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 |
|---|---|---|---|---|
4,044,708 | <p>[3,4] is closed in R <-- R-[3,4] is open</p>
<p>[5,6] is closed in R <-- R-[5,6] is open</p>
<p>Show that [3,4] x [5,6] is closed in R x R by writing it as the complement of the intersection of two open sets in R x R.</p>
<p>(R - [3,4]) x (R - [5,6]) not equal R x R - [3,4] x [5,6]</p>
| Martin R | 42,969 | <p>Using that <span class="math-container">$u$</span> is midpoint-convex works in higher dimensions as well.</p>
<p><span class="math-container">$y \mapsto x - (y-x) = 2x-y$</span> maps the sphere <span class="math-container">$\partial B_r(x)$</span> bijectively onto itself (each point is mapped to the “opposite” point... |
865,293 | <blockquote>
<p>Prove $\ln[\sin(x)] \in L_1 [0,1].$</p>
</blockquote>
<p>Since the problem does not require actually solving for the value, my strategy is to bound the integral somehow. I thought I was out of this one free since for $\epsilon > 0$ small enough, $$\lim_{\epsilon \to 0}\int_\epsilon^1 e^{\left|\ln(... | Community | -1 | <p>We can show this using the fact that $\sin x \sim x$ for small values of $x$; precisely, we have the inequality</p>
<p>$$\frac 1 2 x \le \sin x$$</p>
<p>for all $x \in [0,1]$; this leads to</p>
<p>$$\ln\left(\frac{x}{2}\right) \le \ln \sin x$$</p>
<p>almost everywhere on $[0,1]$. We'll actually use that</p>
<p>... |
865,293 | <blockquote>
<p>Prove $\ln[\sin(x)] \in L_1 [0,1].$</p>
</blockquote>
<p>Since the problem does not require actually solving for the value, my strategy is to bound the integral somehow. I thought I was out of this one free since for $\epsilon > 0$ small enough, $$\lim_{\epsilon \to 0}\int_\epsilon^1 e^{\left|\ln(... | Zarrax | 3,035 | <p>First observe that $\ln \sin x = \ln {\sin x \over x} + \ln x$. </p>
<p>The function ${\sin x \over x}$ is continuous and nonzero on on $[0,1]$ (if you extend it to equal $1$ at $x = 0$), so the same is true for $\ln {\sin x \over x}$ . Thus $\ln {\sin x \over x}$ is in $L^1[0,1]$. </p>
<p>The function $\ln x$ is ... |
4,241 | <p>I was preparing for an area exam in analysis and came across a problem in the book <em>Real Analysis</em> by Haaser & Sullivan. From p.34 Q 2.4.3, If the field <em>F</em> is isomorphic to the subset <em>S'</em> of <em>F'</em>, show that <em>S'</em> is a subfield of <em>F'</em>. I would appreciate any hints on ho... | Martin Brandenburg | 1,650 | <blockquote>
<p>Is there an analogous classification for fields of characteristic zero?</p>
</blockquote>
<p>Yes, but it is somewhat useless and nobody would call it a classification.</p>
<p>Every field of characteristic zero has the form $Quot(\mathbb{Q}[X]/S)$, where $X$ is a set of variables and $S$ is a set of ... |
1,660,794 | <p>Suppose $$a'(x)=b(x)$$ and $$b'(x)=a(x)$$</p>
<p>What is $$\int x \sin (x) a(x) dx$$</p>
<p>Thanks!</p>
| Ian | 83,396 | <p>For constants $a,b$ you have</p>
<p>$$\sum_{i=1}^n ai+b = a \sum_{i=1}^n i + b \sum_{i=1}^n 1 = \frac{a n(n+1)}{2} + bn.$$</p>
<p>You can set this equal to your given number and solve for $n$; if you get an integer then your given number was whatever-gonal.</p>
<p>I'm not sure if this <em>fully</em> answers your ... |
25,337 | <p>If you want to compute crystalline cohomology of a smooth proper variety $X$ over a perfect field $k$ of characteristic $p$, the first thing you might want to try is to lift $X$ to the Witt ring $W_k$ of $k$. If that succeeds, compute de Rham cohomology of the lift over $W_k$ instead, which in general will be much e... | BlaCa | 14,514 | <p>In general the obstruction to lift a scheme $X$ in characteristic zero is in $H^2(X,T_X)$. For examples of $3$-folds in positive characteristic that connot be lifted in characteristic zero you may look at Theorem 22.4 in Hartshorne's "Deformation Theory".</p>
|
113,446 | <p>Suppose a simple equation in Cartesian coordinate:
$$
(x^2+ y^2)^{3/2} = x y
$$
In polar coordinate the equation becomes $r = \cos(\theta) \sin(\theta)$. When I plot both, the one in polar coordinate has two extra lobes (I plot the polar figure with $\theta \in [0.05 \pi, 1.25 \pi]$ so the "flow" of the curve is cle... | Rashid | 39,584 | <p><code>PolarPlot</code> purposely accepts negative radii values as well as angles beyond the range 0 to 2$\pi$. See, for example, the <code>PolarPlot</code> <a href="https://reference.wolfram.com/language/ref/PolarPlot.html" rel="nofollow noreferrer">documentation here</a> showing <code>PolarPlot[Sin[3 t], {t, 0, Pi}... |
1,109,443 | <p>I'm currently learning for an algebra exam and I have some examples of questions from few last years. And I can't find a solution to this one:</p>
<blockquote>
<p>Give three examples of complex numbers where z = -z</p>
</blockquote>
<p>The only complex number I can think of is 0. Because it is a complex number, ... | Alec Teal | 66,223 | <p><strong>Answer:</strong></p>
<p>Base 2 numbers! If someone is $a$ of race A ($a\in[0,1]$) and someone else is $b\in[0,1]$ race A, their offspring is $\frac{1}{2}(a+b)=\frac{1}{2}a+\frac{1}{2}b$.</p>
<p>This immediately screams "base 2" because you'll get this recursive half pattern.</p>
<p>So let's write somethin... |
1,109,443 | <p>I'm currently learning for an algebra exam and I have some examples of questions from few last years. And I can't find a solution to this one:</p>
<blockquote>
<p>Give three examples of complex numbers where z = -z</p>
</blockquote>
<p>The only complex number I can think of is 0. Because it is a complex number, ... | Loren Pechtel | 23,794 | <p>It depends on how you define being Cherokee.</p>
<p>As others have shown, no possible <strong>normal</strong> breeding sequence can produce someone who is 1/12th.</p>
<p>However, we now have three-parent children (26 chromosomes from a male, 26 chromosomes from a female, an ovum with only the mitochondrial DNA fro... |
1,109,443 | <p>I'm currently learning for an algebra exam and I have some examples of questions from few last years. And I can't find a solution to this one:</p>
<blockquote>
<p>Give three examples of complex numbers where z = -z</p>
</blockquote>
<p>The only complex number I can think of is 0. Because it is a complex number, ... | fastmultiplication | 209,236 | <p>In a "continuous inheritance" model, no, because of the answers above.</p>
<p>In a naive chromosomal model, no, because 46 is not divisible by 12.</p>
<p>In a model which includes recombination, yes. Recombination (aka crossing over) between the parent's chromosomes results in a new chromosome type which has part... |
1,350,837 | <p>Find all integer numbers $n$, such that, $$\sqrt{\frac{11n-5}{n+4}}\in \mathbb{N}$$</p>
<p>I really tried but I couldn't guys, help please.</p>
| André Nicolas | 6,312 | <p>If our square root is to be an integer, we need to have $\frac{11n-5}{n+4}$ a non-negative integer. Note that
$$\frac{11n-5}{n+4}=11-\frac{49}{n+4}.$$
So $n+4$ must divide $49$. But $49$ has very few divisors, so there are very few possile integer values of $\frac{49}{n+4}$. Try them all, including the negative ones... |
1,140,212 | <blockquote>
<p>Prove that for every $n\times n$ matrices $A,B$: $$Tr((AB^2)A)=Tr(A^2B^2)$$</p>
</blockquote>
<p>I need a solution that doesn't use expansion. One more question comes into my mind: given $A,B$ are square matrices. For which condition of $A,B$ we can conclude that $Tr(AB)=Tr(BA)$?
Thanks in advance.</... | Brian Fitzpatrick | 56,960 | <p>Note that for $A,B\in M_{n\times n}$ we have
\begin{align*}
\DeclareMathOperator{trace}{trace}\trace(AB)
&= \sum_{k=1}^n [AB]_{kk} \\
&= \sum_{k=1}^n \sum_{j=1}^n[A]_{kj}[B]_{jk} \\
&= \sum_{j=1}^n \sum_{k=1}^n [B]_{jk}[A]_{kj} \\
&= \sum_{j=1}^n [BA]_{jj} \\
&= \trace(BA)
\end{align*}</p>
|
1,140,212 | <blockquote>
<p>Prove that for every $n\times n$ matrices $A,B$: $$Tr((AB^2)A)=Tr(A^2B^2)$$</p>
</blockquote>
<p>I need a solution that doesn't use expansion. One more question comes into my mind: given $A,B$ are square matrices. For which condition of $A,B$ we can conclude that $Tr(AB)=Tr(BA)$?
Thanks in advance.</... | Surender Sharma | 450,906 | <p>$${\rm Tr}(AB^2A) = {\rm Tr} \big((AB^2) A \big) = {\rm Tr} \big(A (AB^2) \big) = {\rm Tr}(A^2B^2)$$</p>
<p>using the property ${\rm Tr}(XY) = {\rm Tr}(YX)$.</p>
|
2,618,273 | <p>In integer-base positional numeral systems, the notation of a number in base $n$ uses $n$ numerals. Base 2 uses the symbols 0 and 1, base 10 uses 0123456789, base 16 uses base 10 + ABCDEF. Although the choice of symbols for the numerals is arbitrary, the number of numerals (unique glyphs) is identical to the base.... | John Bentin | 875 | <p>For base $\tau$, just $\lceil\tau\rceil$ numerals suffice. For example, for $\tau=\phi,\mathrm e,\pi$, correspondingly two, three, and four numerals are required.</p>
|
3,172,693 | <p>Can anybody help me with this equation? I can't find a way to factorize for finding a value of <span class="math-container">$d$</span> as a function of <span class="math-container">$a$</span>:</p>
<p><span class="math-container">$$d^3 - 2\cdot d^2\cdot a^2 + d\cdot a^4 - a^2 = 0$$</span></p>
<p>Another form:</p>
... | Count Iblis | 155,436 | <p>One can immediately see why in this case the partial fraction expansion will lead to a nonzero coefficient for the <span class="math-container">$1/s$</span> term. The asymptotic behavior of the fraction for large <span class="math-container">$s$</span> is <span class="math-container">$\sim 1/s^3$</span>. The singula... |
2,529,533 | <p>Let $f:\mathbf{R}^n \to \mathbf{R}$ be differentiable, $\sum_{i=1}^n y_i \frac{\partial f}{\partial x_i}(y)\geq 0$ for all $y=(y_1,...,y_n)\in \mathbf{R}^n$. How do I show that $f$ is bounded from below by $f(0)$?</p>
| Zach Boyd | 60,023 | <p>Hint:</p>
<p>The expression from your question is the derivative of f with respect to the radius squared. So if that is positive, then...</p>
|
2,515,939 | <p>So, I just need a hint for proving
$$\lim_{n\to \infty} \int_0^1 e^{-nx^2}\, dx = 0$$ </p>
<p>I think maybe the easiest way is to pass the limit inside, because $e^{-nx^2}$ is uniformly convergent on $[0,1]$, but I'm new to that theorem, and have very limited experience with uniform convergence. Furthermore, I don... | Dylan | 135,643 | <p>$$ \int_0^1 e^{-nx^2} dx < \int_0^{\infty} e^{-nx^2} dx = \sqrt{\frac{\pi}{4n}} \to 0 $$</p>
|
1,595,946 | <blockquote>
<p>Let $f:(a,b)\to\mathbb{R}$ be a continuous function such that
$\lim_\limits{x\to a^+}{f(x)}=\lim_\limits{x\to b^-}{f(x)}=-\infty$.
Prove that $f$ has a global maximum.</p>
</blockquote>
<p>Apparently, this is similar to the EVT and I believe the proof would be similar, but I cannot think anything... | Wojowu | 127,263 | <p>Let $x_0\in(a,b)$ be arbitrary. From our assumptions, there exist points $a_1,b_1$ satisfying $a<a_1<x_0<b_1<b$ such that $f(x)<f(x_0)$ for $x\in(a,a_1)$ or $x\in(b_1,b)$. Also, by EVT, $f(x)$ has a maximum on interval $[a_1,b_1]$, say that $f(M)$ is this maximum. I claim this is the global maximum of... |
2,329,542 | <p>I looked up wikipedia but honestly I could not make much sense of what I will basically study in Abstract Algebra or what it is all about .</p>
<p>I also looked up a question here :
<a href="https://math.stackexchange.com/questions/855828/what-is-abstract-algebra-essentially">What is Abstract Algebra essentially?</... | Evargalo | 443,536 | <p>If I had to explain it to my 6yo daughter, I would tell her that "Abstract Algebra" is about how we invented numbers, and why we invented them that way.</p>
<p>She probably wouldn't understand and tell me to let her play.</p>
|
2,329,542 | <p>I looked up wikipedia but honestly I could not make much sense of what I will basically study in Abstract Algebra or what it is all about .</p>
<p>I also looked up a question here :
<a href="https://math.stackexchange.com/questions/855828/what-is-abstract-algebra-essentially">What is Abstract Algebra essentially?</... | user8181651 | 456,229 | <p>You take some kind of set and you define some operations on it so that they have some nice properties that you find interesting . Then you try to learn what happens. Operations, rules and sets are often quite general. You can apply what you've found at things that are not numbers, like set of permutations or rotatio... |
3,828,003 | <blockquote>
<p>Show that <span class="math-container">$G=\{0,1,2,3\}$</span> over addition modulo 4 is isomorphic to <span class="math-container">$H=\{1,2,3,4\}$</span> over multiplication modulo 5</p>
</blockquote>
<p>My solution was to brute force check validity of <span class="math-container">$f(a+b)=f(a)f(b)$</spa... | Community | -1 | <p>In general, you need to prove the existence of a bijective homomorphism between the two groups.</p>
<p>In practice, there is only one cyclic group of each order, <span class="math-container">$\Bbb Z_n$</span>. Here can use that fact to establish the result.</p>
<p>To wit, <span class="math-container">$\Bbb Z_p^\tim... |
43,355 | <p>I recently came across the following formula, which is apparently known as <em>Laplace's summation formula:</em></p>
<p>$$\int_a^b f(x) dx = \sum_{k=a}^{b-1} f(k) + \frac{1}{2} \left(f(b) - f(a)\right) - \frac{1}{12} \left(\Delta f(b) - \Delta f(a)\right) $$
$$+ \frac{1}{24} \left( \Delta^2 f(b) - \Delta^2 f(a) \... | Gerry Myerson | 3,684 | <p>Did you try the Online Encyclopedia of Integer Sequences? </p>
<p><a href="http://oeis.org/A006232" rel="nofollow">http://oeis.org/A006232</a> </p>
<p>Perhaps some of the references there will get you where you want to go. </p>
|
43,355 | <p>I recently came across the following formula, which is apparently known as <em>Laplace's summation formula:</em></p>
<p>$$\int_a^b f(x) dx = \sum_{k=a}^{b-1} f(k) + \frac{1}{2} \left(f(b) - f(a)\right) - \frac{1}{12} \left(\Delta f(b) - \Delta f(a)\right) $$
$$+ \frac{1}{24} \left( \Delta^2 f(b) - \Delta^2 f(a) \... | Tyler Clark | 10,918 | <p>This is a bit late - I could be completely wrong, but I think the issue here is the domain being used.</p>
<p>Laplace's summation formula should be used on the set of integers and will be used for calculations in discrete calculus. I believe that the Euler-Maclaurin summation formula is typically used on the reals ... |
3,450,283 | <p>I confronted with a statement: </p>
<p>Given a ring homomorphism <span class="math-container">$f:A\to B$</span>, with commutative rings with identity <span class="math-container">$A,B$</span>. If <span class="math-container">$A,B$</span> are both subrings of a bigger commutative ring with identity <span class="mat... | Yuyi Zhang | 602,131 | <p>Generally, a ring homomorphsim <span class="math-container">$f:A\to B$</span> within a bigger ring cannot deduce that <span class="math-container">$A$</span> is a subring of <span class="math-container">$B$</span>. But under noetherian hypothesis, if <span class="math-container">$A$</span> and <span class="math-cont... |
441,888 | <p>I should clarify that I'm asking for intuition or informal explanations. I'm starting math and never took set theory so far, thence I'm not asking about formal set theory or an abstract hard answer. </p>
<p>From Gary Chartrand page 216 Mathematical Proofs - </p>
<p>$\begin{align} \text{ range of } f & = \{f(x... | Mauro ALLEGRANZA | 108,274 | <p>About <em>Question 1)</em>, basically, the formula :</p>
<blockquote>
<p>$ \{$ odd numbers $\} = \{ n \in \mathbb{N} : \exists \, k \in \mathbb{N} \; (n = 2k+1 ) \; \}$</p>
</blockquote>
<p>is a shorthand for the formal :</p>
<p>$\{ n : n \in \mathbb{N} \quad \land \quad \exists \, k \; ( k \in \mathbb{N} \land... |
112,096 | <p>Does the inequality $2 \langle x , y \rangle \leqslant \langle x , x \rangle + \langle y , y \rangle $, where $$ \langle \cdot, \cdot \rangle $$ denotes scalar product, have a name? </p>
<p>I've tried looking at several inequalities on wikipedia but I didn't find this one. And of course googling doesn't work for ... | Harald Hanche-Olsen | 23,290 | <p>It is essentially <a href="http://en.wikipedia.org/wiki/Young%27s_inequality" rel="nofollow">Young's inequality</a>.</p>
|
636,391 | <p>Evaluate the following indefinite integral.</p>
<p>$$\int { \frac { x }{ 4+{ x }^{ 4 } } }\,dx$$</p>
<p>In my homework hints, it says let $ u = x^2 $. But still i can't continue.</p>
| Berci | 41,488 | <p><strong>Hint:</strong> If $u=x^2$ then $x=\sqrt u$ and $du=2x\,dx$.</p>
|
636,391 | <p>Evaluate the following indefinite integral.</p>
<p>$$\int { \frac { x }{ 4+{ x }^{ 4 } } }\,dx$$</p>
<p>In my homework hints, it says let $ u = x^2 $. But still i can't continue.</p>
| Dan | 79,007 | <p>Hint: You've substituted $u = x^2$ and found that your original integral becomes</p>
<p>$$
\int\frac{\sqrt u}{4+u^2} \frac{du}{2x},
$$</p>
<p>but you haven't completed the substitution; there's still an $x$ in your integrand. How can you rewrite the $2x$ below the $du$ as a function of $u$? Once you rewrite $2x$ i... |
636,391 | <p>Evaluate the following indefinite integral.</p>
<p>$$\int { \frac { x }{ 4+{ x }^{ 4 } } }\,dx$$</p>
<p>In my homework hints, it says let $ u = x^2 $. But still i can't continue.</p>
| Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}%
\newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}%
\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%
\newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}%
\newcommand{\dd}{{\rm d}}%
\newcommand{\down}{\downarrow}... |
685,567 | <p>For the function, $f(x,y,z)=\sqrt{x^2+y^2+z^2}$, do directional derivatives exist at the origin? If I use the definition $$lim_{h\to 0}\frac{f(x+hv)-f(x)}{h},$$ then I get $$\frac{|h|}{h}$$ which is without limit. But in some places, I keep reading that the directional derivative is 1. </p>
<p>Also, if I were to wr... | Live Free or π Hard | 126,067 | <p>You are not correct. The sum of <em>n</em> Poisson random variables, with parameter <em>k</em>, say, is also a Poisson random variable with parameter <em>nk</em>, <strong>if the random variables are independent</strong>. That is, if X_<em>i</em> are independent Poisson random variables with parameter k_i then their ... |
664,349 | <blockquote>
<p>If $G$ is a finite group where every non-identity element is generator of $G$, what is the order of $G$?</p>
</blockquote>
<p>I know that the order of $G$ must be prime, but I'm not sure how to go about proving this from the problem statement. </p>
<p>Any hints on where to start?</p>
| Nicky Hekster | 9,605 | <p>Using Cauchy's Theorem: let $p$ be a prime dividing $|G|$. Then $G$ has an element $g$ of order $p$. Apparently this element $g$ generates $G$. Hence $|G|=p$.</p>
|
697,336 | <p>Let $ABCD$ be a trapezoid, such that $AB$ is parallel to $CD$. Through $O$, the intersection point of the diagonals $AC$ and $BD$ consider a parallel line to the bases. This line meets $AD$ at $M$ and $BC$ at $N$. </p>
<p>Prove that $OM=ON$ and: $$\frac{2}{MN}=\frac1{AB}+\frac1{CD}$$</p>
| André Nicolas | 6,312 | <p><strong>First problem:</strong> We use an area argument. Draw the trapezoid, with $A,B,C,D$ going counterclockwise, and $AB$ a horizontal line at the "bottom." (We are doing this so we will both be looking at the same picture.)</p>
<p>Note that $\triangle ABC$ and $\triangle ABD$ have the same area. (Same base $AB... |
418,748 | <p>I tried to calculate, but couldn't get out of this:
$$\lim_{x\to1}\frac{x^2+5}{x^2 (\sqrt{x^2 +3}+2)-\sqrt{x^2 +3}}$$</p>
<p>then multiply by the conjugate.</p>
<p>$$\lim_{x\to1}\frac{\sqrt{x^2 +3}-2}{x^2 -1}$$ </p>
<p>Thanks!</p>
| Sujaan Kunalan | 77,862 | <p>Use L'Hospital's Rule. Since plugging in $x=1$, gives you indeterminate form, take the derivative of the numerator and the derivative of the denominator, and try the limit again.</p>
<p>$\lim_{x\to 1}\frac{(x^2+3)^{\frac{1}{2}}-2}{x^2-1}\implies$ (Via L
Hospital's Rule...) $\lim_{x\to 1}\frac{\frac{1}{2}(x^2+3)^{-\... |
1,390,676 | <p>A quasigroup is a pair $(Q,/)$, where $/$ is a binary operation on $Q$, such that (1) for each $a,b\in Q$ there exists unique solutions to the equations
$a/x=b$ and $y/a=b$.</p>
<p>Now I want to extract a class of quasigroups that captures characteristics from $(Q_+,/)$, where $Q_+$ is the set of positive rational ... | Michael Kinyon | 444,012 | <p>Quasigroups constructed from the right division operation in a group (<span class="math-container">$x/y = xy^{-1}$</span>) are called <em>Ward quasigroups</em>. They are characterized by the identity <span class="math-container">$(x/y)/(z/y)=x/z$</span>, that is, if <span class="math-container">$(Q,/)$</span> is a q... |
1,689,923 | <p>I have a sequence $a_{n} = \binom{2n}{n}$ and I need to check whether this sequence converges to a limit without finding the limit itself. Now I tried to calculate $a_{n+1}$ but it doesn't get me anywhere. I think I can show somehow that $a_{n}$ is always increasing and that it has no upper bound, but I'm not sure i... | S.C.B. | 310,930 | <p>Note that if $n \ge 1$ then $$\frac{a_{n+1}}{a_{n}}=\frac{(2n+1)(2n+2)}{(n+1)(n+1)}=2\frac{(2n+1)}{n+1} >2$$ </p>
<p>The series diverges. </p>
|
678,768 | <p>"Let $A$, $B$ be two infinite sets. Suppose that $f: A \to B$ is injective. Show that there exists a surjective map $g: B \to A$"</p>
<p>I am not sure how to go about this proof, I am trying to gather information to help me, and deduce as much as I can:
. Since $f$ is injective we know that $|A| \leq |B|$. </p>
<p... | Asaf Karagila | 622 | <p>You're going at it <strong>all</strong> wrong.</p>
<p>Consider the case $f\colon\Bbb N\to\Bbb R$ defined by $f(n)=n$. That is an injective function, but certainly not surjective. Can you think about a surjection from $\Bbb R$ onto $\Bbb N$? For example, $g(x)=\begin{cases} x&x\in\Bbb N\\ 0&x\notin\Bbb N\end... |
4,489,675 | <p>When saying that in a small time interval <span class="math-container">$dt$</span>, the velocity has changed by <span class="math-container">$d\vec v$</span>, and so the acceleration <span class="math-container">$\vec a$</span> is <span class="math-container">$d\vec v/dt$</span>, are we not assuming that <span class... | WillO | 29,145 | <p>You wrote that you haven't studied calculus.</p>
<p>Okay, then. Do not think of <span class="math-container">$dt$</span> and <span class="math-container">$dv$</span> as numbers. Instead, think of the whole expression <span class="math-container">$dv/dt=a$</span> as an abbreviation for "if <span class="math-co... |
119,589 | <p>I have been given M be an $m\times n$ matrix. I have to show that the matrix $M^TM$ is symmetric positive definite if and only if the columns of the matrix $M$ are linearly independent</p>
<p>My thoughts are as below $M^TM$ looks like a cholesky property and we know it is applicable as the matrix is SPD, still not ... | shuhalo | 3,557 | <p>It is obvious that $M^T M$ is symmetric.</p>
<p>Suppose the matrix $M^T M$ is SPD but $M$ does not have linearly independent columns. Then $M$ is not injective, and consequently $M^T M$ is not definite.</p>
<p>Suppose $M$ has linearly indepedent columns, then $M^T$ has linearly indepedent rows. For any $x$ let $y ... |
119,589 | <p>I have been given M be an $m\times n$ matrix. I have to show that the matrix $M^TM$ is symmetric positive definite if and only if the columns of the matrix $M$ are linearly independent</p>
<p>My thoughts are as below $M^TM$ looks like a cholesky property and we know it is applicable as the matrix is SPD, still not ... | Inquest | 35,001 | <p><em>If and only if</em> proofs go 2 ways:
Since this is homework, I'll throw in some hints:</p>
<p>Given that columns are independent, prove $M^TM$ is symmetric positive definite.</p>
<p>For any matrix (Singular or not), $M^TM$ is always symmetric. (Try transposing it to see what you get).</p>
<p>Now, if columns ... |
2,135,717 | <p>Let $G$ be an Abelian group of order $mn$ where $\gcd(m,n)=1$. </p>
<p>Assume that $G$ contains an element of $a$ of order $m$ and an element $b$ of order $n$. </p>
<p>Prove $G$ is cyclic with generator $ab$.</p>
<hr>
<p>The idea is that $(ab)^k$ for $k \in [0, \dots , mn-1]$ will make distinct elements but do ... | Maxime Ramzi | 408,637 | <p>Let $k\in \mathbb{N}$ and assume $(ab)^k = e$. Then, since $G$ is abelian, $a^k = b^{-k}$. Raising this to the power $n$, we get $a^{nk} = e$, so $m| nk$. But $m,n$ are coprime, so by Gauss's theorem, $m|k$. Witha similar argument, $n|k$, and since again $m,n$ are coprime, we get $nm |k$. Conversely, we obviously ha... |
1,199,912 | <p>What is the optimal (i.e., smallest) constant $\alpha$ such that, given 19 points on a solid, regular hexagon with side 1, there will always be 2 points with distance at most $\alpha$?</p>
<p>This is a reformulation of an <a href="https://math.stackexchange.com/questions/1196787/pigeonhole-problem-about-distance-be... | Jack D'Aurizio | 44,121 | <p>$\alpha=\frac{1}{2}$ is the optimal bound. We just need to prove that for every $\varepsilon>0$ we can take $19$ points in the hexagon such that the distance between any two of them is $\geq\frac{1}{2}-\varepsilon$. Easily done: we place $19$ points in a regular hexagon width side length $1-\delta$ accordingly to... |
3,717,144 | <p>Suppose M is a finitely generated non-zero R-module, where R is a commutative unital ring. Show that the tensor product of M with itself is non-zero.</p>
<p>I know one way to show this is to find an R-bilinear map which is nonzero, but am not sure how to find it.</p>
| GreginGre | 447,764 | <p>This is false. Let <span class="math-container">$R=\mathbb{C}[X]/(X^2)$</span>. Let <span class="math-container">$x=\bar{X}$</span>, so <span class="math-container">$x^2=0$</span>, and let <span class="math-container">$M= R x$</span>. Then <span class="math-container">$M$</span> is a finitely generated nonzero <span... |
9,513 | <p>I'm a very novice user of <em>Mathematica</em> - is there possibility of exporting Mathematica code directly into $\LaTeX$? I'm interested only in exporting mathematical formulas. Also, from which version of program is it possible?</p>
| cormullion | 61 | <p>The Mathematica help system has useful information, and links to videos etc.</p>
<p><img src="https://i.stack.imgur.com/JuOib.png" alt="help screen"></p>
|
3,962,514 | <p>This is just curiosity / a personal exercise.</p>
<p><a href="https://what3words.com/" rel="nofollow noreferrer">What3Words</a> allocates every 3m x 3m square on the Earth a unique set of 3 words. I tried to work out how many words are required, but got a bit stuck.</p>
<p><span class="math-container">$$
Area
= ... | Will Jagy | 10,400 | <p>I like the backwards method: choose the generalized eigenvector(s) with integer elements and see what is forced. Since <span class="math-container">$A-I$</span> gives two genuine eigenvectors, we hold off on that... The minimal polynomial gives the size of the largest Jordan block (always!). That is, <span class=... |
777,863 | <p>Does there exists an $f:\mathbb{R}\rightarrow \mathbb{R}$ differentiable everywhere with $f'$ discontinuous at some point?</p>
| Ross Millikan | 1,827 | <p>Yes. Do you know an example of a continuous function that is not differentiable at some point? (Hint: think of a corner) If you integrate it....</p>
|
184,699 | <p>First, we make the following observation: let $X: M \rightarrow TM $ be a vector
field on a smooth manifold. Taking the contraction with respect to $X$ twice gives zero, i.e.
$$ i_X \circ i_{X} =0.$$
Is there any "name" for the corresponding "homology" group that one can define
(Kernel mod image)? Has this "homo... | Joonas Ilmavirta | 55,893 | <p>If the vector field $X$ never vanishes, the homology corresponding to $i_X$ is trivial.
Suppose $\alpha\in\Omega^p(M)$ is in the kernel of $i_X$.
If we take $\beta=X^\flat\wedge\alpha$, we have $i_X\beta=i_X(X^\flat)\wedge\alpha=|X|^2\alpha$.
Thus if $X$ does not vanish, we have $\alpha=i_X(|X|^{-2}X^\flat\wedge\alp... |
1,728,097 | <p>So i have this integral : $$ \int_0^\infty e^{-xy} dy = -\frac{1}{x} \Big[ e^{-xy} \Big]_0^\infty$$
The integration part is fine, but I'm not sure what i get with the limits, can someone explain this</p>
<p>Thanks </p>
| Doug M | 317,162 | <p>\begin{align}
\int_0^\infty e^{-xy} dy &= \lim_\limits{n\to \infty} \int_0^n e^{-xy} dy
\\ &= \dfrac{e^0}{x} - \lim_\limits{n\to \infty} \dfrac{e^{-xn}}{x}
\end{align}</p>
<p>And that limit is going to $0$.</p>
<p>$\dfrac{1}{x}$</p>
|
341,202 | <p>The divisibility rule for $3$ is well-known: if you add up the digits of $n$ and the sum is divisible by $3$, then $n$ is divisible by three. This is quite helpful for determining if really large numbers are multiples of three, because we can recursively apply this rule:</p>
<p>$$1212582439 \rightarrow 37 \rightar... | Michael Hardy | 11,667 | <p>More generally, a number and the sum of its digits both leave the same remainder on division by $3$. For example: $245$ $\mapsto 2+4+5=11$ $\mapsto1+1=2$, so the remainder when $245$ is divided by $3$ is $2$.</p>
<p>If you know modular arithmetic, this is straightforward:
\begin{align}
245 & = 2\cdot10^2 +4\cd... |
341,202 | <p>The divisibility rule for $3$ is well-known: if you add up the digits of $n$ and the sum is divisible by $3$, then $n$ is divisible by three. This is quite helpful for determining if really large numbers are multiples of three, because we can recursively apply this rule:</p>
<p>$$1212582439 \rightarrow 37 \rightar... | rnjai | 42,043 | <p><strong>Let abc be a 3 digit number divisible by 3.</strong><br>
Then:
$$(100a+10b+c)|3=0$$
or
$$(100|3)(a|3)+(10|3)(b|3)+(c|3)=0$$
Hence
$$(a+b+c)|3=0$$</p>
|
341,202 | <p>The divisibility rule for $3$ is well-known: if you add up the digits of $n$ and the sum is divisible by $3$, then $n$ is divisible by three. This is quite helpful for determining if really large numbers are multiples of three, because we can recursively apply this rule:</p>
<p>$$1212582439 \rightarrow 37 \rightar... | Zubin Mukerjee | 111,946 | <p>For $n \in \mathbb{N}$, let $m = \overline{a_0a_1a_2\cdots a_{n-1}}$ be an $n$-digit natural number, where the $a_i$ are digits (natural numbers between $0$ and $9$, inclusive).</p>
<p>Then</p>
<p>$$ m = \displaystyle\sum\limits_{k=0}^{n-1}10^k \cdot a_k = a_0 + 10a_1 + 100a_2 + \cdots + 10^{n-1}a_{n-1}$$</p>
<p>... |
341,202 | <p>The divisibility rule for $3$ is well-known: if you add up the digits of $n$ and the sum is divisible by $3$, then $n$ is divisible by three. This is quite helpful for determining if really large numbers are multiples of three, because we can recursively apply this rule:</p>
<p>$$1212582439 \rightarrow 37 \rightar... | voldemort | 118,052 | <p>Hint: Represent your number as $10^na_n+\cdots 10a_1+a_0$ where the $a_i's$ are the digit of your number. Now, note that the remained we get when we divide $10^k$ by $3$ is $1$. </p>
|
563,431 | <p>Find the absolute maximum and minimum of $f(x,y)= y^2-2xy+x^3-x$ on the region bounded by the curve $y=x^2$ and the line $y=4$. You must use Lagrange Multipliers to study the function on the curve $y=x^2$.</p>
<p>I'm unsure how to approach this because $y=4$ is given. Is this a trick question?</p>
| user113578 | 113,578 | <p>Forget Sylow theorem. Note $52=2.26$ and consider $$\mathbb Z_2\oplus\mathbb Z_{26}$$</p>
<p>Of course this one is abelian (the components being so). Is it possible to find $(a,b)\in\mathbb Z_2\oplus\mathbb Z_{26}$ such that $l.c.m.\{|a|,|b|\}=52?$</p>
|
2,987,994 | <p>I'm looking for definite integrals that are solvable using the method of differentiation under the integral sign (also called the Feynman Trick) in order to practice using this technique.</p>
<p>Does anyone know of any good ones to tackle?</p>
| Diger | 427,553 | <p><span class="math-container">$I_9$</span> in Zacky's answer is a special case to
<span class="math-container">$$I(s)=\int_0^\infty \frac{x^s}{x^2+1} \, {\rm d}x \tag{1}$$</span> with <span class="math-container">$-1<\Re(s)<1$</span>, i.e.
<span class="math-container">$$I_9 = \int_{2/3}^{4/5} I(s) \, {\rm d}s \... |
1,131,970 | <p>Let $I$ be a proper ideal of a polynomial ring $A$ and $x \in A$ an irreducible element.</p>
<p>In a theorem of commutative algebra I will use the fact that, in this hypothesis, holds the following equality: $$\sqrt{(I,x^k)}=\sqrt{(\sqrt{I},x)}$$</p>
<p>The assert seems to be true, anyone has any counterexample/p... | Lubin | 17,760 | <p>The problem is that if $a=0\in\Bbb Z/p^k\Bbb Z$, then <em>of course</em> $X^2-a$ has a solution in $\Bbb Z/p^k\Bbb Z$. Just avoid this situation, and everything goes smoothly.</p>
<p>So the strongest result is this: If $a$ is prime to $p$ and $X^2-a$ has a solution in $\Bbb Z/p\Bbb Z$, i.e. if $a$ is a nonzero squa... |
2,386,182 | <p>Let $f:[a,b]\to \mathbb{R}$ is an increasing function and for any $y\in[f(a),f(b)],$ there exists a $\xi\in[a,b]$ such that $f(\xi)=y.$ Show that $f(.)$ is continuous on $[a,b]. $</p>
<p>Here is my argument, but I got stuck on the last part.</p>
<p>My goal is to show $$\lim_{n\to \infty}f(x_0+\xi_n)=\lim_{n\to\inf... | Alex Ravsky | 71,850 | <p>Being officially a guy which your are trying hard to become, :-) answering your question I feel obliged to tell you the truth: :-) experience is the best of the teachers. :-) Sometimes you may find a problem solver handbook, helpful for a specific topic, but it won’t be solving problems for you. :-) </p>
<p>Concern... |
4,382,786 | <p>I would like to know what is the following process on the real line called.</p>
<p>Let us fix some <span class="math-container">$X_0$</span> and let <span class="math-container">$X_{i+1} = (1-\gamma)X_i + Y_i$</span> where <span class="math-container">$\gamma$</span> is a fixed real number and <span class="math-cont... | user2316602 | 187,745 | <p>It is called an <em>autoregressive process of order 1</em>. For more information, you can see <a href="https://en.wikipedia.org/wiki/Autoregressive_model" rel="nofollow noreferrer">this Wikipedia page</a> or the textbook "Time Series Analysis: Forecasting and Control".</p>
|
188,150 | <p>I'm reading the paper <em>Loop groups and twisted K-theory I</em> by Freed, Hopkins, and Teleman. They give some examples of computing (twisted) K groups using the Mayer-Vietoris sequence. </p>
<p>I'm a bit confused with some of their computations, for instance $S^3$ (their example 1.4 in the first section). The... | André Henriques | 5,690 | <p>"since $U_\pm$ are non-compact so I believe $K_0(U_\pm)$ should be the reduced $K_0$ of a 1-point compactification"</p>
<p>This is a convention often used in $K$-theory of $C^*$-algebras: by default, people take "$K$-theory" to mean compactly supported $K$-theory.</p>
<p>But that this <i>not</i> the convention use... |
474,568 | <p>In some books I've seen this symbol $\dagger$, next to some theorem's name, and I don't know what it means. I've googled it with no results which makes me suspect it's not standard.</p>
<p>Does anybody know what it means? One example I'm looking at right now is in a probability book, next to a section about Sitrlin... | Cameron Buie | 28,900 | <p>It is often simply used as an alternative to an asterisk, or a footnote notation.</p>
|
597,899 | <p>Let $\epsilon > 0$ be given. Suppose we have that $$a - \epsilon < F(x) < a + \epsilon$$</p>
<p>Does it follow that $a - \epsilon < F(x) \leq a $ ??</p>
| David Holden | 79,543 | <p>strip away the irrelevant context, and the question becomes: if $c \gt 0$ and $a \lt b+c$ does it follow that $a \le b$? why on earth should it???</p>
<p>on the other hand if OP's real question is this:</p>
<p>if $\forall c \gt 0$ we have $a \lt b+c$ does this imply $a \le b$ then the answer is: yes.</p>
|
49,068 | <p>Given lists $a$ and $b$, which represent multisets, how can I compute the complement $a\setminus b$?</p>
<p>I'd like to construct a function <code>xunion</code> that returns the symmetric difference of multisets.
For example, if $a=\{1, 1, 2, 1, 1, 3\}$ and $b=\{1, 5, 5, 1\}$, then their symmetric difference is $\b... | Mr.Wizard | 121 | <p>I believe this question is nearly a duplicate of <a href="https://mathematica.stackexchange.com/q/18100/121">Removing elements from a list which appear in another list</a> but since this one allows other, potentially better, solutions it should not be closed.<br>
To illustrate, using Leonid's <code>unsortedComplemen... |
3,527,785 | <p>I'm reading James Anderson's <em>Automata Theory with Modern Applications. Here:</em></p>
<blockquote>
<p><a href="https://i.stack.imgur.com/sFWNh.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/sFWNh.png" alt="enter image description here" /></a></p>
<p><a href="https://i.stack.imgur.com/k9zne.pn... | MJD | 25,554 | <p>If <span class="math-container">$ab=cd$</span>, and <span class="math-container">$a≠c$</span>, then one of <span class="math-container">$a$</span> and <span class="math-container">$c$</span> is shorter and one is longer. The shorter is a prefix of the longer, therefore the code is not a prefix code.</p>
|
300,753 | <p>Revision in response to early comments. Users of set theory need an <em>implementation</em> (in case "model" means something different) of the axioms. I would expect something like this:</p>
<blockquote>
<p>An <em>implementation</em> consists of a "collection-of-elements" $X$, and a relation
(logical pairing)... | Zuhair Al-Johar | 95,347 | <p>I think what you mean when you said that $x \in A$ must be a logical <em>"function"</em>, is that it is an assignment that sends a pair of sets to a truth value, of course each object in each pair is a set that can substitute the symbol $x$ or the symbol $A$, in this sense $x \in A$ is called a "propositional functi... |
3,382 | <p>A couple of years ago I found the following continued fraction for $\frac1{e-2}$:</p>
<p>$$\frac{1}{e-2} = 1+\cfrac1{2 + \cfrac2{3 + \cfrac3{4 + \cfrac4{5 + \cfrac5{6 + \cfrac6{7 + \cfrac7{\cdots}}}}}}}$$</p>
<p>from fooling around with the well-known continued fraction for $\phi$. Can anyone here help me figure o... | Américo Tavares | 752 | <p>Euler proved in "<em>De Transformatione Serium in Fractiones Continuas</em>" <strong>Reference: The Euler Archive, Index number E593</strong> (On the Transformation of Infinite Series to Continued Fractions) [Theorem VI, §40 to §42] that</p>
<p><span class="math-container">$$s=\cfrac{1}{1+\cfrac{2}{2+\cfra... |
3,382 | <p>A couple of years ago I found the following continued fraction for $\frac1{e-2}$:</p>
<p>$$\frac{1}{e-2} = 1+\cfrac1{2 + \cfrac2{3 + \cfrac3{4 + \cfrac4{5 + \cfrac5{6 + \cfrac6{7 + \cfrac7{\cdots}}}}}}}$$</p>
<p>from fooling around with the well-known continued fraction for $\phi$. Can anyone here help me figure o... | J. M. ain't a mathematician | 498 | <p>Another possibility: remember that the numerators and denominators of successive convergents of a continued fraction can be computed using a three term recurrence.</p>
<p>For a continued fraction</p>
<p>$$b_0+\cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\dots}}$$</p>
<p>with nth convergent $\frac{C_n}{D_n}$, the recurrence</p... |
78,341 | <p>I believe I read somewhere that residually finite-by-$\mathbb{Z}$ groups are residually finite. That is, if $N$ is residually finite with $G/N\cong \mathbb{Z}$ then $G$ is residually finite.</p>
<p>However, I cannot remember where I read this, and nor can I find another place which says it. I was therefore wonderin... | YCor | 14,094 | <p>This is not true if $N$ is not assumed f.g. E.g. the wreath product of a nonabelian finite group $H$ by the integers is not residually finite (Gruenberg 1957, can be checked as a exercise). Here $N$ is an infinite direct sum of copies of $H$ (shifted by the action of the integers) and is residually finite.</p>
|
3,482,476 | <p>For an arbitrary <span class="math-container">$0\leqslant x \leq\frac{\pi^2}6$</span>, can we write <span class="math-container">$x$</span> in the form
<span class="math-container">$$
x = x_0+\sum_{j\in S\subset\mathbb N\setminus\{0\}} \frac1{j^2}, \tag 1
$$</span></p>
<p>where <span class="math-container">$x_0\in... | Milo Brandt | 174,927 | <p>It's neither discrete nor do its possible values cover the entire interval. We can essentially solve this question via the following lemma:</p>
<blockquote>
<p><strong>Lemma:</strong> Let <span class="math-container">$s_n$</span> be any sequence of non-negative real numbers with the property that <span class="mat... |
1,142,624 | <p>Find a generating function for $\{a_n\}$ where $a_0=1$ and $a_n=a_{n-1} + n$</p>
| Samrat Mukhopadhyay | 83,973 | <p><strong>Hint</strong> $a_{n}-a_0=\sum_{i=1}^{n}(a_i-a_{i-1})$</p>
|
1,814,216 | <p>I was trying to show that $\sin(x)$ is non-zero for integers $x$ other than zero and I thought that this result might emerge as a corollary if I managed to show that the result in question is true. </p>
<p>I think it's possible to demonstrate this by looking at the power series expansion of $\sin(x)$ and assuming t... | 5xum | 112,884 | <p>It is not bijective because it is not surjective. There is not $x\in\mathbb N$ such that $\sin(x)=2$.</p>
<p>However, it is true that $\sin(x)=0$ for only one value of $x\in\mathbb N$. This is because </p>
<p>$$\forall x\in \mathbb R:\sin x = 0\iff x=k\pi$$</p>
<p>for some $k\in\mathbb N$, and $k\pi\in\mathbb N\i... |
1,814,216 | <p>I was trying to show that $\sin(x)$ is non-zero for integers $x$ other than zero and I thought that this result might emerge as a corollary if I managed to show that the result in question is true. </p>
<p>I think it's possible to demonstrate this by looking at the power series expansion of $\sin(x)$ and assuming t... | Community | -1 | <p>Bijective implies surjective and injective. $\sin: \Bbb{N} \to \Bbb{R}$ is not surjective since $\sin(x) = 1 \implies x = \pi/2 + 2\pi n \notin \Bbb{N}$.</p>
|
1,814,216 | <p>I was trying to show that $\sin(x)$ is non-zero for integers $x$ other than zero and I thought that this result might emerge as a corollary if I managed to show that the result in question is true. </p>
<p>I think it's possible to demonstrate this by looking at the power series expansion of $\sin(x)$ and assuming t... | Mariano Suárez-Álvarez | 274 | <p>Suppose that we define the function $\def\RR{\mathbb R}f:\mathbb R\to\mathbb R$ putting $$f(x)=\sum_{n=0}^{\infty} \frac{(-1)^nx^{2n+1}}{(2n+1)!}$$ for each $x\in\mathbb R$, after checking that the power series converges for all real $x$. It is easy to compute derivatives of functions defined by power series, and th... |
3,527,004 | <p>As stated in the title, I want <span class="math-container">$f(x)=\frac{1}{x^2}$</span> to be expanded as a series with powers of <span class="math-container">$(x+2)$</span>. </p>
<p>Let <span class="math-container">$u=x+2$</span>. Then <span class="math-container">$f(x)=\frac{1}{x^2}=\frac{1}{(u-2)^2}$</span></p>
... | Axion004 | 258,202 | <p>Your answer is extremely close to the correct derivation. The error occurs when you write</p>
<blockquote>
<p><span class="math-container">$$\frac{d}{dx} \Bigg(\sum_{n=0}^\infty \frac{(x+2)^n}{2^{n+1}}\Bigg)= \sum_{n=0}^\infty \frac{d}{dx} \bigg(\frac{(x+2)^n}{2^{n+1}}\bigg)=\sum_{\color{red}{n=0}}^\infty \frac{n... |
2,702,726 | <p>Find the absolute minimum and maximum values of,</p>
<p>$$f(x) = 2 \sin(x) + \cos^2 (x) \text{ on } [0, 2\pi]$$</p>
<p>What I did so far is</p>
<p>$$f'(x) = 2\cos(x) -2 \cos(x) \sin(x)$$</p>
<p>Could someone please help me get started?</p>
| farruhota | 425,072 | <p>Without differentiation:
$$f(x)=-(1-\sin x)^2+2; \\
f\left(\frac{3\pi}{2}\right)=-2 (\text{min}); \\
f\left(\frac{\pi}{2}\right)=2 (\text{max}).$$</p>
|
850,852 | <p>This one comes from Gilbert Strang's Linear Algebra. Pick any numbers $x+y+z = 0$. Find an angle between $\mathbf v=(x,y,z)$ and $\mathbf w=(z,x,y)$. </p>
<p>Explain why $$\dfrac{\bf v\cdot w}{\bf \Vert v\Vert \cdot\Vert w\Vert}$$ is always $-0.5$. </p>
| JimmyK4542 | 155,509 | <p>If $x+y+z = 0$, then $0 = (x+y+z)^2 = x^2+y^2+z^2+2(xy+yz+zx) = \|v\| \cdot \|w\| + 2 v \cdot w$. </p>
<p>Rearranging gives the result $\dfrac{v \cdot w}{\|v\| \cdot \|w\|} = -\dfrac{1}{2}$.</p>
|
373,510 | <p>I was wondering if there is some generalization of the concept metric to take positive and negative and zero values, such that it can induce an order on the metric space? If there already exists such a concept, what is its name?</p>
<p>For example on $\forall x,y \in \mathbb R$, we can use difference $x-y$ as such... | Ittay Weiss | 30,953 | <p>You will have to lose some of the other axioms of a metric space as well since the requirement that $d(x,y)\ge 0$ in a metric space is actually a consequence of the other axioms: $0=d(x,x)\le d(x,y)+d(y,x)=2\cdot d(x,y)$, thus $d(x,y)\ge 0$. This proof uses the requirements that $d(x,x)=0$, the triangle inequality, ... |
373,510 | <p>I was wondering if there is some generalization of the concept metric to take positive and negative and zero values, such that it can induce an order on the metric space? If there already exists such a concept, what is its name?</p>
<p>For example on $\forall x,y \in \mathbb R$, we can use difference $x-y$ as such... | Eddy | 136,222 | <p>How about a distance defined as relative to an origin? This origin being a point on 1d spaces.</p>
<p>For higher dimensional spaces I'm not sure what would be more interesting, either a finite set of n points or a n-1 dimensional subspace given an n dimensional space (as in 2 points or a line, for 2d planes, a plan... |
1,400,399 | <p>Here is an indefinite integral that is similar to an integral I wanna propose for a contest. Apart from
using CAS, do you see any very easy way of calculating it?</p>
<p>$$\int \frac{1+2x +3 x^2}{\left(2+x+x^2+x^3\right) \sqrt{1+\sqrt{2+x+x^2+x^3}}} \, dx$$</p>
<p><strong>EDIT:</strong> It's a part from the gener... | Harish Chandra Rajpoot | 210,295 | <p>Let $$2+x+x^2+x^3=t^2\implies (1+2x+3x^2)dx=2tdt$$ $$\int \frac{2tdt}{t^2\sqrt{1+t}}$$ $$=2\int \frac{dt}{t\sqrt{1+t}}$$
Let $1+t=x^2\implies dt=2xdx$
$$=2\int \frac{2xdx}{(x^2-1)x}$$
$$=4\int \frac{dx}{x^2-1}$$
$$=4\int \frac{dx}{(x-1)(x+1)}$$
$$=4\frac{1}{2}\int \left(\frac{1}{x-1}-\frac{1}{x+1}\right)dx$$
$$=4\fr... |
360,293 | <p>Calculate $\sum_{n=2}^\infty ({n^4+2n^3-3n^2-8n-3\over(n+2)!})$</p>
<p>I thought about maybe breaking the polynomial in two different fractions in order to make the sum more manageable and reduce it to something similar to $\lim_{n\to\infty}(1+{1\over1!}+{1\over2!}+...+{1\over n!})$, but didn't manage</p>
| Mhenni Benghorbal | 35,472 | <p>First step, we find the Taylor series of $x^4+2x^3-3x^2-8x-3$ at the point $x=-2$ and then use it to write</p>
<p>$$ n^4+2n^3-3n^2-8n-3 = 1-4\, \left( n+2 \right) +9\, \left( n+2 \right)^{2}-6\, \left( n+2
\right) ^{3}+ \left( n+2 \right) ^{4}.$$</p>
<p>Using the above expansion and shifting the index of summatio... |
74,108 | <p>Background: I was trying to convert a MATLAB code (fluid simulation, SPH method) into a <em>Mathematica</em> one, but the speed difference is huge.</p>
<p>MATLAB code:</p>
<pre class="lang-matlab prettyprint-override"><code>function s = initializeDensity2(s)
nTotal = s.params.nTotal; %# particles
h = s.params.... | xzczd | 1,871 | <p>Modify the calculation order a little to avoid ragged array and then make use of <code>Listable</code> and <code>Compile</code>:</p>
<pre><code>computeDistance[pos_] := DistanceMatrix[pos, DistanceFunction -> EuclideanDistance]
liuQuartic = {r, h} \[Function]
15/(7 Pi*h^2) (2/3 - (9 r^2)/(8 h^2) + (19 r^3)/(... |
770,430 | <p>How to find the value of $X$?</p>
<p>If $X$= $\frac {1}{1001}$+$\frac {1}{1002}$+$
\frac {1}{1003}$. . . . $\frac {1}{3001}$</p>
| Hakim | 85,969 | <p>The exact answer is $$\begin{align}
H_{3001}-H_{1000}&=(\gamma+\psi_0(3002))-(\gamma+\psi_0(1001))\\ \,\\&=\dfrac{\Gamma'(3002)\Gamma(1001)-\Gamma'(1001)\Gamma(3002)}{\Gamma(1001)\Gamma(3002)}\\\,\,\\
&\approx 1.09861225...
\end{align}$$ where $H_n$ is the $n$-th Harmonic number defined as: $$H_n:=\sum_{... |
572,125 | <p>How to show this function's discontinuity?<br></p>
<p>$ f(n) = \left\{
\begin{array}{l l}
\frac{xy}{x^2+y^2} & \quad , \quad(x,y)\neq(0,0)\\
0 & \quad , \quad(x,y)=(0,0)
\end{array} \right.$</p>
| user642796 | 8,348 | <p>Yes. Recall that given a cardinal $\kappa$ the <em>cofinality</em> of $\kappa$, $\mathrm{cf} ( \kappa )$, is the least cardinal $\mu$ for which there is an unbounded (cofinal) function $\mu \to \kappa$. Regularity means that $\mathrm{cf} ( \kappa ) = \kappa$, and all successor cardinals are regular.</p>
<hr>
<p>... |
185,478 | <blockquote>
<p>How do I solve for $x$ in $$x\left(x^3+\sin(x)\cos(x)\right)-\big(\sin(x)\big)^2=0$$</p>
</blockquote>
<p>I hate when I find something that looks simple, that I should know how to do, but it holds me up. </p>
<p>I could come up with an approximate answer using Taylor's, but how do I solve this? </... | user 1591719 | 32,016 | <p>Let's do it in a simple way. As N. S. noticed, the function is even. Then it's enough to analyze things on the positive real axis:</p>
<p>$$x\left(x^3+\sin(x)\cos(x)\right)-\big(\sin(x)\big)^2\geq x\left(x^3+x-x^3\right)-\big(\sin(x)\big)^2\geq 0$$
$$ x^2 \geq \big(\sin(x)\big)^2$$</p>
<p>Above I used the fact th... |
49,544 | <p>In reading section 2.2, page 14 of <a href="http://www.gaussianprocess.org/gpml/chapters/" rel="nofollow noreferrer">this book</a>, I came across the term "singular distribution".</p>
<p>Apparently, a multivariate Gaussian distribution is singular if and only if its covariance matrix is singular. One way (... | Tim van Beek | 7,556 | <p>I find only the expression "this Gaussian is singular" on page 14 of your reference, but not the definition of "singular distribution". </p>
<p>But to answer your question:</p>
<p>The delta distribution is not a singular distribution, it is a discrete probability distribution. It does not have a Radon-Nikodym dens... |
396,088 | <p>Let <span class="math-container">$K$</span> be a field and let <span class="math-container">$\Lambda_{1}$</span> and <span class="math-container">$\Lambda_{2}$</span> be two finite-dimensional <span class="math-container">$K$</span>-algebras with Jacobson radicals <span class="math-container">$J_{1}$</span> and <spa... | Mare | 61,949 | <p>We have <span class="math-container">$gldim A \otimes_K B= gldim A + gldim B$</span> if A and B are seperable algebras over the field <span class="math-container">$K$</span>, see <a href="https://www.cambridge.org/core/journals/nagoya-mathematical-journal/article/on-the-dimension-of-modules-and-algebras-viii-dimensi... |
3,264,333 | <p>I am working on my scholarship exam practice and not sure how to begin. Please assume math knowledge at high school or pre-university level.</p>
<blockquote>
<p>Let <span class="math-container">$a$</span> be a real constant. If the constant term of <span class="math-container">$(x^3 + \frac{a}{x^2})^5$</span> is ... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: <span class="math-container">$$(A+B)^5={A}^{5}+5\,{A}^{4}B+10\,{A}^{3}{B}^{2}+10\,{A}^{2}{B}^{3}+5\,A{B}^{4}+
{B}^{5}
$$</span></p>
|
19,148 | <p>I always find the strong law of large numbers hard to motivate to students, especially non-mathematicians. The weak law (giving convergence in probability) is so much easier to prove; why is it worth so much trouble to upgrade the conclusion to almost sure convergence?</p>
<p>I think it comes down to not having a ... | Gjergji Zaimi | 2,384 | <p><a href="http://terrytao.wordpress.com/2008/06/18/the-strong-law-of-large-numbers/">Here</a> is a nice post of T. Tao on SLLN. In the comments section he is asked a very similar question to which he answers the following: (I hope it's ok to reproduce it here, since it is buried down in the comments)</p>
<blockquote... |
19,148 | <p>I always find the strong law of large numbers hard to motivate to students, especially non-mathematicians. The weak law (giving convergence in probability) is so much easier to prove; why is it worth so much trouble to upgrade the conclusion to almost sure convergence?</p>
<p>I think it comes down to not having a ... | Erik Davis | 1,228 | <p>I think it is worth noting that even if real world systems are fundamentally finite (in which case the distinction between WLLN and SLLN gets a bit philosophical), history has shown that it is extremely useful to approximate the discrete with the continuous. Thus we consider limit theorems to approximate statistics ... |
2,962,193 | <p><strong>Q</strong>:If <span class="math-container">$2\cos p=x+\frac{1}{x}$</span> and <span class="math-container">$2\cos q=y+\frac{1}{y}$</span> then show that <span class="math-container">$2\cos(mp-nq)$</span> is one of the values of <span class="math-container">$\left( \frac{x^m}{y^n}+\frac{y^n}{x^m} \right)$</sp... | B. Goddard | 362,009 | <p>Let <span class="math-container">$p,p+2$</span> and <span class="math-container">$q,q+2$</span> be pairs of twin primes.</p>
<p><span class="math-container">$$f(x,y) = \left( \frac{p}{p+2} \right)^x \left(\frac{q}{q+2}\right)^y$$</span> </p>
<p>is between <span class="math-container">$0$</span> and <span class="ma... |
2,609,283 | <p>$u_1 = (2, -1, 3)$ and $u_2 = (0, 0, 0)$</p>
<p>I tried using the cross product of the two but that just gave me the zero vector. I don't know any other methods to get a vector that is orthogonal to two vectors. </p>
<p>The answer is $v = s(1, 2, 0) + t(0, 3, 1)$ , where $s$ and $t$ are scalar values. </p>
| Rodrigo de Azevedo | 339,790 | <p>Using the <a href="https://en.wikipedia.org/wiki/Weinstein%E2%80%93Aronszajn_identity" rel="nofollow noreferrer">Weinstein-Aronszajn determinant identity</a>,</p>
<p><span class="math-container">$$\begin{array}{rl} \det (s \mathrm I_n - 1_n 1_n^\top) &= \det \left( s \cdot \left( \mathrm I_n - s^{-1}1_n 1_n^\top... |
4,149,355 | <p>Are there any stochastic processes <span class="math-container">$(X_t)_{t \in \mathbb{R}^d}$</span> such that</p>
<ol>
<li>almost surely paths are continuous but nowhere differentiable and</li>
<li>sampling of <span class="math-container">$n$</span> points <span class="math-container">$X_{t_n}$</span> on a path can ... | boaz | 83,796 | <p>Note that if <span class="math-container">$x^2+y^2=1$</span>, then by the Arithmetic mean-Geometric mean inequality
<span class="math-container">$$
|xy|=\sqrt{x^2y^2}\leqslant\frac{x^2+y^2}{2}=\frac{1}{2}
$$</span>
Now, if <span class="math-container">$x^2+y^2=1$</span>, then <span class="math-container">$(x+y)^2=1+... |
4,149,355 | <p>Are there any stochastic processes <span class="math-container">$(X_t)_{t \in \mathbb{R}^d}$</span> such that</p>
<ol>
<li>almost surely paths are continuous but nowhere differentiable and</li>
<li>sampling of <span class="math-container">$n$</span> points <span class="math-container">$X_{t_n}$</span> on a path can ... | JMP | 210,189 | <p>Assume <span class="math-container">$x+y>\sqrt2$</span>, so that <span class="math-container">$y>\sqrt2-x$</span>.</p>
<p>Then <span class="math-container">$x^2+y^2>x^2+2-2\sqrt2x+x^2=2x(x-\sqrt2)+2$</span>.</p>
<p>The RHS is minimal when <span class="math-container">$x=\frac1{\sqrt2}$</span>, and equals <s... |
285,548 | <p>I asked the following question on math.SE (<a href="https://math.stackexchange.com/questions/2420298/bvps-for-elliptic-pdos-when-do-green-functions-l2-inverses-define-pseudo-d">https://math.stackexchange.com/questions/2420298/bvps-for-elliptic-pdos-when-do-green-functions-l2-inverses-define-pseudo-d</a>) just over t... | JahvedM | 80,370 | <p>By complementing <a href="https://mathoverflow.net/users/613/deane-yang">Deane Yang</a>'s strategy with <a href="https://mathoverflow.net/users/21051/jochen-wengenroth">Jochen Wengenroth</a>'s observations in the related question <a href="https://mathoverflow.net/questions/287167/composition-of-a-smoothing-operator-... |
3,581,707 | <p>I'm struggling with the following proof and hope some of you can help me:</p>
<p>By <span class="math-container">$H$</span> we denote a real Hilbert space and let be <span class="math-container">$T: H \rightarrow H$</span> be a compact, self-adjoint and linear operator. </p>
<p>(i) Show that <span class="math-cont... | Kavi Rama Murthy | 142,385 | <p>I will assume that <span class="math-container">$H$</span> is separable so that the closed unit ball is a compact metric space in the weak topology. [Please see the remark at the end of the answer]. </p>
<p>Let <span class="math-container">$A$</span> be the supremum of the real numbers <span class="math-container">... |
2,072,473 | <p>I managed to prove the statement:</p>
<blockquote>
<p>If $f: A\to B$ and $g: B\to C$ are surjective, then $g\circ f$ is surjective.</p>
</blockquote>
<p>But now I require a counterexample to the converse of this statement. I am not sure how to formulate the counterexample. Similarly I need a counterexample of th... | Nicolas FRANCOIS | 288,125 | <p>Converse : "If $g\circ f$ is surjective, then $f$ and $g$ are surjective".</p>
<p>Saying this statement is false means $g\circ f$ is surjective, but either $g$ or $f$ is not. But if $g$ is not surjective, $g\circ f$ can't be either (check it). So your counterexample has to be composed of $f:A\to B$ non surjective, ... |
2,119,761 | <p>Asuume that $f:\mathbb{R}\rightarrow \mathbb{R}$ continuous function and $g:\mathbb{R}\rightarrow \mathbb{R}$ uniformly continuous function and $g$ bounded.</p>
<p>I have to prove that $f\circ g$ is uniformly continuous function.
I tried the following:
$f$ continuous function so $\forall \epsilon>0 ~\exists~ \de... | parsiad | 64,601 | <p><strong>Hint</strong>: Did you know that if $k$ is a continuous mapping between metric spaces $X$ and $Y$ and $X$ is compact, $k$ is uniformly continuous?</p>
<p>Since $g$ is bounded, its image is contained in a compact set $K$ (i.e., $g(\mathbb{R}) \subset K$). Letting $k = f|_K$, it is trivially the case that $f\... |
3,432,911 | <p>My argument is as follows:</p>
<p>Let <span class="math-container">$R$</span> be a commutative ring with unity, <span class="math-container">$I$</span> an ideal of <span class="math-container">$R$</span>.
If <span class="math-container">$(R/I)^n\cong (R/I)^m$</span> as <span class="math-container">$R$</span>-module... | lab bhattacharjee | 33,337 | <p>WLOG <span class="math-container">$z-1=\cos2t+i\sin2t$</span></p>
<p><span class="math-container">$$\dfrac1z=\dfrac1{2\cos t(\cos t+i\sin t)}=\dfrac{\cos t-i\sin t}{2\cos t}$$</span></p>
|
2,425,337 | <p>What would be an example of a real valued sequence $\{a_{n}\}_{n=1}^{\infty}$ such that $$\frac{a_{n}}{a_{n+1}} = 1 + \frac{1}{n} + \frac{p}{n \ln n} + O\left(\frac{1}{n \ln^{2}n}\right)\ ?$$</p>
| Wouter | 89,671 | <p>A simple way to construct the desired curve is to start with a <a href="https://en.wikipedia.org/wiki/Sigmoid_function" rel="nofollow noreferrer">sigmoid</a>
$$f(x)=\frac{1}{1+\exp(A(x-1/4))}$$
where $A$ is a positive parameter that influences the steepness.</p>
<p>Add a line through $(1/4,0)$
$$g(x)=\alpha f(x)+\b... |
4,027,604 | <p>Let <span class="math-container">$f(x)$</span> be continuous on <span class="math-container">$[a,b]$</span> and <span class="math-container">$F(x)=\frac{1}{x-a}\int_a^xf(t)dt$</span></p>
<p>Proof: The functions <span class="math-container">$F(x)$</span> and <span class="math-container">$f(x)$</span> have the same mo... | Fred | 380,717 | <p>If <span class="math-container">$x$</span> is fixed, then in <span class="math-container">$\int_a^xf(x)dt$</span> you integrate over the constant <span class="math-container">$f(x)$</span> with respect to <span class="math-container">$t$</span>. Thus <span class="math-container">$\int_a^xf(x)dt =f(x) \int_a^x 1 dt=(... |
8,568 | <p>I'm going to be starting teaching a course called algebra COE, which is for students who didn't pass the required state algebra exam to graduate and are now seniors, to do spaced-out exam-like extended problems after extensive support. </p>
<p>I don't want to start the class out with "getting down to business" beca... | Sue VanHattum | 60 | <p>Eight adults and two kids want to cross a river. There is a boat that can hold one adult or two kids, no more. Can they all cross? How? Extend to <em>n</em> adults.</p>
<p>Use figurines (Lego, Playmobil, ?) or coins to model the situation. </p>
|
8,568 | <p>I'm going to be starting teaching a course called algebra COE, which is for students who didn't pass the required state algebra exam to graduate and are now seniors, to do spaced-out exam-like extended problems after extensive support. </p>
<p>I don't want to start the class out with "getting down to business" beca... | Jon Bannon | 354 | <p>The frog jumping puzzle is nice, in that it is very simple but hides some surprising but manageable complexity. Here's a <a href="http://www.smart-kit.com/s7284/frog-jumping-puzzle/" rel="nofollow">"video game" version</a>. Have the students play with this, and then try to generalize to n male frogs and m female fro... |
287,859 | <p>Prove that $\lim\limits_{x\rightarrow+\infty}\frac{x^k}{a^x} = 0\ (a>1,k>0)$.</p>
<p>P.S. This problem comes from my analysis book. You may use the definition of limits or invoke the Heine theorem for help. <em>It means the proof should only use some basic properties and definition of limits rather than more ... | André Nicolas | 6,312 | <p>Hard to know what is allowed. Let $x^k=y$. We are then looking at $\frac{x}{(a^{1/k})^y}$. </p>
<p>Let $a^{1/k}=b$. We are computing the simpler-looking $\lim_{y\to\infty}\frac{y}{b^y}$. </p>
<p>Assume that we know that $b^y$ is increasing. Let $b=1+d$. </p>
<p>Then $(1+d)^y \ge (1+d)^{\lfloor y\rfloor}$. But by ... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.