qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,778,195 | <p>This is Exercise 4 from Section 2.2 of <em>Topology and Groupoids</em> by Brown.</p>
<p><strong>Exercise:</strong></p>
<blockquote>
<p>Let <span class="math-container">$X$</span> be a topological space and let <span class="math-container">$A \subseteq B \subseteq X$</span>. We
say that <span class="math-container">$... | Joshua P. Swanson | 86,777 | <p>Your attempt has a gap. You've shown separately that every neighborhood of <span class="math-container">$x$</span> intersects <span class="math-container">$U$</span> and also intersects <span class="math-container">$A$</span>. This is not the same as saying that every neighborhood of <span class="math-container">$U$... |
2,129,389 | <p>I'm currently working on reducing the summation on the first line. The three lines below show the steps of the reduction according to the instructor. Unfortunately I'm not following the logic. Can someone help explain the algebraic transformations between each line?</p>
<p>$$\sum_{i=0}^{n-2} n-i-1$$ </p>
<p>$$=n(n... | Hypergeometricx | 168,053 | <p>Put $j=n-i-1$:</p>
<p>$$\sum_{i=0}^{n-2} n-i-1=\sum_{j=1}^{n-1} j=\frac {n(n-1)}2$$</p>
|
1,543,542 | <p><strong>Proposition</strong> If $x \in \mathrm{int}(S \cap T)$, then $x \in \mathrm{int}(S) \cap \mathrm{int}(T)$, where $\mathrm{int}(S)$ is the set of interior points of the set $S$.</p>
<p>I have proven this proposition by demonstrating that $\mathrm{int}(S \cap T) \subseteq \mathrm{int}(S) \cap \mathrm{int}(T)$... | fleablood | 280,126 | <p>You can't just claim being an interior point of the intersection is the same as being in the intersection of the interior points without some justification.</p>
<p>However the justification is simple. If $x \in \text{int}(S \cap T) \implies$ there is an open ball $B$ around x such that $B \subset S \cap T$. As $... |
4,068,439 | <p>Given a circle <span class="math-container">$(O, R)$</span> and a chord <span class="math-container">$AB=2a$</span>. If <span class="math-container">$CA,CB$</span> tangents to it, then what is the area of <span class="math-container">$\triangle ABC$</span> equal to, with respect to a and R?</p>
<p><a href="https://i... | vitamin d | 870,871 | <p>Only if
<span class="math-container">$$0<\lim_{n\to\infty}\frac{a_n}{b_n}<\infty$$</span>
you can conclude that either both series converge or both series diverge. In your example the limit is equal to <span class="math-container">$0$</span>, so the LCT is inconclusive. To show the convergence of <span class=... |
3,575,601 | <p>If the median and bisector of one of its sides of a triangle coincide, then the height also coincides and the triangle is isosceles.</p>
<p>So, to see that is isosceles I used the idea of this question: <a href="https://math.stackexchange.com/questions/3336688/if-the-bisector-of-an-angle-of-a-triangle-also-bisects-... | Mohammad Riazi-Kermani | 514,496 | <p>The two triangles made by the median are congruent and the base is a straight line so each angle made by the base and the median is <span class="math-container">$180/2 =90$</span> which makes the median to be the height as well. </p>
|
1,026,725 | <p>How many even 3 digit numbers contain at least one 7.
I got 126, but it was not an answer choice for the problem. Can anyone help?</p>
| John | 7,163 | <p>The $7$ can't be in the ones place if the number is even. That leaves the tens and hundreds place.</p>
<p>For all hundreds except the $700$s, there are five evens with at least one $7$: $n70, n72, n74, n76, n78$.</p>
<p>That accounts for $40$.</p>
<p>For the $700$s, there are $50$: $700, 702, 704, ... 796, 798$... |
1,026,725 | <p>How many even 3 digit numbers contain at least one 7.
I got 126, but it was not an answer choice for the problem. Can anyone help?</p>
| Fredrik Pihl | 64,541 | <p>A more numerical approach using python</p>
<pre><code>In [2]: import itertools
In [3]: I=itertools.ifilter(lambda x:"7" in str(x), range(100,1000,2))
In [4]: len(list(I))
Out[4]: 90
</code></pre>
|
534,670 | <p>I am a bit unclear about underflowing in terms of binary representation.</p>
<p>Let's say that an unsigned 8-bit variable gets overflown from the addition of $150+150$. </p>
<p>A signed 8-bit variable gets underflown after the subtraction of $-120-60$.</p>
<p>Now my point is let's think of 8-bit variable, we are ... | Post No Bulls | 111,742 | <p>You don't have an overflow here: the result will be <code>01100100</code>. Since the top bit indicates the sign, the addition process is not the same as for unsigned integers. </p>
<p>For example, <code>01100100+01100100</code> overflows for signed integers, because we can't carry from the 7th bit into 8th: the 8th... |
1,991,127 | <p>The angle between two lines is given by </p>
<p>$
\tan(\theta) = \big|\frac{m_2-m_1}{1+m_1m_2}\big|
$</p>
<p>where $m_1$ and $m_2$ are the slopes of the two lines in question.</p>
<p>What is confusing me is the reverse problem. When we try to find the slope of the lines making an acute angle with a line of... | spandan madan | 296,493 | <p>Yes, you're correct in seeing that there are two angles. They are $\theta$ and $\pi - \theta$. This is because, $tan(\pi-\theta)=-tan(\theta)$</p>
<p>If you remember,the angle $\pi$ corresponds to 180$^{\circ}$. </p>
<p>So, in the image below, the angles $\angle e$ and $\angle f$ are both the angles between the li... |
17,914 | <p>How does one compute the cardinality of the set of functions $f:\mathbb{R} \to \mathbb{R}$ (not necessarily continuous)?</p>
| Asaf Karagila | 622 | <p>I guess that you know that $|\mathbb{N}| = |\mathbb{N}\times\mathbb{N}|$ and thus $|\mathbb{R}| = |2^{\mathbb{N}}| = |2^{\mathbb{N}\times\mathbb{N}}| = |2^\mathbb{N}\times 2^\mathbb{N}| = |\mathbb{R}\times\mathbb{R}|$</p>
<p>This means that $|P(\mathbb{R})| = |P(\mathbb{R}\times\mathbb{R})|$. Since $f\colon\mathbb{... |
3,313,216 | <p>How to prove that vectors are parallel iff their unit vectors are equal?</p>
<p><span class="math-container">$$\mathbf{u} \parallel \mathbf{v} \iff \hat{\mathbf{u}} = \hat{\mathbf{v}}$$</span></p>
<p>A vector can be written as a scalar multiple of its magnitude and unit vector in its direction: <span class="math-c... | Mr. Xcoder | 435,694 | <blockquote>
<p><span class="math-container">$$\exists \lambda\in \mathbb{R}^+ \, : \, \mathbf{u}= \lambda\mathbf{v} \iff \hat{\mathbf{u}} = \hat{\mathbf{v}}, \text{ where }\mathbf{u},\mathbf{v}\ne \textbf{0}$$</span></p>
</blockquote>
<p><span class="math-container">$"\implies"$</span> Let <span class="math-contai... |
666,806 | <p>The question I have is mostly on stability analysis but the problem is:<br><br><br>
Consider a nonlinear pendulum. Using a linearized stability analysis, show that the inverted position is unstable. What is the exponential behavior of the angle in the neighborhood of this unstable equilibrium position.<br><br></p>
... | akhil krishnan | 647,960 | <p>"Inverted Pendulum" - this means that the pendulum (system) is in it's upright position. The pendulum in its natural pendent state is stable because that position of the system falls in to a stable attractor. As it swings, up on reaching its upright position the attractor remains no longer stable. The system parame... |
176,714 | <p>For any real number $x$, let's define $Om_{k}(x)$ as the number of positive integers $m$ below $x$ such that $\Omega(m)-\omega(m)=k$, where $\omega(n)$ is the number of distinct primes dividing $n$, and $\Omega(n)$ the total number of prime factors of $n$ counted with multiplicity. Obviously $Om_{0}(x)$ is just the ... | Robert Israel | 13,650 | <p>For the case $k=1$, I get
$$\dfrac{Om_1(x)}{x} \to \dfrac{6}{\pi^2} \sum_p \dfrac{1}{p(p+1)} \approx 0.20076$$
the sum being over all primes.
For $k = 2$, I get
$$ \dfrac{Om_2(x)}{x} \to \dfrac{6}{\pi^2} \left( \sum_p \dfrac{1}{p^2(p+1)} +
\sum_{p,q} \dfrac{1}{p(p+1)q(q+1)}\right) $$
where the second sum is over ... |
3,310,683 | <p>Could you help me please with a question about integrals? Can an integral have more than one answer? For example with this integral:
<span class="math-container">$$\int\sqrt{1+\sqrt{1-x^2}}dx$$</span>
Doing by replacing u=<span class="math-container">$\sqrt{1-x^2}$</span>, I have this solution:
<span class="math-c... | Peter Foreman | 631,494 | <p>The error in one of your solutions lies in the fact that you have to consider the sign of the <span class="math-container">$x$</span> values used when performing the integration. In particular, using <span class="math-container">$u=\sqrt{1-x^2}$</span>, we cannot say that <span class="math-container">$x=\sqrt{1-u^2}... |
142,481 | <ol>
<li><p>Which term is used for model categories whose homotopy categories are triangulated? Stable proper model categories?</p></li>
<li><p>I want $Ho(Pro-M)$ to be triangulated ($Pro-M$ is the category of pro-objects of M) and the functor $Ho(M)\to Ho(Pro-M)$ to be an exact full embedding. Which restrictions on M ... | Timothy Chow | 3,106 | <p>I have heard this concept called, logically enough, "vertex multiplication." The first hit I got when I searched was <a href="http://perso.ens-lyon.fr/eric.thierry/Graphes2010/michele-alberti.pdf" rel="nofollow">Small Survey on Perfect Graphs</a> by Michele Alberti. Be aware, though, that I've also heard the term ... |
3,708,200 | <blockquote>
<p>Evaluation of <span class="math-container">$$\int_{C}xydx+(x+y)dy$$</span> aling the curve <span class="math-container">$y=x^2$</span> from <span class="math-container">$(-2,4)$</span> to <span class="math-container">$(1,1)$</span></p>
</blockquote>
<p>What i try </p>
<p>Let <span class="math-conta... | hdighfan | 796,243 | <p>This is a valid method.</p>
<p>It may be useful to note that instead of taking dot products, one can just write
<span class="math-container">$$\int_C f(x) \, dx + g(x) \, dy = \int_a^b \left(f(x(t)) \frac{dx}{dt} + g(x(t)) \frac{dy}{dt} \right) \, dt$$</span>
after parametrising <span class="math-container">$C$</sp... |
1,212,815 | <p>What is the right algorithm for testing whether the graph is "weakly connected"?</p>
<p>The theory says:</p>
<blockquote>
<p>Oriented graph $G=(V,E)$ is weakly connected graph if and only if for every two vertices $u,v \in V$ exists a directed path from $u$ to $v$ or directed path from $v$ to $u$.</p>
</blockquo... | Landon Carter | 136,523 | <p>Let $X=R\cos A$ and $Y=R\sin A$ where $R>0$ and $A\in (0,2\pi)$. You want to find the distribution of $R$. </p>
<p>So the Jacobian turns out to be $r$.</p>
<p>So the joint distribution of $R$ and $A$ is $$f_{R,A}(r,a)=\dfrac{r}{2\pi}e^{\dfrac{-r^2}{2}}$$.Note that $R$ and $A$ are independent.</p>
<p>Integrate ... |
175,803 | <p>This is one of my old unsolved questions when I reading Novikov's book on homology theory. I do not know how to prove it because standard triangulation, fundamental diagram, etc does not help and it should be easy to prove. </p>
| Community | -1 | <p>This question just floated up again, so let me put some references here.</p>
<p><a href="https://mathoverflow.net/questions/18454/fundamental-groups-of-noncompact-surfaces">This</a> MO question gives several different proofs that for your surface $S$, $\pi_1(S)$ is free. In fact, Lee Mosher's answer gives a direct ... |
547,304 | <p>Let G be a simple graph with n vertices. What is the relation between the number of edges of G and the number of edges of the complement G'? </p>
<p>In the example below, I noticed that by adding the vertices and edges of G and the edges of G' you get a total of 10. Then the number of edges in G' would be 3. It is ... | Brian M. Scott | 12,042 | <p>Assuming that you’re talking about simple graphs (i.e., graphs without loops or multiple edges), there is a nice, straightforward relationship. If there are $n$ vertices, there are $\binom{n}2=\frac12n(n-1)$ pairs of distinct vertices, so there are $\binom{n}2$ possible edges. If $G$ has $m$ edges, then $G'$ must ha... |
547,304 | <p>Let G be a simple graph with n vertices. What is the relation between the number of edges of G and the number of edges of the complement G'? </p>
<p>In the example below, I noticed that by adding the vertices and edges of G and the edges of G' you get a total of 10. Then the number of edges in G' would be 3. It is ... | EuYu | 9,246 | <p>The union of the two graphs would be the complete graph. So for an $n$ vertex graph, if $e$ is the number of edges in your graph and $e'$ the number of edges in the complement, then we have
$$e + e' = \binom{n}{2}$$
If you include the vertex number in your count, then you have
$$e + e' + n = \binom{n}{2} + n = \frac... |
2,660,076 | <p>Okay, so as stated here: <a href="https://math.stackexchange.com/questions/2209755/issue-with-proof-in-follands-real-analysis-theorem-6-18">Issue with proof in Folland's 'Real Analysis' (Theorem 6.18)</a>, the text says: </p>
<blockquote>
<p>(Theorem 6.18.) Let $(X,\mathcal{M},\mu)$ and $(Y,\mathcal{N... | Mason | 752,243 | <p>I had this question too. If <span class="math-container">$K, f \geq 0$</span>, then <span class="math-container">$Tf$</span> is measurable by Tonelli's theorem. To get measurability of <span class="math-container">$Tf$</span> in the general case, take positive and negative parts of <span class="math-container">$K$</... |
42,326 | <p>Does there exist a nowhere monotonic continuous function from some open subset of $\mathbb{R}$ to $\mathbb{R}$?
Some nowhere differentiable function sort of object?</p>
| Ross Millikan | 1,827 | <p>The <a href="http://en.wikipedia.org/wiki/Weierstrass_function" rel="nofollow">Weierstrass function</a> is non-monotonic over any interval. I'm not sure you can prove it non-monotonic at every point.</p>
|
4,578,252 | <p>On the MIT <span class="math-container">$2021$</span> <em>Integration Bee Qualifying Exam</em>, it asked to approximate
<span class="math-container">$$
\int_{0}^{\pi}{\rm e}^{{\rm e}^{x}}\,{\rm d}x
$$</span>
I got <span class="math-container">$\displaystyle{\rm e} + {\rm e}^{\rm e} + {\rm e}^{{\rm e}^{2}} + {\rm e}^... | Gary | 83,800 | <p>Performing the change of variables <span class="math-container">$t=\mathrm{e}^x$</span>, using the fact that the exponential integral grows exponentially, and using the first two terms of its <a href="http://dlmf.nist.gov/6.12.E2" rel="noreferrer">asymptotic expansion</a>,
<span class="math-container">$$
\int_{1}^{{... |
3,304,195 | <p>I have this statement:</p>
<blockquote>
<p>The luggage weight of a commercial aircraft follows normal distribution <span class="math-container">$W(Weight) \sim N(20kg,4kg)$</span>
.If the limit of The total luggage load of an aircraft carrying <span class="math-container">$100$</span>
passengers is <span clas... | Kwabena God | 691,504 | <p>Use X as the sum of 100 iid normals, so X~N(2000kg,400kg). Then use X for the calculation. If the 4kg is the standard deviation, the Var(X) = 100(4^2) =1600 and X ~ N(2000 kg, 40 kg). This gets the correct answer. </p>
|
1,211,751 | <p>I have an induction homework question that I got stuck in the middle.</p>
<p>Prove by induction that if $a + a^{-1} \in \Bbb{Z}$ then for each $n \in \Bbb{N}$ the following is true:
$$a^{n} + a^{-n} \in \Bbb{Z}$$</p>
<p>If possible, I would like to understand the method for solving those kind of questions (I know ... | graydad | 166,967 | <p>This is equivalent to:</p>
<blockquote>
<p>Show for any $a$ where $a+a^{-1} \in \mathbb{Z}$ that $a^n+a^{-n} \in \mathbb{Z}$ for all $n\in\mathbb{Z}$.</p>
</blockquote>
<p>If you need to prove a claim of the form</p>
<blockquote>
<p>If $A$ then $B$</p>
</blockquote>
<p>you start off with "assume $A$." From t... |
470,427 | <p>I am trying to understand the topology on $\{0,1\}^X$, where $X$ is uncountable. The topology on $\{0,1\}$ is the discrete and I am using the product topology on $\{0,1\}^X$.
My question is, who are the basic open sets? From my understanding of the definition of product topology, basic sets should either contain fin... | Ted Shifrin | 71,348 | <p>As always with the product topology, a basis element will look like $\{0\}$ or $\{1\}$ (i.e., a typical open set for the topology of $\{0,1\}$) in <em>finitely</em> many components, and the whole set in all the remaining components.</p>
|
278,357 | <p>How do I prove that</p>
<p>$$ \{\alpha\in\mathsf{On}\,|\,\alpha\prec A\}\in V,$$</p>
<p>assuming $A\in V$? I know that if AC is assumed, this set is equal to $\mbox{card}(A):=\mu_\alpha(\alpha\approx A)$, and hence it exists. But in the absence of the AC, we allow the possibility of sets such that $\forall\alpha\i... | Haskell Curry | 39,362 | <p>The existence of the set that you have presented is an immediate consequence of the result that a <a href="http://en.wikipedia.org/wiki/Hartogs_number" rel="nofollow">Hartogs Number</a> exists for any set $ A $.</p>
<blockquote>
<p><strong>Theorem (ZF)</strong> Let $ \alpha $ be the class of all ordinals that inj... |
3,345,704 | <p>Let <span class="math-container">$X \sim N_p(\mu_1,\Sigma_1)$</span> and <span class="math-container">$Y\sim N_p(\mu_2, \Sigma_2)$</span>, <em>where <span class="math-container">$\Sigma$</span> denotes the covariance matrix</em>, and assume <span class="math-container">$X$</span> and <span class="math-container">$Y$... | J. W. Tanner | 615,567 | <p>The polar coordinate condition corresponding to the rectangular coordinate path <span class="math-container">$y=mx$</span> as <span class="math-container">$x\to0$</span> (where we're taking the limit of a constant times <span class="math-container">$1/x$</span>) would be <span class="math-container">$\theta=\arctan(... |
3,345,704 | <p>Let <span class="math-container">$X \sim N_p(\mu_1,\Sigma_1)$</span> and <span class="math-container">$Y\sim N_p(\mu_2, \Sigma_2)$</span>, <em>where <span class="math-container">$\Sigma$</span> denotes the covariance matrix</em>, and assume <span class="math-container">$X$</span> and <span class="math-container">$Y$... | Allawonder | 145,126 | <p>Apart from the fact that you have no <em>reason</em> in the rectangular calculation to conclude that the limit is <span class="math-container">$\infty$</span> (by the way the limit of <span class="math-container">$1/x$</span> at <span class="math-container">$0$</span> is <span class="math-container">$\pm\infty$</spa... |
2,340,235 | <blockquote>
<p>Which way I can determine whether the polynomial $x^5 + 5x^2 +1$ is irreducible over $\mathbb Q$ or not?</p>
</blockquote>
<p>Mod $p$ Irreducibility Test and Eisenstein's criterion not applicable here.</p>
<p>Which way I should proceed?</p>
| Michael Rozenberg | 190,319 | <p>Try $x=y-1$. It should help by the Eisenstein's criterion.</p>
|
130,496 | <blockquote>
<p>Triangulation is called a planar graph in which every face is a triangle.</p>
<p><span class="math-container">$\bullet$</span> Prove that in every triangulation exists edge <span class="math-container">$\left\{ u,v \right\}$</span> such that <span class="math-container">$\deg(u)+\deg(v)\le 22$</span>.</... | Julián Aguirre | 4,791 | <p>The substitution $e^x=y$ leads to the integral
$$
\int_0^\infty\frac{y^{a-1}}{y+1}\,dy.
$$
This can be computed integrating the function $f(z)=z^{a-1}/(z+1)$ along the <em>keyhole</em> contour. We consider the branch of $z^{a-1}$ defined on $\mathbb{C}\setminus[0,\infty)$ with $f(-1)=e^{\pi i}$. For small $\epsilo... |
2,773,457 | <p>I was doing a sample question and came across this question.</p>
<p>A given surface is defined by the equation:
$3x^2+2y^2-z=0$. Describe the normal vector at a point (x, y, z) on the surface. Calculate the normal vector at the point $(1,-1,5)$ on the surface. </p>
<p>The normal vector is
$(6x, 4y, -1)$</p>
<p>Ho... | Community | -1 | <p>Given an infinitesimal displacement to the point $(x,y,z)$ such that it remains in the surface. By differential calculus, we know that</p>
<p>$$f(x+dx,y+dy,z+dz)=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz=0.$$</p>
<p>This expression can be seen as the dot produc... |
3,167,336 | <p><img src="https://i.stack.imgur.com/FLO6E.jpg" alt="enter image description here">
It took me much time to reach the solution where I find the value of X as 2 but still not sure whether this is correct or not.
please help me with the solution</p>
| DUO Labs | 657,798 | <p>Going off of @Peter Foreman's answer, you can rewrite the second equation as <span class="math-container">$$\dfrac{2bx+d\pm\sqrt{4b^2x^2+4bdx+d^2+4a^2x^2+4acx}}{2a}=y$$</span>. Then, rewrite the first equation as <span class="math-container">$$y=\pm\sqrt{1-x^2}$$</span> Make the two equations equal to each other and... |
1,814,855 | <blockquote>
<p>How many ways are there to hand out 24 cookies to 3 children so that they each get an even number, and they each get at least 2 and no more than 10? Use generating functions. </p>
</blockquote>
<p>So the first couple steps are easy.</p>
<p>The coefficient is $x^{24}$</p>
<p>$g(x) = x^6(1+x^2+x^4+x^... | epi163sqrt | 132,007 | <p>Here is variation of the theme which might be helpful when doing the calculation. It's convenient to use the <em>coefficient of</em> operator $[x^n]$ to denote the coefficient of $x^n$ of a series.</p>
<blockquote>
<p>We obtain
\begin{align*}
[x^{24}]&(x^2+x^4+x^6+x^8+x^{10})^3\tag{1}\\
&=[x^{24}]x^6(1+... |
1,814,855 | <blockquote>
<p>How many ways are there to hand out 24 cookies to 3 children so that they each get an even number, and they each get at least 2 and no more than 10? Use generating functions. </p>
</blockquote>
<p>So the first couple steps are easy.</p>
<p>The coefficient is $x^{24}$</p>
<p>$g(x) = x^6(1+x^2+x^4+x^... | Ashwin Ganesan | 157,927 | <p>Your expression $g(x) = x^6 (1+x^2+\cdots+x^8)^3$ is correct, but the geometric series formula needs to be used correctly. Recall the geometric series formula for the sum of 5 terms: $a+ar+\cdots+ar^4 = \frac{a(1-r^5)}{1-r}$. In our case, $r=x^2$ and $a=1$. So $1+x^2+\cdots+x^8$ simplifies to $\frac{1-(x^2)^5}{1-... |
11,684 | <p>I think it happened dozens of times that I corrected expressions like
$$\huge\mathbb{F_5[x]}$$
to
$$\huge\mathbb F_5[x].$$ </p>
<p>I wonder why this particular misspelling — putting the whole expression as an argument into <code>\mathbb</code> — happens so often.</p>
<p><strong>EDIT</strong>
For an ordinary typo, ... | Carl Mummert | 630 | <p>The fact that MathJax renders <code>$\mathbb{F_5[x]}$</code> as $\mathbb{F_5[x]}$ is just a quirk. If you enter the code into LaTeX, you will get something like this as a result:
$$\mathbb{F}_{\ntrianglelefteq}[\curvearrowleft]$$
This is because the LaTeX mathbb command (which is an AMS extended command) has the eff... |
1,302,732 | <p>This is an example from Stoll's <em>Introduction to Analysis</em>. I'm struggling to understand why there's a contradiction here, though I think I'm on the verge of understanding it, but I'd like to understand it more formally. </p>
<p><img src="https://i.stack.imgur.com/vWRwP.jpg" alt="enter image description here... | Math1000 | 38,584 | <p>If a sequence of continuous functions converges uniformly, then the limit function is continuous. Since the limit function isn't continuous, the convergence isn't uniform.</p>
<p>For a more direct argument though, consider
$$a_n = \sup_{x\in[0,1]}|f_n(x)-f(x)|=\sup_{x\in[0,1)}x^n = 1.$$
Since $\lim_{n\to\infty}a_n\... |
294,946 | <p>Are there any groups (besides $\mathbb{Z}$ itself) which have $\mathbb{Z}$ for their abelianization?</p>
<p>That is, is there any non-abelian group $G$ such that $G/[G,G] \cong \mathbb{Z}$?</p>
<p>I would appreciate some examples (if yes) or some hint as to why not (if no).</p>
| Mariano Suárez-Álvarez | 274 | <p>All <a href="http://en.wikipedia.org/wiki/Knot_group">knot groups</a> have $\mathbb Z$ as abelianization. </p>
<p>That gives a nice supply of examples.</p>
|
100,551 | <p>I need to get a matrix $\{a(x_i-x_j)\}$, where $x_i$ form a partition of an interval, $a(x)$ is a given function. I use </p>
<pre><code>In[67]:= a[x_?NumericQ] := N[Exp[-Abs[x]]];
x = Table[-10 + 0.02 (j - 1), {j, 1, 1001}];
A = Outer[a[#1 - #2] &, x, x]; // AbsoluteTiming
Out[69]= {2.99032, ... | Ali Hashmi | 27,331 | <p>Borrowing from @Leonid Shifrin's approach, we can create a slightly better version than <code>Outer</code> gaining some improvement if not a lot</p>
<p>with built-in <code>Outer</code> and <code>a</code> as <code>Listable</code>:</p>
<pre><code>SetAttributes[a, Listable];
a[x_?NumericQ] := N[Exp[-Abs[x]]];
x = Tab... |
1,050,725 | <p>How might I go about finding the basis of a vector spaced spanned by a set of three vectors? For example, if given the set of vectors <code>{(1,2,3),(4,5,6),(7,8,9)}</code> how would I find the basis?</p>
<p>I've tried looking <a href="http://mathworld.wolfram.com/VectorSpaceBasis.html" rel="nofollow">here</a> but ... | Aaron Maroja | 143,413 | <p>Use definition. </p>
<ul>
<li><p>Take any vector $v \in \mathbb{R}^3 $(or any $K$ you're working with) and see if it can be written as a linear combination of the given vectors. </p></li>
<li><p>See if the given vectors are linearly independent. That is, you need to show that </p></li>
</ul>
<p>$$a(1,2,3) + b(4,5,... |
1,366,462 | <p>The line $x + y − 1 = 0$ intersects the circle $x^2 + y^2 = 13$ at $A(\alpha_1, \alpha_2)$ and $B(\beta_1, \beta_2)$.
Without finding the coordinates of A and B, find the length of the chord AB. </p>
<p><strong>Hint:</strong> Form a quadratic equation in $x$ and evaluate $|\alpha_1 − \beta_1 |$, and similarly find ... | user254437 | 254,437 | <p>How about for a2 and b2? i tried subbing like a2 into euqation to get b2 but it doesn't correspond with the answer, which is 5(2^(1/2))</p>
|
1,751,955 | <p>$ S = \{(1/2)^n : n \in \mathbb{N} \} \cup \{ 0 \} $ is obviously bounded and infinite. It also looks totally disconnected to me (it is not and does not contain as its subset an interval with more than one element). But we know that a compact, totally disconnected set must be finite. Hence we know that $ S $ is not ... | anomaly | 156,999 | <p>Take an open subcover ${\cal U} = \{U_\alpha\}\in S$, and assume without loss of generality that $0\in U_0$. Then all but finitely many points in $S$ lie in $U_0$, so ${\cal U}$ has a finite subcover.</p>
|
3,496,985 | <p>I am confused about a sample solution to a problem (please read until the end, since I am not looking for a solution to the problem itself)</p>
<p>We were asked if the vector </p>
<p><span class="math-container">$\begin{bmatrix} -3 \\ 4\\ 7 \end{bmatrix}$</span></p>
<p>can be written as a linear combination of <... | Aryaman Maithani | 427,810 | <p>Sometimes the condition you've stated is the definition of a linear transform. Since you're confused about that, maybe you've seen an alternate definition which states the following: </p>
<ul>
<li><span class="math-container">$T(av) = aT(v)$</span> for every scalar <span class="math-container">$a$</span> and vecto... |
3,177,596 | <p>I'd be grateful for some help with this problem I am trying to solve.</p>
<p>Let's say that I have an object travelling at a velocity v. I want that object to come to a halt in time t AND travel exactly distance d within that time.</p>
<p>So if we are at t0 when we are at velocity v and apply the brakes, the dista... | Ross Millikan | 1,827 | <p>You have two constraints: the time of the trajectory and the change in velocity over the trajectory. You need two knobs to turn to satisfy these. One way is to imagine you wait time <span class="math-container">$t_1$</span> before you decelerate, then decelerate at <span class="math-container">$a$</span> m/s<span... |
139,125 | <p>This is a variation on an earlier question resolved by <em>user35353</em>: <a href="https://mathoverflow.net/questions/139105/can-a-tangle-of-arcs-interlock">Can a tangle of arcs interlock?</a> In that question, the arcs were restricted to circular arcs, and <em>user35353</em>'s proof that one arc can be removed wit... | Ryan Budney | 1,465 | <p>The Borromean rings have an elliptical form in $\mathbb R^3$ given by the union of the three:</p>
<p>$$ x^2 + y^2/2 = 1, z=0 $$
$$ y^2 + z^2/2 = 1, x=0 $$
$$ z^2 + x^2/2 = 1, y=0 $$</p>
<p>I think if you cut out small chunks from these ellipses appropriately, they will still be locked. If I find a proof I'll edit... |
139,125 | <p>This is a variation on an earlier question resolved by <em>user35353</em>: <a href="https://mathoverflow.net/questions/139105/can-a-tangle-of-arcs-interlock">Can a tangle of arcs interlock?</a> In that question, the arcs were restricted to circular arcs, and <em>user35353</em>'s proof that one arc can be removed wit... | Wlodek Kuperberg | 36,904 | <p>Here's another example of four interlocking ellipses: The large ones are in planes parallel to the screen; the small ones lie in planes perpendicular to it, and they are very narrow.</p>
<p><img src="https://i.imgur.com/bes85NB.png" alt="Four ellipses"></p>
|
139,125 | <p>This is a variation on an earlier question resolved by <em>user35353</em>: <a href="https://mathoverflow.net/questions/139105/can-a-tangle-of-arcs-interlock">Can a tangle of arcs interlock?</a> In that question, the arcs were restricted to circular arcs, and <em>user35353</em>'s proof that one arc can be removed wit... | Cristi Stoica | 10,095 | <p><img src="https://i.stack.imgur.com/dJdBa.gif" alt="Three tangled ellipses">
Three tangled ellipses that can't be unlocked. The major radii of the smaller ellipses have to be a bit smaller than the minor radius of the large ellipse. The minor radii of the small ellipses have to allow them to be linked, but of course... |
13,460 | <p>there're some students, who belive that <span class="math-container">$$\frac10 = \infty $$</span></p>
<p>I need to teach them that this is not true and <span class="math-container">$\frac10 $</span> is undefined, mathematically and give a good picture (for their minds)</p>
<p>what is the proper way to teach them wit... | Dan Christensen | 6,103 | <p>If the students have sufficient algebraic skills, point out that division is defined in terms of multiplication as follows:</p>
<blockquote>
<p>For any real numbers <span class="math-container">$x$</span>, <span class="math-container">$y$</span> and <span class="math-container">$z$</span>, we define <span class="... |
631,586 | <p>$$1.\quad p\quad if\quad q\\ \equiv if\quad q\quad then\quad p\\ \equiv q\rightarrow p\\ \\$$$$2.\quad p\quad only\quad if\quad q\\ \equiv if\quad p\quad then\quad q\\ \equiv p\rightarrow q\\ \\$$$$3.\quad p\quad only\quad if\quad q\\ \equiv if\quad q\quad then\quad p\\ \equiv q\rightarrow p\\ \\$$$$4.\quad p\quad i... | BaronVT | 39,526 | <p>Yes, the third statement is incorrect. It should start "$q$ only if $p$".</p>
<p>The "only" makes all the difference! "$p$ if $q$" means that whenever $q$ is true, $p$ is necessarily true as well. This is the same as "if $q$ then $p$" or "$q\rightarrow p$. </p>
<p>On the other hand, "$p$ only if $q$" means that, u... |
21,801 | <p>I seem to have severe difficulty finding the right tags to add to a post, which is infuriating when I can't post unless I have added some.</p>
<p>For a simple brackets expansion problem I had, I can't use *expansion *brackets or *multiplying.</p>
<p>Why are these Not available?</p>
<p>What should I do to satisfy ... | quid | 85,306 | <p>A way to chose tags is to browse the list of all tags sorted by popularity <a href="https://math.stackexchange.com/tags?tab=popular">https://math.stackexchange.com/tags?tab=popular</a> For all the more popular tags there is a brief description and instruction available. </p>
<p>Take one that fits about alright. Gen... |
3,958,304 | <p>We consider the following Itô-Integral <span class="math-container">$$Z_t := \int_0^t \exp(-\mu s) W_s ds$$</span> for <span class="math-container">$\mu\geq0$</span>. I wonder if I could calculate the limit as <span class="math-container">$t\to\infty$</span> in some convergence, but how should I start? Some ideas?</... | fes | 400,212 | <p>Apply Ito's lemma to <span class="math-container">$-\frac{1}{\mu}\exp(-\mu t)W_t$</span>. You get</p>
<p><span class="math-container">$$d\left(-\frac{1}{\mu}\exp(-\mu s)W_s\right)=-\frac{1}{\mu}\exp(-\mu s)dW_t+\exp(-\mu t)W_sds$$</span></p>
<p>Taking integrals and setting <span class="math-container">$W_0=0$</span>... |
3,237,430 | <p>Suppose there are <code>n</code> voters and <code>k</code> candidates. In how many different ways can the vote be split among the candidates?</p>
<p>To be clear, I am only concerned with the number of votes that each candidate gets, not with how individual voters vote.</p>
<p>Each voter can and must vote for only ... | auscrypt | 675,509 | <p>The tail end of the sequence will be bounded above a geometric progression of ratio less than <span class="math-container">$1$</span>, which is known to converge.</p>
|
2,430,729 | <p>How do you calculate the zeros of $f(x) = x^4-x^2-6$ in best way?
Here are my attempts:</p>
<p>Factorize, but the problem is the $6$:</p>
<p>$$0=x^4-x^2-6 \iff 6 = x^2(x^2-1)$$</p>
<p>This doesn't lead to any good solution...</p>
<p>Here is another attempt, but I don't know if this is allowed:</p>
<p>$$0= x^4-x... | Raffaele | 83,382 | <p>SUbstitute $x^2=u$ so you get a quadratic equation</p>
<p>$u^2-u-6=0$ </p>
<p>then you solve and you get two solutions, one is negative and when you substitute back you have $x^2=negative$ which leads to complex roots</p>
<p>Hope this helps</p>
|
917,500 | <p>We define algebra generated by a subset S of power set of X as intersection of all algebras containing S, Is there a procedure of finding this algebra generated.
Just like we find subspace generated by a subset of a vector space as span of that set i.e. taking all linear combinations of elements of that subset. </p>... | Antoine | 73,561 | <p>Suppose $S$ is a finite set, $S = \{S_1, \dots, S_n\}$. Let be $N =\{1,2\dots, n+1\}$, $S_{n+1} = (\cup_i S_i)^c$ and $S' = S\cup \{S_{n+1}\}$. Define sets
$$T_I = \left(\bigcup_{j\in I} S_j\right)\backslash \left(\bigcup_{j\notin I} S_j\right)$$
for $I\subseteq N$, i. e. $I\in \mathcal{P}(N)$. Make some Venn-diagra... |
186,336 | <p>I have a notebook with grouped cells, where the input cells are hidden and only output cells open. Is it possible in an easy way to evaluate the whole notebook and preventing opening all groups, preserving the group structure?</p>
| SonerAlbayrak | 49,198 | <p>I do not know the answer, however I would like to share what I have observed.</p>
<p>Basically, what we are trying to do is simplifying the expression <code>(1/x) MeijerG[{{}, {3/2}}, {{0, 1, 1}, {}}, x]</code> (upto an overall factor) into some <code>Bessel</code> functions.</p>
<p>I do not now the generally true... |
2,714,190 | <p>Find the last three digits of $6^{2002}$. I did some work and figured out that the last two digits is 36. Can anyone help me with the hundredth digit? By the way, I used modular arithmetic and the recursion method for the tens digit, but it fell short when I attempted to do the hundreds digit. Thank you in advance!<... | KingLogic | 503,528 | <p>I used the formula $\frac {1}{10^k} [n-a_k (n)]$. The last two digits is 36. I obtained the hundreds digit this way: $\frac {1}{100}(6^{2002}-36)=\frac {9}{25}(6^{2000}-1)=\frac {9}{25} (6^{2000}-1)=\frac {9}{25}((6^5)^{400}-1)=\frac {9}{25}((7776)^{400}-1)=\frac {9}{25}(7776-1)(7776^{399}+7776^{398}+...+7776^2+7776... |
143,601 | <p>I am looking for a way to either get or set a vertex property for all vertices of the graph at once. The two functions will be:</p>
<pre><code>getVertexProp[g_?GraphQ, prop_] := ...
setVertexProp[g_?GraphQ, prop_, values_] := ...
</code></pre>
<p><code>getVertexProp</code> must return a list of values. Each valu... | M.R. | 403 | <p>Pardon the rant, but IMO graph related property value infrastructure was poorly designed. The <code>Graph[]</code> api was created before the advent of Associations, and I don't think it has changed since version 8.0.</p>
<p>Anyway, PropertyValue is typically faster than SetProperty according to the docs, maybe thi... |
1,386,677 | <p>Proving that $$\sin x > \frac{(\pi^{2}-x^{2})x}{\pi^{2}+x^{2}}, \qquad\forall x>\pi$$</p>
| TIWARI | 224,623 | <p>Hint: say $$f(x) = \sin(x) - \frac{x(\pi^2 - x^2)}{\pi^2+x^2}$$ now $$f'(x) = \cos(x) + \frac{x^4 + 4\pi^2x^2 - \pi^4}{x^4 + 2\pi^2x^2 + \pi^4} = \cos x + 1 + \frac{2\pi^2\left ( x^2 - \pi^2 \right )}{\left ( x^2 + \pi^2 \right )^2}$$ then it is clear that $f'(x) > 0, \forall x > \pi$ and $f(\pi) = 0$ hence we... |
3,694,658 | <p>How can I prove that?</p>
<p><span class="math-container">$1^3+2^3+\cdots+(n-1)^3<\frac{n^4}{4}$</span> </p>
| Community | -1 | <p>We can proceed by induction on <span class="math-container">$n \ge 2$</span>. The base case <span class="math-container">$n = 2$</span> is clearly true because <span class="math-container">$1^3 = 1 < 4 = \dfrac{2^4}{4}$</span>. Assume it's true for <span class="math-container">$n = k$</span>, you show it's true f... |
1,702,064 | <p>While solving Linear algebra and Its application by Gilbert Strang, I am not getting any idea how to solve the problem 6.4.14, which says </p>
<p>From the zero submatrix decide the signs of the $n$ eigenvalues:
$$\pmatrix{0&.&0&1 \\ .&.&0&2 \\0&0&0&.\\1&2&.&n}$$ </p>
... | the_candyman | 51,370 | <p>Find the eigenvalues using the definition:
$$\pmatrix{0&.&0&1 \\ .&.&0&2 \\0&0&0&.\\1&2&.&n}\pmatrix{v_1\\ v_2\\.\\v_n} = \lambda \pmatrix{v_1\\ v_2\\.\\v_n} \Rightarrow
\\
\begin{cases}
v_n & = \lambda v_1 \\
2v_n & = \lambda v_2 \\
\ldots \\
(n-1)v_n & =... |
2,090,885 | <p>I need to calculate the value of the integral:
$$\int_T\frac 1 {\sqrt{x^2+y^2}} \, dx \, dy$$ where $T=\{(x,y) : x\in[-2,2], x^2<y<4\}$.</p>
<p>Specifically, I need to know how to set integration extremes.</p>
| RGS | 329,832 | <p>The integration extremes are very easy to find:</p>
<p>$$T = \{(x, y): x \in [-2, 2], x^2 < y < 4\}$$</p>
<p>clearly tells you that $-2 \leq x \leq 2$. Therefore $x$ will range from $-2$ to $2$. Now imagine that $x$ is fixed at $x_0$. At that point, $x_0^2 \leq y \leq 4$. This means the integral becomes</p>
... |
385,887 | <p>I'm looking for an example of a mathematical relation that is symmetric but not reflexive. A standard non-mathematical example is siblinghood. </p>
| Marco Vergura | 72,134 | <p>Take the set $A:=\{0,1\} $ (someone might just call it 2) and consider the relation on $A$ given by $R:=\{(0,1),\ (1,0)\}$: it is trivially symmetric but not reflexive. </p>
|
1,967,928 | <p>I came across a question that interested me recently. It asked the following:</p>
<blockquote>
<p>Prove that if <span class="math-container">$\mathbb R$</span> is homeomorphic to <span class="math-container">$X \times Y$</span>, then <span class="math-container">$X$</span> or <span class="math-container">$Y$</span> ... | Forever Mozart | 21,137 | <p>Suppose $X$ and $Y$ each have more than one point and $\mathbb R \simeq X\times Y$. </p>
<p>Using the coordinate projections, we see that $X$ and $Y$ are both connected. </p>
<p>Let $a,b\in X$ and $c,d\in Y$ be distinct points. </p>
<p>Consider the points $p=(a,c)$ and $q=(a,d)$. </p>
<p>Let</p>
<p>$$C_1=\{a\}... |
2,776,912 | <p>Consider the sequences $\{a_n\}_{\forall n \in \mathbb{N}}<0$, $\{b_n\}_{\forall n \in \mathbb{N}}>0$, $\{c_n\}_{\forall n \in \mathbb{N}}>0$ and suppose
$$
\begin{cases}
\lim_{n\rightarrow \infty} (a_n+b_n)=0\\
\lim_{n\rightarrow \infty} c_n=L<\infty
\end{cases}
$$</p>
<p>Could you help me to show that... | marty cohen | 13,079 | <p>In these kind of problems,
I try to get the
terms that go to zero
by themselves.
As long as the other terms
behave
(usually meaning that
they are bounded),
the rest usually follows.</p>
<p>$\begin{array}\\
\exp(a_n)*c_n-L*\exp(-b_n)
&=\exp(a_n)*(c_n-L+L)-L*\exp(-a_n-b_n+a_n)\\
&=\exp(a_n)*(c_n-L)+\exp(a_n)... |
4,616,494 | <p>I have come across an exercise that asks to have “There is only one ball, so you need to have it” translated into predicate logic. Using the predicates Ball(x) for x is a ball and Have(x, y) for x must have y, I translated it this way:</p>
<p><span class="math-container">$$\exists x((Ball(x) \land \forall y(Ball(y) ... | lemontree | 344,246 | <p>"<span class="math-container">$\phi \to \psi$</span>" means "not <span class="math-container">$\phi$</span> or <span class="math-container">$\psi$</span>. So your formula permits for the possibility where there is not exactly one (i.e. no or more than one) ball. But we need to have that there is one b... |
2,891,062 | <p>My textbooks lists the set of values b may take as $b>-4 $ and $b≠4$</p>
<p>One of the roots is given as x=2 in the first part of the question, which asks for the value of a such that, for all values of b, one root of the equation $2x^3+ax+4=b(x-2)$ I calculated this as a=-10</p>
<p>If the RHS of the equation w... | Dr. Sonnhard Graubner | 175,066 | <p>Note that $$\frac{2x^3-10x+4}{x-2}=2x^2+4x-2$$ for $$x\neq 2$$</p>
|
3,126,345 | <p><span class="math-container">$$x= \int_{0}^{f(x)} \frac{du}{\sqrt{1-u^2}\sqrt{1-k^2u^2}\sqrt{1-l^2u^2}}$$</span></p>
<p><span class="math-container">$$f(0)=0$$</span></p>
<p>If we apply derivative operation for both sides, we get:</p>
<p><span class="math-container">$$f'(x)=\sqrt{(1-f^2(x))(1-k^2f^2(x))(1-l^2f^2(... | Yuri Negometyanov | 297,350 | <p><span class="math-container">$\color{brown}{\textbf{Primary transformations.}}$</span></p>
<p>Taking in account the possible substitution in the form of
<span class="math-container">$$u_1= \dfrac um,\quad\text{where}\quad m=\max(1,k,l),$$</span>
the given equation WLOG can be presented in the form of
<span class="m... |
3,126,345 | <p><span class="math-container">$$x= \int_{0}^{f(x)} \frac{du}{\sqrt{1-u^2}\sqrt{1-k^2u^2}\sqrt{1-l^2u^2}}$$</span></p>
<p><span class="math-container">$$f(0)=0$$</span></p>
<p>If we apply derivative operation for both sides, we get:</p>
<p><span class="math-container">$$f'(x)=\sqrt{(1-f^2(x))(1-k^2f^2(x))(1-l^2f^2(... | Paramanand Singh | 72,031 | <p>This is a long comment given in response to the comment by OP regarding algebraic addition theorems.</p>
<hr>
<p>Here are the key theorems taken from <em>Theory of Elliptic Functions</em> by Harris Hancock.</p>
<blockquote>
<p><strong>Theorem 1</strong>: A single valued analytic function with an algebraic addit... |
4,234,559 | <p>We consider closed polygonal chains in the 2-dimensional plane with an even number of sides, say <span class="math-container">$2n$</span>, numbered as <span class="math-container">$A_1B_1A_2B_2\dots A_nB_nE$</span>, where <span class="math-container">$E = A_1$</span>.</p>
<p>We require additionally, that each <span ... | Hagen von Eitzen | 39,174 | <p>For convenience, extend the indices cyclically to all of <span class="math-container">$\Bbb Z$</span>, so that e.g. <span class="math-container">$A_0=A_n$</span>, <span class="math-container">$A_{n+1}=A_1$</span>.</p>
<p>Proof by induction on <span class="math-container">$n$</span>.</p>
<p>The case <span class="math... |
1,558,530 | <p>Let $\alpha, \beta$ be random variables, $P(\alpha = i) = P(\beta = i) = \frac{1}{N}$, $i \in \{1, \ldots, N\}$.</p>
<p>What is the probability that $\alpha^3 + \beta^3 = 3 t, t \in \mathbb{N}$? </p>
| André Nicolas | 6,312 | <p>The answer will depend (but only a little) on the congruence class of $N$ modulo $3$. We want $\alpha+\beta$ to be divisible by $3$.</p>
<p>Assume that the random variables $\alpha$ and $\beta$ are <strong>independent</strong>. Without some assumption about the relationship between $\alpha$ and $\beta$ we cannot so... |
104,492 | <p>Three people A,B,C attend the following game: from 0~100, the Host will come up a number with Uniform, but he doesn't tell them the number, the attendee will guess a number and the closes one will win. A choose the number first and tell the number, B will tell another different number based on A's number, C choose a... | joriki | 6,622 | <p>It turns out that under a plausible assumption about C's strategy, the case where B would need to know C's strategy to decide her own strategy doesn't occur because it's bad for A. Under this assumption, all players have a well-defined optimal strategy independent of the other players' strategies.</p>
<p>Let $S_1$ ... |
2,301,960 | <p>I'm quoting from the <a href="http://stacks.math.columbia.edu/tag/001L" rel="nofollow noreferrer">Stacks Project</a>:</p>
<blockquote>
<p><strong>Definition 4.3.6.</strong> A contravariant functor <span class="math-container">$C \to \text{Sets}$</span> is said to be representable if it is isomorphic to the functor o... | Community | -1 | <p>The general formula gives (after substituting $2x^2$ for $x$)</p>
<p>$$\ln(1+2x^2) = 2x^2-\frac{(2x^2)^2}{2}+\frac{(2x^2)^3}{3}-...+(-1)^{n-1}\frac{(2x^2)^n}{n}+\mathcal{O}({(x^2)^{n+1}}).$$</p>
<p>To get the development to order $3$, take $n=1$ and</p>
<p>$$\ln(1+2x^2) = 2x^2+\mathcal{O}(x^4).$$</p>
|
1,124 | <p>Recently, Oleksandr kindly showed a <a href="https://mathematica.stackexchange.com/questions/1096/list-of-compilable-functions">list of Mathematica commands that can be compiled</a>.
RandomVariate was part of that list. However, whether this can be compiled depends upon the distribution that is being sampled. </p>
... | dr.blochwave | 13,162 | <p>Another option to the answer posted by <a href="https://mathematica.stackexchange.com/users/43/andy-ross">Andy Ross</a> cropped up in a recent question of mine about <a href="https://mathematica.stackexchange.com/questions/56180/speed-up-adding-poisson-noise-to-an-image">corrupting an image with Poisson noise</a>. I... |
1,124 | <p>Recently, Oleksandr kindly showed a <a href="https://mathematica.stackexchange.com/questions/1096/list-of-compilable-functions">list of Mathematica commands that can be compiled</a>.
RandomVariate was part of that list. However, whether this can be compiled depends upon the distribution that is being sampled. </p>
... | Oleksandr R. | 312 | <p>Here is a quick (and dirty?) implementation of the Laplace distribution. Relationship to the uniform distribution as given <a href="http://en.wikipedia.org/wiki/Laplace_distribution#Generating_random_variables_according_to_the_Laplace_distribution" rel="noreferrer">on Wikipedia</a>:</p>
<pre><code>randomLaplace = W... |
2,205,209 | <p>Today I came across a question in which equations of two lines (Which were parallel) were given and it was asked to find their angle bisector.</p>
<p>My answer for this was :</p>
<p>Since there is no point of intersection of Parallel lines, there is no origin of angle bisector. So, answer should be <strong>Doesn't... | David K | 139,123 | <p>In a similar question,
<a href="https://math.stackexchange.com/questions/38665/equation-of-angle-bisector-given-the-equations-of-two-lines-in-2d">Equation of angle bisector, given the equations of two lines in 2D</a>,
someone was given a task similar to yours, except that in their case it was explicitly specified th... |
267,045 | <p>The Fibonacci recurrence $F_n=F_{n-1}+F_{n-2}$ allows values for all indices $n\in\mathbb{Z}$. There is an almost endless list of properties of these numbers in all sorts of ways. The below question might even be known. Yet, if true, I like to ask for alternative proofs.</p>
<blockquote>
<p><strong>Question.</str... | Will Sawin | 18,060 | <p>It follows from the identity</p>
<p>$$F_{n+k} = \sum_{j=0}^k {k \choose j} F_{n-j}$$ </p>
<p>which is obtained by applying the standard recurrence $k$ times to the left side, each time splitting up each term into two terms.</p>
<p>Indeed, this gives</p>
<p>$$\sum_{k=0}^{\infty} \frac{F_{n+k}}{k!} = \sum_{k=0}^{... |
42,864 | <blockquote>
<p>Given a monic polynomial <span class="math-container">$f\in\mathbb{Z}[x]$</span>, how can I determine whether there is <span class="math-container">$k\in\mathbb{Z}^+$</span> such that <span class="math-container">$f\mid x^k-1$</span>?</p>
</blockquote>
<p>For example, <span class="math-container">$x^2-x... | Eric Naslund | 6,075 | <p>Here is the short answer: For monic <span class="math-container">$f(x)\in\mathbb{Z}[x]$</span>, we have <span class="math-container">$f(x) \mid x^k-1$</span> for some <span class="math-container">$k$</span> if and only if each root of <span class="math-container">$f(x)$</span> is distinct, has norm <span class="mat... |
42,864 | <blockquote>
<p>Given a monic polynomial <span class="math-container">$f\in\mathbb{Z}[x]$</span>, how can I determine whether there is <span class="math-container">$k\in\mathbb{Z}^+$</span> such that <span class="math-container">$f\mid x^k-1$</span>?</p>
</blockquote>
<p>For example, <span class="math-container">$x^2-x... | lhf | 589 | <p>You want products of the <a href="https://en.wikipedia.org/wiki/Cyclotomic_polynomial" rel="nofollow noreferrer">cyclotomic polynomials</a> for the divisors of <span class="math-container">$k$</span>.</p>
|
3,820,580 | <p>Given vector <span class="math-container">$\boldsymbol{x} \in \mathbb{R}^n$</span> and real symmetric matrices <span class="math-container">$\boldsymbol{A} \in \mathbb{R}^n \times \mathbb{R}^n$</span> and <span class="math-container">$\boldsymbol{B} \in \mathbb{R}^n \times \mathbb{R}^n$</span>. I am trying to find t... | iarbel84 | 611,127 | <p>Here's an idea: I'm disregarding the <span class="math-container">$A$</span> constraint since for a non-zero <span class="math-container">$\mathbf{x}$</span> it requires <span class="math-container">$\det(A)=0$</span> and nothing more.</p>
<p>So we are left with the second constraint. Since <span class="math-contain... |
2,117,481 | <p>what must be added to $x^3-6x^2+11x-8$ to make a polynomial having factor $x-3$?</p>
<p>If the required expression to be added be $K$ then $x^3-6x^2+11x-8+K$ is exactly divisible by $x-3$ but how do I find $K$??</p>
| Michael Rozenberg | 190,319 | <p>Let $f(x)=x^3-6x^2+11x-8$.</p>
<p>Hence, $f(x)=q(x)(x-3)+f(3)$, where $q$ is a polynomial with degree two.</p>
<p>Thus, we need to add $-f(3)=2$.</p>
|
2,117,481 | <p>what must be added to $x^3-6x^2+11x-8$ to make a polynomial having factor $x-3$?</p>
<p>If the required expression to be added be $K$ then $x^3-6x^2+11x-8+K$ is exactly divisible by $x-3$ but how do I find $K$??</p>
| Kanwaljit Singh | 401,635 | <p>I think you know when to add term and when to subtract term from polynomial.</p>
<p>So continue from that. </p>
<p>Remember that if we have given some factor then you can find value of variable from that factor and can put in polynomial. This is equal to either 0 if any remainder not given else equal to remainder.... |
4,039,336 | <p>I've calculated the probability for rolling a particular value or higher on both dice when rolling 2d6, as follows.</p>
<p>1 or higher: 100%
2 or higher: 69.44%
3 or higher: 44.44%
4 or higher: 25%
5 or higher: 11.11%
6 or higher: 2.778%</p>
<p>If I add a third dice to this, how does it affect the probability if I s... | Captain Lama | 318,467 | <p>The other answers give fine examples, but I think it is worth to first point out that "an affine subset" of a scheme does not truly make sense. A subscheme can be affine or not, but a subset of a scheme does not in general have a natural structure of subscheme: it can have none, and it can have several.</p... |
2,246,777 | <p>I was curious as to whether $$\lim_{x \to 0}\frac{1}{x^2}=\infty $$
Or the limit does not exist? Because doesn't a limit exist if and only if the limit tends to a finite number? </p>
| Chickenmancer | 385,781 | <p>Recall that a limit exists if the left and right limits exist and agree. Just to clarify, yes, $\infty$ or $-\infty$ are suitable limits.</p>
<p>Since $(-x)^2=x^2,$ then </p>
<p>$$\displaystyle\lim_{x\to0^-}\dfrac{1}{x^2}=\displaystyle\lim_{x\to0^+}\dfrac{1}{x^2}=\infty,$$
and so you are correct,
$$\displaystyle\... |
450,686 | <p>I am aware that to check if a linear transformation is injective, then we must simply check if the kernel of that linear transformation is the zero subspace or not. If the kernel is the zero subspace, then the linear transformation is indeed injective.</p>
<p>Is there a similar way to check for surjectivity?</p>
| RghtHndSd | 86,816 | <p>Let $T : V \rightarrow W$ be a linear map and $T^* : W^* \rightarrow V^*$ the map induced on the dual space. Then $T$ is surjective if and only if $T^*$ is injective if and only if the kernel of $T^*$ is the zero subspace.</p>
<p>I think that's about as similar as you can get.</p>
|
1,975,952 | <p>Problem: Suppose that $\mu, \nu$ are $\sigma$-finite (positive) measures on $(X,M)$ with $\nu << \mu$, and let $\lambda = \mu+\nu$. Let $f$ be the Radon-Nikodym derivative $f = d\nu / d\lambda $. Show that $ 0 \leq f < 1$ $\mu $-a.e and $d\nu/ d\mu=f/(1-f)$.</p>
<p>There is something about this that just ... | Ennar | 122,131 | <p>Notice that $a\leq \max\{a,c\}$ and $b\leq \max\{b,d\}$ which means that $$a+b\leq \max\{a,c\}+\max\{b,d\}$$</p>
<p>Can you finish from here?</p>
|
1,975,952 | <p>Problem: Suppose that $\mu, \nu$ are $\sigma$-finite (positive) measures on $(X,M)$ with $\nu << \mu$, and let $\lambda = \mu+\nu$. Let $f$ be the Radon-Nikodym derivative $f = d\nu / d\lambda $. Show that $ 0 \leq f < 1$ $\mu $-a.e and $d\nu/ d\mu=f/(1-f)$.</p>
<p>There is something about this that just ... | Dean C Wills | 366,201 | <p>An alternative is to consider $\max \{x,y\} = \frac{x + y + |x-y|}{2}$ and use the triangle inequality.</p>
|
313,388 | <p>Computing without Taylor series or l'Hôpital's rule </p>
<p>$$\lim_{n\to\infty}\prod_{k=1}^{n}\cos \frac{k}{n\sqrt{n}}$$</p>
<p>What options would I have here? Thanks!</p>
| xpaul | 66,420 | <p>The best way (I think) to solve this problem is to use $\ln(1-x)=-x+O(x^2)$ and $\sin x=x+O(x^3)$. In fact,
\begin{eqnarray*}
\lim_{n\to\infty}\ln\prod_{k=1}^n\cos\frac{k}{n\sqrt{n}}&=&\lim_{n\to\infty}\sum_{k=1}^n\ln\cos\frac{k}{n\sqrt{n}}=\lim_{n\to\infty}\frac{1}{2}\sum_{k=1}^n\ln\cos^2\frac{k}{n\sqrt{n}}... |
507,109 | <p>If we have two homeomorphisms $f:A\to X$ and $g:B\to Y$, then is it true that $f\times g:A\times B\to X\times Y$ defined by $(f\times g)(a,b)=(f(a),g(b))$ is again a homeomorphism?</p>
<p>I think the answer is yes; </p>
<p>It's clearly a bijection. Intuitively it seems to be continuous but I don't know how to show... | Cameron Buie | 28,900 | <p>In general, the Cartesian product of two continuous functions is again continuous. Use the fact that the basic open sets in $X\times Y$ are of the form $U\times V$ where $U$ is open in $X$ and $V$ is open in $Y,$ then show that $$(f\times g)^{-1}[U\times V]=f^{-1}[U]\times g^{-1}[V],$$ which is open in $A\times B$ b... |
1,460,118 | <p>I'm taking a Calculus I course online, which has been going pretty well considering that I haven't done much math since I graduated high school 5 years ago. However, I recently ran into a good bit of trouble with the chapter on delta-epsilon proofs. Specifically, my question is about non-linear equations and bounds.... | Chelton Evans | 60,864 | <p>A change in $x$ implies a change in $f(x)$.</p>
<p>$|x-x_{0}| \lt \delta \Rightarrow |f(x)-L| \lt \epsilon$ </p>
<p>Know $|x--1| = |x+1| \lt \delta$</p>
<p>$f(x) = x^{2}-2x-2$ </p>
<p>$L = 1$</p>
<p>$f(x)-L = x^{2}-2x-2-1 = (x-3)(x+1) \lt (x-3) \delta$ $\lt \epsilon$ </p>
<p>The $(x-3)$ is almost a constant ... |
2,182,371 | <p>This is the most complicated differential equation I've encountered today:</p>
<p>$$y''(x)-\frac{y'^2}{y}+ \frac{\epsilon y(x)}{x^2}=0$$</p>
<p>It is not Euler-Cauchy, I guess the ultimate approach is to do it with variable substitution, but it is hard to make a guess on what to substitute by purely looking at the... | Claude Leibovici | 82,404 | <p><em>Just another way to do it.</em></p>
<p>Considering $$y''-\frac{y'^2}{y}+ \frac{\epsilon y}{x^2}=0$$ let $$y=e^z\implies y'=e^{z} z'\implies y''=e^{z} \left(z''+z'^2\right)$$ Replace in the original equation to get $$\frac{e^{z} \left(x^2 z''+\epsilon \right)}{x^2}=0\implies x^2 z''+\epsilon=0$$ which becomes si... |
1,146,294 | <p>How would one prove that $$(n+1)^{n-1}<n^n \ \forall n>1$$</p>
<p>I have tried several methods such as induction.</p>
| Mathemagician1234 | 7,012 | <p>Let's try induction. Let n = 2. Then $(3)^{1}< 2^{2}=4$, which is clearly true. Now assume the statement is true for some natural number k where k >2. Now consider the case k+1. </p>
<p>$((k+1)+1)^{(k+1)-1}<(k+1)^{k+1}$ = $(k+2)^{k}<(k+1)^{k+1}$ </p>
<p>Let $k= l-1$ where $l$ is some natural number larger... |
1,146,294 | <p>How would one prove that $$(n+1)^{n-1}<n^n \ \forall n>1$$</p>
<p>I have tried several methods such as induction.</p>
| Mr Tsjolder | 190,328 | <p>My try to prove $\forall n>1 : (n+1)^{n-1} < n^n$:</p>
<p>$$(n+1)^{n-1} < n^n \Leftrightarrow (n+1)^{n} < n^n (n+1) \Leftrightarrow \left(\frac{n+1}{n}\right)^n < n+1$$</p>
<p>Now apply induction on this result:</p>
<p><strong>Base case:</strong> $n = 2 \Rightarrow \left(\frac{3}{2}\right)^2 < 3... |
1,146,294 | <p>How would one prove that $$(n+1)^{n-1}<n^n \ \forall n>1$$</p>
<p>I have tried several methods such as induction.</p>
| robjohn | 13,854 | <p>For $n\gt1$, <a href="http://en.wikipedia.org/wiki/Bernoulli%27s_inequality">Bernoulli's Inequality</a> gives
$$
\begin{align}
\frac{n^n}{(n+1)^{n-1}}
&=n\left(1-\frac1{n+1}\right)^{n-1}\\
&\ge n\left(1-\frac{n-1}{n+1}\right)\\
&=\frac{2n}{n+1}\\[6pt]
&\gt1
\end{align}
$$</p>
<hr>
<p>We can also us... |
1,146,294 | <p>How would one prove that $$(n+1)^{n-1}<n^n \ \forall n>1$$</p>
<p>I have tried several methods such as induction.</p>
| achille hui | 59,379 | <p>For $n > 1$, apply GM $\le$ AM to following $n$ numbers $\big(\;\underbrace{n+1,n+1,\ldots,n+1}_{n-1 \text{ copies}}, 1\;\big)$. We have</p>
<p>$$((n+1)^{n-1}\cdot 1)^{1/n} \le \frac{(n-1)(n+1)+1}{n} = n
\quad\implies\quad (n+1)^{n-1} \le n^n$$
Since the $n$ numbers are not the same, the inequality is strict an... |
1,509,917 | <p>Suppose I have an arithmetic progression, e.g. 3,7,11,15,19,....
The sums of the first n elements form tuples of increasing size.
E.g. the sums are:</p>
<ul>
<li>S(n=1)=3</li>
<li>S(n=2)=10</li>
<li>S(n=3)=21</li>
<li>S(n=4)=36</li>
<li>S(n=5)=55</li>
</ul>
<p>So the tuples are:</p>
<ul>
<li>T(1)=[3;10)</li>
<li... | Thomas Andrews | 7,933 | <p>In this case $S(n)=2n(n+1)-n=2n^2+n$. </p>
<p>Given $M$, you want to find a positive root of $2x^2+x-M=0$ and then take the floor.</p>
<p>This is $$x=\frac{-1+\sqrt{1+8M}}{4} = \frac{-1}{4} + \sqrt{\frac{1}{16}+\frac{M}{2}}$$</p>
<p>Not sure if there is a "clever" way to find the floor of this. You always get tha... |
4,122,288 | <p>I have <span class="math-container">$$7^x\bmod {29} = 23 $$</span>
It is possible to get <span class="math-container">$x$</span> by trying out different numbers but that will not be possible if <span class="math-container">$x$</span> is actually big.</p>
<p>Are there any other solutions for this equation?</p>
<p>Kin... | p_square | 920,884 | <p>your question <span class="math-container">$7^x\pmod {29} = 23$</span> can be written as <span class="math-container">$7^x$</span> - <span class="math-container">$29$</span> <span class="math-container">$\lfloor \frac {7^x}{29} \rfloor$</span> = <span class="math-container">$23$</span>. Maybe you can proceed with th... |
1,431,287 | <p>Dear StackExchange users,</p>
<p>I have a little question ... I just don't have a clue how that works</p>
<p>I have the following differential equation</p>
<p>$$\frac{dy}{dx}=xy^2-2\frac{y}{x}-\frac{1}{x^3}$$</p>
<p>My book says that one can substitute $(r,s) = (x^2y,\ln{|x|})$ to get</p>
<p>$$\frac{ds}{dr}=\fr... | David Quinn | 187,299 | <p>You have $$\frac{dr}{dx}=2xy+x^2\frac{dy}{dx}$$ and $$\frac{ds}{dx}=\frac 1x$$</p>
<p>Therefore, $$\frac{dr}{ds}=2x^2y+x^3\frac{dy}{dx}=2r+e^{3s}\frac{dy}{dx}$$</p>
<p>So now you can make the substitution.</p>
|
2,692 | <p>The setting is undergraduate students in Computer Science, a course in Discrete Mathematics (first proof-oriented course they take, they had a mostly computation oriented first course in calculus).</p>
<p>As examples I need nice proofs using contradiction. The typical proof that $\sqrt{2}$ is irrational is standard... | Dennis | 1,708 | <p>If you don't mind probability-related problems, this one works pretty well.</p>
<p>Suppose $A$, and $B$ are sets in $S$. Then $A \cap B$, $A \cap B^C$, $A^C \cap B$ and $A^C \cap B^C$ partition $S$.</p>
<p>The proof by contradiction is in showing that the members of the partition are disjoint. Suppose, for examp... |
314,856 | <p>Does anybody know how to prove this identity?</p>
<p>$$\int_0^\infty \prod_{k=0}^\infty\frac{1+\frac{x^2}{(b+1+k)^2}}{1+\frac{x^2}{(a+k)^2}} \ dx=\frac{\sqrt{\pi}}{2}\frac{\Gamma \left(a+\frac{1}{2}\right)\Gamma(b+1)\Gamma \left(b-a+\frac{1}{2}\right)}{\Gamma(a)\Gamma \left(b+\frac{1}{2}\right)\Gamma(b-a+1)}$$</p>
... | Marko Riedel | 44,883 | <p>While this is fairly advanced we can at least bring it into a manageable form which allows specific values to be calculated, assuming $a$ and $b$ are positive integers. Start by evaluating the inner product, which has two parts.</p>
<p>First, $$f_1(x) = \prod_{k=0}^\infty \frac{1}{1+ \frac{x^2}{(a+k)^2}} =
\prod_{k... |
314,856 | <p>Does anybody know how to prove this identity?</p>
<p>$$\int_0^\infty \prod_{k=0}^\infty\frac{1+\frac{x^2}{(b+1+k)^2}}{1+\frac{x^2}{(a+k)^2}} \ dx=\frac{\sqrt{\pi}}{2}\frac{\Gamma \left(a+\frac{1}{2}\right)\Gamma(b+1)\Gamma \left(b-a+\frac{1}{2}\right)}{\Gamma(a)\Gamma \left(b+\frac{1}{2}\right)\Gamma(b-a+1)}$$</p>
... | Marko Riedel | 44,883 | <p>Responding to the observation/comment by Random Variable from <strong>Thu Aug 8</strong>, what is missing above to turn this into a rigorous proof is an evaluation of the quantity
$$R(a,b) = \sum_{m=a}^b
\operatorname{Res}\left(\prod_{k=a}^b \frac{1}{k^2+x^2};x=im\right).$$</p>
<p>We put
$$g_1(x) = \prod_{k=a}^b \f... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.