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 |
|---|---|---|---|---|
865,598 | <p>How can I calculate this value?</p>
<p>$$\cot\left(\sin^{-1}\left(-\frac12\right)\right)$$</p>
| Ryan | 164,086 | <p>Let arcsin(-1/2)= x
Implies sinx=-1/2
Implies x= 120 degree
Now u need to find value of cot x
So cot x= - underroot 3</p>
|
756,236 | <p>The question is to write the general solution for this recurrence relation:</p>
<p>$y_{k+2} - 4y_{k+1} + 3y_{k} = -4k$.</p>
<p>I first solved the homogeneous equation $y_{k+2} - 4y_{k+1} + 3y_{k} = 0$, by writing the auxiliary equation $r^2 - 4r + 3 = (r-3)(r-1) = 0$. Thus $y_k^{h} = c_1(1)^k + c_2 (3)^k$. The gen... | mercio | 17,445 | <p>You started with the assumption that there existed two constants $a,b$ such that $y_k = a+bk$ was a solution. After a bit of algebra you end up with<br>
$y_k$ is a solution $\iff \forall k, y_{k+2}-4y_{k+1}+3y_k = -4k \iff \forall k, b = 2k$.<br>
So $b$ has to be equal to every even integer at once. Obviously this i... |
3,012,090 | <p>Let <span class="math-container">$x>0$</span>. I have to prove that</p>
<p><span class="math-container">$$
\int_{0}^{\infty}\frac{\cos x}{x^p}dx=\frac{\pi}{2\Gamma(p)\cos(p\frac{\pi}{2})}\tag{1}
$$</span></p>
<p>by converting the integral on the left side to a double integral using the expression below:</p>
<p... | mrtaurho | 537,079 | <p>Your given integral is closely related to the Mellin transform and can be evaluated by using <a href="https://en.wikipedia.org/wiki/Ramanujan%27s_master_theorem" rel="nofollow noreferrer">Ramanujan's Master Theorem</a>.</p>
<blockquote>
<p><strong>Ramanujan's Master Theorem</strong></p>
<p>Let <span class="math-cont... |
2,810,008 | <p>Can I investigate this limit and if yes, how? $${i^∞}$$</p>
<p>I am at a loss of ideas and maybe it is undefined?</p>
| José Carlos Santos | 446,262 | <p>Since the sequence $(i^n)_{n\in\mathbb N}$ is the sequence$$i,-1,-i,1,i,-1,-i,1,\ldots$$your sequence diverges.</p>
|
2,810,008 | <p>Can I investigate this limit and if yes, how? $${i^∞}$$</p>
<p>I am at a loss of ideas and maybe it is undefined?</p>
| C Monsour | 552,399 | <p>The limit doesn't exist in any event, but you should still be specific about <em>what</em> precisely is tending to $\infty$, because you could talk about the limit <em>set</em> (i.e., in this case, the set of subsequential limits), and then it matters. If the exponents are integers, the limit set is $\{1,-1,i,-i\}$... |
1,671,111 | <p>I'm looking for an elegant way to show that, among <em>non-negative</em> numbers,
$$
\max \{a_1 + b_1, \dots, a_n + b_n\} \leq \max \{a_1, \dots, a_n\} + \max \{b_1, \dots, b_n\}
$$</p>
<p>I can show that $\max \{a+b, c+d\} \leq \max \{a,c\} + \max \{b,d\}$ by exhaustively checking all possibilities of orderings am... | carmichael561 | 314,708 | <p>For any index <span class="math-container">$j$</span>, <span class="math-container">$a_j+b_j\leq \max\{a_1,\dots,a_n\}+\max\{b_1,\dots,b_n\}$</span>. Now take the maximum over all <span class="math-container">$j$</span>.</p>
|
3,815,898 | <p>In my stats lecture, my professor introduced this theorem, however I don't quite understand what the theorem means and how he got from step 3 to 4. Also what does the prime symbol mean? Could someone paraphrase what the theorem means or point me to some online resource where I could learn more about this theorem? Th... | lonza leggiera | 632,373 | <p>The proof you cite glosses over a technical point which arises when the distribution function <span class="math-container">$\ F\ $</span> is not <em>strictly</em> increasing. In that case, it is not injective (one-to-one), so it's not altogether clear what the definition of <span class="math-container">$\ F^{-1}\ $... |
4,631,618 | <p>Consider this absolute value quadratic inequality</p>
<p><span class="math-container">$$ |x^2-4| < |x^2+2| $$</span></p>
<p>Right side is always positive for all real numbers,so the absolute value is not needed.</p>
<p>Now consider the cases for the left absolute value</p>
<ol>
<li><span class="math-container">$$... | Anne Bauval | 386,889 | <p><span class="math-container">$4n^2=(2n-2)(2n-3)+5(2n-2)+4$</span> and for <span class="math-container">$n=1,$</span> this reduces to <span class="math-container">$4=0+5\cdot0+4,$</span> hence
<span class="math-container">$$4\sum_{n=1}^{\infty}\frac{n^2}{(2n-2)!}=\sum_{n=2}^\infty\frac1{(2n-4)!}+ \sum_{n=2}^\infty\fr... |
2,248,550 | <p>Will be the value in the form of $\frac{"0"}{"0"}$? Do I have to use the L'Hopital rule? Or can I say, that the limit doesn't exist?</p>
| dxiv | 291,201 | <p>Hint: for $x \gt 0, y \lt 1\,$:</p>
<p>$$\require{cancel}
\frac{x+y-1}{\sqrt{x}-\sqrt{1-y}} = \frac{x-(1-y)}{\sqrt{x}-\sqrt{1-y}} = \frac{(\sqrt{x}+\sqrt{1-y})\cancel{(\sqrt{x}-\sqrt{1-y})}}{\cancel{\sqrt{x}-\sqrt{1-y}}}
$$</p>
|
655,261 | <p>Let meagre subsets be defined as:<br>
$A\text{ meagre}\iff A=\bigcup_{\lvert K\rvert\leq\aleph_0} A_k\text{ with }\overline{A_k}°=\varnothing$<br>
Then it satisfies:<br>
$B\subseteq A\text{ meagre}\Rightarrow B\text{ meagre}$<br>
$A_k\text{ meagre}\Rightarrow\bigcup_{\lvert K\rvert\leq\aleph_0} A_k\text{ meagre}$</p... | Asaf Karagila | 622 | <p>Given a set $X$, we say that $I$ is a $\sigma$-ideal on $X$ if the following holds:</p>
<ol>
<li>$I\subseteq\mathcal P(X)$, is not empty and $X\notin I$.</li>
<li>If $A\in I$ and $B\subseteq A$, then $B\in I$.</li>
<li>If for $n\in\Bbb N$ we have $A_n\in I$ then $\bigcup A_n\in I$. (If we weaken this just for finit... |
2,278,991 | <p>(NOTE: I am new to proof construction. Don't panic if your heart beat increase.)</p>
<p>Proof 1:</p>
<p>suppose $x$ is even and prime, then there is $k$ in $\mathbb N$ such that </p>
<p>$x = 2k$</p>
<p>But x is only divisible by itself and not 2.</p>
<p>$\frac{x}{2} != k$</p>
<p>$x$ can not be even.</p>
<hr>
... | infinitylord | 178,643 | <p>Here's a proof that $2$ is the <strong>only</strong> even prime.</p>
<p>Let $a \in \mathbb{N} $ be prime and even. (We know this is okay to do, as there is at least <em>one</em> even prime)</p>
<p>Assume $a > 2$</p>
<p>Since $a$ is even, it can be written as $a = 2k$ for some $k \in \mathbb{N}$. By assumption,... |
3,075,263 | <p>Let <span class="math-container">$A$</span> be a positive semi-definite matrix. How to show that Frobenius norm is less than trace of the matrix? Formally,
<span class="math-container">$$\sqrt{\text{Tr}(A^2)} \leq \text{Tr}(A)$$</span>
Also, show when <span class="math-container">$A$</span> is an <span class="math-c... | Angina Seng | 436,618 | <p>For the first you can assume that <span class="math-container">$A$</span> is diagonal with diagonal entries
<span class="math-container">$a_1,\ldots,a_n$</span>, all <span class="math-container">$\ge0$</span>. Then your inequality becomes
<span class="math-container">$$\sum_{i=1}^n a_i^2\le\left(\sum_{i=1}^n a_i\rig... |
3,865,607 | <p>Given <span class="math-container">$B\subseteq X$</span> with both <span class="math-container">$B$</span> and <span class="math-container">$X$</span> contractible. How would you prove that the inclusion map <span class="math-container">$i:B \to X$</span> is a homotopy equivalence?</p>
<p>Thank you</p>
| Cornman | 439,383 | <p>Additional to the specific map Tsemo Aristide gave, there is the following theorem:</p>
<p>If <span class="math-container">$Y$</span> is contractible, then any two maps <span class="math-container">$X\to Y$</span> are homotopic (indeed they are nullhomotopic).</p>
<p>Reference: For example 'Introduction to Algebraic... |
2,041,441 | <p>$\binom{74}{37}-2$ is divisible by :</p>
<p>a) $1369$</p>
<p>b) $38$</p>
<p>c) $36$ </p>
<p>d) $none$ $of$ $ these$</p>
<p>I have no idea how to solve this...I tried writing $\binom{74}{37}$ in some useful form but its not helping...any clues?? Thanks in advance!!</p>
| JMoravitz | 179,297 | <p>Note that $\binom{74}{37}=\frac{74!}{37!37!}$</p>
<p>Note also that $n!$ contains $\sum\limits_{k=1}^\infty \left\lfloor\frac{n}{p^k}\right\rfloor$ factors of a prime $p$</p>
<p>Counting the number of factors of $19$ in the numerator, we get $\lfloor\frac{74}{19}\rfloor+\lfloor\frac{74}{19^2}\rfloor+\dots=3+0+0+\d... |
23,268 | <p>I'm the sort of mathematician who works really well with elements. I really enjoy point-set topology, and category theory tends to drive me crazy. When I was given a bunch of exercises on subjects like limits, colimits, and adjoint functors, I was able to do them, although I am sure my proofs were far longer and m... | Steven Gubkin | 1,106 | <p>Are you comfortable with products? Products are just limits of discrete diagrams - the only arrows are the identity arrows. Any small limit will have a unique monic arrow to the product of all of the objects in the diagram you are taking a limit over (Check this!). So in set theory land this means that all limits... |
23,268 | <p>I'm the sort of mathematician who works really well with elements. I really enjoy point-set topology, and category theory tends to drive me crazy. When I was given a bunch of exercises on subjects like limits, colimits, and adjoint functors, I was able to do them, although I am sure my proofs were far longer and m... | Peter LeFanu Lumsdaine | 2,273 | <p>Since you're familiar with the example of the sheaf condition, I think a nice one-liner intuition is:</p>
<blockquote>
<p>A limit of a diagram is an object of <strong>matching families in that diagram</strong>.</p>
</blockquote>
<p>...defined just like how, in the case of (pre)sheaves, you define a matching fami... |
963,503 | <p>Vectors $a$, $b$ and $c$ all have length one. $a + b + c = 0$. Show that
$$
|a-c| = |a-b| = |b-c|
$$
I am not sure how to get started, as writing out the norms didn't help and there is no way to manipulate
$$
|a-c| \le |a-b| + |b-c|
$$
to get an equality. I just need an idea of where to start.</p>
| drhab | 75,923 | <p>Note that $Z\subseteq\left(X\cap Y\right)\cup Z$ so that $X\cap\left(Y\cup Z\right)=\left(X\cap Y\right)\cup Z$
implies that $Z\subseteq X\cap\left(Y\cup Z\right)\subseteq X$.</p>
|
204,842 | <p>A probability measure defined on a sample space $\Omega$ has the following properties:</p>
<ol>
<li>For each $E \subset \Omega$, $0 \le P(E) \le 1$</li>
<li>$P(\Omega) = 1$</li>
<li>If $E_1$ and $E_2$ are disjoint subsets $P(E_1 \cup E_2) = P(E_1) + P(E_2)$</li>
</ol>
<p>The above definition defines a measure that... | Michael Greinecker | 21,674 | <p>Let <span class="math-container">$\mathcal{U}$</span> be a <a href="https://en.wikipedia.org/wiki/Ultrafilter" rel="nofollow noreferrer">free ultrafilter</a> on <span class="math-container">$\mathbb{N}$</span>. Let <span class="math-container">$P(A)=1$</span> if <span class="math-container">$A\in\mathcal{U}$</span> ... |
2,677,134 | <p>Given the definition of Big-O, prove that $f(n) = n^2 - n$ is $O(n^2)$.</p>
<p>When I use the given definition, I get $n^2 - n \leq n^2 - n^2$ which means that $n^2 -n \leq 0$, which is not true. Is there some step I'm missing?</p>
| Siong Thye Goh | 306,553 | <p>We have $$n^2-n \ge n^2-n^2$$ not the opposite.</p>
<p>To prove $f(n)$ is $O(n^2)$.</p>
<p>$$n^2-n \leq n^2$$</p>
|
2,903,359 | <p>I am trying to prove the following:</p>
<p>Given $1 \le d \le n$, a matrix $P \in R^n$ is a rank-$d$ orthogonal projection matrix. Prove that P is projection matrix iff there exists a $n$x$d$ matrix $U$ such that $P =UU^T$ and $U^TU = I$.</p>
<p>I know that this is an obvious fact about projection matrices but I a... | obareey | 111,671 | <p>(<span class="math-container">$\Rightarrow$</span>) Let <span class="math-container">$P=UU^T$</span> and <span class="math-container">$U^TU=I$</span>. Then <span class="math-container">$P^2=UU^TUU^T=UU^T=P$</span> and <span class="math-container">$P^T=P$</span>.</p>
<p>(<span class="math-container">$\Leftarrow$</sp... |
405,783 | <p>I saw the following in my lecture notes, and I am having difficulties
verifying the steps taken.</p>
<p>The question is:</p>
<blockquote>
<p>Assuming $0<\epsilon\ll1$ find all the roots of the polynomial
$$\epsilon^{2}x^{3}+x+1$$ which are $O(1)$ up to a precision of
$O(\epsilon^{2})$</p>
</blockquote>
<... | Danny W. | 77,064 | <p>Ok, here is how this goes: </p>
<ol>
<li><p>This assumption is based on an idea of what the solution is going to look like. That is, by making this assumption, we won't be able to get any solutions where the first term ($x_0$) is not of this order. Since we are apparently not looking for solution of higher order (t... |
2,825,522 | <p>I have this problem:</p>
<p><a href="https://i.stack.imgur.com/blD6N.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/blD6N.png" alt="enter image description here"></a></p>
<p>I have not managed to solve the exercise, but this is my breakthrough:</p>
<p><a href="https://i.stack.imgur.com/0dTdO.j... | Stefan4024 | 67,746 | <p>As noted one of the angle is $30^{\circ}$. Also note that the longer diagonal bisects the $120^{\circ}$ angle. Hence the angles in the top triangle are $30^{\circ}$ and $60^{\circ}$, so the the angle complement to $x$ is $90^{\circ}$, so we must have that $x = 90^{\circ}$</p>
|
1,393,869 | <p>Given a cubic polynomial with real coefficients of the form $f(x) = Ax^3 + Bx^2 + Cx + D$ $(A \neq 0)$ I am trying to determine what the necessary conditions of the coefficients are so that $f(x)$ has exactly three distinct real roots. I am wondering if there is a way to change variables to simplify this problem and... | bubba | 31,744 | <p>By solving <span class="math-container">$f'(x)=0$</span>, you can find out the two turning points of the cubic. Suppose the solutions are <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span>. In fact, we have
<span class="math-container">$$
x_{1,2} = \frac{-B \pm \sqrt{B^2-3AC}}{3A... |
386,172 | <p>The expression was simplified in the answer to <a href="https://math.stackexchange.com/questions/384592/finding-markov-chain-transition-matrix-using-mathematical-induction">this question</a>. I'm trying to simplify it but I got stuck. Multiplying all the factors and regrouping didn't work, but maybe I'm doing the wr... | vadim123 | 73,324 | <p>$$p\cdot (1/2)(2p-1)^n +(1-p) \cdot (-1/2)(2p-1)^n = \\
p \cdot (1/2)(2p-1)^n +(p-1) \cdot(1/2)(2p-1)^{\color{red}{n}}= \\ \left[\frac{p}{2}+\frac{p}{2}-\frac{1}{2}\right](2p-1)^n =\\ (2p-1)\cdot(1/2) (2p-1)^n$$</p>
<p>You are correct that an $n$ was missing; I inserted an extra line which should help clarify.</p>
|
3,982,937 | <p>To avoid typos, please see my screen captures below, and the red underline. The question says <span class="math-container">$h \rightarrow 0$</span>, thus why <span class="math-container">$|h|$</span> in the solution? Mustn't that <span class="math-container">$|h|$</span> be <span class="math-container">$h$</span>?</... | Ben | 754,927 | <p>Don't worry. It takes a while and some practice to wrap your head around this stuff. It's easy to get confused by all the different conditions and variables and what depends on what.</p>
<p>Suppose <span class="math-container">$\lim_{x \to a}f(x) = \ell$</span></p>
<p>This means that for any <span class="math-contai... |
16,749 | <p>I wanted to remove the <code>Ticks</code> in my coding but i can't. Here when i try to remove the <code>Ticks</code> the number also gone. I need numbers without <code>Ticks</code>, <code>Ticks</code> and <code>GridLines</code> should be automatic and don't use<code>PlotRange</code> .</p>
<pre><code>BarChart[{{1,... | Royce | 5,123 | <p>You can keep the numbers as they current show up but remove the ticks by specifying ticks as follows:</p>
<pre><code>Ticks -> {Table[{2 i, 2 i, 0}, {i, 7}], None}
</code></pre>
<p>or </p>
<pre><code>Ticks -> {Table[{2 i + 1, 2 i + 1, 0}, {i, 0, 8}], None}
</code></pre>
<p>(each tick is specified by a tripl... |
1,796,156 | <p>Let $F(n)$ denote the $n^{\text{th}}$ Fibonacci number<a href="http://mathworld.wolfram.com/FibonacciNumber.html" rel="noreferrer">$^{[1]}$</a><a href="http://en.wikipedia.org/wiki/Fibonacci_number" rel="noreferrer">$\!^{[2]}$</a><a href="http://oeis.org/A000045" rel="noreferrer">$\!^{[3]}$</a>. The Fibonacci number... | Ali Caglayan | 87,191 | <p>Extending what Zach466920 said:</p>
<p>$$\partial_t[k(t,n)]=F(n+1)\cdot k(t,n+1)\tag1$$</p>
<p>We can develop a power series solution for this.
Let
$$k(t,n)=\sum_{m=0}^\infty \kappa(n, m)\ t^m$$</p>
<p>We equation $(1)$ as:
$$\sum_{m=1}^\infty m\ \kappa(n, m)\ t^{m-1}
=\sum_{m=0}^\infty(m+1)\ \kappa (n, m) \ t^m
... |
269,548 | <p>I want to know how I can solve this or plot a versus b ?</p>
<pre><code>Solve[ Sqrt[a] Cosh[1.2 Log[1.65 Sqrt[1/a]]] == Sqrt[-b] Sinh[1.2 Log[1.65 Sqrt[-(1/b)]]], b]
</code></pre>
<p>Thanks</p>
| Michael E2 | 4,999 | <p>We can help <code>Solve</code> by converting the equation to a polynomial one.</p>
<pre><code>eqn = Sqrt[a] Cosh[1.2 Log[1.65 Sqrt[1/a]]] ==
Sqrt[-b] Sinh[1.2 Log[1.65 Sqrt[-(1/b)]]];
(* replace constants with symbols *)
coeff = DeleteDuplicates@Cases[eqn, _Real, Infinity] -> {c1, c2} //
Thread
(* {1.2 -... |
269,548 | <p>I want to know how I can solve this or plot a versus b ?</p>
<pre><code>Solve[ Sqrt[a] Cosh[1.2 Log[1.65 Sqrt[1/a]]] == Sqrt[-b] Sinh[1.2 Log[1.65 Sqrt[-(1/b)]]], b]
</code></pre>
<p>Thanks</p>
| Akku14 | 34,287 | <p>One way to get solution in terms of root expression with variable substituion. (Use onesided eq.)</p>
<pre><code>f[a_, b_] =
Sqrt[a] Cosh[
1.2 Log[1.65 Sqrt[1/a]]] - (Sqrt[-b] Sinh[
1.2 Log[1.65 Sqrt[-(1/b)]]]) // Rationalize //
FullSimplify[#, a >= 0 && b <= 0] &
(* Sqrt[a] Co... |
6,355 | <p>My question is located in trying to follow the argument bellow. </p>
<p>Given a normal algebraic variety $X$, and a line bundle $\mathcal{L}\rightarrow X$ which is ample, then eventually such a line bundle will have enough section to define an embedding $\phi:X\rightarrow \mathbb(H^0(X,\mathcal{L}^{\otimes d}))=\ma... | David Lehavi | 404 | <ul>
<li><p>The sequence stabilizes because any increasing sequence of bounded integers (the dimensions of the images of $X$) stabilizes, but
I assume you mean something different.</p></li>
<li><p>Suppose that $X\to|L|$ has already stabilized, then the map $X\to|2L|$ decomposes through the map $|L|\to\mathbb{P}\mathrm{... |
138,921 | <p>If $A\colon H\to S$ is a bounded operator on a Hilbert space $H$, and $S\subset H$. It is known that $\operatorname{trace}(A)=\sum_{n} \langle Af_n,f_n\rangle$ for any orthonormal basis $\{f_{n}\}$. Is there a relation between $\operatorname{trace}(A)$, $\operatorname{rank}(A)$, and dimension of $\operatorname{range... | Gelu | 59,199 | <p>However, if the boundary conditions are for finite $x$, then for the transformed
equation we have boundary conditions depending on time!! For example if $C$ denotes
a poluttant concentration, we see his value at some points.</p>
|
1,592,224 | <p>I need to understand how to find $a \times b = 72$ and $a + b = -17$. Or I am fine with any other example, even general form $a \times b = c$ and $a + b = d$, how to find $a$ and $b$.</p>
<p>Thanks!</p>
| André Nicolas | 6,312 | <p><strong>If</strong> the equations hold, then $(a+b)^2=289$, and therefore
$$(a-b)^2=(a+b)^2-4ab=289-288=1.$$
Thus $a-b=\pm 1$ and $a+b=-17$. </p>
<p>Solving the system of linear equations, we obtain $a=-8$, $b=-9$ or $a=-9$, $b=-8$. It is easy to verify that both of these satisfy the given equations.</p>
<p>The s... |
4,050,855 | <p>On <a href="https://math.stackexchange.com/a/4050373/878105">this answer</a>, the function <span class="math-container">$f_n(x)=x^n$</span> in the interval <span class="math-container">$[0,1]$</span> is given as a pathologic example with pointwise convergence.</p>
<p>Can I say that this Cauchy sequence does not (poi... | Antoni Parellada | 152,225 | <p>After some <a href="https://www.math.ucdavis.edu/%7Ehunter/intro_analysis_pdf/ch9.pdf" rel="nofollow noreferrer">research on the topic</a>, I think that the answer is no (as canonically answered by Elchanan), and the critical bit is his sentence, "<em>Pointwise convergence does not correspond to any metric (thi... |
241,871 | <p>A Lévy measure $\nu$ on $\mathbb R^{d}$ is a measure satisfying
$$\nu\{0\} = 0, \ \int_{\mathbb R^{d}} (|y|^{2}\wedge 1) \nu(dy) <\infty.$$</p>
<p>A Lévy process can be characterized by triples $(b, A, \nu)$ by
Lévy-Itô decomposition, then
$$X_{t} = bt + W_{A}(t) + \int_{B_{1}} x \tilde N(t, dx) + \int_{B_{1}^{c... | R. van Dobben de Bruyn | 82,179 | <p>A reference for birational equivalence of $CH_0$ is Fulton's <em>Intersection Theory</em> [1], <strong>Example 16.1.11</strong>. In the example, he makes the assumption that $k$ is algebraically closed, but he never uses it. Since the argument is fairly short, let me repeat it here.</p>
<blockquote>
<p><strong>Th... |
345,065 | <p>Let f and g be functions of one real variable and define $F(x,y)=f[x+g(y)]$. Find formulas for all the partial derivatives of F of first and second order.</p>
<p>For the first order, I think we have:</p>
<p>$\frac{\partial F}{\partial x}=\frac{\partial f}{\partial x}+ \frac{\partial f}{\partial y}$</p>
<p>$\frac{... | user1337 | 62,839 | <p>$f$ is a function of <strong>one</strong> variable. Therefore the notation $\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}$ is problematic (and I suggest you adapt the prime notation in that case). What you have written is not correct.</p>
<p>The correct formulas are: $$\frac{\partial F}{\partial x}(x,... |
4,144,203 | <p>Let <span class="math-container">$A$</span> and <span class="math-container">$B$</span> be non-empty subsets of <span class="math-container">$\mathbb{R}$</span> with <span class="math-container">$a\leq b$</span> for all <span class="math-container">$a\in A, \space b \in B$</span> and suppose there exists a uniqu... | 311411 | 688,046 | <p>I was initially lost by your first two sentences. It is not super clear with what "we are done" if <span class="math-container">$\sup A$</span> belongs to <span class="math-container">$A$</span>. In any case, we can take the given existence of <span class="math-container">$\alpha$</span>
and apply just the... |
4,144,203 | <p>Let <span class="math-container">$A$</span> and <span class="math-container">$B$</span> be non-empty subsets of <span class="math-container">$\mathbb{R}$</span> with <span class="math-container">$a\leq b$</span> for all <span class="math-container">$a\in A, \space b \in B$</span> and suppose there exists a uniqu... | Hilberto1 | 831,222 | <p>Your argumentation is correct but I would add a bit more detail to the last sentence: Assume that <span class="math-container">$\sup A\neq \alpha \neq \inf B$</span> so we would obtain the inequality <span class="math-container">$\sup A < \alpha < \inf B$</span>. Then we could choose a number</p>
<p><span clas... |
456,826 | <p>I need to find the derivative of this function. I know I need to separate the integrals into two and use the chain rule but I am stuck.</p>
<p>$$y=\int_\sqrt{x}^{x^3}\sqrt{t}\sin t~dt~.$$</p>
<p>Thanks in advance</p>
| Nick Peterson | 81,839 | <p>Let me show you a general method which works in these sorts of situations.</p>
<p>By the Fundamental Theorem of Calculus, we know how to take the derivative of
$$
F(z):=\int_0^z\sqrt{t}\sin(t)\,dt;
$$
in particular, FTC tells us that
$$\tag{1}
F'(z)=\sqrt{z}\sin(z).
$$
Now, note that
$$
\int_{\sqrt{x}}^{x^3}\sqrt{... |
4,163,003 | <p>Let <span class="math-container">$Z \in \mathbb{R}^2$</span> be an i.i.d. Gaussian vector with mean <span class="math-container">$M$</span> where <span class="math-container">$P_{Z\mid M}$</span> is its distribution.</p>
<p>Let <span class="math-container">$g: \mathbb{R}^2 \to \mathbb{R}$</span> and consider the fol... | Luca Ghidelli | 176,416 | <p>That was a very natural question to me, and I really enjoyed the journey of exploring it!
Let me give three different answers, which for me correspond to three different levels of understanding of the problem.</p>
<p>(Major edit on June 18th, 2021)</p>
<p><strong>Three approaches</strong></p>
<p>The three answers ar... |
3,545,548 | <p><span class="math-container">$\def\LIM{\operatorname{LIM}}$</span>
Let <span class="math-container">$(X,d)$</span> be a metric space and given any cauchy sequence <span class="math-container">$(x_n)_{n=1}^{\infty}$</span> in <span class="math-container">$X$</span> we introduce the formal limit <span class="math-cont... | Boka Peer | 304,326 | <p>Without loss of generality, assume <span class="math-container">$x < y< z.$</span> Since <span class="math-container">$x, y,$</span> and <span class="math-container">$z$</span> are in AP, we must have <span class="math-container">$ y-x = z-y.$</span> Therefore, <span class="math-container">$3^{y-x} = 3^{z-y}$... |
2,791,068 | <p>The Laplace transform of a measure $\mu$ on the real line is defined by
$$f_{\mu}(s)= \int_{\mathbb{R}}e^{-st}d\mu(t), \hspace{1cm} \forall s \geqslant 0.$$
My question is ----</p>
<p>1)Does the Laplace transform of a measure (finite or infinite) always exists?</p>
<p>2)If not, can it be said that the Laplace tran... | Kavi Rama Murthy | 142,385 | <p>If $\mu$ has density $\frac 1 {\pi (1+x^{2})}$ the the Laplace transform exists only for $s=0$. </p>
|
66,068 | <p>I have a list like this. </p>
<pre><code>cdatalist = {{1., 0.898785, Failed, Failed, 50., 25., "serial"}, {1., 1.31175,1., Failed, 50., 25., "serial"}, {1., 18.8025, Failed, 0.490235, 50., 25., "serial"}, {1., 19.6628, 0.990079, Failed, 50., 25., "serial"}, {1., 39.547, Failed, Failed, 50., 25., "serial"}, {1., 39.... | wxffles | 427 | <p>Aside from all the other good answers which attack your problem from the beginning, a handy thing to remember is a way to tidy it up at the end:</p>
<pre><code>datalist //. {} :> Sequence[]
</code></pre>
<p>This makes all your empty (sub)lists go away. </p>
|
228,135 | <p>I'm working to understand the Grothendieck topology version of the Zariski topology of schemes. Explained simply, it replaces the notion of "open subschemes" with "open immersions". So instead of $U\subseteq X$, we have $U\hookrightarrow X$.</p>
<p>The intersection $U\cap V$ between two open subschemes is replaced ... | Community | -1 | <p>Maybe the following can help motivate this definition. In the case of $\mathbb{R}$-valued functions on a topological space $X$, if $\mathcal{U}$ is a <em>covering</em> sieve on $X$, so that the union of its open sets is $X$, then defining a function $\mathcal{U}$-locally on X is equivalent to defining a continuous ... |
4,059,489 | <blockquote>
<p>Let <span class="math-container">$ A, B \in M_n (\mathbb{C})$</span> such that <span class="math-container">$(A-B)^2 = A -B$</span>. Then <span class="math-container">$\mathrm{rank}(A^2 - B^2) \geq \mathrm{rank}( AB -BA)$</span>.</p>
</blockquote>
<p>I tried to apply the basic inequalities without resul... | John Bentin | 875 | <p>Write <span class="math-container">$x=1396+t$</span>. Then
<span class="math-container">$$\sqrt{1+\frac t{1388}}\:+\sqrt{1+\frac t{1389}}\:+\sqrt{1+\frac t{1390}}$$</span>
<span class="math-container">$$=\sqrt{1+\frac t8}\:+\sqrt{1+\frac t7}\:+\sqrt{1+\frac t6}.$$</span>
Clearly <span class="math-container">$t=0$</s... |
164,213 | <p>Suppose I have some list with duplicates by some condition and I want to take the duplicates and apply some function to choose which duplicate to keep. Is there an efficient way to apply this transformation?</p>
<p>To clarify here is an example. Consider a list with elements with duplicate first elements:</p>
<pre... | kglr | 125 | <p>Also</p>
<pre><code>KeyValueMap[{#, Max @ #2}&] @ Merge[Rule @@@ list, Identity]
Join @@ TakeLargestBy[Last, 1] /@ GatherBy[list, First]
Join @@ MaximalBy[#, Last, 1]& /@ GatherBy[list, First]
</code></pre>
<blockquote>
<p>{{2, 0.2},{3, 0.4},{4, 0.9},{6, 0.3}}</p>
</blockquote>
|
4,325,373 | <blockquote>
<p>Using Algebraic approach, test the convexity of the set <span class="math-container">$$S=\{(x_1,x_2):x_2^2\geq8x_1\}$$</span></p>
</blockquote>
<p>Definition of convexity: <span class="math-container">$S \in \mathbb R^2$</span> is a convex set if <span class="math-container">$\forall \alpha \in \mathbb ... | tomos | 653,050 | <p>I'm going to ignore the minus sign, the values at 1, and the <span class="math-container">$<$</span> inequality signs. But otherwise, your first sum is
\[ \sum _{de|n}\sum _{c|d}f(c)f^{-1}(e)g(d/c)g^{-1}(n/de)=\sum _{cde|n}f(c)f^{-1}(e)g(d)g^{-1}(n/cde)=\sum _{cde|n}f(c)f^{-1}(e)g(n/ced)g^{-1}(d)=\sum _{cde|n}f(... |
8,307 | <p>I have a program that gives the average distance as the output. When I tried to repeat finding the average distance 100 times using Table, It failed to generate the output. This is the program.</p>
<pre><code>Xarray = A @@@ Tuples[Range[0, 4], 3];
Table[M = RandomSample[Xarray, 7];
energies = RandomVariate[Exponent... | Verbeia | 8 | <p>I see that you are at least using the answers to your previous questions to improve your code, but I do think you need to spend a bit more time with the documentation so that you better understand what you are doing. It really doesn't make people keen to help when you come back with the same problem and ask for help... |
927,188 | <p>This question has been on my mind for a very long time, and I thought I'd finally ask it here. </p>
<p>When I was 6, my dad pulled me out of school. The classes were too easy; the professors, too dull. My father had been man of philosophy his entire life (almost got a PhD in it) and regretted not having a more q... | user164587 | 164,587 | <p>I don't think you can "know" that you aren't making a mistake. This applies to almost everything in life. What I would say is that if you want to do something, then do it. Life is too short to let worries paralyse you into doing nothing. Get into the course, focus your study on the parts that are most interesting, a... |
927,188 | <p>This question has been on my mind for a very long time, and I thought I'd finally ask it here. </p>
<p>When I was 6, my dad pulled me out of school. The classes were too easy; the professors, too dull. My father had been man of philosophy his entire life (almost got a PhD in it) and regretted not having a more q... | tess | 175,323 | <p>As the mother of a high school junior taking all AP math/physics because he wants to be able to choose whatever path he determines is right for him, I wish more teachers were passionate about teaching. It would make a world of difference if a teacher would communicate HOW math and physics are present in our world, g... |
927,188 | <p>This question has been on my mind for a very long time, and I thought I'd finally ask it here. </p>
<p>When I was 6, my dad pulled me out of school. The classes were too easy; the professors, too dull. My father had been man of philosophy his entire life (almost got a PhD in it) and regretted not having a more q... | Gahawar | 129,839 | <p>Your current situation is somewhat similar to what occurred during my adolescent years. I grew up in a rather unstable and meager lifestyle; I never had any opportunities to gain a good education. From what I can remember, I earned first place in a district mathematics competition during the fourth grade, though I d... |
2,339,707 | <p>Suppose I have five bins into which I want to place 15 balls. The bins have capacities $2$, $2$, $3$, $3$, and $7.$ I place the balls one at a time in the bins, randomly and uniformly amongst the bins that are not full (so for example, if after placing four balls, both of the bins with capacity $2$ are already full,... | amd | 265,466 | <p>Let’s untangle your notation a bit and call the eigenvector that you’re using $v$. If your approach were correct, then it should hold for all $n$. In particular, $p_0=A^0v=\lambda^0v=v$, so this clearly can’t work for an arbitrary initial distribution $p_0$.</p>
|
2,227,280 | <p>For every positive number there exists a corresponding negative number. Would that imply that the number of positive numbers is "equal" to the number of negative numbers? (Are they incomparable because they both approach infinity?)</p>
| Guillemus Callelus | 361,108 | <p>There are exactly the same ones.</p>
<p>For each positive number there is one and only one negative,</p>
<p>And for each negative number there is one and only one positive,</p>
<p>So that both sets have the same cardinal.</p>
<p>There is a bijective function of the set of positive numbers to the set of negative ... |
2,894,606 | <p>Please, could you help me with the question below.
Demonstrate that O(log n^k) = O(log n):</p>
| Empy2 | 81,790 | <p>Rearrange your equation to
$$ B=f(B)=\sqrt[S]{\frac{B-1}R+1}$$</p>
<p>You could start with a rough estimate then apply the function $f(B)$ several times until the answer settles down.
$$\begin{split}
B_1&=2\\
B_2&=f(B_1)\\
B_3&=f(B_2)\\
B_4&=f(B_3)\\
B_5&=f(B_4)\\
&\hspace{0.55em}\vdots
\end... |
2,409,268 | <p><strong>Confirm that the identity $1+z+...+z^n=(1-z^{n+1})/(1-z)$ holds for every non-negaive integer $n$ and every complex number $z$, save for $z=1$</strong></p>
<p>I have tried to prove this by induction but I am not sure that I am doing things right, for $ n = 1 $ we have $ (1-z ^ 2) / (1-z) = (1-z) (1+ z) / (1... | BAI | 448,487 | <p>My proof goes like this:
Let $S:=1+z+z^2+...+z^n$
Then
$$zS=z+z^2+z^3+...+z^{n+1}$$
$$zS=S-1+z^{n+1}$$
$$(1-z)S=1-z^{n+1}$$
As $z\neq 1$, it gives the desired result.</p>
|
298,791 | <blockquote>
<p>If a ring $R$ is commutative, I don't understand why if $A, B \in R^{n \times n}$, $AB=1$ means that $BA=1$, i.e., $R^{n \times n}$ is Dedekind finite.</p>
</blockquote>
<p>Arguing with determinant seems to be wrong, although $\det(AB)=\det(BA ) =1$ but it necessarily doesn't mean that $BA =1$.</p>
... | Theon Alexander | 165,460 | <p>Here's a slightly different argument.</p>
<p>The poster's initial argument was by using determinants, which as said above, implies that both $\det(A), \det(B) \in R^\times.$ Hence, both matrices have bilateral inverses, i.e. $\operatorname{adj}(A^T) \det(A)^{-1}$ and resp. for $B$ - recall that the formula
$$A\op... |
1,986,798 | <p>The way I solved the problem is to change the equation to $|x+2|=1-|y-3|$, and then square both sides. But I don't think it is the right way to solve the problem. I hope someone can either give me a hint or show me how to solve the problem.</p>
<blockquote>
<p>$|x+2|+|y-3|=1$ is an equation for a square. How many... | Not a grad student | 36,274 | <p>Yes, that is the Jacobian matrix. Given any differentiable function $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$, the Jacobian matrix is an $m \times n$ matrix.</p>
|
132,266 | <p>I apologize if this question is trivial, but a couple of days of searching for necessary routines have led me here. </p>
<blockquote>
<p>Does there exist software to compute symmetric powers of Schur polynomials? </p>
</blockquote>
<p>I am seeking such software in the hopes of computing the characters of represe... | F. C. | 10,881 | <p>this could be done in sage:</p>
<pre><code>sage: B3 = WeylCharacterRing("B3", style="coroots")
sage: spin = B3(0,0,1)
sage: spin.symmetric_power(6)
B3(0,0,0) + B3(0,0,2) + B3(0,0,4) + B3(0,0,6)
sage: A3 = WeylCharacterRing("A3", style="coroots")
sage: rep = A3(0,0,1)
sage: rep.symmetric_power(6) ... |
132,266 | <p>I apologize if this question is trivial, but a couple of days of searching for necessary routines have led me here. </p>
<blockquote>
<p>Does there exist software to compute symmetric powers of Schur polynomials? </p>
</blockquote>
<p>I am seeking such software in the hopes of computing the characters of represe... | Steven Sam | 321 | <p>This can be done with LiE:
<a href="http://wwwmathlabo.univ-poitiers.fr/~maavl/LiE/" rel="noreferrer">http://wwwmathlabo.univ-poitiers.fr/~maavl/LiE/</a></p>
<p>(In fact it will compute the Schur functor of any irreducible representation.) There is a form interface so you can try LiE on the web: <a href="http://www... |
4,013,065 | <p>I need help interpreting the answer to a question about the base and dimension of a subspace within linear algebra. I have a subspace W of <span class="math-container">$R^5$</span> that is spanned by the vectors:</p>
<p><span class="math-container">$${v_1}=\begin{pmatrix} 1 \\ 2 \\ 3 \\ -1 \\ 1 \end{pmatrix} , {v_2... | Bernard | 202,857 | <p>I would do that transposing, and performing row operations, to make the whole process:
<span class="math-container">\begin{align}
&\begin{bmatrix}
1&2&3&-1&1\cr 0&1&1&-1&2\cr -1&-1&-2&1&2\cr 1&0&1&0&-4
\end{bmatrix}\xrightarrow[v_4\leftarrow v_4-v_... |
986,412 | <p>Let $f(x) = \frac1{(1-x)}$.</p>
<p>Define the function $f^r$ to be $f^r(x) = f(f(f(...f(f(x)))))$.</p>
<p>Find $f^{653}(56)$.</p>
<p><strong>What I've done:</strong> </p>
<p>I started with r=1,2,3 and noticed the following pattern:
$$f^r(x)= \left\{
\begin{array}{c}
\frac1{1-x}, when \ r\equiv 1\pmod 3 \\
\fr... | Vinícius Novelli | 148,344 | <p>You can just calculate the improper integral first:</p>
<p>$$
\int \frac{dx}{1-x^2}=\frac{1}{2}\int{\frac{dx}{1-x}}+\frac{1}{2}\int{\frac{dx}{1+x}} = -\frac{1}{2}\ln|1-x|+\frac{1}{2}\ln|1+x|+C
$$</p>
<p>which holds for $|x|\not= 1$. Given that in the interval $(0,\pi /2)$, $\sin$ or $\cos$ are never $1$ or $-1$, y... |
2,081,792 | <p>I learnt the derivation of the distance formula of two points in first quadrant I.e., $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ where it is easy to find the legs of the hypotenuse (distance between two points) since the first has no negative coordinates and only two axes ($x$ coordinate and $y$ coordinate). while f... | Christopher.L | 347,503 | <p>All you have to realize is that 'crossing' an axis does not change the distance. Try first to just realize this in one dimension, i.e the distance between two points on the real line:</p>
<p>$d(x_1,x_2)=|x_1-x_2|=\sqrt{(x_1-x_2)^2}$.</p>
<p>Take one point $x_1$, on the negative side, and another $x_2$, on the posi... |
2,242,846 | <p>I will try to prove the theorem in the title:</p>
<blockquote>
<p>Suppose S is closed, non-empty then if $b = \sup\{x: x \in S\}$ (least upper bound), $b \in S$.</p>
</blockquote>
<p>I also use the following <strong>Theorem</strong> which we proved in class: S is closed iff every Cauchy sequence in S converges t... | caozi | 581,222 | <p>Here is my solution:</p>
<p>$1.~\exists X~\lnot p(X)\quad $ Premise</p>
<p>$2.~\quad \lnot p(c)\quad$ Existential Elimination:1, Witness Assumed: $[c]$</p>
<p>$3.~\qquad\forall X~p(X)\quad$ Assumption</p>
<p>$4.~\qquad p(c)\quad$ Universal Elimination:3</p>
<p>$5.~\quad \forall X~p(X) \implies p(c)\... |
1,204,566 | <p>I tried asking this on StackOverflow and it was quickly closed for being too broad, so I come here to get the mathematical part nailed down, and then I can do the rest with no help, most likely.</p>
<p>From <a href="http://www.afjarvis.staff.shef.ac.uk/sudoku/sudgroup.html" rel="nofollow">this web page</a>, I learn... | Nick Gill | 211,225 | <p>In response to the OP's request that I post an answer:</p>
<p>I believe that Russell & Jarvis have an explicit list of representatives for the 5.4 billion essentially different types of sudoko grid.</p>
<p>The credit for the answer should go to my office mate, Sian Jones.</p>
|
69,137 | <p>Is there any reference for gluing in the context of Morse homology on Hilbert manifolds?</p>
<p>Gluing is pretty standard in Morse homology for finite-dimensional manifolds. Unfortunately, in the infinite-dimensional case the sources I know avoid gluing. For proving that the Morse boundary operator squares to zero ... | Sesshoumaru | 20,893 | <p>I would say, take a look at the paper of Abbondandolo and Majer :<a href="http://arxiv.org/abs/math/0403552" rel="nofollow">http://arxiv.org/abs/math/0403552</a> or <a href="http://www.dm.unipi.it/~abbondandolo/preprints/montreal.pdf" rel="nofollow">http://www.dm.unipi.it/~abbondandolo/preprints/montreal.pdf</a>
Th... |
1,561,563 | <p>Two circles $\Gamma_1,\Gamma_2$ have centers $O_1,O_2$. Let $\Gamma_1\cap\Gamma_2=A,B$, with $A\neq B$. An arbitrary line through $B$ intersects $\Gamma_1$ at $C$ and $\Gamma_2$ at $D$. The tangents to $\Gamma_1$ at $C$ and to $\Gamma_2$ at $D$ intersect at $M$. Let $N=AM\cap CD$. Let $l$ be a line through $N$ paral... | Kay K. | 292,333 | <p>Finding indefinite integral:</p>
<p>\begin{align}
&x\mapsto\sin u\\
&I=\int \frac{\cos u}{\sin u + \cos u}du\\
&=\frac12\int \frac{\cos u - \sin u + \cos u + \sin u}{\sin u + \cos u}du=\frac12 \int\frac{\cos u-\sin u}{\sin u + \cos u}du+\frac u2\\
&=\frac12 \ln (\cos u + \sin u) + \frac u2+C\\
&... |
1,499,949 | <p>Prove that for all event $A,B$</p>
<p>$P(A\cap B)+P(A\cap \bar B)=P(A)$</p>
<p><strong>My attempt:</strong></p>
<p>Formula: $\color{blue}{P(A\cap B)=P(A)+P(B)-P(A\cup B)}$</p>
<p>$=\overbrace {P(A)+P(B)-P(A\cup B)}^{=P(A\cap B)}+\overbrace {P(A)+P(\bar B)-P(A\cup \bar B}^{=P(A\cap \bar B)})$</p>
<p>$=2P(A)+\un... | Eric Wofsey | 86,856 | <p>There is no universal convention about the meaning of these notations in general. When $x$ is nonnegative, it is fairly common to say that both $x^{1/n}$ and $\sqrt[n]{x}$ refer to the unique nonnegative $n$th root of $x$. Some authors may use other conventions (such as the one mentioned in kilimanjaro's answer, i... |
83,512 | <p>Question: (From an Introduction to Convex Polytopes)</p>
<p>Let $(x_{1},...,x_{n})$ be an $n$-family of points from $\mathbb{R}^d$, where $x_{i} = (\alpha_{1i},...,\alpha_{di})$, and $\bar{x_{i}} =(1,\alpha_{1i},...,\alpha_{di})$, where $i=1,...,n$. Show that the $n$-family $(x_{1},...,x_{n})$ is affinely independe... | Alex Becker | 8,173 | <p>($\Rightarrow$): Suppose $(\bar{x_1},\ldots,\bar{x_n})$ is linearly dependent, so we have $$\left(\sum\limits_{i=1}^n c_i,\sum\limits_{i=1}^n c_i\alpha_{1i},\ldots,\sum\limits_{i=1}^n c_i\alpha_{ni}\right) = \sum\limits_{i=1}^nc_i\bar{x_i} = 0$$
for some set of coefficients $c_i\in\mathbb{R}$, thus
$$\sum\limits_{i=... |
596,005 | <p>Show that $f:\mathbb{R}^2\to\mathbb{R}$, $f \in C^{2}$ satisfies the equation
$$\frac{\partial^2 f}{\partial x^2} - \frac{\partial^2 f}{\partial y^2} = 0$$
for all points $(x,y) \in \mathbb{R}^2$ if and only if for all $(x,y)\in \mathbb{R}^2$ and $t \in \mathbb{R}$ we have:
$$f(x, y + 2t) + f(x, y) = f(x + t,y + t) ... | alexjo | 103,399 | <p>The PDE $f_{xx}-f_{yy}=0$ is the <a href="http://mathworld.wolfram.com/WaveEquation1-Dimensional.html" rel="nofollow">wave equation</a> with general solution
$$
f(x,y)=F(x-y)+G(x+y)
$$
where $F$ and $G$ are any functions.
Forall $t\in \Bbb R$ the function $f(x,y)=F(x-y)+G(x+y)$ satisfies $$f(x, y + 2t) + f(x, y) = f... |
537,228 | <p>I know some things about measures/probabilities and I know some things about categories. Shortly I realized that uptil now I have never encountered something as a category of measure spaces. It seems quite likely to me that something like that can be constructed. I am an amateur however and my scope is small. I have... | user43208 | 43,208 | <p>I'm just going to record some references that you might want to take a look at. They might be a little too heavy on the category theory for your taste, but they also might provide further places to look. </p>
<ul>
<li><p>$n$-Category Café post: <a href="http://golem.ph.utexas.edu/category/2007/02/category_th... |
3,988,808 | <p>I recently got into set theory and i was wondering what is the cardinality of a set of all finite sequences of natural numbers?
I know that it is N for natural numbers and 2^N is for real numbers but how can i prove it?</p>
| user247327 | 247,327 | <p>Every finite sequence can be thought of as the digits in a terminating decimal, a subset of the rational numbers, so the set of all finite sequences of natural numbers is countably infinite.</p>
|
3,988,808 | <p>I recently got into set theory and i was wondering what is the cardinality of a set of all finite sequences of natural numbers?
I know that it is N for natural numbers and 2^N is for real numbers but how can i prove it?</p>
| Gribouillis | 398,505 | <p>Take a finite sequence of natural numbers such as <span class="math-container">$764\ 32\ 87\ 12\ 922$</span>. Write them in base <span class="math-container">$10$</span> and replace the space between the numbers with the letter 'a', giving <span class="math-container">$764a32a87a12a922$</span>. Interprete this as a ... |
1,085,491 | <p>Prove that the following number is an integer:
$$\left( \dfrac{76}{\dfrac{1}{\sqrt[\large{3}]{77}-\sqrt[\large{3}]{75}}-\sqrt[\large{3}]{5775}}+\dfrac{1}{\dfrac{76}{\sqrt[\large{3}]{77}+\sqrt[\large{3}]{75}}+\sqrt[\large{3}]{5775}}\right)^{\large{3}}$$</p>
<p>How can I prove it?</p>
| Peđa | 15,660 | <p><strong>Hint :</strong></p>
<p>$\sqrt[3]{75}=a$</p>
<p>$\sqrt[3]{77}=b$</p>
<p>$\sqrt[3]{5775}=ab$</p>
<p>$(a^3+b^3)/2=76$</p>
<p>Next , try to simplify expression .</p>
|
1,085,491 | <p>Prove that the following number is an integer:
$$\left( \dfrac{76}{\dfrac{1}{\sqrt[\large{3}]{77}-\sqrt[\large{3}]{75}}-\sqrt[\large{3}]{5775}}+\dfrac{1}{\dfrac{76}{\sqrt[\large{3}]{77}+\sqrt[\large{3}]{75}}+\sqrt[\large{3}]{5775}}\right)^{\large{3}}$$</p>
<p>How can I prove it?</p>
| Nick Matteo | 59,435 | <p>Following the simplifications suggested by MathBot, we have:
$$\left( \dfrac{76}{\dfrac{1}{b-a}-ab}+\dfrac{1}{\dfrac{76}{b+a}+ab}\right)^{\large{3}}$$
Let's just take the part inside the parentheses, and put it over a common denominator.
$$\dfrac{\dfrac{76^2}{b+a} + 76 ab + \dfrac{1}{b-a} - ab}{\left(\dfrac{1}{b-a}-... |
2,203,988 | <p>I'm reading the book <i>Heat Transfer</i> by J.P. Holman. On the chapter of unsteady-state conduction, page 140, the author remarks:</p>
<blockquote>
<p>The final series solution is therefore:
$${\theta(x,t) \over \theta_i} =
{4\over \pi} \sum^{\infty}_{n=1} {1\over n} e^{-\left({n\pi/L}\right)^2\alpha \,t}\si... | fleablood | 280,126 | <p>If you're like me the problem is finding a strategy. </p>
<p>And the strategy that figuring each ring has $3$ choices of fingers so there are $3^8$ ways to choose fingers for each ring, is a dead end as order matters and we can't straightforwardly multiply by any choice or permutation value for each finger as we ... |
3,125,093 | <p>Let us remember, the conditions to apply L'Hôpital's Rule:</p>
<p>Let suppose:</p>
<p><span class="math-container">$f(x)$</span> and <span class="math-container">$g(x)$</span> are real and differentiable for all <span class="math-container">$x\in (a,b)$</span> </p>
<p>1-) <span class="math-container">$ \lim_{x\t... | Somos | 438,089 | <p>For this particular group presentation there is a simple way to use cancellation to identify it.
First define <span class="math-container">$\,u := ii = jj = kk = ijk\,$</span> which commutes with <span class="math-container">$\,i,j,k.\,$</span> Now <span class="math-container">$\,(ij)k = u = kk\,$</span> and using c... |
293,026 | <p>The question is to show that $A\sin(x + B)$ can be written as $a\sin x + b\cos x$ for suitable a and b.</p>
<p>Also, could somebody please show me how $f(x)=A\sin(x+B)$ satisfies $f + f ''=0$?</p>
| Ross Millikan | 1,827 | <p>Given $f(x)=\sin(x+B)$ the chain rule gives $f'(x)=(x+B)'\cos(x+B)=\cos (x+B)$. Then another derivative gives $f''(x)=-\sin (x+B)$</p>
|
2,450,007 | <p>Show that if $x\in Q^p$, then there exists $-x\in Q^p$ where $$Q^p=\{a_{-l}p^{-l}+a_{-l+1}p^{-l+1}+...|l\in Z,a_i\in\{0,1,...,p-1\}\}$$ and p is a prime number.</p>
<p>Actually I don't quite understand p-adic numbers and how addition and multiplication work in this number system. For this question, I think I need ... | Henno Brandsma | 4,280 | <p>In fact almost any uniform space is an example: a Tychonoff (so uniformisable) space $X$ is called "almost compact" if there is a unique uniformity that induces its topology. It turns out the there are the spaces $X$ where $\beta(X)\setminus X$ has at most one point. It includes the compact (Hausdorff) spaces, and ... |
656,560 | <p>I'm trying to get a solution for:</p>
<p>$4^{2x+1}-3^{3x+1}=4^{2x+3}-3^{3x+2}$</p>
<p>My main problem is that I don't know how to combine this potencys!</p>
<p>Ive also thought about another function that would bring me same difficulties:</p>
<p>$6^x=36*9.75^{x-2}$</p>
<p>What am I supposed to do? </p>
| Adi Dani | 12,848 | <p>$$4^{2x+1}-3^{3x+1}=4^{2x+3}-3^{3x+2}$$
$$4^{2x+1}-4^{2x+3}=3^{3x+1}-3^{3x+2}$$
$$4\cdot4^{2x}-4^34^{2x}=3\cdot3^{3x}-3^23^{3x}$$
$$60\cdot4^{2x}=6\cdot3^{3x}$$
$$10\cdot4^{2x}=3^{3x}$$
$$10\cdot16^{x}=27^{x}$$
$$10=(27/16)^{x}$$
$$\log_{10} 10=\log_{10} (27/16)^{x}$$
$$1=x\log_{10}(27/16)$$
$$x=\frac{1}{\log_{10}27... |
1,327,644 | <p>Using EM summation formula estimate
$$
\sum_{k=1}^n \sqrt k
$$</p>
<p>up to the term involving $\frac{1}{\sqrt n}$</p>
<p>My attempt is
$$
\sum_{k=1}^n \sqrt k = \frac{2 \sqrt{n^3}}{3} -\frac{2}{3} + \frac 1 2 (\sqrt n -1)+ \frac{1}{24} (\frac{1}{\sqrt n} -1) + \int_1^n P_{2k+1}(x)f^{(2k+1)}(x)dx
$$
I am not s... | robjohn | 13,854 | <p>Using the Euler-Maclaurin Sum Formula, we get
<span class="math-container">$$
\sum_{k=1}^n\sqrt{k}=\tfrac23n^{3/2}+\tfrac12n^{1/2}+\zeta(-\tfrac12)+\tfrac1{24}n^{-1/2}-\tfrac1{1920}n^{-5/2}+\tfrac1{9216}n^{-9/2}+O(n^{-13/2})
$$</span>
where the constant <span class="math-container">$\zeta(-\frac12)=-0.20788622497735... |
1,575,397 | <p>I need help calculating
$$\lim_{n\to\infty}\left(\frac{1}{n^{2}}+\frac{2}{n^{2}}+...+\frac{n}{n^{2}}\right) = ?$$</p>
| Jimmy R. | 128,037 | <p>$$1+2+\ldots+n=\frac{n(n+1)}{2}$$ which implies that $$\frac1{n^2}+\frac2{n^2}+\ldots+\frac{n}{n^2}=\frac{n(n+1)}{2n^2}=\frac{n^2+n}{2n^2}=\frac12+\frac{1}{2n}$$</p>
|
1,526,474 | <p>Find the natural number $k <117$ such that $2^{117}\equiv k \pmod {117}$.</p>
<p>I know $117$ is the product of $3$ and $37$.</p>
<p>$2^{117}\equiv 2 \pmod 3$
$2^{117}\equiv 31 \pmod {37}$.
But $2^{117}\equiv 44 \pmod {117}$.</p>
<p>I can't seem to understand how to get $44$. Can anyone help me understand?</p... | Piquito | 219,998 | <p>Remark that $8^4=4096=35\cdot117+1$ Hence $2^{117}=2^{3\cdot39}=8^{39}=8^{4\cdot9+3}\equiv(1)^9\cdot8^3$(mod$\space 117)$</p>
<p>Therefore $2^{117}\equiv(1)^9\cdot 512$ (mod$\space 117)\equiv(1)^9( 4\cdot117+44)$ (mod$\space 117)\equiv 44$ (mod$\space 117)
$
The answer is $k=44$</p>
|
4,292,618 | <p>I have the following function <span class="math-container">$$\frac{1}{1+2x}-\frac{1-x}{1+x} $$</span>
How to find equivalent way to compute it but when <span class="math-container">$x$</span> is much smaller than 1? I assume the problem here is with <span class="math-container">$1+x$</span> since it probably would b... | Bobby Laspy | 986,676 | <p>You can replace the fractions by geometric progressions, giving</p>
<p><span class="math-container">$$\sum_{k=0}^\infty(-2x)^k-\sum_{k=0}^\infty(-x)^k-\sum_{k=0}^\infty(-x)^{k+1}=
\sum_{k=2}^\infty(-1)^k(2^k-2)x^k\\
=2x^2-6x^3+14x^4-30x^5\cdots.$$</span> Now truncate these sums to some power.</p>
|
1,071,040 | <p>I found <a href="https://math.stackexchange.com/questions/549065/how-exactly-do-you-measure-circumference-or-diameter">How exactly do you measure circumference or diameter?</a> but it was more related to how people measured circumference and diameter in old days.</p>
<p><strong>BUT</strong> now we have a formula, b... | Mietek | 420,053 | <p>Construct the circle of diameter 1/pi.
The circle has circumference =1.
It’s sure exact value.
(You don't escape from pi - it's property of each circle).</p>
|
11,994 | <p>Now that we get to see the SE-network wide list of "hot" questions, I am just shaking my head in disbelief. At the time I am writing this, the two hot questions from Math.SE are titled (get a barf-bag, quick)</p>
<ul>
<li><a href="https://math.stackexchange.com/q/599520/8348">https://math.stackexchange.com/q/599520... | Community | -1 | <p>I've criticized the hot questions algorithm quite heavily on MSO in the past, it does favor certain types of questions without a good reason, and it has a strong self-reinforcing effect where questions that manage to get into the hot questions list get disproportionally more votes, which keeps them there longer.</p>... |
1,992,143 | <p>I'm trying to determine if $\sum \limits_{n=1}^{\infty} \sin(n\pi + \frac{1}{2n})$ absolutely converges or not.</p>
<p>Help me check it. I don't know how to do it. Advance thanks. :)</p>
| DonAntonio | 31,254 | <p>Another approach, perhaps simpler but I think that shorter anyway: the limit comparison theorem with $\;b_n=\frac1{2n}\;$ , then</p>
<p>$$\frac{a_n}{b_n}=\frac{\sin\frac1{2n}}{\frac1{2n}}\xrightarrow[n\to\infty]{}1$$</p>
<p>and thus the series $\;\sum\sin\frac1{2n}\;,\;\;\sum\frac1{2n}\;$ converge or diverge toget... |
749,714 | <p>Does anyone know how to show this preferable <strong>without</strong> using modular</p>
<p>For any prime $p>3$ show that 3 divides $2p^2+1$ </p>
| Community | -1 | <p>For any prime $p>3$ we have $p\equiv 1\mod 3$ or $p\equiv 2\mod 3$ and in the two cases we have
$$2p^2+1\equiv 0\mod 3$$</p>
|
1,980,606 | <p>Let $f:[0, 1] \to \mathbb{R}$ differentiable in $[0, 1]$ and $|f'(x)| \leq\frac{1}{2}$ for all $x \in [0, 1]$. If $a_n = f(\frac{1}{n})$, show that $\lim_{n \to \infty} a_n$ exist (Hint: Cauchy).</p>
<p>Can you help me? Thanks.</p>
| hamam_Abdallah | 369,188 | <p>for every nonzero $n$ and $p$ ,</p>
<p>$|a_{n+p}-a_n|\leq\frac{1}{2}(\frac{1}{n}-\frac{1}{n+p})\leq\frac{1}{2n}$</p>
<p>so $(a_n)$ is a Cauchy sequence.</p>
|
4,941 | <p>I was reviewing my class notes and found the following:</p>
<p>"The name 'torsion' comes from topology and refers to spaces that are twisted, ex. Möbius band"</p>
<p>In our notes we used the following definition for torsion element and torsion module:
An element m of an R-module M is called a torsion element if $r... | Qiaochu Yuan | 232 | <p>The definition of torsion in modules is a generalization of the definition of torsion in $\mathbb{Z}$-modules, e.g. abelian groups. Torsion in abelian groups refers to elements of finite order, and this in turn relates to topology because to any topological space we can associate abelian groups called (integral) <a... |
2,258,697 | <p>I recently encountered this question and have been stuck for a while. Any help would be appreciated!</p>
<p>Q: Given that
$$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{5} \tag{1} \label{eq:1}$$
$$abc = 5 \tag{2} \label{eq:2}$$
Find $a^3 + b^3 + c^3$. It wasn't specified in the question but I think it can be... | spaceisdarkgreen | 397,125 | <p>This matrix has a simple block form $$ \begin{pmatrix}I&I\\I&I\end{pmatrix}$$ where $I$ is the $2\times 2$ identity, so you can eyeball the eigenvalues of the $2\times 2$ all-ones matrix (which are $2$ and $0$) and then realize that they will both contribute twice since each eigenvector of this matrix ($(1,1... |
3,172,008 | <p>Let <span class="math-container">$n \in \mathbb{N}$</span> be the index of sequence <span class="math-container">$\{\frac{1}{n^2}\}_{n=1}^\infty$</span>.</p>
<p>I'm assuming that:
<span class="math-container">$$\displaystyle{\lim _{n\to\infty}} \frac{1}{n^2} = \frac{\displaystyle{\lim _{n\to\infty}} 1}{\displaystyl... | Thorgott | 422,019 | <p>The identity
<span class="math-container">$$\lim_{n\rightarrow\infty}\frac{a_n}{b_n}=\frac{\lim\limits_{n\rightarrow\infty}a_n}{\lim\limits_{n\rightarrow\infty}b_n}$$</span>
does not hold when the limits on the RHS don't exist (or when the limit in the denominator would be <span class="math-container">$0$</span>). T... |
3,172,008 | <p>Let <span class="math-container">$n \in \mathbb{N}$</span> be the index of sequence <span class="math-container">$\{\frac{1}{n^2}\}_{n=1}^\infty$</span>.</p>
<p>I'm assuming that:
<span class="math-container">$$\displaystyle{\lim _{n\to\infty}} \frac{1}{n^2} = \frac{\displaystyle{\lim _{n\to\infty}} 1}{\displaystyl... | fleablood | 280,126 | <p>Just do delta-epsilon. There's no trick. </p>
<p>For any <span class="math-container">$\epsilon> 0$</span> let <span class="math-container">$N \ge \frac 1{\sqrt{\epsilon}}$</span>.</p>
<p>If <span class="math-container">$n > N$</span> then <span class="math-container">$|\frac 1{n^2} - 0| =\frac 1{n^2} <... |
2,076,908 | <blockquote>
<p><strong>Question:</strong> Prove that $e^x, xe^x,$ and $x^2e^x$ are linearly independent over $\mathbb{R}$.</p>
</blockquote>
<p>Generally we proceed by setting up the equation
$$a_1e^x + a_2xe^x+a_3x^2e^x=0_f,$$
which simplifies to $$e^x(a_1+a_2x+a_3x^2)=0_f,$$ and furthermore to
$$a_1+a_2x+a_3x^2=... | symplectomorphic | 23,611 | <p>Though it is overkill and not really in the spirit of the problem (see kobe's answer for the most elementary approach), you can use the fundamental theorem of algebra (or the <a href="https://math.stackexchange.com/questions/25822/how-to-prove-that-a-polynomial-of-degree-n-has-at-most-n-roots">weaker result</a> that... |
1,549,138 | <p>I have a problem with this exercise:</p>
<p>Proove that if $R$ is a reflexive and transitive relation then $R^n=R$ for each $n \ge 1$ (where $R^n \equiv \underbrace {R \times R \times R \times \cdots \times R} _{n \ \text{times}}$).</p>
<p>This exercise comes from my logic excercise book. The problem is that I've ... | Mankind | 207,432 | <p>I think you got the definition of $R^2$ wrong. Here is the correct definition:</p>
<p>Let $R$ be a relation on the set $A$. Then $R^2$ is defined by
$$R^2 = \{(x,y)\ |\ \exists z\in A\text{ such that } (x,y)\in R\text{ and }(y,z)\in R\}.$$</p>
<p>So $R^2$ is not a set of ordered pairs of elements from $R$. It is a... |
241,903 | <p>Suppose $f: \mathbb{D}\to \mathbb{C}$ is a univalent function with $$f(z)=z+a_2z^2+a_3z^3+\cdots.$$ The Bieberbach conjecture/de Branges' theorem asserts that $|a_n|\leq n$ with equality for the Koebe function, which has an unbounded image. Suppose we restrict to the class of univalent functions whose image is actua... | Lasse Rempe | 3,651 | <p>This is a softer answer than Alex's, in line with Christian's comment to that answer.</p>
<p>If you are looking at the family of all bounded conformal maps, then you clearly do <strong>not</strong> get a better bound, as the Koebe function can be approximated by such.</p>
<p>On the other hand, if you look at all ... |
53,188 | <p>Recently I read the chapter "Doctrines in Categorical Logic" by Kock, and Reyes in the Handbook of Mathematical Logic. And I was quite impressed with the entire chapter. However it is very short, and considering that this copy was published in 1977, possibly a bit out of date. </p>
<p>My curiosity has been sparked ... | Harry Gindi | 1,353 | <p>Just as technical texts, <em>Acessible Categories</em> by Makkai and Paré and <em>Locally Presentable and Accessible Categories</em> by Adamek and Rosicky are extremely useful even for non-logicians (they are a natural extension of the material in SGA4.1.i on colimits of functors indexed by "ensembles ordonné grand ... |
53,188 | <p>Recently I read the chapter "Doctrines in Categorical Logic" by Kock, and Reyes in the Handbook of Mathematical Logic. And I was quite impressed with the entire chapter. However it is very short, and considering that this copy was published in 1977, possibly a bit out of date. </p>
<p>My curiosity has been sparked ... | Chris Heunen | 10,368 | <p>I'd like to add Carsten Butz' "<a href="http://www.brics.dk/LS/98/2/" rel="nofollow">Regular categories and regular logic</a>". It is available online in the BRICS lecture series, and is very accessible. Though perhaps a bit too basic at times for readers with some background knowledge, it is very suitable for a fir... |
53,188 | <p>Recently I read the chapter "Doctrines in Categorical Logic" by Kock, and Reyes in the Handbook of Mathematical Logic. And I was quite impressed with the entire chapter. However it is very short, and considering that this copy was published in 1977, possibly a bit out of date. </p>
<p>My curiosity has been sparked ... | none | 13,636 | <p><a href="http://en.wikipedia.org/wiki/Categorical_logic" rel="nofollow">http://en.wikipedia.org/wiki/Categorical_logic</a> has more refs. That's a sufficiently obvious place to look that maybe someone can move this "answer" to a comment.</p>
|
1,891,831 | <p>In linear algebra we have vectors:$$
\mathbf{A}=(x,y,z)=x\mathbf{\hat e}_x+y\mathbf{\hat e}_y+z\mathbf{\hat e}_z$$
We have vector algebra, i.e. <a href="https://en.wikipedia.org/wiki/Euclidean_vector#Basic_properties" rel="nofollow">vector addition, dot product, lines, planes, etc</a>. A vector have a magnitude and ... | Bib-lost | 294,785 | <p>Let's start with vectors in linear algebra. I don't know how abstract your linear algebra class was, but in essence linear algebra focusses on the algebraic aspect of vectors: they live in a structure called a <a href="https://en.wikipedia.org/wiki/Vector_space" rel="nofollow">vector space</a> and can be added, subt... |
383,037 | <p>I was going through "Convergence of Probability Measures" by Patrick Billingsley. In Section 1: I encountered the following problem:</p>
<p><strong>Show that inequivalent metrics can give rise to the same class of Borel sets.</strong></p>
<p>My idea is that the 2 metrics generate different topologies but the Sigma... | joriki | 6,622 | <p>An example is given by the real line, and the real line with the origin replaced by an isolated point.</p>
|
785,188 | <p>I found a very simple algorithm that draws values from a Poisson distribution from <a href="http://www.akira.ruc.dk/~keld/research/javasimulation/javasimulation-1.1/docs/report.pdf" rel="nofollow">this project.</a></p>
<p>The algorithm's code in Java is:</p>
<pre><code>public final int poisson(double a) {
... | PA6OTA | 127,690 | <p>It is related to the Poisson process: suppose $a$ is fixed and $N$ is the number of independent Exponential (mean 1) RV's to be added until the sum exceeds $a$. In this case, $N \sim Poisson(a)$. </p>
<p>In the code above, everything is anti-logged. For example, instead of adding Exponentials, they multiply Uniform... |
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