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 |
|---|---|---|---|---|
27,089 | <p>Off hand, does anyone know of some useful conditions for checking if a ring (or more generally a semiring) has non-trivial derivations? (By non-trivial, I mean they do not squish everything down to the additive identity.) Part of the motivation for this is that I was thinking about it the other day, and had troubl... | Hailong Dao | 2,083 | <p>For your question in the second paragraph, <a href="http://books.google.com/books?id=roEx6lkAp10C&lpg=PA37&ots=jI7AVZExLT&dq=any%20semirings%20can%20be%20embeded%20in%20one%20with%20non%20trivial%20derivations&pg=PA37#v=onepage&q&f=false" rel="nofollow">any semiring can be embedded in one wit... |
2,290,395 | <p>What if in Graham’s Number every “3” was replaced by “tree(3)” instead? How big is this number? Greater than Rayo’s number? Greater than every current named number?</p>
| Owen Wei | 676,645 | <p>The numbers themselves are already so big that doing that would barely change it at all.
It would still be ZERO compared to a number like Rayo's number.
Obviously, if doing so really did make it the largest number ever created, why wouldn't people do it earlier? </p>
|
3,145,973 | <p>Show that for every integer <span class="math-container">$n ≥ 3$</span>, the number <span class="math-container">$n!e$</span> is not an integer.</p>
<p>I have shown the inequality <span class="math-container">$\displaystyle0< \sum_{m=n+1} \frac{1}{m!} < \frac{1}{n!}$</span> for <span class="math-container">$n... | Ari Royce Hidayat | 435,467 | <p>I understand your frustration, I also read several proofs but hard to understand them because somehow and I don't know why they all seem to omit the most important part of the explanation.</p>
<p>First about restriction notation, it's easy to understand and a neat thing write. So instead of writing:</p>
<p><span c... |
3,073,361 | <p>I think I understood 1-forms fairly well with the help of these two sources. They are dual to vectors, so they measure them which can be visualized with planes the vectors pierce.</p>
<ul>
<li><a href="https://the-eye.eu/public/WorldTracker.org/Physics/Misner%20-%20Gravitation%20%28Freeman%2C%201973%29.pdf" rel="nof... | Deane | 10,584 | <p>Below, a tensor is an alternating multilinear function on a vector space, and a form is an assignment of a tensor to the tangent space at each point of a manifold.</p>
<p>A finite dimensional vector space can be viewed as a space with an origin and arrows. Even though the dual space is also a vector space, I don't f... |
1,154,592 | <p>I was doing some basic Number Theory problems and came across this problem :</p>
<blockquote>
<p>Show that if $a$ and $n$ are positive integers with $n\gt 1$ and $a^{n} - 1$ is prime, then $a = 2$ and $n$ is prime</p>
</blockquote>
<p><strong>My Solution : (Sloppy)</strong></p>
<blockquote>
<ul>
<li>$a^{n}-... | Community | -1 | <p>I'm a little late to the party, but here is a way to justify this using a contrapositive statement.</p>
<p>Let <span class="math-container">$a,n\in\mathbb{Z}^+$</span>, where <span class="math-container">$n>1$</span>, and assume <span class="math-container">$a^n-1$</span> is prime. Show <span class="math-containe... |
1,684,741 | <p>I'm able to show it isn't absolutely convergent as the sequence $\{1^n\}$ clearly doesn't converge to $0$ as it is just an infinite sequence of $1$'s. How do I prove the series isn't conditionally convergent to prove divergence!</p>
| Lubin | 17,760 | <p>The only <em>searching</em> that needs to be done is to find where $4$ sits in $\Bbb F_7^\times$ with respect to a generator of this cyclic group of order six.</p>
<p>Now, $\Bbb F_7^\times$ has only the two generators $3$ and $5$, and $4$ is the square of $5$ modulo $7$. That is, calling $5=g$, we have $4\equiv g^2... |
4,050,893 | <p>Given a linear transformation <span class="math-container">$T: V \rightarrow W$</span> where <span class="math-container">$V$</span> and <span class="math-container">$W$</span> are finite dimensional, then is it true that nullity(<span class="math-container">$T$</span>) = nullity(<span class="math-container">$[T]_\b... | user1551 | 1,551 | <p>Here are two simplifications:</p>
<ol>
<li>Up to similarity, there are only six choices of <span class="math-container">$D$</span>. Note that the fourth equation is a Sylvester equation of the form <span class="math-container">$DX+XD=I$</span>. By considering the Jordan form of <span class="math-container">$D$</span... |
1,357,638 | <p>Here is the problem:</p>
<p>Suppose $n$ people are at a party, and some number of them shake hands. At the end of the party, each guest $G_i$, $1 \leq i \leq n$ shares that they shook hands $x_i$ times. Assume there were a total of $h \geq 0$ handshakes at the party. Use induction on $h$ to prove that:</p>
<p>$x_i... | Kopper | 5,218 | <p>This is the right idea. Your base case is correct, and your argument is the right one, modulo some technical points:</p>
<ol>
<li>You don't want to assume $h=k$ $\forall k \in \mathbb{Z}$. That doesn't really make sense and isn't what you are really doing. You need to assume the claim holds for $1\leq h \leq k$, i.... |
51,246 | <p>In undergraduate courses we compute the sum $S$ of some series
of the form $\frac{1}{P(n)}$ where $P(x)$ is some simple
polynomial of degree $2$ with integer coefficients, by the following procedure:</p>
<p>(sketch)</p>
<p>(a) Choose an appropriate periodic function $f(x)$ defined over a domain $D.$</p>
<p>(b) C... | Gerry Myerson | 3,684 | <p>Perhaps it fails because if it worked it would give an answer to a question that doesn't have one. </p>
<p>To be a little less cryptic, if there isn't any evaluation of the sum of the reciprocal cubes in terms of, say, the functions of 1st year calculus, then no method that yields only that kind of function is goin... |
43,611 | <p>I posted this on Stack Exchange and got a lot of interest, but no answer.</p>
<p>A recent <a href="http://people.missouristate.edu/lesreid/POW12_0910.html" rel="nofollow">Missouri State problem</a> stated that it is easy to decompose the plane into half-open intervals and asked us to do so with intervals pointing i... | Jeff Strom | 3,634 | <p>Conway and Croft show it can be done for closed intervals and cannot
be done for open intervals in the paper:</p>
<p><a href="https://doi.org/10.1017/S0305004100038263" rel="nofollow noreferrer">Covering a sphere with congruent great-circle arcs.
Proc. Cambridge Philos. Soc. 60, 1964, pp787–800</a>.</p>
|
4,031,476 | <p>I recently completed a variation of a problem I found from a mathematical olympiad which is as follows:</p>
<p>Prove that, for all <span class="math-container">$n \in \mathbb{Z}^+$</span>, <span class="math-container">$n \geq 1$</span>, <span class="math-container">$$\sum_{k=1}^n \frac{k}{2^k} < 2 $$</span></p>
<... | student | 11,211 | <p>As commented by Albus, the values can be found.</p>
<p>In fact, let <span class="math-container">$p \in (0,1)$</span>, then</p>
<p><span class="math-container">$$\sum_{k=1}^{n}kp^k\le \sum_{k=1}^{\infty}kp^k=p\sum_{k=1}^{\infty}kp^{k-1}=p\frac{d}{dp}\left(\sum_{k=1}^{\infty}p^k\right)=p\frac{d}{dp}\frac{p}{1-p}=\fra... |
10,977 | <p>When I taught calculus, I posted my notes after the lecture. Then I had the students fill out a mid-quarter evaluation, and a lot of them wanted me to post my notes before class.</p>
<p>What I started doing was printing and handing out the notes to them, leaving the examples blank so they can fill those in. Many ... | Marian Minar | 6,845 | <p><strong>My answer</strong></p>
<p>In my experience, I have used guided notes with success, as long as students straddled the "sweet spot" of giving them ample opportunities to interact with the topic and avoiding the creation of mindless, note-taking robots. With guided notes, my guess is that your goal is to accel... |
1,579,528 | <p>You decide to play a holiday drinking game. You start with 100 containers of eggnog in a row. The 1st container contains 1 liter of eggnog, the 2nd contains 2 liters, all the way until the 100th, which contains 100 liters. You select a container uniformly at random and take a one liter sip from it. If the container ... | BGM | 297,308 | <p>You may think there is a group of numbers, with $k$ number $k$, $k = 1, 2, \ldots, n$, a total of $\frac {n(n+1)} {2}$ numbers in the group. Each permutation of numbers corresponding to a sequence of taking the bottles.</p>
<p>The total number of permutation is given by the multinomial coeffcient:</p>
<p>$$ \frac ... |
438,263 | <p>Is there a concrete example of a <span class="math-container">$4$</span> tensor <span class="math-container">$R_{ijkl}$</span> with the same symmetries as the Riemannian curvature tensor, i.e.
<span class="math-container">\begin{gather*}
R_{ijkl} = - R_{ijlk},\quad R_{ijkl} = R_{jikl},\quad R_{ijkl} = R_{klij}, \\
R... | Denis Serre | 8,799 | <p>If you have half-a-dozen open problems in some definite area, why not write an article where you explain them, why there are important, what are the difficulties ? I did that once (<em>Five open problems in compressible mathematical fluid dynamics</em>, Methods and Applications in Analysis, 20 (2013) pp 197-210). Of... |
438,263 | <p>Is there a concrete example of a <span class="math-container">$4$</span> tensor <span class="math-container">$R_{ijkl}$</span> with the same symmetries as the Riemannian curvature tensor, i.e.
<span class="math-container">\begin{gather*}
R_{ijkl} = - R_{ijlk},\quad R_{ijkl} = R_{jikl},\quad R_{ijkl} = R_{klij}, \\
R... | Fedor Petrov | 4,312 | <p>Arnold Mathematical Journal has a problems section.</p>
|
438,263 | <p>Is there a concrete example of a <span class="math-container">$4$</span> tensor <span class="math-container">$R_{ijkl}$</span> with the same symmetries as the Riemannian curvature tensor, i.e.
<span class="math-container">\begin{gather*}
R_{ijkl} = - R_{ijlk},\quad R_{ijkl} = R_{jikl},\quad R_{ijkl} = R_{klij}, \\
R... | Joseph O'Rourke | 6,094 | <p><a href="https://topp.openproblem.net/" rel="nofollow noreferrer">The Open Problems Project (TOPP)</a> is
focussed on discrete and computational geometry.
We (Erik Demaine, Joe Mitchell, and I) started it in 2001 but its <span class="math-container">$78$</span> problems are now only sporadically updated or augmented... |
143,070 | <p>Suppose whole square and the left square in the diagram below are pullbacks, then we may wonder whether the right square is a pullback. It is usually not the case. </p>
<p><img src="https://i.stack.imgur.com/yhrcd.jpg" alt="square"></p>
<p>Now we seek some addition condition on $X\to Y$ that forces the right squar... | Eric Wofsey | 75 | <p>Consider the category consisting of that diagram, together with an extra object $P$ with maps $Y\leftarrow P\to C$ that commute with the maps to $Z$. Then in this category, the whole diagram and the left square are pullbacks, $X\to Y$ is epic, but the right square is not a pullback.</p>
|
754,583 | <p>Write <span class="math-container">$$\phi_n\stackrel{(1)}{=}n+\cfrac{n}{n+\cfrac{n}{\ddots}}$$</span> so that <span class="math-container">$\phi_n=n+\frac{n}{\phi_n},$</span> which gives <span class="math-container">$\phi_n=\frac{n\pm\sqrt{n^2+4n}}{2}.$</span> We know <span class="math-container">$\phi_1=\phi$</span... | Steven Stadnicki | 785 | <p>Let $f_x(t) = x+\frac{x}{t}$ (I'm using $x$ rather than $n$ since we're interested in continuous behavior now, and $f_n$ is a little confusing as a family of functions rather than a sequence); the continued fraction thus corresponds to the sequence $\{t_0 = x, t_n = f_x(t_{n-1})\}$. Then $\frac{df}{dt} = -\frac{x}{... |
418,724 | <p>This question arises in STEP 2011 Paper III, question 2. The paper can be found <a href="http://www.admissionstestingservice.org/our-services/subject-specific/step/preparing-for-step/" rel="nofollow">here</a>. </p>
<p>The first part of the question requires us to prove the result that if the polynomial
$$x^{n}+a_{... | Simone | 77,622 | <p>I think it's not correct, because you are assuming that if $x|5x-7$ then $x|7$. This is not true in general.Think for example to $2+4$. Of course $6|2+4$ but $6$ doesn't divide $2$ nor $4$. What you can say is that $x(x^{n-1}-5)=7$, but $7$ is prime, then you must have $x=1$ and $x^{n-1}-5=7$ or $x=7$ and $x^{n-1}-5... |
445,816 | <p>I have to show that</p>
<blockquote>
<blockquote>
<p>$\mathbb{C}=\overline{\mathbb{C}\setminus\left\{0\right\}}$,</p>
</blockquote>
</blockquote>
<p>what is very probably an easy task; nevertheless I have some problems.</p>
<p>In words this means: $\mathbb{C}$ is the smallest closed superset of $\mathbb{C... | JLA | 30,952 | <p>${\mathbb{C}\setminus\left\{0\right\}}$ is open, $\mathbb{C}$ is closed, and since $\overline{\mathbb{C}\setminus\left\{0\right\}}$ is the smallest closed set containing ${\mathbb{C}\setminus\left\{0\right\}}$, it must be equal to $\mathbb{C}$ (since the only set containing but not equal to ${\mathbb{C}\setminus\lef... |
161,616 | <p>There is a well known result that every one dimensional topological manifold without boundary is homeomorphic either to the circle or to the whole real line. However there is one detail hidden: manifold is understood to be second countable (or paracompact). If we drop this assumption it becomes possible to construct... | Benoît Kloeckner | 4,961 | <p>The one-dimensional case is well known: you have the circle, the line $\mathbb{R}$, the long line $L$ and the long ray $R$. The proof is not that easy to find in the literature since non-metrizable manifolds are (in my opinion) underestimated. It is described in a previous answer, so let me give a few pointers for t... |
1,523,392 | <p>This is question 2.4 in Hartshorne. Let $A$ be a ring and $(X,\mathcal{O}_X)$ a scheme. We have the associated map of sheaves $f^\#: \mathcal{O}_{\text{Spec } A} \rightarrow f_* \mathcal{O}_X$. Taking global sections we obtain a homomorphism $A \rightarrow \Gamma(X,\mathcal{O}_X)$. Thus there is a natural map $\alp... | hm2020 | 858,083 | <p><strong>Answer:</strong> Let <span class="math-container">$A(Y)$</span> be a commutative unital ring and assume you are given a map of commutative unital rings</p>
<p><span class="math-container">$$f: A(Y) \rightarrow \Gamma(X,\mathcal{O}_X).$$</span></p>
<p>Let <span class="math-container">$U:=Spec(A) \subseteq X$<... |
2,416,071 | <p>I have this integral: $\displaystyle \int^{\infty}_0 kx e^{-kx} dx$.</p>
<p>I tried integrating it by parts:</p>
<p>$\dfrac{1}{k}\displaystyle \int^{\infty}_0 kx e^{-kx} dx = ... $. But I'm stuck </p>
<p>now. Can you help me please?</p>
| Ant | 241,144 | <p>Try explicitly writing out what $u$ and $dv$ are in your integration by parts. In this case, you should have $u=kx$, and $dv=e^{-kx}\,dx$. Thus, $du=k\,dx$, and $v=-\frac{1}{k}e^{-kx}$. Now apply the integration-by-parts formula:</p>
<p>$$\int_{0}^{\infty}{u\,dv}={\left.uv\right|}_0^{\infty}-\int_{0}^{\infty}{v\,du... |
3,238,563 | <p>I have a question about a proof I saw in a book about basic algeba rules. The rule to prove is:
<span class="math-container">\begin{eqnarray*}
\frac{1}{\frac{1}{a}} = a, \quad a \in \mathbb{R}_{\ne 0}
\end{eqnarray*}</span></p>
<p>And the proof: </p>
<p><span class="math-container">\begin{eqnarray*}
1 = a \frac{... | Peter Szilas | 408,605 | <p><span class="math-container">$1/a$</span> is the multiplicative inverse <span class="math-container">$a^{-1}$</span> of </p>
<p><span class="math-container">$a (\not =0)$</span>, i .e. <span class="math-container">$a^{-1}a=1$</span>.</p>
<p>Need to show:</p>
<p><span class="math-container">$(a^{-1})^{-1} =a;$</s... |
4,452,885 | <p>Let <span class="math-container">$z = e^{i\theta}, \theta \in \mathbb{R}$</span>. Then, does there exist <span class="math-container">$n \in \mathbb{N}$</span> such that:</p>
<p><span class="math-container">$$1 - z^n = re^{2 \pi i \tau}$$</span></p>
<p>for some <span class="math-container">$\tau \in \mathbb{Q}$</sp... | aschepler | 2,236 | <p>No, on basis of set cardinalities. The possible pairings of <span class="math-container">$n \in \mathbb{N}$</span> and <span class="math-container">$\tau \in \mathbb{Q}$</span> are countable. Each of those equations has at most <span class="math-container">$n$</span> solutions for <span class="math-container">$z$</s... |
4,095,248 | <blockquote>
<p>Suppose that given any <span class="math-container">$\epsilon>0$</span>, <span class="math-container">$$ \sum_{n=1}^\infty P [|X| > n
\epsilon ]< \infty. $$</span> Does this imply that <span class="math-container">$$ E| X| < \infty \quad ?$$</span></p>
</blockquote>
<p>I made the obvious at... | Kavi Rama Murthy | 142,385 | <p>This is false. <span class="math-container">$(0,1)$</span> is totally bounded w.r.t. the usual metric. An equivalent metric is <span class="math-container">$|\frac 1 x-\frac 1 y|$</span> and <span class="math-container">$(0,1)$</span> is not bounded in this metric.</p>
<p>[Definition of equivalent metrics I am usi... |
4,095,248 | <blockquote>
<p>Suppose that given any <span class="math-container">$\epsilon>0$</span>, <span class="math-container">$$ \sum_{n=1}^\infty P [|X| > n
\epsilon ]< \infty. $$</span> Does this imply that <span class="math-container">$$ E| X| < \infty \quad ?$$</span></p>
</blockquote>
<p>I made the obvious at... | Moishe Kohan | 84,907 | <p>My guess about this problem is that there are two issues:</p>
<ol>
<li>There was a typo and the problem should read</li>
</ol>
<blockquote>
<p>Prove a metric space is totally bounded iff it is totally bounded in every equivalent metric.</p>
</blockquote>
<p><strong>and</strong></p>
<ol start="2">
<li>The definition ... |
187,432 | <p>Can we evaluate the integral using <a href="http://en.wikipedia.org/wiki/Jordan%27s_lemma#Application_of_Jordan.27s_lemma">Jordan lemma</a>?
$$ \int_{-\infty}^{\infty} {\sin ^2 (x) \over x^2 (x^2 + 1)}\:dx$$</p>
<p>What de we do if removeable singularity occurs at the path of integration?</p>
| DonAntonio | 31,254 | <p>Taking
$$C_R:=[-R,-\epsilon]\cup\left(\gamma_\epsilon:=\{z=\epsilon e^{it}\;\;|\;\;0\leq t\leq \pi\}\right)\cup [\epsilon,R]\cup\left(\gamma_R:=\{z=Re^{it}\;\;|\;\;0\leq t\leq \pi\}\right)$$
$$f(z)=\frac{1-e^{2iz}}{z^2(z^2+1)}$$<br>
we get
$$\oint_{C_R}\frac{1-e^{2iz}\,dz}{z^2(z^2+1)}=2\pi i\,Res_{z=i}(f)=2\pi i\fr... |
3,700,440 | <p>As stated in the title, it is requested to define a linear transformation <span class="math-container">$T:\Bbb R^3 \to \Bbb R^3$</span> such that the null space of <span class="math-container">$T$</span> is the <span class="math-container">$z$</span>-axis, and the range of <span class="math-container">$T$</span> is ... | Davide Motta | 405,131 | <p>You have to find a basis of that plane: <span class="math-container">$x+y+z=0$</span> then <span class="math-container">$x=-y-z$</span> so you can pick <span class="math-container">$v_1=(1,-1,0), v_2=(1,0,-1)$</span>. The <span class="math-container">$z$</span>-axis is the vector <span class="math-container">$e_3=(0... |
696,511 | <p>Find the absolute maximum and minimum values of the function:</p>
<p>$$f(x,y)=2x^3+2xy^2-x-y^2$$</p>
<p>on the unit disk $D=\{(x,y):x^2+y^2\leq 1\}$.</p>
| Evgeny | 87,697 | <p><strong>Hint</strong>: for the interior of disk you may use usual condition $\nabla f(x,y) = 0$. For the boundary of the disk you may use parametrisation $x = \cos \phi$, $y = \sin \phi$ and minimize it w.r.t. to $\phi$ (remember, it's $2\pi$-periodic, so you should investigate it only on segment $\lbrack 0, 2\pi )$... |
5,528 | <p>Let H be a subgroup of G. (We can assume G finite if it helps.) A complement of H in G is a subgroup K of G such that HK = G and |H∩K|=1. Equivalently, a complement is a transversal of H (a set containing one representative from each coset of H) that happens to be a group.</p>
<p>Contrary to my initial naive... | Alex Collins | 1,713 | <p>A partial answer to this question is known as the Schur-Zassenhaus lemma (or theorem). If N is a normal subgroup of a finite group G whose order is prime to its index (such a subgroup is called a (normal) Hall subgroup of G) then N has a complement in G.</p>
<p>Check the Wikipedia page <a href="http://en.wikipedia.... |
5,528 | <p>Let H be a subgroup of G. (We can assume G finite if it helps.) A complement of H in G is a subgroup K of G such that HK = G and |H∩K|=1. Equivalently, a complement is a transversal of H (a set containing one representative from each coset of H) that happens to be a group.</p>
<p>Contrary to my initial naive... | Marty Isaacs | 9,694 | <p>Given $H \subseteq G$, there are a number of conditions sufficient to guarantee that there exists a $normal$ complement for $G$. One of the more interesting of these is due to Frobenius: Assume that $H \cap H^g = 1$ for all elements $g \in G - H$. Then $H$ has a normal complement in $G$. As yet, there is no proof kn... |
5,528 | <p>Let H be a subgroup of G. (We can assume G finite if it helps.) A complement of H in G is a subgroup K of G such that HK = G and |H∩K|=1. Equivalently, a complement is a transversal of H (a set containing one representative from each coset of H) that happens to be a group.</p>
<p>Contrary to my initial naive... | Gabe Conant | 21,240 | <p>A lot can be said in the finitely generated abelian case, just by using the structure theorem. </p>
<p>Call a group <strong>transversal</strong> if every subgroup has a complement; <strong>non-transversal</strong> of it has proper, nontrivial subgroups, none of which has a complement; and <strong>semi-transversal</... |
508,104 | <p>I want to understand more about this proof from Lang's Algebra:</p>
<p>Let $B$ be a subgroup of a free abelian group $A$ with basis $(x_i)_{i=1...n}$. It has already been shown that $B$ has a basis of cardinality $\leq n$.</p>
<blockquote>
<p>...
We also observe that our proof shows that there exists at leas... | Daniel Fischer | 83,702 | <p>That $S = \{s_1,\,\dotsc,\, s_m\}$ is a basis of $B$ means that every $b \in B$ can be written in a unique way as $b = \sum\limits_{i=1}^m k_i\cdot s_i$ with all $k_i \in \mathbb{z}$. Thus $B$ is the direct sum of $m$ copies of $\mathbb{Z}$,</p>
<p>$$B = \bigoplus_{i=1}^m \mathbb{Z}\cdot s_i.$$</p>
<p>Then we have... |
3,165,460 | <p>I am reading a survey on Frankl's Conjecture. It is stated without commentary that the set of complements of a union-closed family is intersection-closed. I need some clearer indication of why this is true, though I guess it is supposed to be obvious. </p>
| A. Kriegman | 649,089 | <p>An uncountable set does not have to be dense. A simple example would be to take the half interval <span class="math-container">$[0,\frac{1}{2}]$</span> which is uncountable but not dense in <span class="math-container">$[0,1]$</span>. But we can do even better, because we can find a set that is uncountable and not d... |
1,482,644 | <p>I am trying to find the shortest equivalent expression of the following:</p>
<p>((C → D) $\wedge$ (D → C)) $↔$ (C $\wedge$ D ∨ ¬C $\wedge$ ¬D)</p>
<p>I have "simplified" the expression into the following:</p>
<p>(($\neg$C $\vee$ D) $\wedge$ ($\neg$D $\vee$ C)) $↔$ ((C $\wedge$ D) ∨ (¬C $\wedge$ ¬D))</p>
<p>I am ... | Community | -1 | <p>Really, the shortest you can really come up with to preserve meaning is:</p>
<p>$$(c \iff d)\iff ((c\ \wedge d) \vee (\neg c \wedge \neg d))$$</p>
<p>Or if you want to have something that is just equivalent to the above:</p>
<p>$$(c \iff d) \iff (c \iff d)$$</p>
<p>Although this losing some of the intended meani... |
1,652,846 | <p>Let $s$ be any complex number, $t = e^s$ and $z = t^{1/t}$. Define the sequence $(a_n)_{n\in\mathbb{N}}$ by $a_0 = z $ and $a_{n+1} = z^{a_n} $ for $n \geq 0$, that is to say $a_n$ is the sequence $z$, $z^z$, $z^{z^z}$, $z^{z^{z^{z}}}$ and so on.</p>
<p>I want to show that the sequence $(a_n)_{n\in\mathbb{N}}$ con... | cpiegore | 268,070 | <p>I was reading the Wikipedia article on the Lambert W Function and I found this proof that the limit $c$, when it exists, is $c= \frac{w(-ln(z))}{-ln(z)}$</p>
<p>$z^c = c\implies z = c^{1/c} \implies z^{-1} = c^{-1/c} \implies 1/z = (1/c)^{1/c} \implies -ln(z) = \frac{ln(1/c)}{c} \implies -ln(z) = e^{ln(1/c)}ln(1/c)... |
1,119,634 | <p>Find the point on the curve $y=x^2+2$ where the tangent is parallel to the line $2x+y-1=0$</p>
<p>I understand the answer is $(-1,3)$ but I can't find a way to get there... Thanks </p>
| turkeyhundt | 115,823 | <p>Do you know how to find the derivative of a function? The derivative of the curve function will tell you the slope at any point $x$. </p>
<p>So, figure out the slope of $2x+y=1$ and then find the $x$ value of the curve's derivative that returns that same slope. </p>
<p>That will be the $x$ value for your point.... |
316,601 | <p>Can anyone tell me what I am doing wrong? need to prove for $k\ge2$
$$(5-\frac5k )(1+\frac{1}{(k+1)^2}) \le 5 - \frac{5}{k+1}$$$$(5-\frac5k )(1+\frac{1}{(k+1)^2})= 5(1-\frac1k)(1+\frac1{(k+1)^2})$$
$$=5(1+\frac1{k+1)^2}-\frac1k-\frac1{k(k+1)^2})$$
$$= 5(1-\frac{k^2+k+2}{k(k+1)^2})$$
$$=5(1-\frac{k(k+1)}{k(k+1)^2}+\f... | copper.hat | 27,978 | <p>Let $f(x) = (5-\frac{5}{x})(1+\frac{1}{(1+x)^2})-5+\frac{5}{1+x}$. After a little algebra, this gives $f(x) = -\frac{10}{x(x+1)^2}$. Hence $f(x) \leq 0$ when $x>0$.</p>
<p>Hence $f(k) \leq 0$ for $k \geq 2$.</p>
|
316,601 | <p>Can anyone tell me what I am doing wrong? need to prove for $k\ge2$
$$(5-\frac5k )(1+\frac{1}{(k+1)^2}) \le 5 - \frac{5}{k+1}$$$$(5-\frac5k )(1+\frac{1}{(k+1)^2})= 5(1-\frac1k)(1+\frac1{(k+1)^2})$$
$$=5(1+\frac1{k+1)^2}-\frac1k-\frac1{k(k+1)^2})$$
$$= 5(1-\frac{k^2+k+2}{k(k+1)^2})$$
$$=5(1-\frac{k(k+1)}{k(k+1)^2}+\f... | Gigili | 181,853 | <p>$$=5(1+\frac1{(k+1)^2}-\frac1k-\frac1{k(k+1)^2})$$
$$=5(1+\frac{k}{k(k+1)^2}-\frac{(k+1)^2}{k(k+1)^2}-\frac1{k(k+1)^2})$$
$$= 5(1-\frac{k^2+k+2}{k(k+1)^2})$$
$$=5(1-\frac{k(k+1)}{k(k+1)^2}\color{red}{-}\frac2{k(k+1)^2})$$
$$=5(1-\frac{1}{k+1}\color{red}{-}\frac2{k(k+1)^2})$$
$$= 5 - \frac5{k+1}\color{red}{-10}\frac1... |
2,083,460 | <p>While trying to answer <a href="https://stackoverflow.com/questions/41464753/generate-random-numbers-from-lognormal-distribution-in-python/41465013#41465013">this SO question</a> I got stuck on a messy bit of algebra: given</p>
<p>$$
\log m = \log n + \frac32 \, \log \biggl( 1 + \frac{v}{m^2} \biggr)
$$</p>
<p>I n... | Cleisthenes | 393,542 | <p>If by solve you mean isolate $m$ in terms of $n$ and $v$ you can use the one-to-one property of logarithms:
\begin{align*}
\log m & = \log n + \frac{3}{2} \log \left(1 + \frac{v}{m^2}\right) \\
\log\left(\frac{m}{n}\right) & = \log\left(\left(1+\frac{v}{m^2}\right)^\frac{3}{2}\right) \\
\frac{m}{n} & = \... |
2,162,452 | <p>Question: Find the slope of the tangent line to the graph of $r = e^\theta - 4$ at $\theta = \frac{\pi}{4}$.</p>
<p>$$x = r\cos \theta = (e^\theta - 4)\cos\theta$$</p>
<p>$$y = r\sin \theta = (e^\theta - 4)\sin\theta$$</p>
<p>$$\frac{dx}{d\theta} = -e^\theta\sin\theta + e^\theta\cos\theta + 4\sin\theta$$
$$\frac{... | mrnovice | 416,020 | <p>Since $r$ and $s$ are roots of the equation, we have that $(x-r)(x-s)=0$</p>
<p>Then $x^{2} - (r+s)x +rs=0$</p>
<p>Comparing coefficients:</p>
<p>$r+s = 2m$</p>
<p>$rs = m^{2} + 2m + 3$</p>
<p>$\Rightarrow (r+s)^{2} = 4m^{2}$</p>
<p>$\Rightarrow r^{2}+ s^{2} = 4m^{2} - 2rs = 4m^{2} -2m^{2} -4m -6$ </p>
<p>$r^... |
233,238 | <p>I am just practicing making some new designs with Mathematica and I thought of this recently. I want to make a tear drop shape (doesn't matter the orientation) constructed of mini cubes. I am familiar with the preliminary material, I am just having some difficulty getting it to work.</p>
| Ulrich Neumann | 53,677 | <p>Try</p>
<pre><code>list={{Position,{Code}},{1,{0000,0001}},{2,{0100,0011}},{3,{0110,0111}},{4,{1000,1001}},{5,{1100,1011}},{6,{1110,1111}}}
list /. {a_ , b_List } -> Join[{a}, b]
</code></pre>
|
1,873,180 | <p>The final result should be $C(n) = \frac{1}{n+1}\binom{2n}{n}$, for reference.</p>
<p>I've worked my way down to this expression in my derivation:</p>
<p>$$C(n) = \frac{(1)(3)(5)(7)...(2n-1)}{(n+1)!} 2^n$$</p>
<p>And I can see that if I multiply the numerator by $2n!$ I can convert that product chain into $(2n)!$... | Ovi | 64,460 | <p>You can combine to get $\int (f(x)-f(x))dx= \int 0 dx=C$.</p>
|
630,838 | <p>I was woundering if anyone knows any good references about Kähler and complex manifolds? I'm studying supergravity theories and for the simpelest N=1 supergravity we'll get these. Now in the course-notes the're quite short about these complex manifolds. I was hoping someone of you guys might know a good (quite compl... | Michael Albanese | 39,599 | <p>Here are some references that I have used in the past for various reasons. They are listed in no particular order.</p>
<ul>
<li>Huybrechts - <em>Complex Geometry: An Introduction</em></li>
<li>Moroianu - <em>Lectures in Kähler Geometry</em> (pdf version available <a href="http://moroianu.perso.math.cnrs.fr/tex/kg.p... |
4,624,058 | <p>Godsil&Royle <a href="https://doi.org/10.1007/978-1-4613-0163-9" rel="nofollow noreferrer">Algebraic Graph Theory</a> section 2.5 states (slightly paraphrased):</p>
<blockquote>
<p>Let <span class="math-container">$G$</span> be a transitive group acting on a set <span class="math-container">$V$</span>. A nonempt... | colt_browning | 446,709 | <p>Consider <span class="math-container">$h^{-1}(g(S))$</span>. On one hand, its intersection with <span class="math-container">$S$</span> is nonempty because <span class="math-container">$g(S)\cap h(S)$</span> is nonempty. On the other hand, it is not <span class="math-container">$S$</span>: consider <span class="math... |
674,448 | <p>Prove $F: \mathbb{R}\to\mathbb{R}$ where $F(x) = \int_a^x f(t)\, dt$ ($a<x$) is surjective. </p>
<p>$f$ is continuous and bounded below by $m>0$. Also $a$ belongs to $\mathbb{R}$ (reals).</p>
| Paramanand Singh | 72,031 | <p>I suppose it is easier to handle it using derivatives. Clearly $F'(x) = f(x) \geq m > 0$ for all $x$ so that $F(x)$ is strictly increasing. We need to show that range of $F(x)$ is whole of $\mathbb{R}$. Since $F$ is increasing it follows that either $F(x) \to L$ or $F(x) \to \infty$ as $x \to \infty$. If $F(x) \t... |
2,991,366 | <blockquote>
<p>Consider a point <span class="math-container">$Q$</span> inside the <span class="math-container">$\triangle ABC$</span> triangle, and <span class="math-container">$M$</span>, <span class="math-container">$N$</span>, <span class="math-container">$P$</span> the intersections of <span class="math-container... | Blue | 409 | <p>Changing notation a bit, we'll consider a triangle with circumdiameter <span class="math-container">$1$</span> and angles <span class="math-container">$\alpha$</span>, <span class="math-container">$\beta$</span>, <span class="math-container">$\gamma$</span>. Duplicating OP's configuration with <span class="math-cont... |
4,399,371 | <p>According to my textbook, the formula for the distance between 2 parallel lines has been given as below:</p>
<p><a href="https://i.stack.imgur.com/ZQtQk.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ZQtQk.png" alt="enter image description here" /></a></p>
<p>Where PT is a vector from the first l... | bubba | 31,744 | <p>The formula uses only the <em>magnitude</em> of the the cross product. And the magnitude of <span class="math-container">$u \times v$</span> is <span class="math-container">$\|u\|$</span> times <span class="math-container">$\|v\|$</span> times the sine of the angle between <span class="math-container">$u$</span> and... |
119 | <p>One thing I could never wrap my head around is how <code>Flatten</code> works when provided with a matrix as the second argument, and the <em>Mathematica</em> help isn't particularly good on this one.</p>
<p>Taken from the <a href="http://reference.wolfram.com/mathematica/ref/Flatten.html"><code>Flatten</code></a> ... | WReach | 142 | <p>A second list argument to <code>Flatten</code> serves two purposes. First, it specifies the order in which indices will be iterated when gathering elements. Second, it describes list flattening in the final result. Let's look at each of these capabilities in turn.</p>
<p><strong>Iteration Order</strong></p>
<p>... |
3,783,878 | <p>Hey everyone can anyone help me in simplifying the following boolean expression with explanation?</p>
<p><span class="math-container">\begin{equation}[((p\land q)\implies r)\implies((q\land r')\implies r')]\land[(p \land q)\implies(q\iff p)]\end{equation}</span></p>
| Beyond Infinity | 683,570 | <p>Note that <span class="math-container">$(q\land r')\implies r'$</span> is always true. Thus first part of the statement becomes trivial.
Consider the statement <span class="math-container">$(p \land q)\implies(q\iff p)$</span>. If <span class="math-container">$p=q=1$</span>, it is true, otherwise it is vacuous. Henc... |
286,798 | <blockquote>
<p>Find the limit $$\lim_{n \to \infty}\left[\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\cdots\left(1-\frac{1}{n^2}\right)\right]$$</p>
</blockquote>
<p>I take log and get $$\lim_{n \to \infty}\sum_{k=2}^{n} \log\left(1-\frac{1}{k^2}\right)$$</p>
| user 1591719 | 32,016 | <p>Your way works nice if you employ the Euler's infinite product for the sine function. Then </p>
<p>$$\lim_{n \to \infty}\sum_{k=2}^{n} \ln\left(1-\frac{1}{k^2}\right)=\lim_{x\to\pi}\ln\left(\frac{\pi^2\sin x}{x(\pi^2-x^2)}\right)=\lim_{y\to0}\ln\left(\frac{\pi^2\sin y}{y(2\pi-y)(\pi-y)}\right)=\ln(1/2)$$
Thus, your... |
1,103,533 | <p>I am reading John Lee's book <em>Riemannian Manifolds</em>. On page 91, he begins a chapter called "Geodesics and Distance," which is I think the first chapter that seriously addresses geodesics. </p>
<p>I was very surprised when I came across the following sentence: </p>
<p><em>Most of the results of this chapter... | Andrew D. Hwang | 86,418 | <p>For one thing, a Lorentz-signature metric on a compact manifold can fail to be geodesically complete. If memory serves, Chapter 3 of <em>Einstein Manifolds</em> by Besse contains an example of a metric on a torus where a finite-length geodesic "winds" infinitely many times.</p>
<p>Generally, the "unit sphere" in a ... |
1,103,533 | <p>I am reading John Lee's book <em>Riemannian Manifolds</em>. On page 91, he begins a chapter called "Geodesics and Distance," which is I think the first chapter that seriously addresses geodesics. </p>
<p>I was very surprised when I came across the following sentence: </p>
<p><em>Most of the results of this chapter... | Jack Lee | 1,421 | <p>OK, I'll accept the challenge...</p>
<p>The biggest difference in the pseudo-Riemannian case is that curves can have zero length, and the "Riemannian distance function" (the supremum of the lengths of curves between two points) is not a metric in the sense of metric spaces. Thus most of the results in Chapter 6 of ... |
4,058,600 | <p>Please pardon the elementary question, for some reason I'm not grocking why all possible poker hand combinations are equally probable, as all textbooks and websites say. Just intuitively I would think getting 4 of a number is much more improbably than getting 1 of each number, if I were to draw 4 cards. For example,... | Neel Sandell | 405,304 | <p>I can try deriving it. Imagine a classic case of a dealer drawing 5 cards from the top of a shuffled deck one at a time.</p>
<p>Assume that there exists an ordering in a hand, so JQK12 is different from 1JQK2.</p>
<p>This means that the probability of choosing a hand is <span class="math-container">$\frac{1}{52} \ti... |
1,534,724 | <p>Let $U$ be a unitary matrix, show that $r(x,y) := x^*Uy$ is an inner product satisfying </p>
<ol>
<li><p>$(u,v) = \overline{(v,u)}$</p></li>
<li><p>$(u,u)> 0$ for $u\neq0$; $(u,u)=0$ for $u= 0$</p></li>
<li><p>$(u+sv,w)=(u,w)+s(v,w)$</p></li>
</ol>
<p>for a complex vector space $V$</p>
<p>Explain why this woul... | Lutz Lehmann | 115,115 | <p>This should not work, matrices to generate scalar products are self-adjoint positive definite (SPD).</p>
<p>Indeed, property 1) translates to $U^*=U$, property 2) to $U>0$ and 3) is generally true for this type of construction.</p>
|
1,534,724 | <p>Let $U$ be a unitary matrix, show that $r(x,y) := x^*Uy$ is an inner product satisfying </p>
<ol>
<li><p>$(u,v) = \overline{(v,u)}$</p></li>
<li><p>$(u,u)> 0$ for $u\neq0$; $(u,u)=0$ for $u= 0$</p></li>
<li><p>$(u+sv,w)=(u,w)+s(v,w)$</p></li>
</ol>
<p>for a complex vector space $V$</p>
<p>Explain why this woul... | egreg | 62,967 | <p>Let's examine your claim, where I guess you define $x^*$ as the conjugate transpose. Now
$$
r(v,u)=v^*Uu=\overline{u^*U^*v}=\overline{r(u,v)}
$$
if and only if
$$
u^*U^*v=u^*Uv
$$
for all $u$ and $v$. This is the same as requiring that $U^*=U$, so $U$ must be Hermitian, not unitary.</p>
<p>However a Hermitian unita... |
1,677,035 | <p>I'm new to this website so I apologize in advance if what I'm going to ask isn't meant to be posted here.</p>
<p>A bit of background though: I haven't been to school in 6 years and the last level I've graduated was Grade 7 due to financial problems, as well as my mom frequently being in and out of the hospital. I a... | Live Free or π Hard | 126,067 | <p>Good complimentary books to go with whatever resources you chose are the Schaum’s Outline series. You can find some high school level math books in these series with a lot of practice problems and full solutions. Good luck! <a href="https://www.mhprofessional.com/schaum-s" rel="nofollow noreferrer">https://www.mhpro... |
1,643,201 | <p>The spectrum-functor
$$
\operatorname{Spec}: \mathbf{cRng}^{op}\to \mathbf{Set}
$$
sends a (commutative unital) ring $R$ to the set $\operatorname{Spec}(R)=\{\mathfrak{p}\mid \mathfrak{p} \mbox{ is a prime ideal of R}\}$ and a morpshim $f:S\to R$ to the map $\operatorname{Spec}(R)\to \operatorname{Spec}(S)$ with $\m... | John Molokach | 90,422 | <p>I would count the number of three digits numbers (which is 900) and then count how many of them are a multiple of 7 (which is the greatest integer less than or equal to 900/7).</p>
|
3,357,841 | <p>In the diagram (which is not drawn to scale) the small triangles each have the area shown. Find the area of the shaded quadrilateral.</p>
<p><a href="https://i.stack.imgur.com/DK8sn.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/DK8sn.png" alt="enter image description here"></a></p>
| dfnu | 480,425 | <p><strong>A POSSIBLE PATH</strong></p>
<p>Consider the Figure below and let <span class="math-container">$x$</span> be the desired area.</p>
<p><a href="https://i.stack.imgur.com/D9b5B.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/D9b5B.png" alt="enter image description here"></a></p>
<ol>
<li>... |
3,357,841 | <p>In the diagram (which is not drawn to scale) the small triangles each have the area shown. Find the area of the shaded quadrilateral.</p>
<p><a href="https://i.stack.imgur.com/DK8sn.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/DK8sn.png" alt="enter image description here"></a></p>
| albert chan | 696,342 | <p>I tried to build a general formula for the shaded area.<br>
To simplify, <strong>assume base area = 1</strong>. We can scale it back later.</p>
<p>Let left and right triangle area be <span class="math-container">$x, y$</span>. Solving for shaded area, <span class="math-container">$z$</span><br>
Using @dfnu setup, ... |
2,317,625 | <p>How do you compare $6-2\sqrt{3}$ and $3\sqrt{2}-2$? (no calculator)</p>
<p>Look simple but I have tried many ways and fail miserably.
Both are positive, so we cannot find which one is bigger than $0$ and the other smaller than $0$.
Taking the first minus the second in order to see the result positive or negative ge... | Peter Green | 278,485 | <p>I'll use >=< to represent the unknown comparison.</p>
<p>$ 6-2\sqrt{3} >=< 3\sqrt{2}-2$</p>
<p>Lets start by adding two to both sides to reduce the number of numbers. This doesn't change the comparison result.</p>
<p>$ 8-2\sqrt{3} >=< 3\sqrt{2}$</p>
<p>Both sides are clearly positive ( $ 2\sqrt{3}... |
1,114,007 | <p>How to simplify $$\arctan \left(\frac{1}{2}\tan (2A)\right) + \arctan (\cot (A)) + \arctan (\cot ^{3}(A)) $$ for $0< A< \pi /4$?</p>
<p>This is one of the problems in a book I'm using. It is actually an objective question , with 4 options given , so i just put $A=\pi /4$ (even though technically its disallo... | Jack D'Aurizio | 44,121 | <p>Over the given interval we have $\arctan\cot A=\frac{\pi}{2}-A$ and, by setting $t=\tan A$:
$$\begin{eqnarray*}&&\tan\left(\arctan\cot^3 A+\arctan\left(\frac{\tan(2A)}{2}\right)\right)=\frac{\cot^3 A+\frac{1}{2}\tan(2A)}{1-\frac{1}{2}\cot^3 A\tan(2A)}\\&=&\frac{\cot^3 A+\frac{1}{2}\tan(2A)}{1-\frac{1... |
583,030 | <p>I have to show that the following series convergences:</p>
<p>$$\sum_{n=0}^{\infty}(-1)^n \frac{2+(-1)^n}{n+1}$$</p>
<p>I have tried the following:</p>
<ul>
<li>The alternating series test cannot be applied, since $\frac{2+(-1)^n}{n+1}$ is not monotonically decreasing.</li>
<li>I tried splitting up the series in ... | Siméon | 51,594 | <p>It is not convergent. To see this, let
$$
a_n = (-1)^n\frac{2}{n+1},\qquad b_n =\frac{1}{n+1},\qquad c_n = a_n + b_n.
$$
The series $\sum a_n$ is convergent by the alternating test.</p>
<p>We are interested in the convergence of $\sum c_n$. If $\sum c_n$ was convergent, then $\sum b_n = \sum c_n - \sum a_n$ would a... |
4,285,143 | <p>Let's suppose I've got a function <span class="math-container">$f(x)$</span> where I'd like to differentiate with respect to <span class="math-container">$t$</span>, but <span class="math-container">$t$</span> depends on <span class="math-container">$x$</span>: <span class="math-container">$t(x)$</span>. Thus the wh... | Brian Lai | 821,645 | <p>Maybe this is what you're looking for:</p>
<p><span class="math-container">$$ \frac{df}{dt} = \frac{df}{dx} \frac{dx}{dt} = \frac{df}{dx} \frac{1}{\frac{dt}{dx}}. $$</span></p>
<p>For example: Consider <span class="math-container">$f(x) = x^2$</span> and <span class="math-container">$t = e^x$</span>.</p>
<ol>
<li>Di... |
4,434,832 | <p>I have taken this question from molodovian national MO 2008
The question is as follows</p>
<p>The sequence <span class="math-container">$(a_p)_p\ge 0$</span> is defined as <span class="math-container">$$a_p=\sum_{i=0}^p (-1)^i\frac{\binom{p}{i}}{(i+2)(i+4)}$$</span></p>
<p>Now let's find the limit</p>
<p><span clas... | nobody | 1,050,165 | <p>The solution of Stefan Lafon is elegant and shows the usefulness of professional tools. Integral representations (or, moreover, the dominated convergence theorem) are probably not the intended solution of an MO problem, though. So it may be interesting to see how to derive explicit expressions for <span class="math-... |
351,846 | <p>The following problem was on a math competition that I participated in at my school about a month ago: </p>
<blockquote>
<p>Prove that the equation $\cos(\sin x)=\sin(\cos x)$ has no real solutions.</p>
</blockquote>
<p>I will outline my proof below. I think it has some holes. My approach to the problem was to... | Anupriya | 799,968 | <p>Here's how I did it -</p>
<p><span class="math-container">$\cos( \sin x) = \sin(\cos x)$</span> can be written as,
<span class="math-container">$\sin (\frac{\pi}{2} - \sin x) = \sin (\cos x)$</span>, which implies,</p>
<p><span class="math-container">$\cos x + \sin x = \frac{\pi}{2}$</span>, or,
<span class="math-co... |
3,038,965 | <p>Here's the question I'm puzzling over:</p>
<p><span class="math-container">$\textbf{Find the perpendicular distance of the point } (p, q, r) \textbf{ from the plane } \\ax + by + cz = d.$</span></p>
<p>I tried bringing in the idea of a dot product and attempted to get going with solving the problem, but I'm headin... | qualcuno | 362,866 | <p>A different approach, by a manual verification,</p>
<p><span class="math-container">$$
\frac{1}{1-x} = \sum_{n \geq 0}X^n,
$$</span></p>
<p>and so</p>
<p><span class="math-container">$$
\frac{1}{(1-x)^2} = \frac{1}{1-x} \cdot \frac{1}{1-x} = \sum_{n \geq 0}X^n \cdot \sum_{m \geq 0}X^m = \sum_{i \geq 0}\left(\sum... |
2,674,217 | <p>Let $\{ a_{n}\}_{n}$ be a sequence and let $a\in \mathbb{R}$. Define $\{ c_{n}\}_{n}$ as:</p>
<p>$$c_{n}=\frac{a_{1}+...+a_{n}}{n}.$$</p>
<p>I want to prove the following claim: if $\lim\limits_{n\to +\infty}a_{n}=+\infty$ then $\lim\limits_{n\to +\infty}c_{n}=+\infty$</p>
<p>Approach: Suppose $\lim\limits_{n\to ... | hamam_Abdallah | 369,188 | <p>$$\lim_{\infty}(1-\frac {n_0}{n})M=M $$</p>
<p>$$\implies \exists n_1 \in \Bbb N \;: $$
$$n>n_1\implies (1-\frac {n_0}{n})M>\frac {M}{2} $$
on the other hand</p>
<p>$$\lim_{\infty}\frac {a_1+...a_{n_0}}{n}=0\implies $$</p>
<p>$$\exists \; n_2\in \Bbb N \;:$$
$$ n>n_2\implies \frac {a_1+...a_{n_0}}{n}>... |
279,808 | <p>I was working on a way of calculating the square root of a number by the method of x/y → (x+4y)/(x+y) as shown by bobbym at <a href="https://math.stackexchange.com/questions/861509/">https://math.stackexchange.com/questions/861509/</a></p>
<p>I tried to do it via functions on mathematica, everything seems correct. W... | Bob Hanlon | 9,362 | <pre><code>Clear["Global`*"]
a = 1;
b = 4;
</code></pre>
<p>Using <a href="https://reference.wolfram.com/language/ref/RSolve.html" rel="nofollow noreferrer"><code>RSolve</code></a> will provide the general result for arbitrary <code>n</code></p>
<pre><code>Clear[f]; f[n_] = RSolveValue[{
f[n + 1] == (Nu... |
279,808 | <p>I was working on a way of calculating the square root of a number by the method of x/y → (x+4y)/(x+y) as shown by bobbym at <a href="https://math.stackexchange.com/questions/861509/">https://math.stackexchange.com/questions/861509/</a></p>
<p>I tried to do it via functions on mathematica, everything seems correct. W... | bill s | 1,783 | <p>This can also be approached recursively:</p>
<pre><code>Clear[f]; a = 1; b = 4; f[1] = a;
f[n_] := f[n] = (Numerator[f[n - 1]] + b Denominator[f[n - 1]])
/(Numerator[f[n - 1]] + Denominator[f[n - 1]])
</code></pre>
<p>Then the fist 10 values can be calculated</p>
<pre><code>f /@ Range[10]
</code></pre... |
3,561,664 | <p>I did part of this question but am stuck and don't know how to continue</p>
<p>I let <span class="math-container">$x= 2k +1$</span></p>
<p>Also noticed that <span class="math-container">$x^3+x = x(x^2+1)$</span></p>
<p>therefore
<span class="math-container">$4m+2 = 2k+1((2k+1)^2+1)$</span></p>
<p>I simplified th... | Z Ahmed | 671,540 | <p>Let <span class="math-container">$x=2n+1, then
f(x)=x^3+x=(2n+1)^3+(2n+1)=2+8n^3+12n^2+8n \implies \frac{f(x)}{4}=\frac{2}{4}+2n^3+3n^2+2n.$</span> So the remaner is 2.</p>
|
246,071 | <p>How do I solve the following equation?</p>
<p>$$x^2 + 10 = 15$$</p>
<p>Here's how I think this should be solved.
\begin{align*}
x^2 + 10 - 10 & = 15 - 10 \\
x^2 & = 15 - 10 \\
x^2 & = 5 \\
x & = \sqrt{5}
\end{align*}
I was thinking that the square root of 5 is iregular repeating 2.23606797749979 nu... | Henry | 6,460 | <p>You actually want to show </p>
<p>$$\frac{1}{n+2}+\frac{1}{n+3}+\cdots+\frac{1}{(n+1)+(n+1)}+\frac{1}{(n+1)+(n+2)}\le\frac{5}{6}$$</p>
<p>So you take the inductive hypothesis, and subtract $\frac{1}{n+1}$ from and add $\frac{1}{(n+1)+(n+1)}+\frac{1}{(n+1)+(n+2)}$ to the left hand side. Since you can show that cha... |
170,240 | <p>I have the following function of ω</p>
<pre><code>f[ω_] := (2 Sqrt[Γ] (4*g2^2 + (κ1 - 2*I*ω) (κ2 - 2*I*ω)))/(4*g2^2 (Γ -
2*I*ω) + (4*
g1^2 + (Γ - 2*I*ω) (κ1 -
2*I*ω)) (κ2 - 2*I*ω))
</code></pre>
<p>And I wish to obtain the poles for the denominator of the function:</p>
<pre><code>wroots1 = x /. Solve[(Denomina... | Roman | 26,598 | <p>If you <code>Conjugate</code> the equation before <code>Solve</code>, you'll get the complex-conjugated solutions (complex analysis is nice!):</p>
<pre><code>wroots1 = w /. Solve[Denominator[f[w]] == 0, w];
wroots1C = w /. Solve[ComplexExpand[Conjugate[Denominator[f[w]]]] == 0, w];
</code></pre>
<p>This way you do... |
3,703,981 | <p>If we consider an equation <span class="math-container">$x=2x^2,$</span> we find that the values of <span class="math-container">$x$</span> that solve this equation are <span class="math-container">$0$</span> and <span class="math-container">$1/2$</span>. Now, if we differentiate this equation on both sides with res... | J. W. Tanner | 615,567 | <p>There are no solutions, because <span class="math-container">$x^3,y^3\equiv0$</span> or <span class="math-container">$\pm1\pmod7$</span>, but <span class="math-container">$2020\equiv4\bmod7$</span>.</p>
|
194,547 | <p>I know the definition of a linear transformation, but I am not sure how to turn this word problem into a matrix to solve:</p>
<p>$T(x_1, x_2) = (x_1-4x_2, 2x_1+x_2, x_1+2x_2)$</p>
<p><strong>Find the image of the line that passes through the origin and point $(1, -1)$.</strong></p>
| Brian M. Scott | 12,042 | <p>HINT: A linear transformation sends straight lines to straight lines. If $T$ sends the origin and the point $\langle 1,-1\rangle$ to the points $P$ and $Q$, it must send the line through the origin and the point $\langle 1,-1\rangle$ to the line through $P$ and $Q$.</p>
|
2,736,426 | <p>Let's imagine a point in 3D coordinate such that its distance to the origin is <span class="math-container">$1 \text{ unit}$</span>.</p>
<p>The coordinates of that point have been given as <span class="math-container">$x = a$</span>, <span class="math-container">$y = b$</span>, and <span class="math-container">$z = ... | sirous | 346,566 | <p>Suppose angle of vector related to x axis is $\alpha$, related to y axis is $\beta$ and related to z axis is $\gamma$ then we have:</p>
<p>Due to presumption; $\sqrt {a^2+b^2+c^2}=1$ </p>
<p>$1\times \ cos \alpha=a$</p>
<p>$1\times \ cos \beta=b$</p>
<p>$1 \times \ cos \gamma=c$</p>
|
3,757,213 | <blockquote>
<p>Prove that the maximum area of a rectangle inscribed in an ellipse <span class="math-container">$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$</span> is <span class="math-container">$2ab$</span>.</p>
</blockquote>
<p><strong>My attempt:</strong></p>
<p>Equation of ellipse: <span class="math-container">$\dfrac{x^... | Quanto | 686,284 | <p>Let one vertex of the rectangle be <span class="math-container">$(a\cos t, b\sin t)$</span>. Then, the other three are known as well and the area is</p>
<p><span class="math-container">$$A= 4ab|\sin t \cos t |\le 2ab (\cos^2t+\sin^2t)=2ab$$</span></p>
<p>where the inequality <span class="math-container">$2uv\le u^2+... |
1,015,498 | <p>I am merely looking for the result of the convolution of a function and a delta function.
I know there is some sort of identity but I can't seem to find it. </p>
<p>$\int_{-\infty}^{\infty} f(u-x)\delta(u-a)du=?$</p>
| JohnD | 52,893 | <p>It's called the <a href="http://en.wikipedia.org/wiki/Dirac_delta_function#Translation" rel="noreferrer">sifting property</a>:</p>
<p>$$
\int_{-\infty}^\infty f(x)\delta(x-a)\,dx=f(a).
$$</p>
<p>Now, if
$$
f(t)*g(t):=\int_0^t f(t-s)g(s)\,ds,
$$
we want to compute
$$
f(t)*\delta(t-a)=\int_0^t f(t-s)\delta(s-a)\,ds.... |
22,839 | <p>Is it possible to have the text generated by <code>PlotLabel</code> (or any other function) aligned to the left side of the plot instead of in the center?</p>
| Verbeia | 8 | <p>You can specify your <code>PlotLabel</code> to be a construct that takes a <code>TextAlignment</code> or <code>Alignment</code> option, such as <code>Pane</code>. For example:</p>
<pre><code>Plot[Sin[x], {x, 0, 5},
PlotLabel -> Framed@Pane["This is the title", Alignment -> Left,
ImageSize -> 270], I... |
22,839 | <p>Is it possible to have the text generated by <code>PlotLabel</code> (or any other function) aligned to the left side of the plot instead of in the center?</p>
| Carl Woll | 45,431 | <p>Here is a refinement of @DavidC's approach.</p>
<p>We can use <code>PlotRangeClipping->False</code> and then stick the label outside of the plot range and still have it show up. In order to do this we need to know the <a href="http://reference.wolfram.com/language/ref/ImagePadding" rel="nofollow noreferrer"><cod... |
336,834 | <p>It is a well known theorem, that every signed measure can be split into its positive and negative parts (Hahn-Jordan-Decomposition). My question is, if something similar is possible for functionals on Sobolev spaces.</p>
<p>To be precise, let $\Omega \subset \mathbb{R}^n$ be some open domain and $\mu \in H^{-1}(\Om... | gerw | 58,577 | <p>Just for reference, I post a different answer (which doesn't rely on the fact that positive distributions are measures).</p>
<p>Define $\Omega = (-1, 1)$, and
$$
f = \begin{cases} 0 & x \le 0, \\ |x|^{-1/3} & x > 0. \end{cases}
$$
Then, $f \in L^2(\Omega)$ and hence, $\mu$ defined by
$$
\langle \mu, v \r... |
92,983 | <p><strong>Does every polyhedron in $\mathbb{R}^3$ with $n$ triangular facets have a <em>topological</em> triangulation with complexity $O(n)$?</strong></p>
<p>Suppose $P$ is a non-convex polyhedron in $\mathbb{R}^3$ with $n$ triangular facets, possibly with positive genus. A <em>topological</em> triangulation of $P$... | Sam Nead | 1,650 | <p>I've been thinking about the main question in the original post on and off for a few days. All of my efforts have been in the direction of finding enough examples to prove a super-linear lower bound, following Misha's suggestion to use hyperbolic volume. This hasn't worked yet - the problem appears to be tricky! ... |
1,431,464 | <p>Does anyone know a good reference where it is shown that the Schwartz class $\mathcal{S}(\mathbb R)$ is a dense subset of $L^2(\mathbb R)$?</p>
<p>Many thanks</p>
| Dan | 79,007 | <p>The most frequent/easiest way I've seen this proved is to show instead that $C_c^\infty(\mathbb R)$ is dense $L^2(\mathbb R)$ and then just note $C_c^\infty(\mathbb R) \subset S(\mathbb R)$. This can be found in anything from <s>big Rudin to</s> Folland's <em>Real Analysis</em> to Trèves's <em>Topological Vector Spa... |
183,077 | <p>A complex Lie group may have several real forms.
Are there any duality/trinity... between them?
Maybe a trivial question to ask, is $SL(3,\mathbb{C})$ a real form of $SL(3,\mathbb{C})\times SL(3,\mathbb{C})$ ?</p>
| Amritanshu Prasad | 9,672 | <p>According to <a href="http://dx.doi.org/10.1016/0021-8693(75)90041-1">Djokovic and Maizan</a>, the Specht module $V_{(3, 1, 1)}$ of $S_5$ is monomial. This is a representation of dimension $6$, induced from a representation of dimension $3$ of $A_5$. Since $A_5$ has no subgroup of index $3$ (see <a href="http://grou... |
155,455 | <p>I want to find the maximum of a function (f) over a variable (t). The function is huge and it's not possible to maximize f(t) directly. So I want to create f inside a Table and then find the highest value over a small range of t. How can I add the steps to construct f into a Table? It seems "/." is not working. </p... | danielsmw | 24,976 | <p>You don't actually need to create a variable called <code>f</code> in each element of the table; just return the value you need. You also want rules, rather than assignments. Try</p>
<pre><code>v = Table[ TR1 + TR2 /. Flatten[{
Solve[TR1 - t == 0, TR1],
TR2 -> t + 1}],
{t, 1, 3, 1}];
tstar... |
20,314 | <p>Hi all.
I'm looking for english books with a good coverage of distribution theory.
I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions.
Thanks in advance.</p>
| Anweshi | 2,938 | <p>Lieb and Loss, "Analysis" quickly starts with measure theory and after a short break with Fourier transforms, gets on to Distributions. I would imagine this is the fastest way to learn distributions. </p>
|
20,314 | <p>Hi all.
I'm looking for english books with a good coverage of distribution theory.
I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions.
Thanks in advance.</p>
| O.R. | 5,506 | <p>I liked Functional Analysis by Kosaku Yosida. It is book on functional analysis but oriented to get the applications of it to differential equations. </p>
|
20,314 | <p>Hi all.
I'm looking for english books with a good coverage of distribution theory.
I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions.
Thanks in advance.</p>
| Anand | 36,814 | <p>Why don't people mention about Rudin's book, <em>Functional Analysis</em>. Chapter 1-8 are pretty good for the theory of distribution. The problem is that this book is quite dry, no much motivations behind. So you might have a difficult time in the beginning. It is good to read the book Strichartz, R. (1994), <em>A ... |
842,266 | <p>I have a tiny little doubt related to one proof given in Ahlfors' textbook. I'll copy the statement and the first part of the proof, which is the part where my doubt lies on.</p>
<p><strong>Statement</strong>
The stereographic projection transforms every straight line in the $z$-plane into a circle on $S$ which pas... | Alucard | 167,097 | <p>i was searching right this question for my complex analysis course because i was stuck on the same page ( guess we are studying the same thing) but my doubt was on ${\alpha_1}^2+{\alpha_2}^2+{\alpha_3}^2=1$. Anyway, now that i have understood this equation i think i can give you another answer for why $ 0 &l... |
1,048,668 | <p>Let $f\colon (a,b) \to \mathbb{R}$ a non constant differentiable function. </p>
<p>Is the following statement true:</p>
<p>If $f$ has a local maximum <em>and</em> a local minimum then $f$ also does have an inflection point.</p>
<p>If so, how to prove it, if not, what would be a counterexample?</p>
<p><em>Remark<... | Narasimham | 95,860 | <p>Definition of inflection point: when $ f''(x)=0. $</p>
<p>Between two extrema with second derivatives of opposite sign there will always be at least one inflection point. The graph of second derivative must pass through zero as consequence of Rolle's theorem.</p>
<p>Non-strict concave/convex situation is when $ f'... |
1,048,668 | <p>Let $f\colon (a,b) \to \mathbb{R}$ a non constant differentiable function. </p>
<p>Is the following statement true:</p>
<p>If $f$ has a local maximum <em>and</em> a local minimum then $f$ also does have an inflection point.</p>
<p>If so, how to prove it, if not, what would be a counterexample?</p>
<p><em>Remark<... | Nate Eldredge | 822 | <p>Consider the bump function
$$\phi(x) = \begin{cases} e^{-1/(1-x^2)}, & -1 < x < 1 \\ 0, & \text{else.} \end{cases}$$
It's a standard exercise to verify that $\phi$ is $C^\infty$ and has a local maximum at $x=0$. Set $g(x) = \phi(x-2) - \phi(x+2)$, so that $g$ is $C^\infty$, strictly negative on $(-3,-... |
3,752,455 | <blockquote>
<p><strong>Problem.</strong> Show that for <span class="math-container">$n\ge 2$</span> there are no solution <span class="math-container">$$x^n+y^n=z^n$$</span> such that <span class="math-container">$x$</span>, <span class="math-container">$y$</span>, <span class="math-container">$z$</span> are prime num... | poetasis | 546,655 | <p>Fermat's last theorem, <a href="https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem" rel="nofollow noreferrer">now proven</a>, shows that <span class="math-container">$A^x+B^x=C^x$</span> cannot be true for <span class="math-container">$x\ne2$</span> so it is not an issue here.</p>
<p>If we use ... |
3,761,689 | <p>I was watching a YouTube video where it showed how length of daylight changes depending on the time of year, and I was curious and wanted to try calculating the value of how long the daylight is in the Tropic of Cancer (23.5 degrees latitude) during the winter solstice, apparently 10 hours and 33 minutes or so accor... | David K | 139,123 | <blockquote>
<p>That means the orange leg is <span class="math-container">$0.173381r/0.917060r$</span> fraction of the yellow and orange leg, about <span class="math-container">$0.189061784.$</span> This represents how much extra darkness there is along the line.</p>
</blockquote>
<p>Yes, that's how much extra darkness... |
873,992 | <p>I am having problem with the onto part of this problem.</p>
<p>$\mathbb{N}\rightarrow \mathbb{E}$</p>
<p>My function or pattern is </p>
<p>$x \rightarrow f(x)=2x$ </p>
<p>Which take my natural to even.</p>
<p><strong>One to One</strong></p>
<p>$f(x)=f(y)$</p>
<p>$2x=2y$</p>
<p>$x=y$</p>
<p><strong>Onto</str... | ant11 | 110,047 | <p>Let $Nx$ be "$x$ is naive" and $Bx$ be "$x$ is bad".</p>
<p>Well, if $x$ is naive, $x$ can't be bad. So
$$\forall x(Nx\rightarrow \neg Bx)$$</p>
|
44,562 | <p>The question is motivated from the definition of $C^r(\Omega)$ I learned from S.S.Chern's <em>Lectures on Differential Geometry</em>:</p>
<p>Suppose $f$ is a real-valued function defined on an open set $\Omega\subset{\bf R}^m$. If all the $k$-th order partial derivatives of $f$ exist and are continuous for $k\leq r... | JDH | 413 | <p>Now that your question has been answered, let me point out that it may be interesting to observe furthermore that all the countable well-orderings are in fact represented by suborders of $\langle\mathbb{R},\lt\rangle$, and even of $\langle\mathbb{Q},\lt\rangle$. Let me give two proofs. </p>
<p>The first proof is an... |
4,528,629 | <p>When doing an exercise about linear representations of finite groups I stumbled upon this Isomorphism in the comments of another <a href="https://math.stackexchange.com/questions/308680/basic-identity-of-characters?rq=1">post</a> which I was not aware of.</p>
<p>In this context <span class="math-container">$V$</span... | bluemaster | 460,565 | <p>Part of the answer to your question is about notation. If <span class="math-container">$\mathbf{Y}$</span> is a matrix <span class="math-container">$n\times m$</span> such that <span class="math-container">$\mathbf{Y}=[Y_{ij}]$</span>, with <span class="math-container">$Y_{ij}$</span> being a r.v. in the position <s... |
1,663,113 | <p>I'm having a mind wrenching question that I just cannot answer. It's been a while since I was at the school bench so I wonder if anyone can help me out? :)</p>
<p>We have 10 students with 5 cakes each to be shared amongst each other.
The students can give the cakes out, but they can’t give a piece to a person who g... | Steve Kass | 60,500 | <p>The first solution that comes to mind is for no one to give any cakes to anyone, so everyone "gets" (to keep) the five cakes they brought. But I assume you want "gets" to mean "gets from someone else."</p>
<p>If this is the idea, no one can give out all five cakes and get five back, because that would require there... |
3,676,284 | <p><a href="https://i.stack.imgur.com/9xfxz.png" rel="nofollow noreferrer">This is link to question</a>
[Here is my attempt, but the answer key is convergent. I dont think I count it wrong.<a href="https://i.stack.imgur.com/nAcEs.jpg" rel="nofollow noreferrer">][1]</a></p>
| Henno Brandsma | 4,280 | <p>Maybe working in the <a href="https://en.wikipedia.org/wiki/Constructible_universe" rel="nofollow noreferrer">constructible universe</a>, or Gödel's model of ZF would be interesting for you. There we restrict our universe (to so-called constructible sets), so that Choice becomes a theorem and we also have a "constru... |
3,676,284 | <p><a href="https://i.stack.imgur.com/9xfxz.png" rel="nofollow noreferrer">This is link to question</a>
[Here is my attempt, but the answer key is convergent. I dont think I count it wrong.<a href="https://i.stack.imgur.com/nAcEs.jpg" rel="nofollow noreferrer">][1]</a></p>
| Asaf Karagila | 622 | <p>Well, the question is what is a list. If a list is just a well ordered set, then in principle you are correct.</p>
<p>In some sense this is similar to working with Global Choice, where every set has a distinguished well ordering.</p>
<p>The problem is formalizing lists is somehow more elaborate than formalizing se... |
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