qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,601,095 | <p>Let
$$f_n(x) = \begin{cases}
\sin nx & 0 \leq x \leq \frac\pi n\\
0 & x \geq \frac\pi n
\end{cases}$$</p>
<p>Then my book says that $f_n \to f \equiv 0$ on the interval $[0, +\infty)$.<br>
I don't understand why $0$ is included. I think that the interval of convergence should be $(0, +\infty)$ because $f(0)... | Aloizio Macedo | 59,234 | <p>You are talking about <em>pointwise convergence</em>.</p>
<p>$$f_n(0)=\sin(n\cdot0)=0 \quad \forall n$$</p>
<p>Hence, $\lim f_n(0)=0$.</p>
<p>It doesn't matter what the function does near the point, as you implicitly assume in the comments. What matters is the values at the point you've fixed.</p>
|
55,022 | <p>Why does <code>DiscretizeGraphics</code> seems to work on one <code>GraphicsComplex</code> and not the other? Here is an example that works:</p>
<pre><code>v = {{1, 0}, {0, 1}, {-1, 0}, {0, -1}};
p1 = Graphics[GraphicsComplex[v, Polygon[{1, 2, 3, 4}]]];
DiscretizeGraphics[p1]
</code></pre>
<p>But this does not</p>... | RunnyKine | 5,709 | <p>Here is a workaround for this I've been using:</p>
<pre><code>p2 = Graphics3D@First@ParametricPlot3D[{Cos[t], Sin[u], Sin[t]}, {u, 0, 2 Pi}, {t, 0, 2 Pi}]
</code></pre>
<p><img src="https://i.stack.imgur.com/gvAjb.png" alt="Mathematica graphics"></p>
<p>Now we discretize:</p>
<pre><code>DiscretizeGraphics[Normal... |
459,553 | <p>How do I compute this integral?
$$\int_{\mathbb {R}}\sin \left(\frac {\pi x}{x^2+1}\right)\frac{1}{x^2+1} dx$$</p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>$x^3-8=x^3-2^3=(x-2)\{x^2+2x+2^2\}=(x-2)\{(x+1)^2+3\}$</p>
<p>So assuming $x-2\ne0,$ we can safely cancel $x-2$ and utilize $|x^2+2x+4|=x^2+2x+4$</p>
|
459,553 | <p>How do I compute this integral?
$$\int_{\mathbb {R}}\sin \left(\frac {\pi x}{x^2+1}\right)\frac{1}{x^2+1} dx$$</p>
| DonAntonio | 31,254 | <p>$$\left|\;\frac3{x^3-8}\;\right|=\left|\;\frac1{x-2}\;\right|\iff3|x-2|=|x-2||x^2+2x+4|$$</p>
<p>Now, it obviously <em>has</em> to be $\,x\ne 2\,$ , so...</p>
|
11,294 | <p>Let $f(x)= \displaystyle \sum \limits_{n=1}^\infty \frac{\sin(nx)}{n^3}.$ Show that $f(x)$ is differentiable and that the derivative $f'(x)$ is continuous.</p>
<p>In class we solved a similar problem, and I think we had to show that both $f(x)$ and $f'(x)$ converge uniformly, but I am not really sure <em>wh... | Jonas Meyer | 1,424 | <p>To show uniform convergence of the original series and the series of derivatives you can use the facts that $\sin$ and $\cos$ are bounded and that $\sum_{n=1}^\infty \frac{1}{n^3}$ and $\sum_{n=1}^\infty \frac{1}{n^2}$ converge. </p>
<p>The uniform limit of continuous functions is continuous. To see that $f$ is d... |
11,294 | <p>Let $f(x)= \displaystyle \sum \limits_{n=1}^\infty \frac{\sin(nx)}{n^3}.$ Show that $f(x)$ is differentiable and that the derivative $f'(x)$ is continuous.</p>
<p>In class we solved a similar problem, and I think we had to show that both $f(x)$ and $f'(x)$ converge uniformly, but I am not really sure <em>wh... | kahen | 1,269 | <p>You had to show that both $(f_n)$ and $(f_n')$ converge uniformly because you want to apply a theorem like this:</p>
<p><strong>Theorem 10.7</strong> (from <a href="http://rads.stackoverflow.com/amzn/click/0521497566" rel="nofollow">"Real Analysis"</a> by N.L. Carothers):</p>
<p>Suppose that $(f_n)$ is a seque... |
4,548,865 | <p>I wish to determine whether the limit <span class="math-container">$L = \lim_{z \rightarrow i} \frac{z^3 + i}{|z| - 1}$</span> exists. Noticing it to be of the form <span class="math-container">$0/0$</span>, I separate the expression into its real and imaginary parts:
<span class="math-container">$$L = \lim_{(r, \th... | José Carlos Santos | 446,262 | <p>If <span class="math-container">$t\in\Bbb R$</span>,<span class="math-container">$$\operatorname{Re}\left(\frac{(t+i)^3+i}{|t+i|-1}\right)=\operatorname{Re}\left(\frac{t^3+3it^2-3t}{\sqrt{t^2+1}-1}\right)=\frac{t^3-3t}{\sqrt{t^2+1}-1},$$</span>and<span class="math-container">$$\lim_{t\to0^+}\frac{t^3-3t}{\sqrt{t^2+1... |
24,321 | <p>I'm trying to figure out the probability of a 3rd failure occurring on the 5th attempt of doing something. Let's just call the probability of success of failure P(S) or P(F), I won't put numbers as I want to actually learn.</p>
| Matt E | 221 | <p>Regarding the third part of your question: The intersection of two lines in $\mathbb P^3$ is generically empty (if you write down two random lines, they will be skew), but sometimes the lines will be coplanar (i.e. lie in a common plane), and then they will meet in a point (as any two lines in $\mathbb P^2$ do).</p>... |
3,694,053 | <p>Considering the set of real numbers:</p>
<p><span class="math-container">$$A = \left\{\ln\left(\frac{2n+\sqrt{n}}{2n-\sqrt{n}}\right): n \in \mathbb{N} \right\}.$$</span></p>
<p>I must prove that <span class="math-container">$0$</span> is the greatest lower bound of <span class="math-container">$A$</span>.</p>
<p... | sai-kartik | 736,802 | <p>We have the well known result:
<span class="math-container">$$\sin^2\theta+\cos^2\theta=1$$</span>
From which:
<span class="math-container">$$\cos^2\theta=1-\sin^2\theta$$</span>
Now we have to be careful while taking the square root both sides:
<span class="math-container">$$\cos\theta=\left|\sqrt{1-\sin^2\theta} \... |
3,325,953 | <p>Once you have reached perhaps 10 decimal places, calculators can make rounding errors and so on. Is it possible to build a calculator that makes none of these errors? For example, it could work out each decimal place of an irrational number – as you click a button it gives, say, 10 more digits. (Obviously, it wouldn... | hmakholm left over Monica | 14,366 | <p>As long as we're only talking about rational numbers, one can easily program a computer to represent them all exactly -- just store them as a pair of arbitrary-precision numerator and denominators.</p>
<p>This is much slower (and uses much more memory) than the usual approximate "floating-point" representation, and... |
271,252 | <p>Let $R$ be a local UFD of Krull dimension 2. Let $a\in R$ be a nonzero, non-unit. I am trying to show that the ring $R[1/a]$ is a principal ideal domain. Does anyone have any suggestions as to how this can be done?</p>
| Community | -1 | <p>From this <a href="https://math.stackexchange.com/questions/78006/prove-that-a-ufd-r-is-a-pid-if-and-only-if-every-nonzero-prime-ideal-in-r-is">topic</a> it follows that you have to prove that the Krull dimension of $R[1/a]$ is $1$. But the maximal ideal of $R$ is the only ideal of height $2$ and this explodes in $R... |
1,640,740 | <p>I have the following topology over $\mathbb R$
$$ T = \{\emptyset\} \cup \{G\subseteq \mathbb R: \mathbb Q \setminus G \text{ is finite}\} $$
How could I study the closure of $\mathbb Q$ and $\mathbb R\setminus \mathbb Q$? Thanks in advance</p>
| Brian M. Scott | 12,042 | <p>HINTS: The answers to the following questions are what you need.</p>
<ul>
<li>Are there any non-empty open sets that do <em>not</em> intersect $\Bbb Q$? </li>
<li>Is $\Bbb Q$ an open set?</li>
</ul>
|
1,158,292 | <p>I'm totally lost. I've been trying to figure this out.
This is what I've figured out:</p>
<p>$dy/dx = 1/x$</p>
<p>$y$-intercept $= 1$</p>
<p>So I try to do $y-y_1 = m(x-x_1)+b,$ which I get as $y-1 = 1/x(x-0)+1,$ simplified to $y = 3.$</p>
<p>But I feel like that is totally wrong and well, obviously it isn't eve... | abel | 9,252 | <p>let us pick a general point $(a, \ln a)$ on the graph of $y = \ln x.$ the tangent line at this point has slope $\dfrac1a.$ the equation of the tangent line therefore is $$y - \ln a = \dfrac1{a}(x-a)$$ suppose this linen goes through $(0,1).$ that requires $$1-\ln a=-1 $$ that is $$a = e^2.$$
the tangent at the p... |
164,683 | <p>I am trying to figure out the elements of the $2\times 2$ matrix</p>
<p>$$B=A_nA_{n-1}A_{n-2}\cdots A_1,\;\; n=1,2,3,\ldots$$</p>
<p>where</p>
<p>$$A_k=\begin{bmatrix}a-2k&-k(k-1)-b\\1&0\end{bmatrix},$$</p>
<p>with $a,b>0$ fixed. I wrote the following script to see what's going on:</p>
<pre><code>$As... | halirutan | 187 | <p>Not an answer, but I had an idea. You can write your sequence of matrix multiplications as a recursive system of equations</p>
<pre><code>af[k_] := a - 2 k;
bf[k_] := -k (k - 1) - b;
req = Thread[
Flatten /@ ({{m11[n + 1], m12[n + 1]}, {m21[n + 1],
m22[n + 1]}} == {{af[n + 1], bf[n + 1]}, {1, 0}}.{{m11[n... |
2,622,311 | <p>Just need a bit of help with this.</p>
<p>Number Theory: Show for all positive integers σ(2n)>2σ(n)</p>
<p>Sigma being the function of total numbers of factors including 1 and itself from number theory</p>
<p>I know a good starting point would be to consider n=2^r m, but I'm really stuck on how to apply this to a... | jgon | 90,543 | <p>An alternative solution (again assuming $\sigma(n)$ is the sum of the divisors of $n$):</p>
<p>$\sigma$ is multiplicative. Let $n=2^io$ where $o$ is odd. Then
$$\sigma(2n)=\sigma(2^{i+1})\sigma(o) = (2^{i+2}-1)\sigma(o),$$ but
$$2\sigma(n)=2\sigma(2^i)\sigma(o)=2(2^{i+1}-1)\sigma(o)=(2^{i+2}-2)\sigma(o).$$</p>
<p>... |
1,320,727 | <p>Construct an example in which a field $F$ is of degree 2 over two distinct subfields $A$ and $B$, but so that $F$ is not algebraic over $A\cap B$. </p>
<p>I'm having trouble thinking of an explicit example here.</p>
| Lubin | 17,760 | <p>Here’s another example, purely geometric. Consider $\Bbb Q$ and the field $\Bbb Q(t)$ of rational functions in one variable. Your $F$ will be $\Bbb Q(t)$. It has two obvious automorphisms of order two, namely $\sigma(t)=-t$ and $\tau(t)=2-t$. The fixed field of $\sigma$ is “quadratic beneath” F, in fact this fixed f... |
2,208,814 | <p>Let $X$ be a nonempty set. Fix two metrics $d: X\times X \to [0,1]$ and $d^\prime: X\times X \to [0,1]$ such that the topology $\tau$ generated by $d$ is finer than the topology $\tau^\prime$ generated by $d^\prime$, i.e.,
$
\tau^\prime \subseteq \tau.
$</p>
<blockquote>
<p><strong>Question.</strong> Is it true t... | Arthur | 15,500 | <p>No, it is not true, and here is a counterexample: Let $X$ be the subset $\{1/n\mid n\in \Bbb N\}\subseteq\Bbb R$ and let $d$ be the metric inherited from the usual metric on the real line. Then $\tau$ is the discrete topology, and therefore the finest one there is.</p>
<p>Let $f:X\to \Bbb Q$ be a bijection, and let... |
2,427,747 | <p>I want to prove that $2^{n+2} +3^{2n+1}$ is divisible by $7$ for all $n \in \mathbb{N}$ using proof by induction.</p>
<p>Attempt</p>
<p>Prove true for $n = 1$</p>
<p>$2^{1+2} + 3^{2(1) +1} = 35$</p>
<p>35 is divisible by 7 so true for $n =1$</p>
<p><em>Induction step</em>: Assume true for $n = k$ and prove true... | Abhyudaya Sharma | 417,279 | <p>Hint: Write the equation for $n=k$ equal to $7(c)$ where $c$ is some constant. Now, look hard at the equation where $n=k+1$.</p>
|
2,598,289 | <p><a href="https://i.stack.imgur.com/eCeNh.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/eCeNh.jpg" alt="enter image description here"></a></p>
<p>This is the definition of the real projective space in John Lee's book. However what I know is that the real projective space is defined by the quotie... | Aloizio Macedo | 59,234 | <p>You essentially just need to apply the universal property of quotients. </p>
<p>Consider the inclusion map $S^n \hookrightarrow \mathbb{R}^{n+1}$. This is of course continuous, and thus the composition
$$\begin{array}{ccccccccc} S^n & \xrightarrow{i} & \mathbb{R}^{n+1} \backslash \{0\} &\\
& \searr... |
4,068,439 | <p>Given a circle <span class="math-container">$(O, R)$</span> and a chord <span class="math-container">$AB=2a$</span>. If <span class="math-container">$CA,CB$</span> tangents to it, then what is the area of <span class="math-container">$\triangle ABC$</span> equal to, with respect to a and R?</p>
<p><a href="https://i... | Trevor Gunn | 437,127 | <p>Sure it works. The limit comparison test says that if <span class="math-container">$a_n, b_n \ge 0$</span> and</p>
<p><span class="math-container">$$ \lim \frac{a_n}{b_n} = L$$</span></p>
<p>and <span class="math-container">$0 < L < \infty$</span> then <span class="math-container">$\sum a_n$</span> converges i... |
3,575,601 | <p>If the median and bisector of one of its sides of a triangle coincide, then the height also coincides and the triangle is isosceles.</p>
<p>So, to see that is isosceles I used the idea of this question: <a href="https://math.stackexchange.com/questions/3336688/if-the-bisector-of-an-angle-of-a-triangle-also-bisects-... | Rushabh Mehta | 537,349 | <p>Call the triangle <span class="math-container">$\triangle ABC$</span> and let <span class="math-container">$AD$</span> be such that <span class="math-container">$\angle BAD=\angle CAD$</span> and <span class="math-container">$\overline{BD}=\overline{CD}$</span>.</p>
<p>By SSA triangle congruence for non-obtuse tria... |
1,991,127 | <p>The angle between two lines is given by </p>
<p>$
\tan(\theta) = \big|\frac{m_2-m_1}{1+m_1m_2}\big|
$</p>
<p>where $m_1$ and $m_2$ are the slopes of the two lines in question.</p>
<p>What is confusing me is the reverse problem. When we try to find the slope of the lines making an acute angle with a line of... | Camille | 383,840 | <p>Where I'm from, the convention for angles between two undirected lines refers to the acute angle between them.</p>
<p>The negative angle you find using a calculator should be between $-90^\circ$ and $0^\circ$.</p>
<p>One way to think about it is instead of this range, you have an angle $\theta$ which corresponds t... |
3,304,441 | <p>Let <span class="math-container">$G$</span> be any locally compact group and <span class="math-container">$H$</span> be a compact group. </p>
<p>We know that a map <span class="math-container">$F: G \rightarrow G$</span> is called affine if there exists some <span class="math-container">$\alpha \in G$</span> and an... | Tsemo Aristide | 280,301 | <p>every irreducible polynomial over <span class="math-container">$\mathbb{Q}$</span> is separable since its characteristic is zero</p>
|
17,914 | <p>How does one compute the cardinality of the set of functions $f:\mathbb{R} \to \mathbb{R}$ (not necessarily continuous)?</p>
| Arturo Magidin | 742 | <p>All you need is a few basics of cardinal arithmetic: if $\kappa$ and $\lambda$ are cardinals, none of them zero, and at least one of them is infinite, then $\kappa+\lambda = \kappa\lambda = \max\{\kappa,\lambda\}$. And cardinal exponentiation satisfies some of the same laws as regular exponentiation; in particular, ... |
2,050,385 | <p>Solve $15x$ "congruent to" $20\mod 88$</p>
<p>So far I think I know $15\mod 88$ is $-41$ or if positive $47$`</p>
| Kaj Hansen | 138,538 | <p><strong>A general approach for problems of this type:</strong></p>
<p>Notice that, if you can find an $y$ such that $15y \equiv 1 \pmod{88}$, then we can multiply both sides of this congruence by $y$ to yield $15xy \equiv 20y \pmod{88}$. This becomes $x \equiv 20y \pmod{88}$.</p>
<p>So how do we find this $y$? N... |
1,597,186 | <blockquote>
<p>In $a^n \equiv b^n \pmod m$, does $n$ have to be an integer?</p>
</blockquote>
<p>I just came upon this rule and am wondering its limits. Thank you</p>
| Ofir | 2,125 | <p>Usually in equations of the form $a^n \equiv_m b^n$, the $n$ is an integer. In general it is not well defined if $n$ is not an integer. If for example you take $n=\frac{1}{2}$ then for all odd prime $m$, an element $a$ will either have 2 roots, so that $a^{1/2}$ is two elements, or that $a$ has no roots at all. </p>... |
266,283 | <p>Recall that a space $X$ is a locally compact if for every $x\in X$ there exists a neighbourhood $U$ of the point of $x$ such that $cl U$ is compact subspace of $X$. How we can show that compact $cl U$ is $T_1$?</p>
| Brian M. Scott | 12,042 | <p>$\newcommand{\cl}{\operatorname{cl}}$Engelking defines <em>compact</em> to mean what I would call <em>compact and Hausdorff</em>. It is true that if each point of a space $X$ has a nbhd $U$ such that $\cl U$ is compact and Hausdorff, then $X$ is $T_1$. More generally, if $X$ is any space, and $x\in X$ has a nbhd $U$... |
2,522,125 | <blockquote>
<p>Is it true that if $A$ and $B$ are $2\times 2$ matrices and $AB=0$ then $A=0$ or $B=0$. Prove it, or prove the contrary. </p>
</blockquote>
<p>I tried saying that if:
$$A= \begin{pmatrix}
0 & 0 \\
0 & 0 \\
\end{pmatrix}\quad\text{and}\quad B= \begin{pmatrix}
... | Mark Bennet | 2,906 | <p>The answer to your question is "no". </p>
<p>If the first matrix has rank $1$ it annihilates a subspace of dimension $1$. All you need then is for a second matrix of rank $1$ to annihilate the subspace that the first preserves, and preserve the subspace that it annihilates (because once the first matrix annihilates... |
42,326 | <p>Does there exist a nowhere monotonic continuous function from some open subset of $\mathbb{R}$ to $\mathbb{R}$?
Some nowhere differentiable function sort of object?</p>
| Damian Sobota | 12,690 | <p>Every monotonic function is almost everywhere differentiable (<a href="http://www.math.uiuc.edu/~mjunge/54004-diffmon.pdf" rel="nofollow">Theorem 4.3</a> - it's due to Lebesgue), so as an example of nowhere monotonic function you can just take any nowhere differentiable function (for example mentioned above <a href=... |
42,326 | <p>Does there exist a nowhere monotonic continuous function from some open subset of $\mathbb{R}$ to $\mathbb{R}$?
Some nowhere differentiable function sort of object?</p>
| Andrew D. Hwang | 86,418 | <p>I was writing up a "continuous" construction for another question that got deleted. Though very belated, this idea seemed sufficiently simple and self-contained to be worth posting.</p>
<hr>
<p>Fix an arbitrary interval $[a, b]$, and let $f_{0}$ be an arbitrary non-constant affine function on $[a, b]$.</p>
<p>Ind... |
3,973,885 | <p>To add some context, let <span class="math-container">$(\Omega,\mathcal{F},\mu)$</span> be a measure space. The statement of the Extended Dominated Convergence Theorem(EDCT) is:</p>
<blockquote>
<p><span class="math-container">$\lbrace f_{n}\rbrace_{n\geq 1}$</span> be a sequence of functions from <span class="math-... | J. W. Tanner | 615,567 | <p>You are correct that the three vectors are not linearly independent;</p>
<p>in fact, <span class="math-container">$(2,3,5)=3(1,1,2)-(1,0,1)$</span>.</p>
<p>But if you take any two of them, you should find that one is not a multiple of the other,</p>
<p>so they are linearly independent, so they are a basis for the sp... |
1,984,849 | <p>Can you please clarify whether, for the following question, I need to use the definition of linear transformation, or something else?</p>
<blockquote>
<p>Compute the inverse of the function
$f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$, where $f(x_1,x_2,x_3) := (x_2+x_3, x_1+x_3, x_1+x_2)$.</p>
</blockquote>
| hamam_Abdallah | 369,188 | <p>Given a vector $v=(y_1,y_2,y _3)$,</p>
<p>$f^{-1}(v)$ is the vector $u=(x_1,x_2,x_3)$ such that</p>
<p>$f(u)=v$ which means</p>
<p>$x_2+x_3=y_1$</p>
<p>$x_1+x_3=y_2$</p>
<p>$x_1+x_2=y_3$</p>
<p>the sum of the three equations gives</p>
<p>$x_1+x_2+x_3=\frac{1}{2}(y_1+y_2+y_3)$</p>
<p>thus by substraction, we ... |
202,235 | <p>I'm not quite sure how to do this problem: </p>
<p>Calculate the the proportion of values that is 3 in the following data set:</p>
<p>2, 3, 3, 6, 9</p>
| P.K. | 34,397 | <p>The proportion, in general, is <strong>just the ratio of the frequency of a datum you are given to the number of elements in your data. For example</strong>, in the set <code>1,2,3,3,4</code>, the proportion of $1$ to the rest of the data is,$${\text{frequency of 1}\over \text{total number of data}} = {1 \over 5}$$G... |
3,205,317 | <p>(a) <span class="math-container">$x(v)= 3, y(v)= 4, z(v)= v$</span> for <span class="math-container">$−\infty < v < \infty$</span>,</p>
<p>(b) <span class="math-container">$x(t)= 3\cos(t), y(t)= 2\sin(t), z(t)= 3t−1$</span> for <span class="math-container">$0 \leq t < 2\pi$</span>.</p>
<p>I have no idea w... | lab bhattacharjee | 33,337 | <p>Hint</p>
<p><span class="math-container">$\binom nk\cos(k+1)x$</span> is the real part of</p>
<p><span class="math-container">$e^{ix}\binom nk(e^{ix})^k$</span></p>
<p>which is </p>
<p><span class="math-container">$=e^{ix}(1+e^{ix})^n$</span></p>
<p><span class="math-container">$=e^{ix(1+n/2)}\left(2\cos \dfrac... |
130,496 | <blockquote>
<p>Triangulation is called a planar graph in which every face is a triangle.</p>
<p><span class="math-container">$\bullet$</span> Prove that in every triangulation exists edge <span class="math-container">$\left\{ u,v \right\}$</span> such that <span class="math-container">$\deg(u)+\deg(v)\le 22$</span>.</... | Ahmed Hejazi | 690,585 | <p>there is a nice question related with this problem , and we can evaluate it by using Real analysis</p>
<p><span class="math-container">$$I=\int_{-\infty }^{\infty }\frac{e^{ax}}{1-e^{x}}dx,\ \ \ \ \ \ \ \ (*)\ \ \ \ \ 0<a<1\\
\\
let \ x\rightarrow -x \ \ then\ \ I=\int_{-\infty }^{\infty }\frac{e^{-ax}}{1... |
1,379,849 | <blockquote>
<p>Find the number of seven digit whole numbers in which only $2$ and $3$ are present as digits if no two $2$'s are consecutive in any number?</p>
</blockquote>
<p><strong>My Approach</strong>:
We can make numbers and see like: $2323232$, $2333333$, $2332332$, etc. Please suggest alternate solution of t... | Anonymous Pi | 80,245 | <p>One can see that $128^{3}\equiv 1\mod 7)$ (you can check this by hand). Now, this implies $128^{n}\equiv 128^{n\mod 3}\mod 7$. To check what $128^{128}\mod 3$ is, we note that $128^{2}\equiv 1\mod 7$, so this implies $128^{n}\equiv 128^{n\mod 2}\mod 3$ Therefore, the number you are looking for is $128^{128^{128}}\mo... |
3,167,336 | <p><img src="https://i.stack.imgur.com/FLO6E.jpg" alt="enter image description here">
It took me much time to reach the solution where I find the value of X as 2 but still not sure whether this is correct or not.
please help me with the solution</p>
| Peter Foreman | 631,494 | <p>Write <span class="math-container">$y=\pm\sqrt{1-x^2}$</span> and use this definition in the second equation. This problem then becomes solving a quartic for which there is a quartic formula.</p>
|
521,928 | <p>I would appreciate help showing $e^{D}(f(x)) = f(x+1)$</p>
<p>Where $D$ is the linear operator $D: \mathbb{C}[x] \rightarrow \mathbb{C}[x]$ where (in the context where this statement arose) $x \in \mathbb{N}$; </p>
<p>$f(x) \mapsto \frac{d}{dx} f(x)$</p>
<p>By the Taylor series expansion $e^{D} = \sum_{n=0}^{\inf... | zyx | 14,120 | <p>The Taylor series is $$ f(x+h) = \displaystyle \sum_{n \geq 0} \frac{f^{(n)}(x)}{n!}h^n $$ and here $e^A$ is nothing more than a notation for the power series $\sum A^n/n!$. The series is a well-defined operator for any operator $A$ that is "locally nilpotent" (only a finite number of terms are nonzero when applie... |
1,234,217 | <p>Why do we choose Natural number to describe whether a set is countable or not? How can we say that Natural Number is countable?</p>
| bird | 214,486 | <p>It's just the definition. Cardinality is the size of the set - meaning the number of elements in the set. When we count the elements we call the first element number one, the second number two etc.
So we use the natural numbers for counting and hence it's called a countable set. </p>
<p>E.g. when you decide for ex... |
1,895,248 | <p><a href="http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/exams/prac1a.pdf" rel="nofollow noreferrer">Problem 5. a)</a> Find the area of the space triangle with vertices $P_0, P_1, P_2$:</p>
<p>$$ P_0 = (2,1,0),\ \ P_1=(1,0,1),\ \ P_2=(2,-1,1) $$</p>
<p>My current solution is to use $\... | shardulc | 140,607 | <p>You have $s \lor t$. Assume $s$. If you make the definitions a little more precise, you find that $s \implies p$ and thus $s \implies p \implies (q \lor r)$. We already know that it was sunny on Tuesday, so it was not raining or snowing: $u \implies \neg (q \lor r)$. There we have a contradiction, so our assumption ... |
1,302,732 | <p>This is an example from Stoll's <em>Introduction to Analysis</em>. I'm struggling to understand why there's a contradiction here, though I think I'm on the verge of understanding it, but I'd like to understand it more formally. </p>
<p><img src="https://i.stack.imgur.com/vWRwP.jpg" alt="enter image description here... | Ian | 83,396 | <p>If there were uniform convergence, then for sufficiently large $n$, you would have $|f_n(x)-f(x)|<\varepsilon$ for all $x \in [0,1]$. The problem is that near $1$ the convergence rate is very slow: to get $|f_n(x)-f(x)|<\varepsilon$ for $x \in (0,1)$, you need $x^n<\varepsilon$, hence $n>\log(\varepsilon... |
100,551 | <p>I need to get a matrix $\{a(x_i-x_j)\}$, where $x_i$ form a partition of an interval, $a(x)$ is a given function. I use </p>
<pre><code>In[67]:= a[x_?NumericQ] := N[Exp[-Abs[x]]];
x = Table[-10 + 0.02 (j - 1), {j, 1, 1001}];
A = Outer[a[#1 - #2] &, x, x]; // AbsoluteTiming
Out[69]= {2.99032, ... | Pillsy | 531 | <p>Yes, two things help. The first is that <code>Subtract</code> is going to execute faster than <code>#1 - #2 &</code>, and the other is that all the operations involved in <code>a</code> are <code>Listable</code>, so getting rid of that <code>_?NumericQ</code> restriction speeds things up greatly. On my computer,... |
3,419,850 | <p>Does <span class="math-container">$\displaystyle\sum_{n\geq 0} \dfrac{n!k^n}{(n+1)^n}$</span> converge or diverge for <span class="math-container">$k=\dfrac{19}{7}$</span>?</p>
<p>I'm not sure what convergence test I should use for this one. <span class="math-container">$k=\dfrac{19}{7}$</span> also seems randomly ... | Simon Fraser | 717,270 | <p>You should learn more about convergence tests and when to use them. For this problem, you can apply the ratio test. </p>
<p>Let <span class="math-container">$a_n = \dfrac{n!k^n}{(n+1)^n}.$</span> So we have that <span class="math-container">$\lim\limits_{n\to \infty} \left|\dfrac{a_{n+1}}{a_{n}}\right|=\lim\limits_... |
3,517,019 | <p>Let <span class="math-container">$k_0$</span> be a field, <span class="math-container">$k$</span> its algebraic closure, and <span class="math-container">$K$</span> a field extension of <span class="math-container">$k_0$</span>.
Let <span class="math-container">$P_1$</span> and <span class="math-container">$P_2$</s... | GreginGre | 447,764 | <p>Too long for a comment.</p>
<p>Let <span class="math-container">$L_i$</span> the field of fractions of <span class="math-container">$K\otimes_{k_0}/P_i$</span>. Let <span class="math-container">$M$</span> be a maximal ideal of <span class="math-container">$L_1\otimes_{k_0} L_2$</span>, and set <span class="math-con... |
26,256 | <p>One can define the Euler characteristic χ for a graph as the number of vertices minus the number of edges. Thus an <span class="math-container">$n$</span>-cycle has <span class="math-container">$\chi = 0$</span> and <span class="math-container">$K_4$</span> has <span class="math-container">$\chi=-2$</span>.
Is t... | Oliver | 13,903 | <p>There are different type of curvatures for graphs. In two
dimensions its not the degree of the point which matters but
the length of the circles at the point like in differential geometry.
This is different from the degree if graphs with boundary are considered.
The simplest curvature for two dimensional graphs
(I ... |
139,125 | <p>This is a variation on an earlier question resolved by <em>user35353</em>: <a href="https://mathoverflow.net/questions/139105/can-a-tangle-of-arcs-interlock">Can a tangle of arcs interlock?</a> In that question, the arcs were restricted to circular arcs, and <em>user35353</em>'s proof that one arc can be removed wit... | Wlodek Kuperberg | 36,904 | <p>Here is a construction of five interlocking elliptical arcs.</p>
<p>In Stage 1, a large elliptical arc $a$ is braced by two very narrow elliptical arcs $b$ and $c$. The distance between the braces can be very small. The braces can be moved outwards and slip off of a, but they can be moved towards the middle only a ... |
2,176,656 | <p>C.T. is comparison test</p>
<p>TYPE II is when a improper integral is improper but not at $\infty$. </p>
<p>a)</p>
<p>$$\int_{1}^{\infty} \frac{\sin\left(\frac{\pi}{x}\right)}{x^2}dx$$</p>
<p>Let g(x) = $\frac{1}{x^2}$ because $|sin(\frac{\pi}{x})| \leq 1$. Since the numerator has been maximized and denominator ... | Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\n... |
4,493,926 | <blockquote>
<p>Suppose <span class="math-container">$p(x)=ax^n + b_1x^{n-1}+\cdots~$</span> and <span class="math-container">$g(x)=ax^n + b_2x^{n-1}+\cdots~~$</span> (basically only the leading coefficients are same).</p>
<p>I am required to find/proof: <span class="math-container">$$\lim_{x \to \infty}{p(x)^{1/n}-g(x... | egreg | 62,967 | <p>Let's write
<span class="math-container">$$
p(x)=ax^n+bx^{n-1}+p_0(x)
\qquad
g(x)=ax^n+cx^{n-1}+g_0(x)
$$</span>
where <span class="math-container">$p_0$</span> and <span class="math-container">$g_0$</span> have degree <span class="math-container">$<n-1$</span>. Now a trick is to replace <span class="math-contain... |
631,586 | <p>$$1.\quad p\quad if\quad q\\ \equiv if\quad q\quad then\quad p\\ \equiv q\rightarrow p\\ \\$$$$2.\quad p\quad only\quad if\quad q\\ \equiv if\quad p\quad then\quad q\\ \equiv p\rightarrow q\\ \\$$$$3.\quad p\quad only\quad if\quad q\\ \equiv if\quad q\quad then\quad p\\ \equiv q\rightarrow p\\ \\$$$$4.\quad p\quad i... | Mauro ALLEGRANZA | 108,274 | <p>From a just published book by Jan von Plato, <a href="https://rads.stackoverflow.com/amzn/click/com/110761077X" rel="nofollow noreferrer" rel="nofollow noreferrer">Elements of Logical Reasoning</a> (Cambridge UP, 2013), pag 11:</p>
<blockquote>
<p>The two sentences <em>if A, then B</em> and <em>B if A</em> seem to e... |
917,500 | <p>We define algebra generated by a subset S of power set of X as intersection of all algebras containing S, Is there a procedure of finding this algebra generated.
Just like we find subspace generated by a subset of a vector space as span of that set i.e. taking all linear combinations of elements of that subset. </p>... | Rustyn | 53,783 | <p>A good example of an algebra is the <em>sigma algebra</em> of Borel sets. For a description of Borel Sets, see <a href="http://en.wikipedia.org/wiki/Borel_hierarchy" rel="nofollow">Borel Hierarchy</a>. If you're working in a separable, completely metrizable metric space, (Polish space), then this is the smallest sig... |
180,672 | <p>Let $f: C \to C$ be a smooth function and $C$ be a compact set, subset of $\mathbb{R}^n$.
We assume that all the fixed points are hyperbolic. Is it true that the number of fixed points is finite or countable?</p>
| Tadashi | 22,389 | <p>As Robert Israel has pointed out: by the definition of hyperbolic fixed point, the number of fixed points is finite (or countable). </p>
<p>A natural generalization of hyperbolicity for non-isolated equilibria is that of <a href="https://en.wikipedia.org/wiki/Normally_hyperbolic_invariant_manifold" rel="nofollow">n... |
1,156,513 | <p>Let $0<x<1$. How can i prove the following identity:</p>
<p>$$\frac{-\log(1-x)}{x(1-x)}=1+(1+1/2)x+(1+1/2+1/3)x^3+...\ \ .$$</p>
| Pedro | 23,350 | <p>In general, let $f(x)=\sum a_nx^n$. Show that $$(1-x)^{-1}f(x)=\sum \sum_{k=0}^n a_k x^n$$</p>
|
1,156,513 | <p>Let $0<x<1$. How can i prove the following identity:</p>
<p>$$\frac{-\log(1-x)}{x(1-x)}=1+(1+1/2)x+(1+1/2+1/3)x^3+...\ \ .$$</p>
| Khosrotash | 104,171 | <p>$$0<x<1 \\1+x+x^2+x^3+x^4+...=\frac{1}{1-x}\\\int(1+x+x^2+x^3+x^4+...)dx=\int \frac{1}{1-x}dx\\x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+...=-ln(1-x)\\$$now divide by x $$\frac{-ln(1-x)}{x}=\frac{x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+...}{x}=1+\frac{x}{2}+\frac{x^2}{3}+\frac{x^3}{4}+...$$now multiply b... |
593,438 | <p>I was reading Mathematics for Economists by Simon and Blume.</p>
<blockquote>
<p>The level set $x^2+y^2+z^2=1$ is a two-dimensional sphere of radius $1$.</p>
</blockquote>
<p>How to actually know that it is two dimensional?</p>
| Slugger | 59,157 | <p>First of all, it is a surface, namely a sphere. Also we can see that we can change two variables arbitrarily, say $x$ and $y$ and then $z$ changes accordingly. We cant arbitrarily change all three variables. This means there are two degrees of freedom.</p>
|
35,023 | <p><code>Simplify</code> and <code>FullSimplify</code> do not simplify this kind of expression: $(r^\frac{1}{x})^x$. Consider</p>
<pre><code>FullSimplify[r^2+(r^(2/x))^x+(r^(2/(x+y)))^(x+y)]
</code></pre>
<p><em>Mathematica</em>'s output is the same as my input, instead of <code>3 r^2</code>. Is there a way to simpli... | Jason B. | 9,490 | <p>When I want an expression like this to simplify, I'll use <code>PowerExpand</code> after <code>FullSimplify</code></p>
<p>Try this:</p>
<pre><code>PowerExpand[r^2+(r^(2/x))^x+(r^(2/(x+y)))^(x+y)]
</code></pre>
|
35,023 | <p><code>Simplify</code> and <code>FullSimplify</code> do not simplify this kind of expression: $(r^\frac{1}{x})^x$. Consider</p>
<pre><code>FullSimplify[r^2+(r^(2/x))^x+(r^(2/(x+y)))^(x+y)]
</code></pre>
<p><em>Mathematica</em>'s output is the same as my input, instead of <code>3 r^2</code>. Is there a way to simpli... | bill s | 1,783 | <p>To expand on rm -rf's hint, the variables need to be positive and real in order for the result you wish to see to hold. You can tell Mathematica to do this using the <code>Assumptions</code> option:</p>
<pre><code>FullSimplify[r^2 + (r^(2/x))^x + (r^(2/(x + y)))^(x + y),
Assumptions -> {x > 0, y... |
4,616,494 | <p>I have come across an exercise that asks to have “There is only one ball, so you need to have it” translated into predicate logic. Using the predicates Ball(x) for x is a ball and Have(x, y) for x must have y, I translated it this way:</p>
<p><span class="math-container">$$\exists x((Ball(x) \land \forall y(Ball(y) ... | Bram28 | 256,001 | <p>I like neither translation.</p>
<p>Yours is certainly not right, since your statement would be rather vacuously true if there exists something that is not a ball: that something would make <span class="math-container">$Ball(x)$</span> False, hence the whole antecedent of the conditional false, and therefore the whol... |
4,234,559 | <p>We consider closed polygonal chains in the 2-dimensional plane with an even number of sides, say <span class="math-container">$2n$</span>, numbered as <span class="math-container">$A_1B_1A_2B_2\dots A_nB_nE$</span>, where <span class="math-container">$E = A_1$</span>.</p>
<p>We require additionally, that each <span ... | Jukka Kohonen | 880,212 | <p>Yes. For simplicity let us assume that "polygon" means that consecutive edges cannot be parallel (no 180-degree angles at vertices). In particular this implies that the outgoing edges from <span class="math-container">$B_i$</span> are <em>not</em> vertical.</p>
<p>If the polygon does not intersect itself, ... |
863,101 | <p>Can anyone help me to find the derivative of this function? I know I have to use the quotient rule: $\dfrac{f(x)}{g(x)}=\dfrac{f′(x)g(x)−f(x)g′(x)}{(g(x))^2}$, but I don't know how I use this when the function is:</p>
<blockquote>
<p>$$f(x,y) = \frac{7y + x^2}{1+y^2}$$ </p>
</blockquote>
<p>$f_x (x,y) = ?$ </p... | draks ... | 19,341 | <p>$$
\frac16 n(n+1)(2n+1) \bmod 4 =0
$$
if $n=8k$ of $n=8k+7$. Using Aleksander's hint: $\sum\limits_{i = 1}^ni^2 = \frac{n(n+1)(2n+1)}{6}$ you can easily check that if you sum up the first eight values, you get $4$ odd and $4$ even values, which makes the overall sum divisble by $4$. The last value is $8^2$, so you ... |
863,101 | <p>Can anyone help me to find the derivative of this function? I know I have to use the quotient rule: $\dfrac{f(x)}{g(x)}=\dfrac{f′(x)g(x)−f(x)g′(x)}{(g(x))^2}$, but I don't know how I use this when the function is:</p>
<blockquote>
<p>$$f(x,y) = \frac{7y + x^2}{1+y^2}$$ </p>
</blockquote>
<p>$f_x (x,y) = ?$ </p... | Gerry Myerson | 8,269 | <p>Can you see that it's the same as asking for $n(n+1)/2$ to be divisible by 4? And can you see how to solve that problem?</p>
|
4,278,192 | <p>That's the definition we got for the symplectic form:</p>
<p>Let <span class="math-container">$$\omega : \: \mathbb{C}^n \times \mathbb{C}^n \rightarrow \mathbb{C}$$</span> be a bilinear, anti-symmetric and non-degenerate (<span class="math-container">$\forall_{y \in \mathbb{C}^n} \: \omega(x,y)=0 \: \Rightarrow \:... | Martin Peters | 185,067 | <p>Here is a recommendation: Have a look in Vladimir Arnold´s book <em>Mathematical Methods of Classical Mechanics</em>. There the geometric meaning behind the formulae is explained.</p>
|
4,278,192 | <p>That's the definition we got for the symplectic form:</p>
<p>Let <span class="math-container">$$\omega : \: \mathbb{C}^n \times \mathbb{C}^n \rightarrow \mathbb{C}$$</span> be a bilinear, anti-symmetric and non-degenerate (<span class="math-container">$\forall_{y \in \mathbb{C}^n} \: \omega(x,y)=0 \: \Rightarrow \:... | Ricardo Buring | 23,180 | <p>Here are some informal sources of intuition and motivation:</p>
<ul>
<li><p><em><a href="https://math.mit.edu/%7Ecohn/Thoughts/symplectic.html" rel="nofollow noreferrer">Why symplectic geometry is the natural setting for classical mechanics</a></em> by Henry Cohn.</p>
</li>
<li><p><em><a href="https://sbseminar.word... |
1,018,292 | <p>A Markov chain with discrete time dependence and stationary transition probabilities is defined as follows. Let $S$ be a countable set, $p_{ij}$ be a nonnegative number for each $i,j\in S$ and assume that these numbers satisfy $\sum_{j\in S}p_{ij}=1$ for each $i$. A sequence $\{X_j\}$ of random variables with ranges... | Suzu Hirose | 190,784 | <p>Any probability distribution at all will satisfy your equation 2. The point of a Markov chain is that in the equation
$$
P[X_{n+1}=j|X_0=i_0,\ldots,X_n=i_n]=p_{i_nj},
$$
the probability only depends on the previous element:
$$
P[X_{n+1}=j|X_0=i_0,\ldots,X_n=i_n]=P[X_{n+1}=j|X_n=i_n].
$$</p>
|
3,664,433 | <p>I'm reading Peter Scott's <em><a href="https://homepages.warwick.ac.uk/~masgar/Teach/2012_MA4J2/geometry.pdf" rel="nofollow noreferrer">The Geometry of 3-Manifolds</a></em> and am trying to understand the argument behind this statement, which arises in the proof of Corollary 3.3:</p>
<blockquote>
<p>If <span clas... | Adele Jackson | 1,091,284 | <p>I know this is an old question, but just wanted to note that Kyle Miller's answer misses a subtlety -- the correct statement is that an <em>orientable</em> closed embedded incompressible surface in an orientable irreducible Seifert fibered space is vertical or horizontal. It isn't true in the one-sided case.</p>
<p>... |
3,528,227 | <blockquote>
<p>How to transform the <span class="math-container">$\tanh$</span> sigmoid function so that it starts from <span class="math-container">$f(0)=0$</span>, goes asymptotically to <span class="math-container">$1$</span>, and has <span class="math-container">$f(0.1)=a$</span> and <span class="math-container"... | Christian Blatter | 1,303 | <p>I understand that you want
<span class="math-container">$$\lim_{t\to-\infty}f(t)=0,\quad f(0.1)=a,\quad f(0.9)=b,\quad \lim_{t\to\infty}f(t)=1\ .$$</span>
The limit conditions can be fulfilled with
<span class="math-container">$$f(t)={1+\tanh\bigl(c(t-t_0)\bigr)\over2},\quad c>0,\quad t_0\in{\mathbb R}\ .$$</span... |
2,205,209 | <p>Today I came across a question in which equations of two lines (Which were parallel) were given and it was asked to find their angle bisector.</p>
<p>My answer for this was :</p>
<p>Since there is no point of intersection of Parallel lines, there is no origin of angle bisector. So, answer should be <strong>Doesn't... | quasi | 400,434 | <p>Your objection is valid.
<p>
Unless there's some special definition in force (which is why I asked for the textbook), there's no vertex, hence no angle, hence no angle bisector.
<p>
Thus, assuming the standard definition, the answer you quoted is simply wrong. </p>
|
2,117,481 | <p>what must be added to $x^3-6x^2+11x-8$ to make a polynomial having factor $x-3$?</p>
<p>If the required expression to be added be $K$ then $x^3-6x^2+11x-8+K$ is exactly divisible by $x-3$ but how do I find $K$??</p>
| Sarah | 410,125 | <p>Plug in $x=3$ into the polynomial you get $-2$, so you must add $2$ to it to make the polynomial zero at $3$. </p>
|
2,963,964 | <p>I'm usually good at determining divergence and using the comparison test, but I can't figure out what function I can use to determine if
<span class="math-container">$$ \int_{0}^{1} \frac{e^{x^2}}{x^2} \, dx$$</span>
is divergent. If anyone can help me, that would be greatly appreciated. </p>
| Szeto | 512,032 | <p><span class="math-container">$$x^2\ge 0\implies e^{x^2}\ge 1\implies \int^1_0\frac{e^{x^2}}{x^2}dx\ge\int^1_0\frac1{x^2}dx=\infty$$</span></p>
<p>Thus the integral is divergent.</p>
|
1,184,963 | <p>Toss two fair dice. There are $36$ outcomes in the sample space $\Omega$, each with probability $\frac{1}{36}$. Let:</p>
<ul>
<li>$A$ be the event '$4$ on first die'.</li>
<li>$B$ be the event 'sum of numbers is $7$'.</li>
<li>$C$ be the event 'sum of numbers is $8$'.</li>
</ul>
<p>It says here $A$ and $B$ are ind... | ryang | 21,813 | <p>Here's an analogous example from <a href="https://math.stackexchange.com/q/3897127/21813">my previous post</a>:</p>
<p>Pairwise independence can be characterised as:</p>
<blockquote>
<p>Let <span class="math-container">$P(A)\neq0.$</span></p>
<p>Events <span class="math-container">$A$</span> and <span class="math-co... |
1,279,829 | <p>I know that other groups like $\mathbb{Z}_4 \times \mathbb{Z}_3 \times \mathbb{Z}_5$ are isomorphic to $\mathbb{Z}_{60}$. </p>
| Buzi | 86,244 | <p>Notice that $\mathbb{Z}_{60}$ is cyclic, while $\mathbb{Z}_2\times\mathbb{Z}_{30}$ is not.<br>
That is for example because $\gcd(30,2)\neq 1$.</p>
|
1,279,829 | <p>I know that other groups like $\mathbb{Z}_4 \times \mathbb{Z}_3 \times \mathbb{Z}_5$ are isomorphic to $\mathbb{Z}_{60}$. </p>
| KON3 | 43,773 | <p>Several answers have been provided already. Another one is here: $\mathbb Z_{60}$ contains an element of order 60 but $\mathbb Z_2\times \mathbb Z_{30}$ does not contain any element of order 60. Infact, the maimum possible order of any element in $\mathbb Z_2\times \mathbb Z_{30}$ is lcm$(2,30)=30$.</p>
|
3,652,180 | <blockquote>
<p>Let <span class="math-container">$D$</span> be an integral domain which is not a field and <span class="math-container">$Q=\text{Frac}(D)$</span> the field of fractions of <span class="math-container">$D$</span>. Then <span class="math-container">$Q$</span> as a <span class="math-container">$D$</span>... | Zeek | 573,067 | <p>As a hint, note that <span class="math-container">$Q=\text{Frac}(D)$</span> is also injective as an <span class="math-container">$R$</span>-module. Therefore to show that it is not projective you can show that there are no non-zero projective-injective modules over <span class="math-container">$D$</span>. </p>
<p>T... |
3,110,664 | <p>Given the equation <span class="math-container">$k = (p - 1)/2$</span> where <span class="math-container">$p$</span> is any prime number, what is the chance that a randomly chosen element from the set of all <span class="math-container">$k$</span>s will be divisible by 3? Or rather, how can this probability calculat... | Myunghyun Song | 609,441 | <p>This is a supplement of @Mindlack's comment. Let us consider for each <span class="math-container">$n\ge 1$</span>,
<span class="math-container">$$
\mathcal C_n =\{x\in\Bbb R^n: f(x) = 0,\ \|x\|\le n\}.
$$</span> By inverse mapping theorem, for each <span class="math-container">$x\in \mathcal C_n$</span>, there exis... |
2,176,080 | <p>'$\Leftrightarrow$' Is very much important in this question . Actually, it seems very obvious to me.</p>
<p>We say a function is differentiable at $x=a$ iff </p>
<p>$\lim_{ h\rightarrow 0 }{ \frac { f(a+h)-f(a) }{ h } } = lim_{ h\rightarrow 0 }{ \frac { f(a-h)-f(a) }{ -h } }$</p>
<p>Now, let</p>
<p>$f'(x)=g(x)... | Kenny Wong | 301,805 | <p>In addition to the previous answer, let me try to convince you that your $f$ is differentiable at $x = 0$.</p>
<p>I claim that $f'(0) = 0$. To prove this, I have to demonstrate that, for any $\epsilon > 0$ that you give me, I can find a $\delta > 0$ such that
$$ | x| < \delta \implies | \ \frac{f(x) - f(0... |
2,246,777 | <p>I was curious as to whether $$\lim_{x \to 0}\frac{1}{x^2}=\infty $$
Or the limit does not exist? Because doesn't a limit exist if and only if the limit tends to a finite number? </p>
| TonyK | 1,508 | <p>Infinity is a special case. We write $$\lim_{x\to a}f(x)=\infty$$ to mean:</p>
<p>For all $M\in\mathbb R$ there exists $\delta > 0$ such that $0<|x-a|<\delta\implies f(x)>M$.</p>
<p>As you noticed, this is different from the definition used for a finite limit.</p>
|
277,331 | <p>One can write functions which depend on the type of actual parameter before they are actually called. E.g.:</p>
<pre><code>Clear[f,g,DsQ];
DsQ[x_]:=MatchQ[x,{String__}];
f[i_Integer, ds_?DsQ] :=Print["called with integer i and DsQ[ds]==True"];
f[i_String, ds_?DsQ] :=Print["called with String i and Ds... | Ulrich Neumann | 53,677 | <p><strong>modified</strong></p>
<p>Try <code>RegionDistance</code></p>
<pre><code>elli1 = ImplicitRegion[((x - 200)/10)^2 + ((y - 110)/8)^2 == 1, {x, y}]
elli2 = ImplicitRegion[((x - 210)/25)^2 + ((y - 100)/30)^2 == 1, {x,y}]
mini1 = NMinimize[{RegionDistance[elli1, {x, y}],Element[{x, y}, elli2]}, {x,y}]
mini2 = NMi... |
551,994 | <p>I want to prove this statement:</p>
<p>$$(A_1 \cup A_2)^c = {A_1}^c \cup {A_2}^c$$</p>
<p>where the $c$ means the complement.</p>
<p>Any help would be greatly appreciated.</p>
| 1233dfv | 102,540 | <p>This is one of De Morgan's Laws. We want to prove that $(A\cup B)^c=A^c \cap B^c$. </p>
<p>Let $x\in(A\cup B)^c$. Then $x\notin A\cup B$. So $x\notin A$ and $x\notin B$. Therefore, $x\in A^c$ and $x\in B^c$. It follows that $x\in A^c \cap B^c$. Thus $(A\cup B)^c\subseteq A^c \cap B^c$. Now, let $x\in A^c \cap B^c$.... |
497,344 | <p>I would like to ask for a little help or a hint about a set theory exercise i am stuck in.</p>
<blockquote>
<p>Let $f: \mathbb{N}\rightarrow \mathbb{P}(\mathbb{N})$, $\mathbb{P}(\mathbb{N})$ is the power set of the natural numbers, be a map. Consider the subset $A\subset \mathbb{N}$ defined by $A=\{ m\in \mathbb{... | Lord_Farin | 43,351 | <p>Your ideas are correct.</p>
<p>What is interesting to note, is that we didn't use that we were talking about $\Bbb N$. The same proof works for any set $X$.</p>
<p>This result, often denoted as a statement about cardinalities:</p>
<p>$$|X| < |\mathcal P(X)|$$</p>
<p>meaning that there is an injection $X \hook... |
871,526 | <p>Prove that $$\textbf{r}''' = [s'''-\kappa^2(s')^3]\textbf{ T } + [3\kappa s's''+\kappa'(s')^2]\textbf{ N }+\kappa \hspace{1mm}\tau (s')^3\textbf{B}.$$</p>
<p>What is $\tau$, I can't figure that part out.</p>
<p>All ideas are welcome.</p>
| RasoolNaseer | 418,344 | <p>Actually its easy. You just need to start with an equation r'= s' t and take its two further derivaties w.r.t s, then you just need to use all the three Ferent-Serret equations in the whole process, wherever required.</p>
<p>Note: there will be a use of chain rule in the initial process.</p>
|
219,965 | <p>I am currently reading up on nuclear spaces in Jarchow, "Locally Convex Spaces", but I got confused and don't seem to find my mistake. In said book, theorem 21.5.9 states:</p>
<p>Let $F$ be a nuclear Frechet space. Then, $F'_\beta \otimes_{\pi} F = L_\beta (F,F)$.
i.e. the projective tensor product of $F$ with its ... | Jochen Wengenroth | 21,051 | <p>It is indeed true that for a nuclear Frechet space $F$, the <em>complete</em> projective tensor product $F'_\beta \tilde{\otimes}_\pi F$ is <em>isomorphic</em> to $L_\beta(F,F)$.
The mistake is your claim that the complete projective tensor product $E\tilde{\otimes}_\pi F$ consists of convergent series $\sum\limits_... |
2,262,467 | <p>Hi guys i have two questions i´m struggling with.</p>
<p>1)The mean duration of education for a population is 12 years and
the standard deviation is 2 years. What is the maximum probability that a randomly selected individual will have had less than
9 or more than 15 years of education?</p>
<p>2)Limit Theorem: A f... | MvG | 35,416 | <p>I agree with the comment by Blue: the property you describe appears to be universal; the choice of $A$ does not matter.</p>
<p><a href="https://i.stack.imgur.com/clx0L.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/clx0L.png" alt="Figure"></a></p>
<p>So let's prove this. I like to use coordinat... |
1,460,118 | <p>I'm taking a Calculus I course online, which has been going pretty well considering that I haven't done much math since I graduated high school 5 years ago. However, I recently ran into a good bit of trouble with the chapter on delta-epsilon proofs. Specifically, my question is about non-linear equations and bounds.... | yurnero | 178,464 | <p><strong>Intuition</strong>: $x$ being close to $-1$ will give you a bound on $x+1=x-(-1)$. But this in fact also gives you a bound on $x-3$ in the sense that $x-3$ is close to $-1-3=-4$. Say, if $x$ is within $1$ unit from $-1$, then $x-3$ is between $-5$ and $-3$.</p>
<p>Let $\delta=\min\{1,\epsilon/5\}$. Then if ... |
3,988,837 | <p>Let <span class="math-container">$A, B$</span> two rings and <span class="math-container">$I_A: {}_A\text{Mod} \to{}_A\text{Mod}$</span> the identity functor. I am trying to show that if <span class="math-container">$A, B$</span> are Morita equivalent, then <span class="math-container">$\underline{\text{Nat}}(I_A, I... | Jeroen van der Meer | 874,176 | <p>Your approach is correct, and in fact it's a special case of a more abstract category-theoretic observation. Suppose that <span class="math-container">$\mathcal{C}$</span> and <span class="math-container">$\mathcal{D}$</span> are equivalent categories, say <span class="math-container">$F \colon \mathcal{C} \to \math... |
1,192,137 | <p>Define a linear transformation $T\colon \mathbb{R}^3\to\mathbb{R}^3$, such that $T(x) = [x]_B$ ($B$-coordinate vector of $x$). </p>
<p>$B = \{b_1, b_2, b_3\}$, which is a basis for $\mathbb{R}^3$.</p>
<p>$b_1 = (1, 1, 0)$
$b_2 = (0, 1, 1)$
$b_3 = (1, 1, 1)$</p>
<p>$T$ is a matrix transformation $T(x) = Ax$ for e... | Sry | 112,128 | <p>Your answer is not correct otherwise A operated over b1,b2,b3 must give e1,e2,e3(the standard basis).
What you can do is write e1,e2,e3 in linear combination of b1,b2,b3 and from that combination find out A.</p>
|
483,392 | <p>For the real axis, the Mandelbrot set consists of points from $[-2,0.25]$. Some of these points are in the interior of the m-set, and some are on the boundary. Those points in the interior are inside hyperbolic regions of one of the bulbs on the real axis. What is the probability that a random point on the real a... | Michael Albanese | 39,599 | <p>Note that $n = \sum_{i=0}^kn_ib^i$ and $b \equiv 1 \operatorname{mod} (b - 1)$. Now use the fact that if $a \equiv b \operatorname{mod} m$ and $c \equiv d \operatorname{mod} m$, then $a + c \equiv b + d \operatorname{mod} m$ and $ac \equiv bd \operatorname{mod} m$; in particular, $a^n \equiv b^n \operatorname{mod} m... |
4,122,288 | <p>I have <span class="math-container">$$7^x\bmod {29} = 23 $$</span>
It is possible to get <span class="math-container">$x$</span> by trying out different numbers but that will not be possible if <span class="math-container">$x$</span> is actually big.</p>
<p>Are there any other solutions for this equation?</p>
<p>Kin... | Thomas Andrews | 7,933 | <p>Trial and error is your only option for this type of problem.</p>
<p>The general problem is called the <a href="https://en.wikipedia.org/wiki/Discrete_logarithm" rel="nofollow noreferrer">discrete logarithm,</a> and it is hard.</p>
<p>Give prime <span class="math-container">$p$</span> and <span class="math-container... |
2,967,366 | <p>I need to integrate function <span class="math-container">$\int_0^1 pur\mathrm{d}r$</span>, where I only have discrete values for <span class="math-container">$p$</span>,<span class="math-container">$u$</span> and <span class="math-container">$r$</span>. So, if I multiply these values, would it be correct to integra... | whpowell96 | 516,072 | <p>Assuming that <span class="math-container">$p$</span> and <span class="math-container">$u$</span> are evaluated at the same points as <span class="math-container">$r$</span>, the MATLAB code to approximate this integral would be "trapz(r,p.*u.*r)". This integrates the vector of products of <span class="math-containe... |
1,691,963 | <p>Given the differential equation $dy/dx = f(x,y)$ with initial condition $y(x_0)=y_0$. Let $f$ be a continuous function in $x$ and $y$ and Lipschitz-continuous in $y$ with Lipschitz constant L. Let F be a mapping on the space $C^0(I)$ of continuous functions $u:I\rightarrow \mathbb{R}$. $I = [x_0,M]$.
$$Fu(x) = y_0 +... | Eric Towers | 123,905 | <p>If $x>1$, $x(x-1)^{(x-1)} = x \left(\frac{x-1}{x}\right)^{x-1} x^{x-1} < x 1^{x-1} x^{x-1} = x^x$.</p>
|
64,265 | <p>I've been using a DateList plot to visualise property information but I don't think it's the best way display my data. My data is formatted as {time (hours), property} where property is an integer between 1 and 20</p>
<pre><code>data = {{0, 0}, {0.2187, 3}, {0.25, 1}, {0.3715, 15}, {0.868,
1}, {1.261, 15}, {1.4595... | kglr | 125 | <p>In addition to the methods suggested in previosly posted answers and in <a href="https://mathematica.stackexchange.com/q/58604/125">this related Q/A</a>, there is also ...</p>
<p><code>Graphics</code> with thick <code>Line</code>s (instead of <code>Rectangle</code>s):</p>
<pre><code>colorRules = Thread[# -> Cha... |
3,913,856 | <p>we have analytically calculated distance between the centers of a big circle and a small circle in <strong>mm</strong>.</p>
<p><a href="https://i.stack.imgur.com/7WJEv.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/7WJEv.png" alt="enter image description here" /></a></p>
<p>We also have coordinat... | Hagen von Eitzen | 39,174 | <p>For your <span class="math-container">$S=\{1,2,3,\ldots, 300\}$</span> write
<span class="math-container">$S=A\cup B\cup C$</span> with
<span class="math-container">$$A=\{3,6,9,\ldots, 300\}=S\cap 3\Bbb Z $$</span>
<span class="math-container">$$B=\{1,4,7,\ldots, 298\}=S\cap(3\Bbb Z+1) $$</span>
<span class="math-... |
2,397,837 | <p>Please help! I don't remember what happened last night but I woke up this morning in a prison cell with nothing but my pile of game theory lecture notes. The guard came in and says they have another rational person in the cell next door, and we both have to play a one-shot nonzero-sum matrix game without communicat... | mlc | 360,141 | <p>Most of game theory concerns <em>interacting agents</em>: what is optimal for you to do depends on what your opponent does (and vice versa). Thus, most of game theory focuses on equilibria, interpreted as profiles of strategies were all agents are playing optimally <em>given how their opponents are playing</em>.</p... |
194,745 | <p>I'm trying to create a program which converts latitude/longitude (wgs84) to UPS (Universal Polar stereographic) coordinates, and then UPS(x, y) to WGS 84. I mean Given UPS(x, y), be able to compute latitude/longitude and vice versa.</p>
<p>Is there any easy way to do this?</p>
| jose | 10,552 | <p>GeoProjectionData has the "UPSNorth" and "UPSSouth" projections. For example, take the same point chosen by Carl:</p>
<pre><code>In[1]:= alert = GeoPosition[{82.5307536, -62.2750895}]
Out[1]= GeoPosition[{82.5308, -62.2751}]
</code></pre>
<p>Then you can do:</p>
<pre><code>In[2]:= GeoGridPosition[alert, "UPSNorth... |
1,011,564 | <p>How can I prove the following inequality using mean value theorem?
<span class="math-container">$$1.997<129^{1/7}<2.003$$</span></p>
<h3>Progress</h3>
| Christian Blatter | 1,303 | <p>We shall use the MVT in the following form: If $f$ is differentiable on the interval $[0,x]$ then there is a $\xi\in\ ]0,x[\ $ with
$$f(x)=f(0)+x\>f'(\xi)\ .\tag{1}$$
Apply this to the function
$$f(x):=(1+x)^{1/7}\qquad(x\geq0)$$
and obtain
$$f(x)=1+x\cdot {1\over 7}(1+\xi)^{-6/7}\leq1+{x\over7}\ .$$
It follows ... |
1,011,564 | <p>How can I prove the following inequality using mean value theorem?
<span class="math-container">$$1.997<129^{1/7}<2.003$$</span></p>
<h3>Progress</h3>
| Barry Cipra | 86,747 | <p>Let $f(x)=x^{1/7}$, and note that $f(128)=2$. Note also that $f'(x)={1\over7}x^{-6/7}={1\over7f(x)^6}$. By the Mean Value Theorem,</p>
<p>$${f(129)-f(128)\over129-128}=f'(c)$$</p>
<p>for some $128\lt c\lt 129$. Thus $129^{1/7}=2+f'(c)$ with $|f'(c)|={1\over7f(c)^6}\lt{1\over7\cdot2^6}={1\over7\cdot64}\lt{3\over... |
12,544 | <blockquote>
<p>Can <span class="math-container">$n!$</span> be a perfect square when <span class="math-container">$n$</span> is an integer greater than <span class="math-container">$1$</span>?</p>
</blockquote>
<p>Clearly, when <span class="math-container">$n$</span> is prime, <span class="math-container">$n!$</span> ... | user3628041 | 154,386 | <ul>
<li><p>If $n$ is prime, then for $n!$ to be a perfect square, one of $n-1, n-2, ... , 2$ must contain n as a factor. But this means one of $n-1, n-2, ... , 2 \geq n$, which is impossible. </p></li>
<li><p>If $n$ is not prime, then the first prime less than $n$ will be $p = n-k$, $0<k<n-1, 2\leq p<n$. No n... |
3,682,277 | <p>By integral test I found its converges to <span class="math-container">$\frac\pi4$</span> but thats the only thing I can find :( Hope somebody can give me a clue about how can I handle this question. </p>
<p>Show that
<span class="math-container">$$\frac{\pi}{4}\leq\sum_{n=1}^{\infty}\frac{1}{n^2+1}\leq\frac{1}{2}+... | saulspatz | 235,128 | <p><strong>HINT:</strong></p>
<p><span class="math-container">$$\int_{n-1}^n\frac{\mathrm{d}x}{x^2+1}\leq\frac1{n^2+1}\leq
\int_{n}^{n+1}\frac{\mathrm{d}x}{x^2+1}$$</span></p>
|
1,915,560 | <p><a href="https://math.dartmouth.edu/archive/m105f13/public_html/m105f13notes1.pdf" rel="nofollow">These notes on harmonic sum</a> present the following inequality:</p>
<p>$$\frac{1}{n} < \int_{n-1}^n \frac{dt}{t}$$ for $n≥2$.</p>
<p>How can this be shown? By induction and by evaluation the integral?</p>
| user90369 | 332,823 | <p>Integral with $n\in\mathbb{N}$. </p>
<p>$\displaystyle \int\limits_n^{n+1}\frac{dt}{t} =\ln(1+\frac{1}{n})=(\frac{1}{n}-\frac{1}{2n^2})+\sum\limits_{k=1}^\infty (\frac{1}{(2k+1)n^{2k+1}}-\frac{1}{(2k+2)n^{2k+2}})$</p>
<p>It is $\displaystyle \frac{1}{n+1}\leq \frac{1}{n}-\frac{1}{2n^2}$ and therefore $\displaystyl... |
1,915,560 | <p><a href="https://math.dartmouth.edu/archive/m105f13/public_html/m105f13notes1.pdf" rel="nofollow">These notes on harmonic sum</a> present the following inequality:</p>
<p>$$\frac{1}{n} < \int_{n-1}^n \frac{dt}{t}$$ for $n≥2$.</p>
<p>How can this be shown? By induction and by evaluation the integral?</p>
| robjohn | 13,854 | <p>For $t\in[n-1,n]$, we have $\frac1n\le\frac1t$. Therefore,
$$
\frac1n=\int_{n-1}^n\frac1n\,\mathrm{d}t\le\int_{n-1}^n\frac1t\,\mathrm{d}t
$$</p>
|
3,677,964 | <p>High school student here.
I'm trying to find the maximum of this function:
<span class="math-container">$$f(x)=\frac{2x-1}{2-x}.$$</span>
where <span class="math-container">$0 \leq x \leq 1$</span>.
The standard process would involve finding the values of <span class="math-container">$x$</span> such that <span clas... | Matt Samuel | 187,867 | <p>Does <span class="math-container">$i^4=i$</span>? I'd say that <span class="math-container">$i^4=1$</span>. Note that <span class="math-container">$i=e^{\frac {i\pi} 2+2k\pi}$</span>, so <span class="math-container">$$i^{1/4}=e^{\frac{i\pi}8+k\frac\pi 2}$$</span>
for <span class="math-container">$k=0,1,2,3$</span>.<... |
926,581 | <p>I find the <a href="https://en.wikipedia.org/wiki/Surreal_number" rel="nofollow noreferrer">surreal numbers</a> very interesting. I have tried my best to work through John Conway's <em>On Numbers and Games</em> and teach myself from some excellent <a href="http://www.tondering.dk/claus/sur16.pdf" rel="nofollow noref... | Community | -1 | <p>Consider the sequence
$$x_n=\sum_{i=1}^n\frac{9}{10^i}$$</p>
<p>whose 'limit' we understand to be the thing we call $.9\bar{9}$. When we consider this sequence in the surreal numbers, it does not converge to anything.</p>
<p>Let $I(1)$ represent the neighborhood of all infinitesimal numbers around $1$. The sequenc... |
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