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 |
|---|---|---|---|---|
1,707,675 | <p>How can I find the indefinite integral which is $$\int \frac{\ln(1-x)}{x}\text{d}x$$</p>
<p>I tried to use substitution by assigning $$\ln(1-x)\text{d}x = \text{d}v $$ and $$\frac{1}{x}=u$$ but, it is meaningless but true, the only thing I came up from integration by part is that $$\int \frac{\ln(1-x)}{x^2}\text{d... | Robert Israel | 8,508 | <p>This is a "well-known" special function: $$\int \dfrac{\ln(1-x)}{x} \; dx = - \text{dilog}(1-x) $$
It is (provably) not an elementary function. In particular, there is no closed-form expression for it in terms of the functions familiar to the typical calculus student.</p>
|
97,131 | <p>I have the following problem:</p>
<p>I have a convex hull $\Omega$ defined by a set of n-dimensional hyperplanes $S = [(n_1,d_1), (n_2,d_2),...,(n_k,d_k)]$ such that a point $p \in \Omega$ if $n_i^T p \geq d_i \quad \forall (n_i,d_i) \in S $. Now I have a "joining" hyperplane $(n_{k+1},d_{k+1})$ and I want to know ... | Yoav Kallus | 20,186 | <p>There are two ways to represent convex polytopes: as the convex hull of its vertices, or as the intersection of the half-spaces whose boundaries contain the faces. If you store both of these representations, checking if a new constraint is redundant is easy: if all current vertices satisfy it, then so do all points ... |
48,746 | <p>Let's assume that I have some particular signal on the finite time interval which is described by function <span class="math-container">$f(t)$</span>. It could be, for instance, a rectangular pulse with amplitude <span class="math-container">$a$</span> and period T; Gauss function with <span class="math-container">$... | Community | -1 | <p>Sounds like what you actually need after your edit is a way to smooth a list of data while keeping the endpoints fixed. Here's a dumb approach that will work with any "symmetrical" smoothing filter, including <code>GaussianFilter</code>, <code>MeanFilter</code>, even <code>MedianFilter</code>. It won't work with <co... |
3,418,810 | <p>So my question is what does it mean to be <span class="math-container">$0$</span> in <span class="math-container">$S^{-1} M$</span>, where <span class="math-container">$S$</span> is a multi-closed subset of a ring <span class="math-container">$A$</span>, <span class="math-container">$M$</span>, lets assume to be a f... | egreg | 62,967 | <p>The idea is to make equivalence classes from pairs <span class="math-container">$(m,s)$</span> with <span class="math-container">$m\in M$</span> and <span class="math-container">$s\in S$</span> and denote the equivalence class of <span class="math-container">$(m,s)$</span> by <span class="math-container">$m/s$</span... |
4,351,497 | <p>Northcott Multilinear Algebra poses a problem. Consider R-modules <span class="math-container">$M_1, \ldots, M_p$</span>, <span class="math-container">$M$</span> and <span class="math-container">$N$</span>. Consider multilinear mapping</p>
<p><span class="math-container">$$
\psi: M_1 \times \ldots \times M_p \righta... | Jan Eerland | 226,665 | <p>Well, we are trying to find:</p>
<p><span class="math-container">$$\text{y}_\text{k}\left(\text{n}\space;x\right):=\mathscr{L}_\text{s}^{-1}\left[-\sqrt{\frac{\text{k}}{\text{s}}}\cdot\exp\left(-\text{n}\cdot\sqrt{\frac{\text{s}}{\text{k}}}\right)\right]_{\left(x\right)}\tag1$$</span></p>
<p>Using the linearity of t... |
194,191 | <p>Test the convergence of $\int_{0}^{1}\frac{\sin(1/x)}{\sqrt{x}}dx$</p>
<p><strong>What I did</strong></p>
<ol>
<li>Expanded sin (1/x) as per Maclaurin Series</li>
<li>Divided by $\sqrt{x}$</li>
<li>Integrate</li>
<li>Putting the limits of 1 and h, where h tends to zero</li>
</ol>
<p>So after step 3, I get somethi... | DonAntonio | 31,254 | <p>$$y:=\frac{1}{x}\Longrightarrow dy=-\frac{dx}{x^2}\Longrightarrow \int_0^1\frac{\sin 1/x}{x}\,dx=\int_\infty^1\frac{\sin y}{1/y}\left(-\frac{dy}{y^2}\right)=$$</p>
<p>$$=\int_1^\infty\frac{\sin y}{y}\,dy$$</p>
<p>And since </p>
<p>$$\int_0^\infty\frac{\sin x}{x}\,dx=\frac{\pi}{2}$$</p>
<p>we're done</p>
|
2,600,679 | <p>Provided two real number sequences: $a_1,a_2,...,a_n$;$b_1,b_2,...,b_n$, define their means respectively:
$$\bar a=\frac{1}{n}\sum_{i=1}^n a_i,\bar b=\frac{1}{n}\sum_{i=1}^n b_i$$
and define their variances and covariance respectively:
$$var(a)=\frac{1}{n}\sum_{i=1}^n (a_i-\bar a)^2,var(b)=\frac{1}{n}\sum_{i=1}^n (b... | Community | -1 | <p>Let $$y(x)=\sum\frac{x^j}{j!^2}.$$</p>
<p>We have
$$y'(x)=\sum\frac{x^{j-1}}{j!(j-1)},$$</p>
<p>$$xy'(x)=\sum\frac{x^j}{j!(j-1)!},$$
and
$$(xy')'(x)=\sum\frac{x^{j-1}}{(j-1)^2}=y(x).$$</p>
<p>This finally leads us to the differential equation</p>
<p>$$xy''+y'-y=0.$$</p>
<p>By a change of variable $t=2\sqrt x$, ... |
987,054 | <p>Prove that the sequence
$$b_n=\left(1+\frac{1}{n}\right)^{n+1}$$
Is decreasing.</p>
<p>I have calculated $b_n/b_{n-1}$ but it is obtain:
$$\left(1-\frac{1}{n^2}\right)^n \left(1+\frac{1}{n}\right)^n$$
But I can't go on.</p>
<p>Any suggestions please?</p>
| orangeskid | 168,051 | <p>My 2¢: consider the function defined apriori for $x>0$
$$f(x)=\log(1+x)\cdot (\frac{1}{x}+1)= \frac{\log(1+x)\cdot (1+x)}{x}$$</p>
<p>$f$ extends analytically to $(-1, \infty)$, and continuously to $[-1, \infty)$. We have $f(-1)=0$ and $f(0)=1$. </p>
<p>We calculate: $$f'(x)= \frac{x - \log(1+x)}{x^2}$$
so $f'... |
2,618,675 | <p>I've seen the nice proof of this using spheres, but I'm looking for a way to prove it parametrically if possible. Using a cylinder $x^2+y^2=r^2$ and a plane $ax+by+cz+d=0$ I got:</p>
<p>$x=r\cos(\theta), y=r\sin(\theta), z=\dfrac{-ar\cos(\theta)+br\sin(\theta)+d}{c}$</p>
<p>But after this I'm stuck trying differen... | Mathematical | 524,351 | <p>Without loss of generality let's assume the cylinder has radius $1$, so that it has equation $x^2 + y^2 = 1$. Assuming that the plane is not parallel to the cylinder, we can always rearrange the coordinate system so that the plane goes through the origin, or even better, make the plane go through the $x$-axis after ... |
2,618,675 | <p>I've seen the nice proof of this using spheres, but I'm looking for a way to prove it parametrically if possible. Using a cylinder $x^2+y^2=r^2$ and a plane $ax+by+cz+d=0$ I got:</p>
<p>$x=r\cos(\theta), y=r\sin(\theta), z=\dfrac{-ar\cos(\theta)+br\sin(\theta)+d}{c}$</p>
<p>But after this I'm stuck trying differen... | amd | 265,466 | <p>W.l.o.g. we can take the cylinder to have unit radius and the cutting plane to be the $x$-$y$ plane rotated through an angle of $\theta$ about the $x$-axis. This rotation is represented by the homogeneous transformation matrix $$R = \begin{bmatrix} 1&0&0&0 \\ 0& \cos\theta & -\sin\theta &0 \\... |
70,582 | <p>For which n can $a^{2}+(a+n)^{2}=c^{2}$ be solved, where $a,b,c,n$ are positive integers?
I have found solutions for $n=1,7,17,23,31,41,47,79,89$ and for multiples of $7,17,23$...
Are there infinitely many prime $n$ for which it is solvable? </p>
| poetasis | 546,655 | <p>There are infinitely many Pythagorean triples where the difference between two legs is either <span class="math-container">$1$</span> or any of the infinite prime numbers <span class="math-container">$P\equiv\pm 1 \mod 8\quad$</span> taken to any non-zero power.</p>
<p>Under <span class="math-container">$100\quad
P... |
3,807,708 | <p>I was asked to prove the following identity (starting from the left-hand side):
<span class="math-container">$$(a+b)³(a⁵+b⁵)+5ab(a+b)²(a⁴+b⁴)+15a²b²(a+b)(a³+b³)+35a³b³(a²+b²)+70a⁴b⁴=(a+b)^8.$$</span>
I'm trying to solve it by a sort of "inspection", but I haven't made it yet. Of course I could try to expan... | Fawkes4494d3 | 260,674 | <p><span class="math-container">$$(a+b)^3(a^5+b^5)+5ab(a+b)^2(a^4+b^4)+15a^2b^2(a+b)(a^3+b^3)+35a^3b^3(a^2+b^2)+70a^4b^4=(a+b)^8$$</span></p>
<p>Note that <span class="math-container">$a+b|(a+b)^3$</span>, so except the last two terms in the LHS, every term is divisible by <span class="math-container">$(a+b)^2$</span>,... |
3,850,320 | <p>If a graph is Eulerian (i.e. has an Eulerian tour), then do we immediately assume for it to be connected?</p>
<p>The reason I ask is because I came across this question:</p>
<p><a href="https://math.stackexchange.com/questions/1689726/graph-and-its-line-graph-that-both-contain-eulerian-circuits">Graph and its line G... | Brian M. Scott | 12,042 | <p>A graph <span class="math-container">$G$</span> with an Euler circuit need not be connected, but the subgraph induced by the vertices that are on the Euler circuit must be a connected component of <span class="math-container">$G$</span>, and any other components must be isolated vertices. In the question to which yo... |
148,160 | <p>While writing a response to a <a href="https://mathematica.stackexchange.com/q/147679/34008">certain MSE question</a> I made a function that tabulates code and comments. (See the definition below.) </p>
<p>Here is an example:</p>
<pre><code>code = "
FoldList[(* reduction function *)
Plus,(* function to apply... | CElliott | 40,812 | <p>This is a problem in design, and the chief difficulty in design is understanding the problem. Suppose you were charged with automating an algorithm in some complex subject area, say finite automata, so your organization could have fairly low-level workers give a it set of inputs and return a nicely formatted correc... |
148,160 | <p>While writing a response to a <a href="https://mathematica.stackexchange.com/q/147679/34008">certain MSE question</a> I made a function that tabulates code and comments. (See the definition below.) </p>
<p>Here is an example:</p>
<pre><code>code = "
FoldList[(* reduction function *)
Plus,(* function to apply... | Anton Antonov | 34,008 | <p>This answer is for a less general question:</p>
<ul>
<li><strong><em>How to improve the creation of tables of code and comments for monadic pipelines?</em></strong></li>
</ul>
<p>As I mentioned in the formulation of the original question post, I am interested in making tables of code and comments in order <a href=... |
148,160 | <p>While writing a response to a <a href="https://mathematica.stackexchange.com/q/147679/34008">certain MSE question</a> I made a function that tabulates code and comments. (See the definition below.) </p>
<p>Here is an example:</p>
<pre><code>code = "
FoldList[(* reduction function *)
Plus,(* function to apply... | b3m2a1 | 38,205 | <p>Here's another possibility. Since your problem is fundamentally a problem of the comments being stripped, we can define an invisible wrapper <code>Commented</code> that evaluates away to nothing when operated on, and formats like a comment.</p>
<p>Here's a possible imp.</p>
<p>First make the formatting right:</p>
... |
89,845 | <p>first,I think we can avoid set theory to bulid the first order logic , by the operation of the finite string.but I have The following questions:</p>
<p>How does "meta-logic" work. I don't really know this stuff yet, but from what I can see right now, meta-logic proves things about formal languages and logics in g... | Mauro ALLEGRANZA | 42,676 | <p>The issue of language/metalanguage and logic/metalogic seems easy to grasp after a careful study on modern math log textbooks (like Shoenfiled) but it can have (for me) interesting "philosophical" aspects.
Please, take a look to Russell & Whithead Intro to their monumental book Principia Mathematica (written 100... |
89,845 | <p>first,I think we can avoid set theory to bulid the first order logic , by the operation of the finite string.but I have The following questions:</p>
<p>How does "meta-logic" work. I don't really know this stuff yet, but from what I can see right now, meta-logic proves things about formal languages and logics in g... | Noam Zeilberger | 1,015 | <blockquote>
<p>If I want to prove that two formal languages are equivalent in some respect, aren't I presupposing a "background" formal language? </p>
</blockquote>
<p>Yes -- but the distinction between object language and meta language can be studied carefully. This is an important part of proof theory, as well as... |
293,921 | <p>The problem I am working on is:</p>
<p>An ATM personal identification number (PIN) consists of four digits, each a 0, 1, 2, . . . 8, or 9, in succession.</p>
<p>a.How many different possible PINs are there if there are no restrictions on the choice of digits?</p>
<p>b.According to a representative at the author’s... | Ben | 93,875 | <p>Just my two cents, </p>
<p>10^4 possibilities. </p>
<p>There are 14 ascending and descending groups of 4.</p>
<p>Keyspace 9,876</p>
<p>If the badguy knows two of the four spaces, he only has to guess through entropy 10^2.
(None of the restrictions meet up with the range 8xx1)</p>
<p>3 Tries in 100</p>
<p>:)</... |
293,921 | <p>The problem I am working on is:</p>
<p>An ATM personal identification number (PIN) consists of four digits, each a 0, 1, 2, . . . 8, or 9, in succession.</p>
<p>a.How many different possible PINs are there if there are no restrictions on the choice of digits?</p>
<p>b.According to a representative at the author’s... | Mr.Young | 140,361 | <p>For (c) he has 3 tries, and there are a total of 100 choices. There are 10 choices for the 2nd number and 10 choices for the 3rd number. Since the PIN begins with an 8 and ends with a 1, none of the restrictions apply.</p>
<ul>
<li>For his first try, there is a <span class="math-container">$1/100 = 0.0100$</span> pr... |
2,519,623 | <p>How do I calculate the side B of the triangle if I know the following:</p>
<p>Side $A = 15 \rm {cm}
;\beta = 12^{\circ}
;\gamma= 90^{\circ}
;\alpha = 78^{\circ}
$</p>
<p>Thank you.</p>
| Kyky | 423,726 | <p>We need to find the probability of not getting a ball above $17$ first. In the urn, there are $4$ balls that are equal or larger than $17$ ($17,18,19,20$). Since there are $20$ balls, there are $20-4=16$ number of balls that are below $17$ in the urn. That means there is a $\frac{16}{20}$ chance that the first ball ... |
1,437,979 | <p>The given equation is $\dfrac{d^2y}{dx^2}+y=f(x)$.
I know that the C.F. is $~a\sin{x}+b\cos{x}$ but i stuck on P.I.. For non-homogeneous eautions, the theorem stating the methods to find P.I.s are not helpful for me in this case. The answer given is $~y(x)=a\sin{x}+b\cos{x}+\int_{0}^{x}f(t)\sin{(x-t)}dt$. How can ... | mickep | 97,236 | <p>They have probably used the following general result (which I state without giving any conditions):</p>
<blockquote>
<p><strong>Theorem</strong> If $Y$ is a solution to the homogeneous differential equation
$$
y''(x)+y(x)=0
$$
with conditions $y(0)=0$ and $y'(0)=1$. Then the function
$$
u(x)=\int_0^x Y(x-t... |
4,527,880 | <p>Suppose</p>
<p><span class="math-container">$$ R = \begin{bmatrix} A & B\\ C & D\end{bmatrix} $$</span></p>
<p>is a <span class="math-container">$2 \times 2$</span> block matrix of real numbers, where <span class="math-container">$A$</span> and <span class="math-container">$D$</span> are squared diagonal mat... | aitzkora | 860,148 | <p>The previous answer is off course correct and give a nice solution to your question. However there are some problems in the question. What is THE orthonormal basis of <span class="math-container">$\mathbb{R}^d$</span> ? There are lots of orthonormal basis in <span class="math-container">$\mathbb{R}^d$</span>, but I ... |
67,124 | <p>Let $S$ be a "rich enough" theory such as Peano arithmetic or ZFC ; assume that we have
a complete formalization of the theory of $S$ so that we may talk about Godel numbers and
the length of a proof.</p>
<p>Godel's sentence is constructed so that it says "I am not provable from S". Now let $n$
be a fixed integer, ... | Jakub Konieczny | 10,674 | <p>I suppose that $\phi_n$ is always provable, unless I am making some basic mistake. Given a string of length at most $n$, you can determine if it happens to be a proof for $\phi_n$. Now, just brute-seach <em>all</em> possible strings of length at most $n$ to see that none of them is a proof of $\phi_n$ (none can be, ... |
67,124 | <p>Let $S$ be a "rich enough" theory such as Peano arithmetic or ZFC ; assume that we have
a complete formalization of the theory of $S$ so that we may talk about Godel numbers and
the length of a proof.</p>
<p>Godel's sentence is constructed so that it says "I am not provable from S". Now let $n$
be a fixed integer, ... | hmakholm left over Monica | 14,366 | <p>Every true statement of the form "Such-and-such $\phi$ is not provable by a proof that contains at most $n$ symbols" is in fact provable -- the proof can consist of simply listing all strings of $n$ symbols or less and noting that neither of them is a valid proof of $\phi$.</p>
<p>I am fairly sure that the same hol... |
2,353,193 | <p>I've recently been learning some homological algebra, mainly out of Northcott and some other sources, and I'm having trouble with the notion of projective dimension. In particular, I have a question (not from Northcott) that says</p>
<blockquote>
<p>Let $R = k[x,y]$ for a field $k$ and $M$ a finitely generated $R... | MooS | 211,913 | <p>One direction is <strong>false</strong>. Let me investigate this problem:</p>
<p>Over $k[x,y]$ free and projective is the same for finitely generated modules by Quillen-Suslin.</p>
<p>We have an exact sequence $$0 \to C \to F \to M \to 0,$$ where $F$ is free and $C$ is free if and only if the projective dimension ... |
150,472 | <p>Let $h\in C_0([a,b])$ arbitrary, that is $h$ is continuous and vanishes on the boundary.
I want to show that
$\int\limits_a^b h(x)\sin(nx)dx \rightarrow 0$.</p>
<p>If $h\in C^1$, integration by parts immediately yields the claim, since $h'$ is continuous and thence bounded on the compact interval, using also the ze... | Davide Giraudo | 9,849 | <p>We can apply Stone-Weierstrass: polynomial are dense in $C_0([a,b])$ endowed with the supremum norm. We can also choose such a sequence vanishing at the boundary. Indeed, if $\{P_n\}$ is a sequence of polynomial converging uniformly to $h$, then $Q_n(x)=P_n(x)-P_n(a)-\frac{x-a}{b-a}(P_n(b)-P_n(a))$, we have $Q_n(a)=... |
3,746,597 | <p>My assumption would be</p>
<p><span class="math-container">$$\int_{-a}^a x\ dx=0$$</span></p>
<p>Am I on the right track here? Also, for indefinite integrals</p>
<p><span class="math-container">$$\int (f)x\ dx$$</span></p>
<p>would this be correct as well?</p>
<p><strong>Background</strong></p>
<p>My professor raise... | QC_QAOA | 364,346 | <p>We have</p>
<p><span class="math-container">$$\int_{-a}^ax^{2n+1}dx=\frac{1}{2n+2}x^{2n+2}\bigg\vert_{-a}^a=\frac{1}{2n+2}\left(a^{2n+2}-(-a)^{2n+2}\right)=\frac{a^{2n+2}}{2n+2}\left(1-(-1)^{2n+2}\right)$$</span></p>
<p>But <span class="math-container">$2n+2$</span> is always even. This implies <span class="math-con... |
1,420,277 | <p>I have to solve this:</p>
<p>$$[(\nabla \times \nabla)\cdot \nabla](x^2 + y^2 + z^2)$$</p>
<p>But I am really drowning in the sand..</p>
<p>Can anybody help me please?</p>
| gt6989b | 16,192 | <p><strong>HINT</strong></p>
<p>Let $E$ denote the number of voters from Estrada and $A$ - from Arrayo. Then, the total is $A+E = 8600$. Translate the second sentence into an equation and solve them together.</p>
|
4,126,470 | <p><span class="math-container">$$\sum_{n=2}^{∞} \frac{1}{n\left(\left(\ln\left(n\right)\right)^3+\ln\left(n\right)\right)}$$</span></p>
<p>I know that there are several methods of finding the convergence of a series. The ratio test, the comparison test, the limit comparison test. There is also this theorem: If a serie... | DonAntonio | 31,254 | <p>Use for example <a href="https://en.wikipedia.org/wiki/Cauchy_condensation_test" rel="nofollow noreferrer">Cauchy's Condensation Test</a> for <span class="math-container">$\;a_n=\frac1{n\log^2n}\;$</span> after a first comparison (why can you? Check carefully the conditions to apply this test!):</p>
<p><span class="... |
3,325,250 | <p>Here is the proof that every Hilbert space is refexive:</p>
<p>Let <span class="math-container">$\varphi\in\mathcal{H^{**}}$</span> be arbitrary. By Riesz, there is a unique <span class="math-container">$f_\varphi\in\mathcal{H^*}$</span> with </p>
<p><span class="math-container">$\varphi(f)=\langle\,f,f_\varphi\ra... | Ben Grossmann | 81,360 | <p>Let <span class="math-container">$\Phi:\mathcal H \to \mathcal H^{**}$</span> denote the canonical injection, AKA the evaluation map (in the notation of the proof, <span class="math-container">$\Phi(x) = \hat x$</span>)t. We want to prove that <span class="math-container">$\Phi$</span> is surjective. In other word... |
315,457 | <p>I am trying to evaluate $\cos(x)$ at the point $x=3$ with $7$ decimal places to be correct. There is no requirement to be the most efficient but only evaluate at this point.</p>
<p>Currently, I am thinking first write $x=\pi+x'$ where $x'=-0.14159265358979312$ and then use Taylor series $\cos(x)=\sum_{i=1}^n(-1)^n\... | kingpin | 324,094 | <p>There is another simple criterion for the irreducibility of a matrix with nonnegative entries. Such an $n\times n$-matrix $A$ is irreducible if and only if all entries of
$$\sum\limits_{i=0}^{n}A^i$$
are greater than $0$.</p>
<p>Since I do not have a reference, I will briefly sketch a proof, using the definition th... |
300,105 | <p>I want to find the proof of the spectrum of the hypercube</p>
| Chris Godsil | 16,143 | <p>There is a proof here: <a href="http://www.cs.yale.edu/homes/spielman/eigs/lect12.ps" rel="nofollow">http://www.cs.yale.edu/homes/spielman/eigs/lect12.ps</a></p>
<p>Or you can look up eigenvalues of Cartesian products and then follow Marion's hint.</p>
|
300,105 | <p>I want to find the proof of the spectrum of the hypercube</p>
| achille hui | 59,379 | <p>This is not an independent answer but filling in what Mariano didn't bother to prove.</p>
<p>Start with Mariano's hint:
$$A_{n+1}=\begin{pmatrix}A_{n}&I_{2^{n-1}}\\I_{2^{n-1}}&A_{n}\end{pmatrix}$$
Let $\chi_n(\lambda) = \det(\lambda I_{2^{n}} - A_{n+1})$ be the characteristic polynomial of the $n$-dim hyper... |
291,684 | <p>Linear ODE systems $x'=Ax$ are well understood. Suppose I have a quadratic ODE system where each component satisfies $x_i'=x^T A_i x$ for given matrix $A_i$. What resources, textbooks or papers, are there that study these systems thoroughly? My guess is that they aren't completely understood, but it would be good to... | MzF | 389,000 | <p>For quadratic systems there is the famous paper by Larry Markus
"Quadratic Differential equations and non-associative algebras" in Contributions to the Theory of Nonlinear Oscillations Vol V (1960) pages 185 - 213.</p>
<p>The PhD thesis: "Quadratic differential Equations a Study in Nonlinear Systems Theory" by M. ... |
1,453,010 | <p>A certain biased coin is flipped until it shows heads for the first time. If the probability of getting heads on a given flip is $5/11$ and $X$ is a random variable corresponding to the number of flips it will take to get heads for the first time, the expected value of $X$ is:
$$E[x] = \sum_{x=1}^\infty{x\frac{5}{1... | Carlos H. Mendoza-Cardenas | 274,058 | <p>$X$ is a geometric random variable with parameter $p$. A way to compute its expected value is through the total expectation theorem:</p>
<p>\begin{align}
E[X] &= E[X\mid X=1]P(X=1) + E[X\mid X>1]P(X>1)\\
\end{align}</p>
<p>When you already know that $X=1$, its expected value is 1, therefore $E[X \mid X=1... |
184,601 | <p>A user on the chat asked how could he make something that would cap when it gets a specific value like 20. Then the behavior would be as follows:</p>
<p>$f(...)=...$</p>
<p>$f(18)=18$</p>
<p>$f(19)=19$</p>
<p>$f(20)=20$</p>
<p>$f(21)=20$</p>
<p>$f(22)=20$</p>
<p>$f(...)=20$</p>
<p>He said he would like to pe... | Community | -1 | <p>We can also get a bit (unnecessarily) fancier:
$$ f(x) = x + (20 - x) \int\limits_{-\infty}^{x-20} \delta(t)\ dt $$
where
$$ \int\limits_{-\infty}^{x-20} \delta(t)\ dt = \begin{cases} 0 & x < 20 \\ 1 & x \ge 20 \end{cases} $$
(See <a href="http://en.wikipedia.org/wiki/Unit_step_function" rel="nofollow">He... |
4,572,517 | <p>Given two 3x3 matrix:</p>
<p><span class="math-container">$$
V=
\begin{bmatrix}
1 & 0 & 9 \cr
6 & 4 & -18 \cr
-3 & 0 & 13 \cr
\end{bmatrix}\quad
W=
\begin{bmatrix}
13 & 9 & 3 \cr
-14 & -8 & 2 \cr
5 & 3 & -1 \cr
\end{bmatrix}
$$</span></p>
<p>Is there any way to predic... | fusheng | 880,505 | <p>I think <span class="math-container">$\mathscr{dim}~V<\infty$</span> guarantees the existence of the polynomial <span class="math-container">$p$</span>.</p>
<p>Consider <span class="math-container">$V=\mathbb{R}^{N}$</span>,<span class="math-container">$ \forall ~x=(x_n)\in \mathbb{R}^N$</span>, we define <span c... |
2,893,568 | <p>I need some help finding the standard deviation using Chebyshev's theorem. Here's the problem:</p>
<blockquote>
<p>You have concluded that at least $77.66\%$ of the $3,075$ runners took between $60.5$ and $87.5$ minutes to complete the $10$ km race. What was the standard deviation of these $3,075$ runners?</p>
... | Henry | 6,460 | <p>There is no upper or lower bound on the standard deviation (apart from $0$) given that information. Perhaps</p>
<ul>
<li>everybody finished in exactly $61$ minutes to give a standard deviation of $0$</li>
<li>$2389$ people finished in $61$ minutes and $686$ finished in $100000$ minutes (almost $10$ weeks) to give ... |
2,090,790 | <p>We are given $g(x)=\frac{x \sin x}{x+1}$, and as I said we need to show it has no maxima in $(0,\infty)$.</p>
<p><strong>My attempt</strong>: assume there is some $x_0>0$ that yields a maxima. then for all $x$</p>
<p>$$-1+\frac{1}{x+1}\leq \frac{x \sin x}{x+1}\leq \frac{x_0 \sin x_0}{x_0+1}\leq 1-\frac{1}{x_0+1... | A. Salguero-Alarcón | 405,514 | <p>Notice that, for $x_n=\frac{(4n+1)\pi}{2}n$, $n\in\mathbb N$, $\sin(x_n)=1$, so $f(x_n)=\frac{x_n}{x_n+1}$.</p>
<p>We also have $f(x)<1$ for all $x\in(0,+\infty)$.
The sequence $f(x_n)$ is strictly increasing and gets closer and closer to $1$, so that means $f$ cannot have a maxima.</p>
|
34,724 | <h3>Overview</h3>
<p>For integers n ≥ 1, let T(n) = {0,1,...,n}<sup>n</sup> and B(n)= {0,1}<sup>n</sup>. Note that |T(n)|=(n+1)<sup>n</sup> and |B(n)| = 2<sup>n</sup>.
A certain set S(n) ⊂ T(n), defined below, contains B(n). The question is about the growth rate of |S(n)|. Does it grow exponentially, like |B(n)... | Roland Bacher | 4,556 | <p>This is no answer but a description of an efficient way for computing the cardinality
of $S(n)$. I have to post it as an answer since it is too long for a comment.</p>
<p>Consider the subsets $U_a(n),C_a(n),L_a(n)$ of $S(n)$ defined as follows:
all coefficients of $U_a(n),C_a(n),L_a(n)$ are $\leq a$ and satisfy
th... |
3,277,555 | <p>For a math class I was given the assignment to make a game of chance, for my game the person must roll 4 dice and get a 6, a 5, and a 4 in a row in 3 rolls or less to qualify. the remaining dice must be over 3 for you to win. my question though is how can I find out the probability of rolling the 6,5, and 4 in a sin... | N. F. Taussig | 173,070 | <p>Let's say the dice are blue, green, red, and yellow. Then an outcome may be specified by <span class="math-container">$(b, g, r, y)$</span>. There are six possible outcomes for each die, so our sample space has <span class="math-container">$6^4$</span> possible outcomes.</p>
<p>For the favorable cases, there are ... |
4,005,522 | <p>Let <span class="math-container">$U$</span> be connected open set of <span class="math-container">$\mathbb{R}^{n}$</span>. Consider <span class="math-container">$C^{\infty}(U)$</span> with sup norm, we said <span class="math-container">$A\subset C^{\infty}(U)$</span> is a minimal dense subalgebra of <span class="mat... | Henno Brandsma | 4,280 | <p>Let <span class="math-container">$Y=\beta \Bbb N \setminus \{p\}$</span> for <span class="math-container">$p \in \Bbb N^\ast$</span>.
Then theorem 6.4/6.7 from Gillman and Jerrison tells us that <span class="math-container">$\beta Y=\beta \Bbb N$</span> and another standard theorem tells us that the one-point compac... |
979,144 | <p>I am searching for a formula of sum of binomial coefficients $^{n}C_{k}$ where $k$ is fixed but $n$ varies in a given range? Does any such formula exist?</p>
| Hypergeometricx | 168,053 | <p>Is this what you're looking for?
$$\sum_{n=k}^m {n\choose k}={m+1\choose {k+1}}$$</p>
|
2,771,059 | <p>This question is similar to a question I posted earlier.<br/>
<span class="math-container">$$z=\cos\frac{\pi}{3}+j\sin\frac{\pi}{3}$$</span>
<br/>
This time I have to do the sum <span class="math-container">$z^4+z$</span><br/>
<br/>
I have used the approach I was shown in my previous question. Here is what I've done... | José Carlos Santos | 446,262 | <p>The answer is $0$ because$$\cos\left(\frac{4\pi}3\right)=\cos\left(\pi+\frac\pi3\right)=-\cos\left(\frac\pi3\right)\text{ and }\sin\left(\frac{4\pi}3\right)=\sin\left(\pi+\frac\pi3\right)=-\sin\left(\frac\pi3\right).$$</p>
|
1,524,615 | <p>I'm trying to solve some task and I'm stuck. I suppose that I will be able to solve my problem, if I'll find elementary way to calculate $\lim_{x \to \infty}\sqrt[x-1]{\frac{x^x}{x!}}$ for $x \in \mathbb{N}_+$.<br>
My effort: I had prove, that $x! \geq (\frac{x+1}{e})^x$, so (cause $x^x>x!$):</p>
<p>$$
\left(\fr... | mahbubweb | 289,201 | <p>Say $y=\sqrt[x-1]{\frac{x^x}{x!}}$ </p>
<p>Then, $\ln y=\frac{\ln(\frac{x^x}{x!})}{x-1}=-\frac{x}{x-1} \cdot \frac{\ln(\frac{x!}{x^x})}{x} =-\frac{x}{x-1} \cdot \sum \limits_{i=1}^{x} \ln{\frac{i}{x}}\cdot\frac{1}{x} \xrightarrow{x \to \infty} (-1) \cdot \int \limits_{0}^{1}\ln x ~dx=(-1) \cdot(-1)=1$</p>
<p>But, ... |
1,987,480 | <p>This question has been bugging me for a while now and I want to know where I'm going wrong. </p>
<blockquote>
<p>There are $20$ tickets in a raffle with one prize. What should each ticket cost if the prize is \$80 and the expected gain to the organizer is \$30?</p>
</blockquote>
<p>Now I can get the right answer... | Graham Kemp | 135,106 | <p>If $p$ is the price per ticket, then $\frac 1{20} (p−\$80)+\frac{19}{20} p$ is the expected return for selling <em>one</em> ticket.</p>
<p>You want the expected return for selling <em>twenty</em> tickets to equal $\$30$. Fortunately the Linearity of Expectation means this is:</p>
<p>$$20\times(\frac 1{20} (... |
16,584 | <p>In the definition of vertex algebra, we call the vertex operator state-field correspondence, does that mean that it is an injective map??
Are there some physical interpretations about state-field correspondence ? Or why we need state-field correspondence in physical viewpoint??
Does it have some relations to highe... | David Ben-Zvi | 582 | <p>I want to elaborate a little on Pavel's excellent answer.</p>
<p>We can think (very schematically) of local operators in an n-dimensional
field theory the following way. We have an n-1 manifold M with some additional
structures (topological, conformal, metric etc), to which our field theory assigns a vector space... |
1,697,206 | <p>In the figure, $BG=10$, $AG=13$, $DC=12$, and $m\angle DBC=39^\circ$.</p>
<p>Given that $AB=BC$, find $AD$ and $m\angle ABC$.</p>
<p>Here is the figure:</p>
<p><a href="https://i.stack.imgur.com/u05wa.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/u05wa.jpg" alt="enter image description here">... | Justin Benfield | 297,916 | <p>You can use the law of sines with the given angle and length of $DC$ to find the length $BD$, then since you're given $BG$, you can find $GD$, and via pythagorean theorem, find $AD$. (Is it exactly 12 as you predicted?)</p>
|
152,336 | <p>Let $G$ be an algebraic group. Choose a Borel subgroup $B$ and
a maximal Torus $T \subset B$. Let $\Lambda$ be the set of weights wrt $T$ and let $\mathfrak{g}$ be the lie algebra of $G$.
Now, consider the following two sets,</p>
<p>1) $\Lambda^+$, the set of dominant weights wrt $B$,</p>
<p>2) The set $N_{o,r}$ ... | Jay Taylor | 22,846 | <p>This isn't even vaguely an answer to your question but is more of a clarifying remark concerning the canonical quotient. Throughout I will write [Lus84] for Lusztig's orange book "Characters of reductive groups over a finite field", Princeton University Press, 1984.</p>
<p>In what follows I will assume that $\mathb... |
747,789 | <p>I've been reading some basic classical algebraic geometry, and some authors choose to define the more general algebraic sets as the locus of points in affine/projective space satisfying a finite collection of polynomials $f_1, \dots, f_m$ in $n$ variables without any more restrictions. Then they define an algebraic ... | Jared | 65,034 | <p>It is true that every algebraic set is a finite union of algebraic varieties (irreducible algebraic sets), and this union is unique up to reordering. These irreducible pieces of an algebraic set are called the irreducible components. This all follows from the fact that a polynomial ring over a field is Noetherian,... |
3,112,682 | <p>I was looking at</p>
<blockquote>
<p><em>Izzo, Alexander J.</em>, <a href="http://dx.doi.org/10.2307/2159282" rel="nofollow noreferrer"><strong>A functional analysis proof of the existence of Haar measure on locally compact Abelian groups</strong></a>, Proc. Am. Math. Soc. 115, No. 2, 581-583 (1992). <a href="htt... | Aniket Sharma | 639,028 | <p><img src="https://latex.codecogs.com/gif.latex?x%5E%7B2%7D-2xcos%5CTheta&space;+1&space;=&space;0&space;%5CRightarrow&space;cos%5E%7B2%7D%5CTheta" title="x^{2}-2xcos\Theta +1 = 0 \Rightarrow cos^{2}\Theta" /> - 1 has to be positive for solving the quadratic equation with real roots</p>
<p>this gives <img src="... |
3,112,682 | <p>I was looking at</p>
<blockquote>
<p><em>Izzo, Alexander J.</em>, <a href="http://dx.doi.org/10.2307/2159282" rel="nofollow noreferrer"><strong>A functional analysis proof of the existence of Haar measure on locally compact Abelian groups</strong></a>, Proc. Am. Math. Soc. 115, No. 2, 581-583 (1992). <a href="htt... | Swapnil Rustagi | 182,381 | <p>Simply consider it a quadratic equation in <span class="math-container">$x$</span>.</p>
<p><span class="math-container">$x^2 + 1 = 2x\cos\theta $</span></p>
<p><span class="math-container">$x^2 - 2x\cos\theta + 1 = 0 $</span>
By quadratic formula, </p>
<p><span class="math-container">$$x = \frac{2\cos\theta \pm \... |
1,811,612 | <p>We have $5$ normal dice. What is the chance to get five $6$'s if you can roll the dice that do not show a 6 one more time (if you do get a die with a $6$, you can leave it and roll the others one more time. Example: first roll $6$ $5$ $1$ $2$ $3$, we will roll $4$ dice and hope for four $6$s or if we get $6$ $6$ $2$... | Felicity | 332,295 | <p>In a very long but straightforward way, we can calculate the probability by breaking it down into scenarios by how many $6$'s appear on the first roll and calculate this probability.</p>
<p><strong>Case 1:</strong> Five $6$'s appear on the first roll: Event $A$</p>
<p>$$P(A)=\left(\dfrac{1}{6}\right)^5$$</p>
<p><... |
221,729 | <p>Till now, I have proved followings;</p>
<p>Suppose $X,Y$ are metric spaces and $E$ is dense in $X$ and $f:E\rightarrow Y$ is uniformly continuous. Then,</p>
<ol>
<li><p>$Y=\mathbb{R}^k \Rightarrow \exists$ a continuous extension.</p></li>
<li><p>$Y$ is compact $\Rightarrow \exists$ a continuous extension.</p></li>... | Austin Mohr | 11,245 | <p><a href="http://books.google.com/books/about/Concrete_mathematics.html?id=pntQAAAAMAAJ" rel="nofollow">Concrete Mathematics</a> is a good place to start with asymptotics and related ideas, particularly if you are in computer science (which your tags suggest).</p>
|
978,114 | <p>From $ax\geq 0$ for $a>0$, we have $x\geq 0$. So I suggest that if $Ax\geq 0$ for $A$ positive definite matrix, $x$ a column vector, $0$ is the column vector with $0$ as elements, then $x\geq 0$, that is, the coordinate of $x$ is greater than $0$.</p>
<p>However, I could not prove it...</p>
| Alfred Chern | 42,820 | <p>If $Ax\geq0$ for arbitrary positive definite matrix $A$, then the conclusion is right just by take $A=I$.
If $Ax\geq0$ for a fixed positive definite matrix $A$, then
$A=\left(
\begin{array}{cc}
1 & 1 \\
1 & 4 \\
\end{array}
\right)$
$x=\left(
\begin{array}{c}
-1 \\
1 \\
\end{a... |
2,721,992 | <p>I would like to see the fact that the components of a vector transform differently (controvariant transformation) than the unit bases vectors (covariant transformation) for the specific case of cartesian to polar coordinate transformation. </p>
<p>The polar unit vectors $\hat{r}$ and $\hat{\theta}$ can be expressed... | Ash | 114,080 | <p>I think that your answer is unnecessarily complicated for this question. In matrix notation equation (1) in the question is</p>
<p><span class="math-container">\begin{equation}
\begin{pmatrix}
\hat{r} && \hat{\theta}
\end{pmatrix}
=\begin{pmatrix}
\hat{x} && \hat{y}
\end{pmatrix}
\begin{pmatrix}
cos\... |
163,296 | <p>For positive real numbers $x_1,x_2,\ldots,x_n$ and any $1\leq r\leq n$ let $A_r$ and $G_r$ be , respectively, the arithmetic mean and geometric mean of $x_1,x_2,\ldots,x_r$.</p>
<p>Is it true that the arithmetic mean of $G_1,G_2,\ldots,G_n$ is never greater then the geometric mean of $A_1,A_2,\ldots,A_n$ ?</p>
<p>... | Andrew | 11,265 | <p>It's a special case ($r=0$, $s=1$) of the mixed means inequality
$$
M_n^s[M^r[\bar a]]\le M_n^r[M^s[\bar a]], \quad r,s\in \mathbb R,\ r<s,
$$
where $M^s$ is the power mean with exponent $s$, see <a href="http://books.google.ru/books?id=ycExBYeCnu4C&printsec=frontcover&hl=ru#v=onepage&q&f=false">S... |
1,444,820 | <p>I want to solve the following funktion for $x$, is that possible? And how woult it look like?</p>
<p>$y = xp -qx^{2}$</p>
<p>Thanks for Help!</p>
| Jack D'Aurizio | 44,121 | <p>There is no solution. Polynomials are a dense subset of $L^2(0,1)$ or $C^0(0,1)$. The only working choice is $f(x)=\delta(x-1/2)$, but it is a distribution, not a polynomial.</p>
|
1,780,253 | <p>If I have two points $p_1, p_2$ uniformly randomly selected in the unit ball, how can I calculate the probability that one of them is closer to the center of the ball than the distance between the two points?</p>
<p>I know how to calculate the distribution of the distance between two random points in the ball, same... | Amit Bikram | 509,096 | <p>For one of the points to be closer to the center than the other point, both points should lie outside the region of sphere which subtends a solid angle of
2*pi*(1-cos(a)).</p>
<p>Where a= 1 radian (180/pi).</p>
<p>and solid angle subtended at the center by the entire sphere is 4pi.
Hence, required probability is ... |
129,530 | <p>This book, which needs to be returned quite soon, has a problem I don't know where to start. How do I find a 4 parameter solution to the equation</p>
<p>$x^2+axy+by^2=u^2+auv+bv^2$</p>
<p>The title of the section this problem comes from is entitled (as this question is titled) "Numbers of the Form $x^2+axy+by^2$"... | zyx | 14,120 | <p>Over a field the space of rational solutions is three dimensional. Integer solutions can be formed as multiples of rational solutions and maybe this multiplication factor is the fourth parameter but it is not clear what the problem is asking. The number of parameters can always be increased from a known parametriza... |
3,056,616 | <p><span class="math-container">$P(x) = 0$</span> is a polynomial equation having <strong>at least one</strong> integer root, where <span class="math-container">$P(x)$</span> is a polynomial of degree five and having integer coefficients. If <span class="math-container">$P(2) = 3$</span> and <span class="math-containe... | Bill Dubuque | 242 | <p><a href="https://math.stackexchange.com/a/617426/242"><strong>Key Idea</strong> <span class="math-container">$\ $</span> (Kronecker)</a> <span class="math-container">$ $</span> How polynomials can factor is constrained by how their <em>values</em> factor, <span class="math-container">$ $</span> e.g. as below, in som... |
2,917,896 | <p>I think my proof is wrong but I don't know how to approach the statement differently. I hope you can help me identify where I'm mistaken/incomplete.</p>
<p>Proof:
$$\text{We need to prove: } \bigcup_{n=1}^{\infty}[3 - \frac{1}{n}, 6] = [2, 6] $$</p>
<p>$$\text{Thus, } x \in \bigcup_{n=1}^{\infty}[3 - \frac{1}{n... | xbh | 514,490 | <p>Not to the question but to the updated proof: </p>
<p>You have not yet proved
$$
\bigcup_1^\infty \left[3 -\frac 1n, 6\right] = [2,6],
$$
so in your proof such equation is definitely not allowed to appear. Also I still do not clearly get your logic inferences in your proof, i.e. I do not see the reasoning part. He... |
2,773,515 | <p>Given $X_1 \sim \exp(\lambda_1)$ and $X_2 \sim \exp(\lambda_2)$, and that they are independent, how can I calculate the probability density function of $X_1+X_2$? </p>
<hr>
<p>I tried to define $Z=X_1+X_2$ and then: $f_Z(z)=\int_{-\infty}^\infty f_{Z,X_1}(z,x) \, dx = \int_0^\infty f_{Z,X_1}(z,x) \, dx$.<br>
An... | TheSimpliFire | 471,884 | <p><strong>HINT:</strong></p>
<p>Assuming independence.</p>
<p>We are given that the P.D.F. of $X$ is $f_X(x)=\lambda e^{-\lambda x},\, x\ge0$ and the P.D.F. of $Y$ is $f_Y(Y)=\lambda e^{-\lambda y},\, y\ge0$. </p>
<p>Then using convolution, $$\begin{align}f_{X+Y}(x+y)&=\int_{-\infty}^\infty f_X(x+y-y)f_Y(y)\,dy... |
1,691,306 | <p>Find all pairs of values $a$ and $b$ that satisfy $(a+bi)^2 = 48 + 14i$</p>
<p>Here's what I have so far:</p>
<p>$$\begin{align}
z^2 &= 48 + 14i = 50 \operatorname{cis} 0.2837\\
z &= \sqrt{50} \operatorname{cis} 0.1419 = 7 + i \\
z &= \sqrt{50} \operatorname{cis} 3.2834 = -7 - i\\
a &= ± 7 \\
b &am... | Jan Eerland | 226,665 | <p>$$\left(a+bi\right)^2=48+14i\Longleftrightarrow$$
$$\left(a+bi\right)\left(a+bi\right)=48+14i\Longleftrightarrow$$
$$a^2-b^2+2abi=48+14i$$</p>
<p>Now, see that:</p>
<ul>
<li>$$\Re\left(a^2-b^2+2abi\right)=48\Longleftrightarrow a^2-b^2=48$$</li>
<li>$$\Im\left(a^2-b^2+2abi\right)=14\Longleftrightarrow 2ab=14\Longle... |
518,140 | <p>What is the relation between the definition of homotopy of two functions</p>
<blockquote>
<p>"A homotopy between two continuous functions $f$ and $g$ from a topological space $X$ to a topological space $Y$ is defined to be a continuous function $H : X × [0,1] → Y$ from the product of the space $X$ with the unit i... | Mikhail Katz | 72,694 | <p>The Cusanus (Nicolas of Cusa) (1401-1464) made some observations that proved to be instrumental in the development of the calculus. Historians of mathematics credit him with the "bridge of continuity", or the "principle of continuity", an instance of which is his view of a circle as an infinite-sided polygon. This i... |
105,614 | <p>I have the following problem: Let $A, B\subset R^3$, $A$ is homeomorphic to a ball, while $B$ is a standard Euclidean ball. Can it happen that the fundamental group of $A\setminus B$ is a perfect group? I am interested in answers for $A$ and $B$ both closed and open, so in fact this is 4 questions. </p>
<p>I am awa... | Roberto Frigerio | 6,206 | <p>Let me deal with the following very special case: the closure of $B$ is contained in the internal part of $A$, and $A$ is bounded. In this case, using that $\partial A$ is compact one can show that there exists a standard open ball $B'$ which has the same center as $B$ but a strictly larger radius, and is such that ... |
105,614 | <p>I have the following problem: Let $A, B\subset R^3$, $A$ is homeomorphic to a ball, while $B$ is a standard Euclidean ball. Can it happen that the fundamental group of $A\setminus B$ is a perfect group? I am interested in answers for $A$ and $B$ both closed and open, so in fact this is 4 questions. </p>
<p>I am awa... | Ian Agol | 1,345 | <p>Consider a smooth properly embedded surface $P\subset \mathbb{R}^3$. Then $\mathbb{R}^3= X\cup Y$, where $X\cap Y=P$ and $X, Y$ are properly embedded submanifolds with $\partial X=\partial Y=P$. By Mayer-Vietoris, we have an exact sequence $0=H_2(\mathbb{R}^3)\to H_1(P)\to H_1(X)\oplus H_1(Y)\to H_1(\mathbb{R}^3)=0... |
230,887 | <p>Let $(F^\bullet,d_F)$ and $(G^\bullet,d_G)$ be two complexes in an abelian category $\mathbf{A}$.</p>
<p>The complex cone $Cone(\varphi)^\bullet$ of a morphism of complexes $\varphi:F^\bullet \to G^\bullet$ is defined as</p>
<p>$$Cone(\varphi)^i=G^i\oplus F^{i+1},$$</p>
<p>and its differential is</p>
<p>$$d(g^i,... | Piotr Achinger | 3,847 | <p>Short answer: Otherwise it wouldn't depend on $\phi$!</p>
<p>Longer answer: Think about it this way: write $F$ and $G$ vertically side by side (in the 0-th and 1st column, respectively), with horizontal maps $\phi$. Since $\phi$ commutes with $d$, you get a double complex, call it $C$. The projection to $F$ and the... |
3,896,562 | <p>Suppose <span class="math-container">$f:\mathbb{S}^n\rightarrow Y$</span> is a continuous map null homotopic to a constant map <span class="math-container">$c$</span>. In other words: <span class="math-container">$F: f\simeq c$</span> , where <span class="math-container">$c(x)=y$</span></p>
<p>Now, we may extend <sp... | Especially Lime | 341,019 | <p>Suppose not. Then for every <span class="math-container">$\delta>0$</span> there is a point <span class="math-container">$x\in[a,b]$</span> where <span class="math-container">$g(x)-f(x)<\delta$</span>. In particular, there is a sequence <span class="math-container">$x_n\in[a,b]$</span> with <span class="math-c... |
3,931,672 | <p>Is there any bounded continuous map f:A to R (A is open) which can not be extended on whole R?</p>
<p>This is a question posed by myself.
My attempt: Let A=(1,2) then we can extend it. If A is finitely many intervals it can be extended. If A is countable many intervals then it can also be extended.</p>
<p>But the la... | Bio | 568,617 | <p><span class="math-container">$\sin(1/x)$</span> is continuous and bounded on <span class="math-container">$(0,1)$</span>, but if you extend it, what is the limit at <span class="math-container">$x =0$</span>?</p>
|
1,563,205 | <p>I have 3 points: $A(0;0;0), B(0;0;1), C(2;2;1) $. They exist on the plane.
I assumed that scalar product of the normal vector and a line which exists on the same plane will be equal to 0. Scalar product equals to $x*2+y*2+z*1=0$ where $x,y,z$ are coordinates of the normal vector. Finally i can get $x,y,z$ using a se... | zickens | 258,840 | <p>As long as there are enough points to define a plane, there <strong>will</strong> exist a normal vector to the plane; as a matter of fact, a plane is defined by
$$
\vec{N}\cdot\left( \vec{a} + \vec{b} \right) = 0
$$
Where $\vec{a}$ and $\vec{b}$ are vectors that go from one point in space to another. We call the ve... |
2,463,565 | <p>I want to use the fact that for a $(n \times n)$ nilpotent matrix $A$, we have that $A^n=0$, but we haven't yet introduced the minimal polynomials -if we had, I know how to prove this.</p>
<p>The definition for a nilpotent matrix is that there exists some $k\in \mathbb{N}$ such that $A^k=0$.</p>
<p>Any ideas?</p>
| Joppy | 431,940 | <p>Let $T: V \to V$ be any linear transformation. Then the following facts are true:</p>
<ol>
<li>For all $k \in \mathbb{N}$, $\operatorname{ker}(T^k) \subseteq \operatorname{ker}(T^{k+1})$.</li>
<li>If $\operatorname{ker}(T^k) = \operatorname{ker}(T^{k+1})$, then $\operatorname{ker}(T^k) = \operatorname{ker}(T^{k+m})... |
3,735,904 | <p><span class="math-container">$\mathbf{Question:}$</span> Prove that <span class="math-container">$(A\cap C)-B=(C-B)\cap A$</span></p>
<p><span class="math-container">$\mathbf{My\ attempt:}$</span></p>
<p>Looking at LHS, assuming <span class="math-container">$(A\cap C)-B \neq \emptyset$</span></p>
<p>Let <span class=... | Graham Kemp | 135,106 | <p>Aside from the typo, yes. In short.</p>
<p><span class="math-container">$$\begin{align}&(A\cap C)\smallsetminus C \\ =~&\{x:(x\in A\wedge x\in C)\wedge x\notin B\}\\=~&\{x:(x\in C\wedge x\notin B)\wedge x\in A\}\\=~&(C\smallsetminus B)\cap A\end{align}$$</span></p>
|
4,551,674 | <p>The question is:</p>
<blockquote>
<p><span class="math-container">$f: A\to R$</span> is a continuous, real-valued function, where <span class="math-container">$A\subseteq\mathbb{R}^n$</span>.</p>
<p>If <span class="math-container">$f(x)\to\infty$</span> as <span class="math-container">$\|x\|\to\infty,$</span> show t... | Drew Brady | 503,984 | <p>If <span class="math-container">$f$</span> is continuous on <span class="math-container">$\mathbb{R}^n$</span> then the claim is true.</p>
<p>It suffices to show that there exists a compact set <span class="math-container">$K$</span> such that
<span class="math-container">$$\inf_K f = \inf_{\mathbb{R}^n} f$$</span>.... |
3,933,851 | <p>Suppose <span class="math-container">$N$</span> is called a magic number if it is a positive integer and when you stick <span class="math-container">$N$</span> on the end of any positive integer, the resulting integer is divisible by <span class="math-container">$N.$</span> How many magic numbers are there less than... | lonza leggiera | 632,373 | <p><strong>Hint:</strong></p>
<p>If <span class="math-container">$\ N\,\big|\,10^rM+N\ $</span> for <em>any</em> positive integer <span class="math-container">$\ M\ $</span>, where <span class="math-container">$\ 10^r\ $</span> is the smallest power of <span class="math-container">$\ 10\ $</span> exceeding <span class=... |
4,069,120 | <p>I am confused with the definition of 'basis'. <br/>
A basis <span class="math-container">$\beta$</span> for a vector space <span class="math-container">$V$</span> is a linearly independent subset of <span class="math-container">$V$</span> that generates <span class="math-container">$V$</span>. And span(<span class="... | esoteric-elliptic | 425,395 | <p>Yes, it is true that <span class="math-container">$$V = \text{span}\ \beta$$</span>
To address your concern, suppose <span class="math-container">$\beta = \{v_1,v_2,\ldots,v_n\}$</span>. If <span class="math-container">$b\in\text{span}\ \beta$</span>, then <span class="math-container">$b = \alpha_1v_1 + \ldots + \al... |
101,526 | <p>Is there a notion of <i>"smooth bundle of Hilbert spaces"</i> (the base is a smooth finite dimensional manifold, and the fibers are Hilbert spaces) such that:</p>
<blockquote>
<p><b>1•</b> A smooth bundle of Hilbert spaces over a point is the same thing as a Hilbert space.</p>
<p><b>2•</b> If <span class="... | Peter Michor | 26,935 | <p>The answer is yes: </p>
<p>Let me sketch the proof. So $p:E\to M$ is the fiber bundle with typical fiber $F$ which is compact, connected (and oriented, for simplicity's sake), and you are given a vertical volume form $\mu$; so $\mu_x$ is a volume form on each fiber $E_x$ which depends smoothly on $x\in M$. First I ... |
3,414,208 | <blockquote>
<p>In the beginning A=0. Every time you toss a coin, if you get head, you increase A by 1, otherwise decrease A by 1. Once you tossed the coin 7 times or A=3, you stop. How many different sequences of coin tosses are there?</p>
</blockquote>
<p>The tricky part of this problem is the combination of the r... | Parcly Taxel | 357,390 | <p>The sequences where <span class="math-container">$A=3$</span> is reached early can only be of length <span class="math-container">$3$</span> or <span class="math-container">$5$</span>, since <span class="math-container">$A$</span> changes parity at every flip. It is easy to list out these early stops:
<span class="m... |
1,376,659 | <p>Let $5=\frac ab$
$\forall\ a,b\ \epsilon\ N$. And $(a,b)=1$ <Br>
Squaring both sides, <Br>
$25b^2=a^2$ <Br>
Thus, $25|a^2$; $25|a$ <Br>
So $a=25m$ <Br>
Substituting, $25b^2=25^2m^2$ <Br>
So $b^2=25m^2$ <Br>
So $25|b$ (By the same logic used before). <Br>
But are assumption is proved to be wrong, because $25$ comes t... | MJD | 25,554 | <p>The answer is no; John can't even fill up the topmost $7\times 11\times 1$ slice of the $7\times 11\times 9$ box. Consider just the top $7\times 11$ face of this box; look just at this face and ignore the rest of the box. A solution to the problem would fill up this $7\times 11$ rectangle with large $3\times3$ rec... |
2,936,269 | <p>How do you simplify: <span class="math-container">$$\sqrt{9-6\sqrt{2}}$$</span></p>
<p>A classmate of mine changed it to <span class="math-container">$$\sqrt{9-6\sqrt{2}}=\sqrt{a^2-2ab+b^2}$$</span> but I'm not sure how that helps or why it helps.</p>
<p>This questions probably too easy to be on the Math Stack Exc... | user587054 | 587,054 | <p>Try to use the formula your classmate gave. In this situation, <span class="math-container">$$9-6\sqrt2={\sqrt3}^2-2{\sqrt{3\times6}}+{\sqrt6}^2\Rightarrow(1)$$</span> That is because <span class="math-container">$6{\sqrt2}=2{\sqrt{3\times6}}$</span> Expression (1) now looks similar to <span class="math-container">$... |
126,983 | <p>I am working on an integral on the following trigonometric functions</p>
<p>$$\int_{-\pi}^\pi \frac{\cos[(4m+2)x] \cos[(4m+1)x]}{\cos x}dx$$</p>
<p>where $m$ is positive integer. I am running the following code in mathematica </p>
<pre><code>Assuming[Element[m, Integers] && (m > 0),
Integrate[
Cos[(... | Dr. Wolfgang Hintze | 16,361 | <p><strong>Solution #2: proof</strong></p>
<p>Here's the missing proof that the integral</p>
<p>$$f(\text{m})=\int_{-\pi }^{\pi } \sec (x) \cos ((4 m+1) x) \cos ((4 m+2) x) \, dx$$</p>
<p>is zero for m = 0, 1, 2, ...</p>
<p>Following the idea of Yarchik in <a href="https://mathematica.stackexchange.com/questions/12... |
126,983 | <p>I am working on an integral on the following trigonometric functions</p>
<p>$$\int_{-\pi}^\pi \frac{\cos[(4m+2)x] \cos[(4m+1)x]}{\cos x}dx$$</p>
<p>where $m$ is positive integer. I am running the following code in mathematica </p>
<pre><code>Assuming[Element[m, Integers] && (m > 0),
Integrate[
Cos[(... | mikado | 36,788 | <p>When I execute the code, I get an error. Have we found some sort of bug?</p>
<pre><code>$Version
Assuming[Element[m, Integers] && (m > 0),
Integrate[
Cos[(4 m + 2) x] Cos[(4 m + 1) x]*1/Cos[x], {x, -π, π}]]
(* "11.0.0 for Linux x86 (64-bit) (July 28, 2016)" *)
</code></pre>
<blockquote>
<pre><code>... |
3,671,608 | <p>Find the number of ways to distribute <span class="math-container">$7$</span> red balls, <span class="math-container">$8$</span> blue ones and <span class="math-container">$9$</span> green ones to two people so that each person gets <span class="math-container">$12$</span> balls. The balls of one color are indisting... | h-squared | 728,189 | <p>Without any restrictions,</p>
<p><span class="math-container">$$r+b+g=12$$</span></p>
<p>The number of ways to distribute balls are <span class="math-container">$$\binom{4}{2}$$</span></p>
<p>But we have counted ways in which <span class="math-container">$g\gt 9$</span>.</p>
<p>Fix <span class="math-container">$... |
4,105,812 | <p>could someone help me check if my proof is valid?</p>
<p>Use direct proof to prove the following theorem: <span class="math-container">$$ A \lor (B \rightarrow A), B \vdash_R A $$</span></p>
<p>We aren't allowed to use proof by resolution, we can only use logic axioms and inference rules such as hypothetical and dis... | user577215664 | 475,762 | <p><span class="math-container">$$5x^3(y')^2+5x^2yy'-3=0$$</span>
<span class="math-container">$$xy' + y - \dfrac{3}{5x^2y'} = 0$$</span>
Change the variable <span class="math-container">$u=1/x$</span>
<span class="math-container">$$y'=\dfrac {dy}{d1/x}\dfrac{d1/x}{dx}=-x^{-2}\dfrac {dy}{d1/x}$$</span>
<span class="mat... |
2,648,549 | <p>Let $\tau_{ij}$ be a transposition if degree n. What does it mean when one says that $\tau_{ij}=\tau_{ji}$? Thanks in advance!</p>
| D F | 501,035 | <p>We know that the product of all eigenvalues is equal to the $\det$ and the sum of eigenvalues is equal to the trace of a matrix. Hence $\lambda_1 \lambda_2 = 1 - p - q$ and $\lambda_1+\lambda_2 = 2 -p - q$. I think you are able to continue</p>
|
2,648,549 | <p>Let $\tau_{ij}$ be a transposition if degree n. What does it mean when one says that $\tau_{ij}=\tau_{ji}$? Thanks in advance!</p>
| amd | 265,466 | <p>A stochastic matrix always has $1$ as an eigenvalue. If it’s row-stochastic, all of its rows sum to $1$, but summing rows is equivalent to right-multiplying by the column vector that consists entirely of $1$s, hence it’s a right eigenvector with eigenvalue $1$. (If the matrix is column-stochastic, you can apply the ... |
52,874 | <p>Consider a coprime pair of integers $a, b.$ As we all know ("Bezout's theorem") there is a pair of integers $c, d$ such that $ac + bd=1.$ Consider the smallest (in the sense of Euclidean norm) such pair $c_0, d_0$, and consider the ratio $\frac{\|(c_0, d_0)\|}{\|(a, b)\|}.$ The question is: what is the statistics of... | Aaron Meyerowitz | 8,008 | <p>This is more a few comments than an answer (since the question seems well answered). I assume that the pair $(a,b)=(1,1)$ was discarded, it would give a value $\frac{\sqrt{2}}{2}$ outside the range of the rest. </p>
<p>Taking instead the $10^6$ points in a quarter disk of radius 1128 gives almost the same bin size... |
18,659 | <p>I'm looking for a fast algorithm for generating all the partitions of an integer up to a certain maximum length; ideally, I don't want to have to generate <em>all</em> of them and then discard the ones that are too long, as this will take around 5 times longer in my case.</p>
<p>Specifically, given <span class="math... | Peter Taylor | 5,676 | <p>You can do it recursively. Let $f(n, maxcount, maxval)$ return the list of partitions of $n$ containing no more than $maxcount$ parts and in which each part is no more than $maxval$.</p>
<p>If $n = 0$ you return a single list containing the empty partition.</p>
<p>If $n > maxcount * maxval$ you return the empty... |
18,659 | <p>I'm looking for a fast algorithm for generating all the partitions of an integer up to a certain maximum length; ideally, I don't want to have to generate <em>all</em> of them and then discard the ones that are too long, as this will take around 5 times longer in my case.</p>
<p>Specifically, given <span class="math... | Joseph Malkevitch | 1,369 | <p>This article about Gray codes includes partitions. The idea behind a Gray code is to enumerate a cyclic sequence of some combinatorial collection of objects so that the "distance" between consecutive items in the list are "close." <a href="http://linkinghub.elsevier.com/retrieve/pii/0196677489900072" rel="nofollow... |
18,659 | <p>I'm looking for a fast algorithm for generating all the partitions of an integer up to a certain maximum length; ideally, I don't want to have to generate <em>all</em> of them and then discard the ones that are too long, as this will take around 5 times longer in my case.</p>
<p>Specifically, given <span class="math... | Community | -1 | <p>This can be done with a very simple modification to the ruleAsc algorithm at <a href="http://jeromekelleher.net/category/combinatorics.html" rel="nofollow">http://jeromekelleher.net/category/combinatorics.html</a></p>
<pre><code> def ruleAscLen(n, l):
a = [0 for i in range(n + 1)]
k = 1
a[0] = 0
a[1... |
18,659 | <p>I'm looking for a fast algorithm for generating all the partitions of an integer up to a certain maximum length; ideally, I don't want to have to generate <em>all</em> of them and then discard the ones that are too long, as this will take around 5 times longer in my case.</p>
<p>Specifically, given <span class="math... | Fred Schoen | 428,361 | <p>I was looking for an algorithm that generates all partitions of <span class="math-container">$L$</span> into <span class="math-container">$N$</span> parts in the multiplicity representation, Knuth calls it the "part-count form". I only found algorithm Z from A. Zoghbi's 1993 thesis <a href="http://dx.doi.o... |
1,567,152 | <blockquote>
<p>Theorem: $X$ is a finite Hausdorff. Show that the topology is discrete.</p>
</blockquote>
<p>My attempt: $X$ is Hausdorff then $T_2 \implies T_1$ Thus for any $x \in X$ we have $\{x\}$ is closed. Thus $X \setminus \{x\}$ is open. Now for any $y\in X \setminus \{x\}$ and $x$ using Hausdorff property, ... | nlmath | 876,389 | <p>Let <span class="math-container">$(X,\tau)$</span> be a finite topological space. Let <span class="math-container">$x\in X$</span>. If the singleton <span class="math-container">$\{x\}$</span> is not an open set, then <span class="math-container">$(X,\tau)$</span> cannot be Hausdorff. This is shown as follows. Le... |
2,796,694 | <p>So for my latest physics homework question, I had to derive an equation for the terminal velocity of a ball falling in some gravitational field assuming that the air resistance force was equal to some constant <em>c</em> multiplied by $v^2.$ <br> So first I started with the differntial equation: <br>
$\frac{dv}{dt}... | Phil H | 554,494 | <p>Write the differential equation as a rate of change of velocity with respect to just aerodynamic drag. Then solve for the time it takes for the drag to equal $mg$. </p>
<p>$$\frac{dV}{dt} = \frac{cv^2}{m}$$
$$\frac{v^{-2}}{c}dV = \frac{dt}{m}$$
$$-\frac{1}{cv} = \frac{t}{m} + C$$
Assuming $t=0, v=0$ then.......</p>... |
3,235,300 | <p>I tried with , whenever <span class="math-container">$x > y$</span> implies <span class="math-container">$p(x) - p(y) =( 5/13)^x (1-(13/5)^{(x-y)}) + (12/13)^x (1- (13/12)^{(x-y)}) > 0 $</span>.
But here I don't understand why the answer is no.</p>
| Peter Szilas | 408,605 | <p>It suffices to show that <span class="math-container">$y=a^x$</span> , <span class="math-container">$0<a<1$</span>, is strictly decreasing(why?)</p>
<p>Set <span class="math-container">$e^{-b}:=a$</span>, <span class="math-container">$b>0$</span>.</p>
<p><span class="math-container">$y= e^{-bx}$</span> is... |
2,405,505 | <p>How to prove that the infinite product $\prod_{n=1}^{+\infty} \left(1-\frac{1}{2n^2}\right)$ is positive ?</p>
<p>Thanks</p>
| Bernard | 202,857 | <p>Just prove the associated log series:
$$\sum_{n=1}^{\infty}\log\Bigl(1-\frac1{2n^2}\Bigr)$$
converges. Observe the general term of this series
$$\log\Bigl(1-\frac1{2n^2}\Bigr)\sim_\infty -\frac1{2n^2}.$$</p>
|
2,405,505 | <p>How to prove that the infinite product $\prod_{n=1}^{+\infty} \left(1-\frac{1}{2n^2}\right)$ is positive ?</p>
<p>Thanks</p>
| H. H. Rugh | 355,946 | <p>Hint: Only the lower limit is a problem. Look e.g. at the product for $n\geq 2$, you could prove:
$$ \prod_{n\geq 2} (1-\frac{1}{2 n^2}) \geq \prod_{n\geq 2} (1-\frac{1}{ n^2})=\prod_{n\geq 2} \frac{(n-1)(n+1)}{ n \cdot n}=\frac12$$
(the last being a telescopic product). So your product is $\geq \frac14$.</p>
|
2,405,505 | <p>How to prove that the infinite product $\prod_{n=1}^{+\infty} \left(1-\frac{1}{2n^2}\right)$ is positive ?</p>
<p>Thanks</p>
| Jack D'Aurizio | 44,121 | <p>The exact value of such product can be derived from the Weierstrass product for the sine function, as already shown by Raffaele. As an alternative approach, we may notice that
$$ \prod_{n\geq 1}\left(1-\frac{1}{2n^2}\right)^2 = \frac{1}{4}\prod_{n\geq 2}\left(1-\frac{1}{n^2}+\frac{1}{4n^4}\right)\geq\frac{1}{4}\prod... |
1,898,803 | <p>So, I looked up this question from G.H. Hardy's <em>A Course of Pure Mathematics</em> and found one examination question from the Cambridge Mathematical Tripos and it has baffled me ever since. I am supposed to sketch</p>
<p>$\lim_{n\to\infty}\dfrac{x^{2n}\sin{(\pi x/2)}+x^2}{x^{2n}+1}$</p>
<p>I have found out tha... | iamvegan | 118,029 | <p>\begin{align*}
\lim \limits_{n \rightarrow \infty} \frac{x^{2n}\sin(\pi x/2)+x^2}{x^{2n}+1} &= \lim \limits_{n \rightarrow \infty} \frac{\sin(\pi x/2)+x^{2-2n}}{1+x^{-2n}}\\
&=\lim \limits_{n \rightarrow \infty} \frac{\sin(\pi x/2)+0}{1+0}
\end{align*}</p>
|
4,332,812 | <p>I came across this series.</p>
<p><span class="math-container">$$\sum_{n=1}^\infty \frac{n!}{n^n}x^n$$</span></p>
<p>I was able to calculate its radius of convergence. If my calculations are OK, it is the number <span class="math-container">$e$</span>. Is that correct?</p>
<p>Then I started wondering if the series i... | Koro | 266,435 | <p>Note the limit <span class="math-container">$\frac{{n!}^{\frac 1n}}{n}\to \frac 1e$</span>.</p>
<p>The above limit by <a href="https://en.wikipedia.org/wiki/Cauchy%E2%80%93Hadamard_theorem" rel="nofollow noreferrer">Cauchy- Hadamard theorem</a> gives the radius of convergence of the series <span class="math-containe... |
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