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 |
|---|---|---|---|---|
3,408,082 | <blockquote>
<p><span class="math-container">$\textbf{Definition}$</span>: We say <span class="math-container">$f:\mathbb{R} \to \mathbb{R}$</span> is <em>intersecting</em> if for every nonempty <span class="math-container">$A \subset \mathbb{R}$</span>, <span class="math-container">$f[A] \cap A \neq \varnothing$</span... | Misha Lavrov | 383,078 | <p>In <a href="https://math.stackexchange.com/a/3408428/383078">antkam's answer</a>, it's conjectured that an <span class="math-container">$\ell$</span>-intersecting function has deviation at most <span class="math-container">$3(\ell-1)$</span>. I will prove that in this answer. It's tight, by considering a permutation... |
4,524,554 | <p>Consider we have an <span class="math-container">$n \times n$</span> matrix, <span class="math-container">$A$</span>. This matrix represents a linear function from <span class="math-container">$\Bbb R^n$</span> to <span class="math-container">$\Bbb R^n$</span>. Let's say we found a sub-space spanned by the vectors <... | lisyarus | 135,314 | <p><em>I'll omit the surjectivity condition, because the analysis isn't much harder in this case.</em></p>
<p>First, let's deal with <span class="math-container">$f(1)$</span>.</p>
<p>If <span class="math-container">$f(1) = 0$</span>, then <span class="math-container">$\forall x\,\, f(x)=f(x\cdot 1)=f(x)+f(1)+f(x)f(1)=... |
3,082,779 | <p>I love watches, and I had an idea for a weird kind of watch movement (all of the stuff that moves the hands). It is made up of a a central wheel, with one of the hands connected to it (in this case, it will be the hour hand). This hand goes through a pivot, and then displays the time. I attached a video of a 3d mock... | Ivo Terek | 118,056 | <p>Not quite. Although the index balance is correct, in <span class="math-container">${\rm d}x^\mu = g^{\mu\nu}\partial_\nu$</span> you have that the left side is a <span class="math-container">$1$</span>-form, while the right side is a vector field. What happens here is that if <span class="math-container">$(M,g)$</sp... |
1,938 | <p>I no longer want to help a particular user (picakhu) due to repeated comments that my considered answers are not helpful, changes in questions after I answer, and the user's pattern of not voting things up (26 questions, only 25 up votes) which I think is antisocial. I have deleted several of my answers to that user... | Willie Wong | 1,543 | <p>@Douglas: please don't do what you are requesting the moderators to do. Remember that this is a Q and A website, and it seeks to document questions and answers <strong>is a way that is easily searchable and useful to other people</strong> besides just the OP and the answerer. Deliberately removing content from the s... |
1,938 | <p>I no longer want to help a particular user (picakhu) due to repeated comments that my considered answers are not helpful, changes in questions after I answer, and the user's pattern of not voting things up (26 questions, only 25 up votes) which I think is antisocial. I have deleted several of my answers to that user... | Alex B. | 3,212 | <p>This got too long for a comment.</p>
<p>I find the logic advocated by Willie Wong quite strange and even alarming. It is effectively saying that a user loses control over his own contribution as soon as he posts it, since a moderator reserves the right to suspend him, should he choose to exercise this control. Ther... |
887,473 | <p>I have been struggling with the following claim:</p>
<p>Let $A_n$ be a sequence of compact sets and $A$ a compact set. $A=\lim\sup_n A_n=\lim\inf_n A_n$ iff $d_H(A_n,A)\to 0$ where $d_H(.,.)$ is the Hausdorff metric.</p>
<p>$\lim\inf$ and $\lim\sup$ are defined by $\lim\inf_nA_n=\left\{y\in Y:\forall \varepsilon&... | GEdgar | 442 | <p>Let's think about this. Some directions don't work. In space $A=[0,1]$, which is compact, let $A_n = \{ k/2^n : k=0,1,\dots,2^n\}$. Then $d_H(A_n,A) \to 0$. But $\limsup A_n = \liminf A_n = \bigcup A_n$ is not compact at all.</p>
|
2,715,374 | <p>We know that \begin{equation*}
a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cfrac{1}{\ddots+\cfrac{1}{a_n}}}}}=[a_0,a_1, \cdots, a_n]
\end{equation*}</p>
<p>If $\frac{p_n}{q_n}=[a_0,a_1, \cdots, a_n]$.</p>
<blockquote>
<p>How to prove that $$
\begin{pmatrix}
p_n & p_{n-1} \\
q_n & q_{n-1} \... | StuartMN | 439,545 | <p>But with x=0 ,0x=1x (=0) so line 2 is a fallacy (false) </p>
|
367,204 | <p>I'm trying to prove that $\mathbb Z_p^*$ ($p$ prime) is a group using the Fermat's little theorem to show that every element is invertible.</p>
<p>Thus using the Fermat's little theorem, for each $a\in Z_p^*$, we have $a^{p-1}\equiv1$ (mod p). The problem is to prove that p-1 is the least positive integer which $a^... | Mr.Lilly | 67,969 | <p>Please check my inverse solution.</p>
<p>Let $\overline{x} \in \mathbb{Z}_{p}^*$. Then $(x,p)=1$, that is there exist $k, q \in \mathbb{Z}$ such that $kx + qp=1$. This implies $\overline{k}\overline{x}=\overline{1}$. Suppose that $\overline{k}=\overline{0}$. Then $\overline{0}=\overline{1}$, a contradict. Thus $\ov... |
3,516,241 | <p>Consider the equation:</p>
<p><span class="math-container">$$ x ^ 4 - (2m - 1) x^ 2 + 4m -5 = 0 $$</span></p>
<p>with <span class="math-container">$m \in \mathbb{R}$</span>. I have to find the values of <span class="math-container">$m$</span> such that the given equation has all of its roots real.</p>
<p>This is ... | Quanto | 686,284 | <p>One approach is to express <span class="math-container">$m$</span> as a function of <span class="math-container">$x$</span>,</p>
<p><span class="math-container">$$m(x)=\frac{x^4+x^2-5}{2x^2-4}
=\frac12\left(x^2+3+\frac1{x^2-2}\right)$$</span></p>
<p>Then, set <span class="math-container">$m’(x)=0$</span> to get </... |
2,892,342 | <p>Given two adjacent sides and all four angles of a quadrilateral, what is the most efficient way to calculate the angle that is made between a side and the diagonal of the quadrilateral that crosses (but does not necessarily bisect) the angle in between the two known sides?</p>
<p>Other known information:</p>
<ul>
... | Jack D'Aurizio | 44,121 | <p>Le us assume that $D$ lies at the origin and $\widehat{CDA}=\theta$. Then the coordinates of $A$ are $(x\cos\theta,x\sin\theta)$ and the line through $A$ which is orthogonal to $DA$ has equation $y(t)=-\frac{\cos\theta}{\sin\theta}t+\left(x\sin\theta+x\frac{\cos^2\theta}{\sin\theta}\right) $. It follows that the len... |
441,374 | <p>Let $K_{\alpha}(z)$ be the <a href="https://en.wikipedia.org/wiki/Bessel_function#Modified_Bessel_functions:_I.CE.B1_.2C_K.CE.B1" rel="nofollow noreferrer">modified Bessel function of the second kind of order $\alpha$</a>.</p>
<p>I need to compute the following integral:</p>
<p>$$\int_0^\infty\;\;K_0\left(\sqrt{a(... | doraemonpaul | 30,938 | <p>$\int_0^\infty K_0\left(\sqrt{a(k^2+b)}\right)dk$</p>
<p>$=\int_0^\infty\int_0^\infty e^{-\sqrt{a(k^2+b)}\cosh t}~dt~dk$</p>
<p>$=\int_0^\infty\int_0^\infty e^{-(\sqrt{a}\cosh t)\sqrt{k^2+b}}~dk~dt$</p>
<p>$=\int_0^\infty\int_0^\infty e^{-(\sqrt{a}\cosh t)\sqrt{(\sqrt{b}\sinh u)^2+b}}~d(\sqrt{b}\sinh u)~dt$</p>
... |
598,635 | <p>Prove the two Identities for
$-1 < r < 1$</p>
<p>$$\sum_{n=0}^{\infty} r^n\cos n\theta =\frac{1-r\cos\theta}{1-2r\cos\theta+r^2}$$</p>
<p>$$\sum_{n=0}^{\infty} r^n\sin{n\theta}=\frac{r \sin\theta }{1-2r\cos\theta+r^2}$$</p>
<p>Sorry could not figure out how to format equations</p>
| Igor Rivin | 109,865 | <p>Hint: $\exp(i x) = \cos(x ) + i \sin(x).$</p>
|
3,271,675 | <p>Let <span class="math-container">$p$</span> be a prime of the form <span class="math-container">$p = a^2 + b^2$</span> with <span class="math-container">$a,b \in \mathbb{Z}$</span> and <span class="math-container">$a$</span> an odd prime. Prove that <span class="math-container">$(a/p) =1$</span></p>
<p>Could anyon... | user10354138 | 592,552 | <p><strong>Hint</strong>: Note that <span class="math-container">$p\equiv 1\pmod 4$</span>, so quadratic reciprocity gives <span class="math-container">$\left(\frac{a}p\right)=\left(\frac{p}a\right)$</span>.</p>
|
1,714,902 | <p>(Question edited to shorten and clarify it, see the history for the original)</p>
<p>Suppose we are given two $n\times n$ matrices $A$ and $B$. I am interested in finding the closest matrix to $B$ that can be achieved by multiplying $A$ with orthogonal matrices. To be precise, the problem is</p>
<p>$$\begin{align}... | Community | -1 | <p>Independently of @loup blanc's answer, I found a more elementary partial solution showing that the $2\times2$ case implies the general $n\times n$ case.</p>
<p>The desired proposition is equivalent to the following: For any $n\times n$ matrix $A$ and diagonal $n\times n$ matrix $B$, a matrix $\tilde A=UAV^T$ which ... |
737,915 | <p>I'm reading Calculus: Basic Concepts for High School Students and am trying to digest the definition of 'limit of function'. There are two details that I am struggling to fully accept:</p>
<ol>
<li><p>If you are supposed to pick an interval $(a - \delta, a + \delta)$ but $a$ can be an undefined point at the end of ... | T_O | 84,702 | <ol>
<li><p>Yes, if your function is defined on $[a,b]$ and you want to compute the limit in $b$, you can only consider what is happening in $[b-\delta, b]$</p></li>
<li><p>Look at the function floor(x) which is defined as the highest integer that is less than equal to x. Here is the graph :</p></li>
</ol>
<p><img src... |
3,121,361 | <p>Given <span class="math-container">$G$</span> has elements in the interval <span class="math-container">$(-c, c)$</span>. Group operation is defined as:
<span class="math-container">$$x\cdot y = \frac{x + y}{1 + \frac{xy}{c^2}}$$</span></p>
<p>How to prove closure property to prove that G is a group?</p>
| user90369 | 332,823 | <p><em>Without</em> using <span class="math-container">$\,b_n\,$</span> you can use of course what the others have written!</p>
<p>There is nothing to add.</p>
<hr>
<p>Given: <span class="math-container">$\,a_n\,$</span> monotone increasing and <span class="math-container">$\,b_n\,$</span> monotone decreasing and <s... |
2,555,499 | <p>Let $v_1=(1,1)$ and $v_2=(-1,1)$ vectors in $\mathbb{R}^2$. They are <strong>clearly linearly independent</strong> since each is not an scalar multiple of the other. The following information about a linear transformation $f: \mathbb{R}^2 \to \mathbb{R}^2$ is given: $$f(v_1)=10 \cdot v_1 \text{ and } f(v_2)=4 \cdot ... | wendy.krieger | 78,024 | <p>My trick is to make the large operators × and ÷ and the small operators (mn and m/n) separate. The small operators are applied as multiplication then division.</p>
<p>So $ab/cd=\frac {ab}{cd}$. In essence the small operators are parts of the same 'word'. </p>
<p>The large operators are always applied to the fir... |
1,071,564 | <p>Let's a call a directed simple graph $G$ on $n$ labelled vertices <strong>good</strong> if every vertex has outdegree 1 and, when considered as if it were undirected, it is connected. How many good graphs of size $n$ are there?</p>
<p>Here's my work so far. Let's call this number $T(n)$. Clearly, $T(2) = 1$: there'... | jschnei | 113,308 | <p>The graphs you are describing are known as simple (directed) pseudotrees; see <a href="http://en.wikipedia.org/wiki/Pseudoforest" rel="nofollow">http://en.wikipedia.org/wiki/Pseudoforest</a>. There doesn't appear to be a 'nice' closed form for these trees. Wikipedia/OEIS gives the number of undirected connected grap... |
206,780 | <p>Let $f:X\to Y$ is a measurable function. Banach indicatrix
$$
N(y,f) = \#\{x\in X \mid f(x) = y\}
$$
is the number of the pre-images of $y$ under $f$. If there are infinitely many pre-images then $N(y,f) = \infty$. </p>
<p>Let $X\subset\mathbb R^n$, $Y\subset\mathbb R^m$ with Lebesgue measure.</p>
<p><em>I am inte... | R W | 8,588 | <p>This is a corollary of Rokhlin's theorem on classification of measurable partitions in Lebesgue spaces from his famous paper "On the fundamental ideas of measure theory". The Russian original is <a href="http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=5995&option_lang=rus" rel="nofol... |
2,611,656 | <p>Suppose $x = 1/t$. So now $x$ is a function of $t$, i.e., $x(t)$.</p>
<p>So $$\frac{dx(t)}{dt} = -t^{-2} \Rightarrow dx(t) = -t^{-2}dt$$</p>
<p>This problem is from the textbook: <code>advanced mathematical methods for scientists and engineers</code></p>
<p><a href="https://i.stack.imgur.com/SqKnQ.png" rel="nofol... | David Reed | 444,890 | <p>You use the inverse-function theorem and the chain rule:</p>
<p>Inverse function theorem says $$\frac{dx}{dt} = -t^{-2} \to \frac{dt}{dx} = \frac{1}{\frac{dx}{dt}}= -t^2
$$</p>
<p>The chain rule says : $$\frac{d}{dx} = \frac{dt}{dx}\frac{d}{dt} = -t^2\frac{d}{dt}$$</p>
|
1,010 | <p>For periodic/symmetric tilings, it seems somewhat "obvious" to me that it just comes down to working out the right group of symmetries for each of the relevant shapes/tiles, but its not clear to me if that carries over in any nice algebraic way for more complicated objects such as a <a href="http://en.wikipedia.org/... | Alex Fink | 30 | <p>For periodic tilings, Bill Thurston and JH Conway would say that it's better to think about the orbifolds of tilings than their symmetry groups: this is the approach to the classification of plane symmetry groups and several other things Conway and Burgiel and Goodman-Strauss take in the beautiful <em>The symmetries... |
737,692 | <p>I'm working on this question:</p>
<blockquote>
<p>Rewrite the following summation using sigma notation and then compute it
using the technique of telescoping summation.
$$\frac{1}{2*5}+\frac{1}{3*6}+\frac{1}{4*7}+...+\frac{1}{(n-2)(n+1)}+\frac{1}{(n-1)(n+2)} $$</p>
</blockquote>
<p>My work:
I replaced the ... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>$$\frac1{(i-2)(i+1)}=\frac13\cdot\frac{i+1-(i-2)}{(i-2)(i+1)}=\frac13\left(\frac1{i-2}-\frac1{i+1}\right)$$</p>
<p>Similarly, $$\frac1{(i-1)(i+2)}=\cdots=\frac13\left(\frac1{i-1}-\frac1{i+2}\right)$$</p>
<p>Set $i=1,2,\cdots,n-1,n$ find the surviving terms </p>
|
2,137,706 | <p>$$\sum_{k=0}^m {n \choose k} {n-k \choose m-k} = 2^m {n \choose m}, m<n$$</p>
<p>I know that $2^m$ represents the number of subsets of a set of length $m$, which I can see there being a connection to the ${n \choose k}$ term, but I can't see how the combination it's multiplied by affects this.</p>
| Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\n... |
1,310,233 | <p>Fundamental theorem of calculus states that the derivative of the
integral is the original function, meaning that
$$
f(x)=\frac{d}{dx}\int_{a}^{x}f(y)dy.\tag{*}
$$
To motivate the statement of the Lebesgue differentiation theorem, observe
that (*) may be written in terms of symmetric differences as
$$
f(x)=\lim_{r\t... | Community | -1 | <p>The last equality as stated is not true. But we don't need that. </p>
<p>In general we have </p>
<p>$$g'(x) = \lim_{r\to 0}\frac{g(x+r) - g(x-r)}{2r}$$</p>
<p>for any function differentiable at $x$ (Can you show that?). Put $g(x) = \int_a^x f(s) \, ds$, by $(*)$, </p>
<p>\begin{equation}
\begin{split}
f(x) &... |
690,465 | <p>So we are learning trigonometry in school and I would like to ask for a little help with these. I would really appreciate if somebody can explain me how I can solve such equations :)</p>
<ul>
<li><p>$\sin 3x \cdot \cos 3x = \sin 2x$</p></li>
<li><p>$2( 1 + \sin^6 x + \cos^6 x ) - 3(\sin^4 x + \cos^4 x) - \cos x = 0... | Alan | 54,910 | <p>For the second:</p>
<p>$$
2\left(1+\sin^6 x+\cos^6 x\right)-3\left( \sin^4 x + \cos^4 x\right) - \cos x= 0 \\
2\left(1+\left( \sin^2 x + \cos^2 x \right) \left(\sin^4 x - \sin^2 x \cos^2 x + \cos^4 x\right)\right)-3\left( \sin^4 x + \cos^4 x\right) - \cos x = 0 \\
2\left(1+\sin^4 x - \sin^2 x \cos^2 x + \cos^4 x\... |
4,485,550 | <p>I'm interested in playing with nonwell-founded variants of set theory and weaker/different axioms of induction/extensionality.</p>
<p>I have a hunch coalgebraic methods could better handle weirdness like modelling homotopy type theory.</p>
<p>I also have just been interested in the idea of coinduction as primitive a... | Peter Smith | 35,151 | <p>One version of non-well founded set theory arises when we replace the Axiom of Foundation with the Anti-Foundation Axiom (AFA), explored in an influential book written by Peter Aczel in 1988.</p>
<p>The obvious place to start finding out more about this is the <em>Stanford Encyclopaedia</em> article <a href="https:/... |
4,804 | <p>Using the Unanswered tab, I was surprised to find a large number of questions that were already answered <em>in the answer box</em>, correctly, and thoroughly. But if the answer(s) is never upvoted, the question remains in the tab where it does not belong. To make things worse, the Community user occasionally bumps ... | Community | -1 | <p>Another reason seems to be that many people won't recognize (or are not interested in) a correct answer the more specialized or advanced the question is. I got many upvotes and answer tags on absolutely trivial statements while sometimes I (quite obviously) put lots of work into answers on special topics for which n... |
2,405,767 | <p>This is Velleman's exercise 3.6.8.b (<strong>And of course not a duplicate of</strong> <a href="https://math.stackexchange.com/questions/253446/uniqueness-proof-for-forall-a-in-mathcalpu-existsb-in-mathcalpu-f">Uniqueness proof for $\forall A\in\mathcal{P}(U)\ \exists!B\in\mathcal{P}(U)\ \forall C\in\mathcal{P}(U)\ ... | Raffaele | 83,382 | <p>$x=r \cos\phi;\;y=r\sin\phi$</p>
<p>$r=\sqrt{x^2+y^2}$</p>
<p>The equations can be written as</p>
<p>$r\cos\phi+ r(r\sin\phi)=2$ and then</p>
<p>$x+y\sqrt{x^2+y^2}=2$</p>
|
97,946 | <p>I want to prove the following:</p>
<p>Let $G$ be a finite abelian $p$-group that is not cyclic.
Let $L \ne {1}$ be a subgroup of $G$ and $U$ be a maximal subgroup of L then there exists a maximal subgroup $M$ of $G$ such that $U \leq M$ and $L \nleq M$.</p>
<p>Proof.
If $L=G$ then we are done.Suppose $L \ne G$ . ... | Geoff Robinson | 14,450 | <p>This is false in general. Consider the case (which you can reduce to by isomorphism theorems) that $U =1$ and $L$ then necessarily has order $p.$ You are asking for the existence of a complement to $L$ in $G$, for you would have $G = L \times M$ if there were such a maximal subgroup $M.$ There is such a subgroup $M$... |
2,725,455 | <p>Probably this is pretty simple (or even trivial), but I'm stucked.</p>
<p>If $H\leq G$ is a subgroup, does it follow that $hH=Hh$, if $h\in H$ ? I can't prove or find a counter-example. If anyone could help me, I'd be grateful!</p>
| GNUSupporter 8964民主女神 地下教會 | 290,189 | <p>$$hH=\{hh' \mid h' \in H\}, \color{red}{H}h = \{h'h \mid h' \in H\}$$</p>
<p>Note that $hh' = \color{red}{hh'h^{-1}}\cdot h$, so $hH \subseteq Hh$. A similar trick shows the reverse inclusion.</p>
|
2,725,455 | <p>Probably this is pretty simple (or even trivial), but I'm stucked.</p>
<p>If $H\leq G$ is a subgroup, does it follow that $hH=Hh$, if $h\in H$ ? I can't prove or find a counter-example. If anyone could help me, I'd be grateful!</p>
| Joaquim Nabuco | 549,679 | <p>They are Always equal to $H$, then ir Always holds</p>
|
4,425,234 | <p>Let <span class="math-container">$f:[1,\infty)\rightarrow [1,\infty)$</span> be a function such that for every <span class="math-container">$x\in [1,\infty)$</span>, <span class="math-container">$f(f(x))=2x^{2}-3x+2$</span>. I am required to show that <span class="math-container">$f$</span> is bijective and also to ... | FlipTack | 396,317 | <p>I would say Euler's method is actually the most intuitive of the numerical methods!</p>
<p>We have a function <span class="math-container">$f$</span> that tells us the derivative of a solution to the ODE passing through our current point <span class="math-container">$(t_n, y_n)$</span>. What is the derivative? In a ... |
2,156,331 | <p>Consider the discrete topology $\tau$ on $X:= \{ a,b,c, d,e \}$. Find subbasis for $\tau$ which does not contain any singleton sets.</p>
<p>The definition of subbasis is as follows: </p>
<blockquote>
<p><strong>Definition:</strong> A <em>subbasis</em> $S$ for a topology on $X$ is a collection of subsets of $X$ w... | Ibrahim | 696,130 | <p>The correct singleton sets should be <span class="math-container">$\{(a,b),(b,c),(c,d),(d,e),(a,e)\}$</span></p>
|
499,171 | <p>Let $\{x_n\}$ be "any" sequence containing all rationals. I have to prove that every real number is the limit of some subsequence. I know that rationals are dense in real. But, are not the order of the rationals in the sequence creating problem here ? How to pick rationals from this sequence. </p>
| Daniel Montealegre | 24,005 | <p>Pick any real number you want, call it $r$. Then let pick the first rational in your sequence that has difference with $r$ of less than $1$, say your number was $x_{n_1}$. Then pick the next rational that appears after $n_1$ in your sequence that has difference with $r$ of less than $1/2$. This can be done since you... |
297,036 | <p>If $f'(x) = \sin{\dfrac{\pi e^x}{2}}$ and $f(0)= 1$, then what will be $f(2)$?</p>
<p>This is what I tried to find the antiderivative of $f'(x)$ with u-substitution, </p>
<p>$$
\begin{align}
u &=\frac{\pi e^x}{2} \\
\frac{2}{\pi}du &=e^x dx
\end{align}
$$</p>
<p>I don't know what to do next.</p>
| ILoveMath | 42,344 | <p>Some thoughts: Apply mean value theorem to $f$ on the interval $[0,2]$ to obtain:</p>
<p>$$ \frac{f(2) - f(0)}{2} = \sin (\frac{\pi e^c}{2}) $$</p>
<p>for some $c$ in the interval. Then, we have that $f(2) = 2 \sin (\frac{\pi e^c}{2}) + 1$</p>
<p><img src="https://i.stack.imgur.com/iaMYy.png" alt="enter image des... |
297,036 | <p>If $f'(x) = \sin{\dfrac{\pi e^x}{2}}$ and $f(0)= 1$, then what will be $f(2)$?</p>
<p>This is what I tried to find the antiderivative of $f'(x)$ with u-substitution, </p>
<p>$$
\begin{align}
u &=\frac{\pi e^x}{2} \\
\frac{2}{\pi}du &=e^x dx
\end{align}
$$</p>
<p>I don't know what to do next.</p>
| Mhenni Benghorbal | 35,472 | <p>Note that</p>
<p>$$ {f'(x) = \sin{\frac{\pi e^x}{2}}}\implies \int_{0}^{x}f'(t)dt = \int_{0}^{x}\sin\left(\frac{\pi e^t}{2}\right) $$</p>
<p>$$\implies f(x)=f(0)+\int_{0}^{x}\sin\left(\frac{\pi e^t}{2} \right)dt $$</p>
<p>$$ \implies f(2)=1+\int_{0}^{2}\sin\left(\frac{\pi e^t}{2} \right)dt \longrightarrow (*)$$</... |
1,256,460 | <p>I want to solve the following problem: </p>
<p>$$u_{xx}(x,y)+u_{yy}(x,y)=0, 0<x<\pi, y>0 \\ u(0,y)=u(\pi, y)=0, y>0 \\ u(x,0)=\sin x +\sin^3 x, 0<x<\pi$$ </p>
<p>$u$ bounded </p>
<p>I have done the following: </p>
<p>$$u(x,y)=X(x)Y(y)$$ </p>
<p>We get the following two problems: </p>
<p>$$X''... | MaxV | 464,821 | <p>First of all that formula is wrong, because after few minutes a coffee is always cold, and here after 30 minutes a cup of coffe is at <strong>61°C</strong>!!! </p>
<p><strong>I mean this is a killer coffee!!!</strong> <em>(normal people over 49°C get permanent injuries!)</em></p>
<p>However, here we have 2 solutio... |
2,414,472 | <blockquote>
<p>Let $(a_n)_{n\geq2}$ be a sequence defined as
$$
a_2=1,\qquad a_{n+1}=\frac{n^2-1}{n^2}a_n.
$$
Show that
$$
a_n=\frac{n}{2(n-1)},\quad\forall n\geq2
$$
and determine $\lim_{n\rightarrow+\infty}a_n$.</p>
</blockquote>
<p>I cannot show that $a_n$ is $\frac{1}{2}\frac{n}{n-1}$. Some helps? </p>
... | SC30 | 352,208 | <p>$a_{n+1}=(1-\frac{1}{n^2})a_n=(1-\frac{1}{n^2})(1-\frac{1}{(n-1)^2}) a_{n-1}=\prod_{k=2}^n (1-\frac{1}{k^2}) $ where the last step follows from the fact that $a_2=1$. Now $\frac{k^2-1}{k^2}=\frac{(k-1)(k+1)}{k^2}$, so writing down the first few terms you'll convince yourself that everything cancels except $\frac{1}{... |
2,414,472 | <blockquote>
<p>Let $(a_n)_{n\geq2}$ be a sequence defined as
$$
a_2=1,\qquad a_{n+1}=\frac{n^2-1}{n^2}a_n.
$$
Show that
$$
a_n=\frac{n}{2(n-1)},\quad\forall n\geq2
$$
and determine $\lim_{n\rightarrow+\infty}a_n$.</p>
</blockquote>
<p>I cannot show that $a_n$ is $\frac{1}{2}\frac{n}{n-1}$. Some helps? </p>
... | farruhota | 425,072 | <p>Hint: You can substitute the common term formula into the difference equation to verify it.</p>
|
2,569,267 | <p><a href="https://gowers.wordpress.com/2011/10/16/permutations/" rel="nofollow noreferrer">This</a> article claims:</p>
<blockquote>
<p>we simply replace the number 1 by 2, the number 2 by 4, and the number 4 by 1</p>
<p>....I start with the numbers arranged as follows: 1 2 3 4 5 6. After doing the permutation (124) ... | xxxxxxxxx | 252,194 | <p>You seem to be reading it as saying "1 goes to position 2", but the convention is that it should be read as "1 gets replaced by 2", or "object 1 becomes object 2". It helps to view these permutations in an alternate form (where for each number, we write the image underneath).
$$\begin{pmatrix}
1 & 2 & 3 &am... |
499,652 | <p>I saw this a lot in physics textbook but today I am curious about it and want to know if anyone can show me a formal mathematical proof of this statement? Thanks!</p>
| Eric Auld | 76,333 | <p>The one-sentence answer is that the Taylor series for tangent at zero is $x + O(x^3)$. So it is actually quite a good approximation.</p>
|
2,761,151 | <p>In the formula below, where does the $\frac{4}{3}$ come from and what happened to the $3$? How did they get the far right answer? Taken from Stewart Early Transcendentals Calculus textbook.</p>
<p>$$\sum^\infty_{n=1} 2^{2n}3^{1-n}=\sum^\infty_{n=1}(2^2)^{n}3^{-(n-1)}=\sum^\infty_{n=1}\frac{4^n}{3^{n-1}}=\sum_{n=1}^... | MPW | 113,214 | <p>$$\frac{4^n}{3^{n-1}}=\frac{4^{1+(n-1)}}{3^{n-1}}$$</p>
<p>$$=\frac{4^1\cdot 4^{n-1}}{3^{n-1}}$$</p>
<p>$$=4\cdot \frac{4^{n-1}}{3^{n-1}}$$</p>
<p>$$=4\cdot\big(\frac{4}{3}\big)^{n-1}$$</p>
|
281,288 | <p><img src="https://i.imgur.com/0C2Jl.jpg" alt="curved line graph"></p>
<p>In this curved line graph, I need to be able to make a formula that can tell me the interpolated value at any point on the curved path given one Data input.</p>
<p>So for example if I wanted to know what value the line was at exactly half way... | bubba | 31,744 | <p>If you just have four data points (or some smallish number, anyway), then the easiest approach is probably something called "Lagrange interpolation". </p>
<p>The wikipedia page has a bunch of formulas that might be hard to understand, but the examples should be pretty clear:</p>
<p><a href="http://en.wikipedia.org... |
2,878,448 | <p>What could be a possible approach to find the proof of:</p>
<blockquote>
<p>$\binom{2k+1}{k}$ is odd when $k=2^m-1$, otherwise $\binom{2k+1}{k}$ is even.</p>
</blockquote>
<p>I have seen some similar problems in <a href="https://math.stackexchange.com/questions/317163/prove-if-n-2k-1-then-binomni-is-odd-for-0-le... | Batominovski | 72,152 | <p>Let
$$k=2^{r_1}+2^{r_2}+\ldots+2^{r_n}$$
where $r_1,r_2,\ldots,r_n$ are nonnegative integers such that $r_1<r_2<\ldots<r_n$. Thus.
$$2k+1=2^0+2^{r_1+1}+2^{r_2+1}+\ldots+2^{r_n+1}\,.$$
If there exists $j\in\{1,2,\ldots,n\}$ such that $r_j\neq r_{j-1}+1$ (here, $r_{0}:=-1$), then the bit corresponding to $2... |
2,031,699 | <p>Let $A,B$ be open subsets of $\mathbb{R}^n$. </p>
<p>Does the following equality hold?</p>
<p>$$\partial(A\cap B)= (\bar A \cap \partial B) \cup (\partial A \cap \bar B)$$</p>
<p>Edit: Thanks for showing me in the answers that above formula fails if $A$ and $B$ are disjoint but their boundaries still intersect. I... | Eugene Zhang | 215,082 | <p>This is not true generally unless <span class="math-container">$\overline{A\cap B}=\overline{A}\cap \overline{B}$</span>.
<span class="math-container">\begin{align}
\partial (A\cap B)&= \overline{A\cap B}-(A\cap B)^{o}
\\
&=(\overline{A}\cap \overline{B})-(A^{o}\cap B^{o})
\\
&=(\overline{A}\cap \overlin... |
902,592 | <p>Consider,</p>
<p>$$ \displaystyle x\frac{\partial u}{\partial x}+\frac{\partial
u}{\partial t} = 0 $$</p>
<p>with initial values $ t = 0 : \ u(x, 0) = f(x) $ and calculate the
solution $ u(x,t) $ of the above Cauchy problem using the method of
characteristics.</p>
<p>And here is the solution, I will point out ... | atomteori | 156,639 | <p>If you'll abide the strangeness of it, group theory (Lie theory) provides an answer. Use this stretching group:
$$
G(x,t,u)=(\lambda x,\lambda^\beta t, \lambda^\alpha u)\lambda_o=1
$$$\lambda_o=1$ is the unit transformation without which no group is complete. Plug these transformed variables into the equation and ... |
1,895,323 | <p>Recently, I had a mock-test of a Mathematics Olympiad. There was a question which not only I but my friends too were not able to solve. The question goes like this: </p>
<p>If,
$$ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{a+b+c} $$<br>
Then what is the value of<br>
$$ \frac{1}{a^5} + \frac{1}{b^5} + \fra... | Doug M | 317,162 | <p>$\frac 1a + \frac 1b + \frac 1c = \frac 1{a+b+c}\\
\frac {ab +ac + bc}{abc}= \frac 1{a+b+c}\\
(a+b+c)(ab +ac + bc) = abc\\
a^2b + a^2c + ab^2 +ac^2 + b^2c + bc^2 + 2abc = 0\\
(a^2b + ab^2) + c(a^2 + b^2 + 2ab + cb + ca)= 0\\
ab (a+b) + c((a+b)^2 + c(a+b))= 0\\
(a+b)(ab + c(a+b) + c^2) = 0\\
(a+b)(a(b + c) + c (b+c))... |
1,423,449 | <p>Find all extrema for the function $f(x)=-\frac{x^{3}}{3}+x^{2}-x+4$ on the domain $x \in [-3.3]$.</p>
<p><strong>Solution:</strong> $f'(x)=-x^{2}+2x-1 = 0 \implies (x-1)^{2}=0 \implies x^{*}=1$. </p>
<p>Is that it? </p>
| MathAdam | 266,049 | <p>Hint:</p>
<p>Next, I think you need to show that this isn't merely an infection point, such as (0,0) on $y=x^3$ </p>
<p>To do this, try taking the second derivative of $f'(x)=-x^{2}+2x-1$ </p>
<p>$$f''(x)=-2x+2$$ </p>
<p>then substitute $x:=1$ to test the point you discovered. Is it a relative maximum, a mini... |
2,842,217 | <p>im looking to understand the tangent taylor series, but im struggling to understand how to use long division to divide one series (sine) into the other (cosine). I also can't find examples of the Tangent series much beyond X^5 (wikipedia and youtube videos both stop at the second or third term), which is not enough ... | farruhota | 425,072 | <p>An alternative and straightforward method is:
$$\begin{align}y&=\tan x \ (=0)\\
y'&=\frac{1}{\cos^2 x}=1+\tan^2x=1+y^2 \ (=1)\\
y''&=2yy'=2y(1+y^2)=2y+2y^3 \ (=0) \\
y'''&=2y'+6y^2y'=2+8y^2+6y^4 \ (=2)\\
y^{(4)}&=16yy'+24y^3y'=16y+40y^3+24y^5 \ (=0)\\
y^{(5)}&=16+120y^2y'+120y^4y'=16+136y^2+2... |
3,991,691 | <p>I'm having some trouble proving the following:</p>
<blockquote>
<p>Let <span class="math-container">$d$</span> be the smallest positive integer such that <span class="math-container">$a^d \equiv 1 \pmod m$</span>, for <span class="math-container">$a \in \mathbb Z$</span> and <span class="math-container">$m \in \math... | J. W. Tanner | 615,567 | <p>It's <span class="math-container">$(5y-2x)^2-21y^2=-20$</span>, which is a Pell type equation.</p>
<hr />
<p>I got that by completing the square:</p>
<p><span class="math-container">$x^2-5xy+y^2=-5\implies \left(x-\frac52y\right)^2-\frac{21}4y^2=-5\implies (5y-2x)^2-21y^2=-20$</span>.</p>
<p>So it's <span class="mat... |
3,991,691 | <p>I'm having some trouble proving the following:</p>
<blockquote>
<p>Let <span class="math-container">$d$</span> be the smallest positive integer such that <span class="math-container">$a^d \equiv 1 \pmod m$</span>, for <span class="math-container">$a \in \mathbb Z$</span> and <span class="math-container">$m \in \math... | Will Jagy | 10,400 | <p>The question as given is perfect for a technique from contest mathematics called <strong>Vieta Jumping</strong>. This is a special case of automorphism of quadratic forms. It has the virtue that it can be justified using nothing worse that the quadratic formula, and it does not require the use of square roots eithe... |
3,713,395 | <p>I would like the definition of a modular form with complex multiplication and if possible a reference.
Thank you ! </p>
| Homieomorphism | 553,656 | <p>A newform <span class="math-container">$f=\sum_{n=1}^\infty a(n)q^n$</span> of level N and weight k has complex multiplication if there is a quadratic imaginary field K such that <span class="math-container">$a(p)=0$</span> as soon as p is a prime which is inert in K. The field K is then unique (if the weight k≥2), ... |
96,799 | <pre><code>NDSolve[{ (-y''[r]/1880) + (470 (0.04077)^2 r^2 - 48 + 1/(1880 r^2)) y[r] == 0,
y[0] = 0, y'[0] = 0}, y, {r, -4, 4}]
</code></pre>
<p>I use this but get errors and am unable to get a plot.</p>
<hr>
<p><em>Update</em></p>
<p>Even after I fix the syntax error,</p>
<pre><code>NDSolve[{(-y''[r]/18... | user21 | 18,437 | <p>There is a singularity at <code>r==0</code> you have to deal with.</p>
<pre><code>{(-y''[r]/1880) + (470 (0.04077)^2 r^2 - 48 + 1/(1880 r^2)) y[r] == 0,
y[0] == 0, y'[0] == 0} /. r -> 0
{False, y[0] == 0, Derivative[1][y][0] == 0}
Power::infy : "\"Infinite expression \[NoBreak]1/0^2\[NoBreak] \
encountered.\... |
96,799 | <pre><code>NDSolve[{ (-y''[r]/1880) + (470 (0.04077)^2 r^2 - 48 + 1/(1880 r^2)) y[r] == 0,
y[0] = 0, y'[0] = 0}, y, {r, -4, 4}]
</code></pre>
<p>I use this but get errors and am unable to get a plot.</p>
<hr>
<p><em>Update</em></p>
<p>Even after I fix the syntax error,</p>
<pre><code>NDSolve[{(-y''[r]/18... | Michael E2 | 4,999 | <p><em>[Update notice: I had left in the previous code initial conditions for <code>NDSolve</code> from working through the OP's problem and forgot to generalize them. They are now fixed.]</em></p>
<h3>Introduction</h3>
<p>The <a href="http://mathworld.wolfram.com/FrobeniusMethod.html" rel="nofollow noreferrer">meth... |
3,613,950 | <blockquote>
<p>Given the set <span class="math-container">$S$</span> that is the set of all subsets of <span class="math-container">$\{1, 2, \ldots, n\}$</span>. Two different sets are chosen at random from <span class="math-container">$S$</span>. What is the probability that
the two subsets share exactly two eq... | mathcounterexamples.net | 187,663 | <p><strong>Hint</strong></p>
<p>Fréchet derivative of <span class="math-container">$f(X) =\mathbf{a}^TX^2\mathbf{a}$</span> is given by</p>
<p><span class="math-container">$$\partial_{X_0}f(h) = \mathbf{a}^TX_0 h\mathbf{a} + \mathbf{a}^Th X_0 \mathbf{a}$$</span></p>
|
400,926 | <p>Maybe you can help here. There is kind of a lengthy setup to understand what the question is asking. There is a paper I'm reading, and in one section of it I can't make heads or tails of the result. The reference is "Global Carleman Estimates for Waves and Applications" by Baudouin, Buhan, Ervedoza. </p>
<hr>
... | guacho | 77,946 | <p>Consider $g(\epsilon)_{z,v}=K_{s,p}(z+\epsilon v)$ for $v$ an arbitrary function in the appropriate space. Now take the derivative in $\epsilon$ and evaluate at $\epsilon =0$. This is the directional derivative of the functional along the line given by $v$.</p>
|
100,739 | <p>Let $a\in (1,e)\cup(e,\infty).$ I'd like to show that the equation $a^x=x^a$ has exactly two positive solutions, and one is larger and one smaller than $e.$ Is it even possible to show? I think I've tried everything.</p>
| Ilya | 5,887 | <p>For positive numbers, your equation is equivalent to $\sqrt[a]{a} = \sqrt[x]{x}$, so you have to consider the graph of the function
$$
y = \sqrt[x]{x}
$$
with sections of the form $y = \mathrm{const}.$</p>
|
1,645,361 | <p>I am aware that the union of subspaces does not necessarily yield a subspace. However, I am confused about the following question: </p>
<blockquote>
<p>(i) Let $U, U'$ be subspaces of a vector space $V$ (both not equal to $V$). Prove that the union of $U$ and $U'$ does not equal $V$.<br>
(ii) Find an example o... | Bernard | 202,857 | <p>This is perfectly impossible if the base field is infinite, and is known as the <em>avoidance lemma for subspaces</em>: </p>
<blockquote>
<p>If a subspace of a vector space over an infinite field is contained in a finite union of subspaces, it is contained in one of them.</p>
</blockquote>
<p>You can find a pro... |
3,354,566 | <p>I see integrals defined as anti-derivatives but for some reason I haven't come across the reverse. Both seem equally implied by the fundamental theorem of calculus.</p>
<p>This emerged as a sticking point in <a href="https://math.stackexchange.com/questions/3354502/are-integrals-thought-of-as-antiderivatives-to-avo... | symplectomorphic | 23,611 | <p>In a sense your question is very natural. Let's take an informal approach to it, and then see where the technicalities arise. (That's how a lot of research mathematics works, by the way! Have an intuitive idea, and then try to implement it carefully. The devil is always in the details.)</p>
<p>So, one way to tell t... |
553,845 | <p>Could we assert that if $H$ is a subgroup of $G$, then the factor group $N_G(H)/C_G(H)$ is isomorphic to a subgroup of ${\rm Inn}(H)$ instead of ${\rm Aut}(H)$?</p>
| Nicky Hekster | 9,605 | <p>There is another approach that is less known than the "N/C" theorem. <br>The <a href="http://en.wikipedia.org/wiki/Outer_automorphism_group" rel="nofollow">outer automorphisms</a> of a group $H$, ${\rm Out}(H)$, are defined as the quotient group ${\rm Aut}(H)/{\rm Inn}(H)$. Note that the elements of ${\rm Out}(H)$ a... |
395,685 | <p>I recall seeing a quote by William Thurston where he stated that the Geometrization conjecture was almost certain to be true and predicted that it would be proven by curvature flow methods. I don't remember the exact date, but it was from after Hamilton introduced the Ricci flow but well before Perelman's work. Unfo... | Carlo Beenakker | 11,260 | <p>This 1994 paper by Thurston may or may not be the source you are thinking of, but it is a thoughtful essay that conveys the confidence Thurston had in his conjecture (albeit without referring to curvature flow):</p>
<blockquote>
<p>The full geometrization conjecture is still a conjecture. It has been
proven for many... |
897,756 | <p>How can I solve the following trigonometric inequation?</p>
<p>$$\sin\left(x\right)\ne \sin\left(y\right)\>,\>x,y\in \mathbb{R}$$</p>
<p>Why I'm asking this question... I was doing my calculus homework, trying to plot the domain of the function $f\left(x,y\right)=\frac{x-y}{sin\left(x\right)-sin\left(y\right... | Cookie | 111,793 | <p>\begin{align*}
\lim_{h \rightarrow 0} \frac{\sin(x+h)-\sin x}h &=\lim_{h \rightarrow 0} \frac{(\sin x \cos h + \cos x \sin h)-\sin x}h & \text{trigonometric sum formula} \\
&=\lim_{h \rightarrow 0} \frac{\sin x(\cos h-1) + \cos x \sin h}h &\text{shuffle terms in numerator} \\
&=\lim_{h \rightar... |
1,708,900 | <p>Does a closed form exist for </p>
<blockquote>
<p>$$\sum \limits_{n=0}^{\infty} \frac{1}{(kn)!}$$</p>
</blockquote>
<p>in terms of $k$ and other functions? The best that I have been able to do is solve the case where $k=1$, since the sum is just the infinite series for $e$. I would guess that any closed form mus... | robjohn | 13,854 | <p>This is a different approach to the idea in David Ullrich's answer.</p>
<hr>
<p>As long as $\frac nk\not\in\mathbb{Z}$,
$$
\begin{align}
\sum_{j=0}^{k-1}e^{2\pi ij\frac nk}
&=\frac{e^{2\pi in}-1}{e^{2\pi i\frac nk}-1}\\
&=0
\end{align}
$$
if $\frac nk\in\mathbb{Z}$, then
$$
\begin{align}
\sum_{j=0}^{k-1}e^... |
1,708,900 | <p>Does a closed form exist for </p>
<blockquote>
<p>$$\sum \limits_{n=0}^{\infty} \frac{1}{(kn)!}$$</p>
</blockquote>
<p>in terms of $k$ and other functions? The best that I have been able to do is solve the case where $k=1$, since the sum is just the infinite series for $e$. I would guess that any closed form mus... | Mariusz Iwaniuk | 276,773 | <p>Sum expressed by a special function:</p>
<p>$$\color{red}{\sum _{n=0}^{\infty } \frac{1}{(k n)!}}=\sum _{n=0}^{\infty } \frac{1}{\Gamma (k
n+1)}=\color{red}{E_{k,1}(1)}$$</p>
<p>where: $\color{red}{E_{k,1}(1)}$ is generalized <a href="http://mathworld.wolfram.com/Mittag-LefflerFunction.html" rel="nofollow noref... |
38,731 | <p>The <a href="http://en.wikipedia.org/wiki/Ramanujan_summation">Ramanujan Summation</a> of some infinite sums is consistent with a couple of sets of values of the Riemann zeta function. We have, for instance, $$\zeta(-2n)=\sum_{n=1}^{\infty} n^{2k} = 0 (\mathfrak{R}) $$ (for non-negative integer $k$) and $$\zeta(-(2n... | Anixx | 2,513 | <p>You should note that the Cauchy principlal value of $\zeta(1)$ is $\gamma$:</p>
<p>$$\lim_{h\to0}\frac{\zeta(1+h)+\zeta(1-h)}2=\gamma$$</p>
<p>Saying $\zeta(1)=\infty$ is wrong because zeta has no limit at that point (except for directional limits).</p>
|
1,748,001 | <p>I need to find a relation between $\sqrt{x+ia}$ and $\sqrt{\sqrt{x^2+a^2}+x}$</p>
<p>where $a>0$ $x\in \mathbb{R}$</p>
<p>Thank you</p>
| Brian | 331,755 | <p>You can treat $\frac{1}{3x^{2/3}}$ as $3x^{-2/3}$</p>
<p>Similarly, you can treat $\frac{2}{3x^{5/3}}$ as $2*3x^{-5/3}=6x^{-5/3}$</p>
|
1,251,914 | <p>I do not understand how to set up the following problem:</p>
<p>"Forces of 20 lb and 32 lb make an angle of 52 degrees with each other. find the magnitude of the resultant force."</p>
<p>An actually picture would really help.</p>
| ParaH2 | 164,924 | <p>This man can ! <a href="https://www.youtube.com/watch?v=M9sbdrPVfOQ" rel="nofollow">https://www.youtube.com/watch?v=M9sbdrPVfOQ</a> :) </p>
<p>But I never understand I think we have to have a brain in 4D to understand. </p>
<p>You can also see that :</p>
<p>A conference really interesting <a href="https://www.you... |
1,251,914 | <p>I do not understand how to set up the following problem:</p>
<p>"Forces of 20 lb and 32 lb make an angle of 52 degrees with each other. find the magnitude of the resultant force."</p>
<p>An actually picture would really help.</p>
| David K | 139,123 | <p>There have been people who reportedly can visualize things in four dimensions
as easily as other people can in three. It's rare, however.
Moreover, visualizing four dimensions may not help much when you
want to solve a problem in five dimensions or more.
So as Henning Makholm's answer states, to do anything really u... |
4,581,539 | <p>Consider the task of proving that <span class="math-container">$|z+w|\leq |z|+|w|$</span>, where <span class="math-container">$z$</span> and <span class="math-container">$w$</span> are complex numbers.</p>
<p>We can consider three cases:</p>
<ol>
<li><span class="math-container">$|z|$</span> or <span class="math-con... | Siong Thye Goh | 306,553 | <p>A sequence of continuous functions need not converge to a continuous function.</p>
<p>We can consider a simpler examples:</p>
<p><span class="math-container">$$f_n(x)=x^n, x\in [0,1]$$</span></p>
<p>The limit is not a continuous function.</p>
<p>Forier-series allow us to construct many more such examples. Just take ... |
4,581,539 | <p>Consider the task of proving that <span class="math-container">$|z+w|\leq |z|+|w|$</span>, where <span class="math-container">$z$</span> and <span class="math-container">$w$</span> are complex numbers.</p>
<p>We can consider three cases:</p>
<ol>
<li><span class="math-container">$|z|$</span> or <span class="math-con... | Jam | 161,490 | <blockquote>
<p>"Each term in the summation is obviously continuous… Thus, I would expect the infinite sum to be continuous as well."</p>
</blockquote>
<p>Err… why?</p>
<p>Properties that hold for all <em>members</em> of a sequence need not hold for the <em>limit</em> of the sequence (see <a href="https://mat... |
2,559,564 | <blockquote>
<p>A nonempty subfamily $\mathcal{F}$ of $Z(X)$ is called $z$-filter on $X$ provided that</p>
<ol>
<li>$ \emptyset \not \in \mathcal{F}$ </li>
<li>If $z_{1} , z_{2} \in \mathcal{F}$ , then $z_{1} \cap z_{2} \in \mathcal{F}$ </li>
<li>If $ z \in \mathcal{F} , z^{*} \in Z(X) , z^{*} \sup... | Peter Elias | 392,689 | <p>2 $\Rightarrow$ 3: If $z_1,z_2\in Z(X)$ then we have $f_1,f_2\in C(X)$ such that $Z(f_1)=z_1$ and $Z(f_2)=z_2$. For $x\in X$, let $g_1(x)=\max\{0,|f_1(x)|-|f_2(x)|\}$, $g_2(x)=\max\{0,|f_2(x)|-|f_1(x)|\}$. Then $g_1,g_2\in C(X)$ and $Z(g_1)\cup Z(g_2)=X$, hence 2. implies that there is some $z\in\{Z(g_1),Z(g_2)\}\ca... |
305,166 | <p>If two undirected graphs are identical except that one has an additional loop at vertex $A$, do they actually have the same complement?</p>
| Mathemagician1234 | 7,012 | <p>Well, technically,by the terminology I know, these are <em>multigraphs</em> and not graphs. In this particular case, I don't think it applies since I think only simple graphs have complements.Think about it: the complement of this multigraph would have loops on the adjacent vertices without edges in the complement a... |
4,475,082 | <p>Problem:</p>
<ul>
<li>Three-of-a-kind poker hand: Three cards have one rank and the remaining two cards have
two other ranks. e.g. {2♥, 2♠, 2♣, 5♣, K♦}</li>
</ul>
<p>Calculate the probability of drawing this kind of poker hand.</p>
<p>My confusion: When choosing the three ranks, the explanation used <span class="mat... | utobi | 220,145 | <p>Here is another way to solve it through unordered samples.</p>
<p>We are looking for hands of the kind <span class="math-container">$x_1$</span>-<span class="math-container">$x_2$</span>-<span class="math-container">$x_2$</span>-<span class="math-container">$y$</span>-<span class="math-container">$z$</span>, where <... |
1,190,083 | <p>A positive element x of a C*-algebra A is a self-adjoint element whose spectrum is contained in the non-negative reals. If there's a faithful finite-dimensional representation of A where the involution is conjugate transposition, I think the second condition just means that x can be thought of as a matrix with posit... | Qiaochu Yuan | 232 | <p>Sure. For example, any $n \times n$ nilpotent matrix has all eigenvalues zero, so has non-negative real spectrum as an element of $M_n(\mathbb{C})$, but no nonzero nilpotent matrix can be self-adjoint by the spectral theorem. In fact the first reason that occurs to me to not consider these positive is precisely that... |
2,521,331 | <p>I need to show, that when we have $X,Y$ - any metric spaces and
<br>
$f:X \ni x \to a \in Y$ is constant , then $f$ is continuous . </p>
<p>$(X,\tau_{1}),(Y,\tau_{2}) $ - topological spaces : $f: X\to Y$.
I know a definition : $f: X\to Y $ is continuous if $ \forall_{W \in \tau_{2}}\ f^{-1}[W] \in \tau_{1} $ .... | William Elliot | 426,203 | <p>Let U be an open set of y.<br>
Case one. If a in U, what is the image of U?<br>
Case two. If a not in U, what is the image of U? </p>
<p>Do not guess but use the definition of inverse image to caculate the inverse image of a set by a function. Also look up the fact that a function is continuous iff for every op... |
2,366,610 | <p>Let $U$ be an $n \times n$ unitary matrix and $X$ an $n \times n$ real symmetric matrix. Suppose that $$U^\dagger X U = X$$ for all real symmetric $X$, then what are the allowed unitaries $U$? It seems that the only possible $U$ is some phase multiple of the identity $U=aI$ where $|a|=1$ but I'm not able to show th... | G.H.lee | 445,037 | <p>Let $\sin x = y $ , $ \sinh x = y + z$</p>
<p>Then $L = \lim_{x\rightarrow 0}\frac{\sinh^{-1}(y+z) - \sinh^{-1}(y)}{z} = {\sinh^{-1}}' (0) = 1 $</p>
<p>($z \rightarrow 0 $ as $ x \rightarrow 0$)</p>
|
172,058 | <p>I'm wondering whether there is certain relationship between the largest eigenvalue of a positive matrix(every element is positive, not neccesarily positive definite) $A$, $\rho(A)$ and that of $A∘A^T$, $\rho( A∘A^T)$, where $∘$ denotes hadamard product.</p>
<p>Here's a result I find for many numerical cases. I crea... | ofer zeitouni | 35,520 | <p>If I understand correctly the question, the answer is that no reasonable such function exists. Take the matrix that is zero everywhere except that $A_{i,i+1}=1$, $i=1,\ldots,n-1$, and $A_{n,1}=1$. Then $\rho(A)=1$ but
$\rho(A\circ A^T)=0$.</p>
|
2,761,509 | <p>I hope it's not a duplicate but I've been searching about this problem for some time on this site and I couldn't find anything. My problem is why a number $\in(-1,0)$ raised to $\infty$ is $0$. For example let's take
$$\lim_{n\to \infty} \left(\frac{-1}{2}\right)^n$$
Which is equivalent to
$$\left(\frac{-1}{2}\righ... | fleablood | 280,126 | <p>If $\lim a_n = L$ exist and $a_n = b_n*c_n$ it does not follow that $\lim b_n$ exists and, indeed, we can <em>ALWAYS</em> find counter examples. (Ex: $\lim \frac 1{2^n} = 0$ but $\frac 1{2^n} = 2^n\frac 1{2^{2n}}$ but $\lim 2^n$ is not finite.)</p>
<p>So $\lim (\frac {-1}2)^n = (-1)^{n}*(\frac {1}2)^n$ but $\lim ... |
69,902 | <p>I'm VERY new to Mathematica programming (and by new I mean two days), and was solving Project Euler question 12, which states:</p>
<blockquote>
<p>Which starting number, under one million, produces the longest [Collatz] chain?</p>
</blockquote>
<p>Now don't take this question wrong. <strong>I am not asking for a... | mgamer | 19,726 | <p>You´ll find a lot of Mathematica Code on the internet regarding this problem. Your code generates the collate sequence for every number without taking into account, that there are a lot of duplicate calculations. You can approach it via </p>
<pre><code>collatz[n_] := collatz[n] = If[EvenQ[n], n/2, 3*n + 1]
</code><... |
69,902 | <p>I'm VERY new to Mathematica programming (and by new I mean two days), and was solving Project Euler question 12, which states:</p>
<blockquote>
<p>Which starting number, under one million, produces the longest [Collatz] chain?</p>
</blockquote>
<p>Now don't take this question wrong. <strong>I am not asking for a... | DumpsterDoofus | 9,697 | <p>For extra brute force, just <code>Compile</code> it to C code:</p>
<pre><code>collatzLength =
Compile[{{x, _Integer}},
Module[{c,
n}, (For[n = x; c = 1, n != 1, c += 1,
If[EvenQ[n], n = Round[n/2], n = 3*n + 1]]); c],
CompilationTarget -> "C", RuntimeAttributes -> {Listable}]
</code></pre>
... |
8,052 | <p>I wonder how you teachers walk the line between justifying mathematics because of
its many—and sometimes surprising—applications, and justifying it as the study
of one of the great intellectual and creative achievements of humankind?</p>
<p>I have quoted to my students G.H. Hardy's famous line,</p>
<bl... | Gerald Edgar | 127 | <p>Maybe teachers should have some of <a href="http://www.ams.org/samplings/posters/posters" rel="nofollow">these posters</a> (available free from the AMS) hanging around the place.</p>
|
4,281,028 | <p>I have the following problem, which was asked in a <a href="https://math.stackexchange.com/questions/584375/cumulative-distribution-function-word-problem">similar question</a> but it doesn't help me.</p>
<p><strong>A dart is equally likely to land at any point inside a circular target of unit radius.
Let <span class... | tommik | 791,458 | <p>reading your question, the joint distribution is UNIFORM on the unit disk, thus the joint pdf is simply the reciprocal of the area thus</p>
<p><span class="math-container">$$f_{XY}(x,y)=\frac{1}{\pi}$$</span></p>
<p>in order to find the pdf of the vector <span class="math-container">$(R;\Theta)$</span> indicating r... |
2,798,847 | <p>I would like to prove that $\|e^A-e^B\| \leq \|A-B\|e^{max\{\|A\|,\|B\|\}}$, where $A,B \in \mathbb{R}^{n \times n}$.</p>
<p>So far I was able to create the first difference term, but I have no idea how to incorporate the max norm.
I've read <a href="https://math.stackexchange.com/questions/2262000/inequality-norm-... | Áron Fehér | 565,498 | <p>So after some sleepless nights I came up with, what I hope is the answer.
First let's use the Taylor series expansion:
$\|e^A-e^B\| = \|\displaystyle\sum_{k=0}^\infty\frac{A^k-B^k}{k!}\|$</p>
<p>We can then use the property of the binomial polynoms $(x-y)^n=(x-y)(x^{n-1}+x^{n-2}y+\cdots+xy^{n-2}+y^{n-1})$ as
$\|e^A... |
1,379,188 | <p>The Riemann distance function $d(p,q)$ is usually defined as the infimum of the lengths of all <strong>piecewise</strong> smooth paths between $p$ and $q$.</p>
<p><strong>Does it change if we take the infimum only over smooth paths?</strong>
(Note that if a smooth manifold is connected, <a href="https://math.stacke... | mlk | 155,406 | <p>If you already reduced this problem to the $\mathbb{R}^n$ case, then we should be able to tackle it with the usual analytical methods. The following is probably a bit of technical overkill but should work.</p>
<p>As far as I can see, the only problem is to smoothly connect two pieces with an arbitrarily small loss ... |
1,257,193 | <p>Let $f:[0,\infty)\to\mathbb{R}$ continuous and $\lim\limits_{x\to\infty}f(x)=a$. Claim: $\lim\limits_{x\to\infty}\frac{1}{x}\int_0^xf(t)dt=a$.</p>
<p>My try: It is $\int_0^xf(t)dt=F(x)-F(0)$ (because of the fundamentaltheorem of calculus) and $\lim\limits_{x\to\infty}F(x)=ax+b$, because $\lim\limits_{x\to\infty}f(x... | Tom-Tom | 116,182 | <p>Let us go back to the rigorous definition of the limit and it works straightforwardly.
For any $\epsilon>0$, there is an $X>0$ such that $\forall x>X$, $|f(x)-a|<\epsilon/2$. For such an $x$, let us compute
$$\begin{split}
\left\lvert-a+\frac1x\int_0^xf(t)\mathrm dt\right\rvert&=\frac1x\left\lvert-... |
2,737,869 | <p>Determine the value of real parameter $p$ </p>
<p>in such a way that the equation</p>
<p>$$\sqrt{x^2+2p} = p+x $$ </p>
<p>has just one real solution</p>
<p>a. $p \ne 0$</p>
<p>b. There is no such value of parameter$p$</p>
<p>c. None of the remaining possibilities is correct.</p>
<p>d. $p\in [−2,\infty)$</p>
... | Angina Seng | 436,618 | <p>By the Heine-Borel theorem, if an infinite sequence of open intervals
cover the compact set $[0,1]$, then a finite subcollection do.
It is not hard to prove that if intervals $I_1,\ldots,I_n$
cover $[0,1]$ then the sum of their lengths is at least $1$.</p>
|
258,205 | <p>I want to know if $\displaystyle{\int_{0}^{+\infty}\frac{e^{-x} - e^{-2x}}{x}dx}$ is finite, or in the other words, if the function $\displaystyle{\frac{e^{-x} - e^{-2x}}{x}}$ is integrable in the neighborhood of zero.</p>
| Ron Gordon | 53,268 | <p>In general, when $f$ is "well-behaved" at zero and infinity:</p>
<p>$$\int_0^{\infty} dx \frac{f(a x) - f(b x)}{x} = (f(\infty)-f(0)) \log{\frac{a}{b}}$$</p>
<p>You can see this from this (rough) "proof":</p>
<p>$$\begin{align}\int_0^{\infty} dx \frac{f(a x) - f(b x)}{x} &= \int_0^{\infty} dx \: \int_b^a du \... |
3,576,026 | <p>I was solving a problem and got down to this:
<span class="math-container">$$\lim_{n \to \infty} \arctan\left(\frac{\sum_{k=0}^n-\frac{1}{1+k^2}}{\sum_{k=0}^n \frac{k}{1+k^2}}\right)$$</span>
After this, I said that, since the bottom series diverges and the upper one converges, the result is <span class="math-contai... | Community | -1 | <p>You can swap the limit and the function because the arc tangent function is <em>continuous</em>. Then you have the limit of a ratio such that the numerator is bounded, while the denominator diverges to infinity.</p>
<p>Hence your limit is <span class="math-container">$$\arctan(0)=0.$$</span></p>
<hr>
<p>You might... |
1,572,954 | <p>What is the only ordered pair of numbers $(x,y)$ which, for all $a$ and $b$, satisfies </p>
<p>$$x^a y^b=\left(\frac34\right)^{a-b} \text{and } x^b y^a=\left(\frac34\right)^{b-a}$$</p>
<p>I started off with the trivial cases, $a=0$ and $b=0$ and you get $1=1$ on both sides, so that works.</p>
<p>I can't seem to f... | Brian Tung | 224,454 | <p>By inspection,</p>
<p>$$
x = x^1y^0 = \left(\frac{3}{4}\right)^{1-0} = \frac{3}{4}
$$</p>
<p>and</p>
<p>$$
y = x^0y^1 = \left(\frac{3}{4}\right)^{0-1} = \frac{4}{3}
$$</p>
<p>Note that this only suffices to identify $x$ and $y$ provided the condition stated actually holds; it does not prove that this $x$ and $y$... |
7,871 | <p>I'm trying to make a demonstration of how rounding to different numbers of digits affects things but I can't find a way to round numbers to a specified number of digits. </p>
<p>The <code>Round</code>function only round to the nearest whole integer, and that is not what I always want. Other ways seems to only chang... | Artes | 184 | <pre><code>round1[x_, n_] := Ceiling[10^n x]/10^n // N
round2[x_, n_] := Floor[10^n x]/10^n // N
round1[3.4647, 1]
round2[3.4647, 2]
</code></pre>
<blockquote>
<pre><code>3.5
3.46
</code></pre>
</blockquote>
|
7,871 | <p>I'm trying to make a demonstration of how rounding to different numbers of digits affects things but I can't find a way to round numbers to a specified number of digits. </p>
<p>The <code>Round</code>function only round to the nearest whole integer, and that is not what I always want. Other ways seems to only chang... | hlren | 54,486 | <p>Another solution is</p>
<pre><code>Round1[x_, n_] := With[{m = Round[Log10[Abs[x]]]}, Round[x 10^(n - m)] 10.^(m - n)];
(*m estimates the scale of x, n sets the number precision, Abs function enables negative number.*)
Round1[-3.46473*10^-15, 4] // InputForm
(*-3.465*^-15*)
</code></pre>
|
4,473,632 | <p>Let <span class="math-container">$f:\mathbb R^+\to\mathbb R$</span> be a continuous function satisfied <span class="math-container">$f(a)+f(b)\ge f(2\sqrt{ab})$</span> for all <span class="math-container">$a,b>0$</span> , is <span class="math-container">$f$</span> differentiable?</p>
<p>Morever, if for all <span ... | Chris Sanders | 309,566 | <p>Easy example: find <span class="math-container">$f$</span> such that <span class="math-container">$2\leq f(x)\leq 3$</span> for all <span class="math-container">$x\in\mathbb{R}$</span> and <span class="math-container">$f$</span> is everywhere continuous but nowhere differentiable.</p>
<p>You may refer to <a href="ht... |
2,653,829 | <blockquote>
<p>How can I show $(x^2+1, y^2+1)$ is not maximal in $\mathbb R[x,y]$?</p>
</blockquote>
<p>I know I can mod out the ideal one piece at a time and show $\mathbb C[x]/(x^2+1)$ is not a field since $(x^2+1)$ is not maximal in $\mathbb C[x]$, <strong>but is there another way of showing this?</strong></p>
| Angina Seng | 436,618 | <p>The zeros of these polynomials are the complex points $(i,i)$, $(i,-i)$,
$(-i,i)$ and $(-i,-i)$. These fall into two orbits under the action
of complex conjugation, viz., $(i,i)$ and $(-i,-i)$ which also satisfy
$x=y$, and $ (i,-i)$ and $(-i,i)$ which also satisfy
$x=-y$. It follows that say, $\left<x^2+1,y^2+1,x... |
141,655 | <blockquote>
<p>What is the chance that at least two people were born on the same day
of the week if there are 3 people in the room?</p>
</blockquote>
<p>I'm wondering if my solution is accurate, as my answer was different than the solution I found:</p>
<p>Probability that there are at least 2 people in the room ... | Brian M. Scott | 12,042 | <p>Label the three people $A,B$, and $C$. Suppose that no two were born on the same day of the week. $A$ can be born on any day of the week. The probability that $B$ was born on a different day is $\frac67$. (We are of course assuming that the seven days are equally likely, though I believe that in fact this isn't the ... |
3,168,662 | <p>How do you evaluate <span class="math-container">$\int_{|z|=1} \frac{\sin(z)}{z^2+(3-i)z-3i}dz$</span> ? </p>
<p>Here is my thought process: </p>
<p>I want to use <a href="http://mathworld.wolfram.com/CauchyIntegralFormula.html" rel="nofollow noreferrer">Cauchy's Integral Formula</a>, but in order to use it I need... | jmerry | 619,637 | <p>Since <span class="math-container">$\sin(i)\neq 0$</span>, we're integrating across a pole. That is, we're integrating something comparable to <span class="math-container">$\frac{c}{z-i}$</span> on a path that approaches <span class="math-container">$i$</span> on two sides. The integral diverges, by comparison to <s... |
2,838,037 | <p>For the set $A=\{0\} \cup \{\frac 1n \mid n \in \mathbb N\}$, I understand that $\{\frac 1n \mid n \in \mathbb N\}$ is open and closed in $A$ because it is a union of all the connected components $\{\frac 1n\}$ in $A$ for all $n \in \mathbb N$. Even though $\{0\}$ is also a connected component of $A$, why is $\{0\}$... | Henno Brandsma | 4,280 | <p>Let $X$ be an uncountable set with the cocountable topology. As all convergent sequences in $X$ are eventually constant and then have that constant as its limit, all subsets are sequentially closed. But only $X$ and at most countable subsets are closed. </p>
|
1,746,363 | <p>I got maybe easy problem. I am not sure if it is true that [$\mathbb Z_2[x]/f\mathbb Z_2[x]: \mathbb Z_2$]=deg $f$ where $f \in \mathbb Z_2[x]$ irreducible. Can anybody help me ? Thanks</p>
| Sharky | 332,069 | <p><strong>Trig by Reference Triangles</strong>:</p>
<p>The angle $\frac{5 \pi}{6}$ in radians will be given by $150^o$. This simple conversion can be done by remembering that $180^o = \pi$ $rad $.</p>
<p>To solve for</p>
<p>$h'(x)=12 sin^2x cosx$
$h'( \frac{ 5 \pi}{6} )=12 sin^2 ( \frac{5 \pi}{6}) cos( \frac{5 \pi... |
3,087,570 | <p>The "school identities with derivatives", like
<span class="math-container">$$
(x^2)'=2x
$$</span>
are not identities in the normal sense, since they do not admint substitutions. For example if we insert <span class="math-container">$1$</span> instead of <span class="math-container">$x$</span> into the identity abov... | ncmathsadist | 4,154 | <p>The domain of the derivative is functions, not numbers. That's what at the bottom of this.</p>
|
3,787,167 | <p>Let <span class="math-container">$\{a_{jk}\}$</span> be an infinite matrix such that corresponding mapping <span class="math-container">$$A:(x_i) \mapsto (\sum_{j=1}^\infty a_{ij}x_j)$$</span> is well defined linear operator <span class="math-container">$A:l^2\to l^2$</span>.
I need help with showing that this ope... | Bart Michels | 43,288 | <p>This follows from the fact that the pointwise limit of bounded operators is bounded, which follows from the uniform boundedness principle:</p>
<p><a href="https://math.stackexchange.com/questions/2542884">If a sequence of bounded operator converges pointwise, then it is bounded in norm</a></p>
|
888,101 | <p>Suppose I am asked to show that some topology is not metrizable. What I have to prove exactly for that ?</p>
| Alexander Golys | 789,572 | <p>Another possible way is to find homeomorphic embedding of known not-metrizable space. This is for instance a nice way of proving not-metrazibility of ordered square (by embedding of the real line with lower-limit topology)</p>
|
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