3,\dots)\\
R(x_1,x_2,\dots) = (0,x_1,x_2,\dots)
$$
Then $LR = \text{id}_{\ell^p}$ is invertible, but neither $L$ nor $R$ are bijective.</p>
<p>You might be able to say something in the case that both $AB$ and $BA$ are invertible.</p>
|
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