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 |
|---|---|---|---|---|
59,567 | <p>I am looking for a way to add a legend showing the identity of various atoms (with different colours) to this picture. Any Clues?</p>
<pre><code>Import["ExampleData/1PPT.pdb", "Rendering" -> "BallAndStick"]
</code></pre>
<p><img src="https://i.stack.imgur.com/FSFoH.png" alt="enter image description here"></p>
| rcollyer | 52 | <p>Bob Hanlon's <a href="https://mathematica.stackexchange.com/a/59587/52">answer</a> works very well, but in some ways it is the hard way of doing things. If you have v9 or v10, then it is arguably easier to use the legend constructs within it. Similar to his answer, we get the image and element names:</p>
<pre><code... |
4,415,907 | <p>I need to evaluate the Fourier inverse integral</p>
<p><span class="math-container">$\displaystyle \int_{-\infty}^{\infty}\frac{\sinh\left(y\sqrt{\alpha^2-\omega^2}\right)}{\sinh\left(H\sqrt{\alpha^2-\omega^2}\right)}e^{i\alpha x}d\alpha \tag*{}$</span></p>
<p>which arose while solving a PDE.</p>
<p>Here, <span clas... | Kaira | 691,829 | <p>For completeness, I will provide the complete solution I can imagine.</p>
<p>We calculate the integral for <span class="math-container">$x>0$</span>. The case <span class="math-container">$x<0$</span> can be solved with a similar method.</p>
<p>We firstly note that the poles of the integrand are located at <sp... |
47,926 | <p>Is there any known two-dimensional Conway's game of life variation where each cell can not be just on/off but able to hold more states, maybe 4 or 5?</p>
| paul garrett | 12,291 | <p>I wrote a "spatial ecology" variant
<A Href="http://www.math.umn.edu/~garrett/a05/Life1.html" rel="nofollow"> here </A>
in which different populations (with various birth and death rates) in-effect compete for "space". It's in Java, with source available (tho' was written a long time ago, with the 'original' window... |
1,343,909 | <p>I was reading some examples about linear functionals from the book Introductory functional analysis with applications of Kreysig and one of the examples states the following </p>
<p>Let <span class="math-container">$L:C[0,1]\rightarrow C[0,1]$</span></p>
<p><span class="math-container">$$L[f(x)]=\int_{0}^{x}f(s)ds... | mich95 | 229,072 | <p>The function is not bijective, as for any function $f$ in $C[0,1]$ $L(f(0))=\int\limits_{0}^{0}f(s)ds=0$, so the function $x \to x+1$ does not have a preimage, as $0+1=1$</p>
|
88,363 | <p>It is easy to truncate Series upto some order, say $n$. My question is how do I remove low orders? Let us say my series is a power series in $x$. I want to remove the terms with negative powers because they diverge at $x = 0$. I can simply write</p>
<p>s1-s2, where</p>
<p>s1=Normal[Series[blah, {x, 0, n}]</p>
<p>... | Dr. belisarius | 193 | <p>If everything else fails, you can always do</p>
<pre><code>Total[SeriesCoefficient[f@x, {x, 0, #}] x^# & /@ Range[0, 10]]
</code></pre>
|
3,583,117 | <p>I would like to understand clearly why the following equality is true</p>
<p><span class="math-container">$P[X+Y \leq z] = E_Y[P[X+Y] \leq z | Y]]$</span></p>
<p>I wrote the left part of the equation as follows:</p>
<p><span class="math-container">$E_Y[P[X+Y] \leq z | Y]] = \sum_y y P[X+y \leq z]P(y)$</span></p>
... | Masoud | 653,056 | <p>It is clear that <span class="math-container">$E(1_A)=P(A)$</span> so
<span class="math-container">$$P(X+Y\leq z)=P(A)=E(1_A)\overset{(1)}{=}EE(1_A|Y)=E\bigg( E(1_A|Y)\bigg)=E\bigg(P(X+Y\leq z |Y)\bigg)$$</span>
when <span class="math-container">$A=\{X+Y\leq z\}$</span></p>
<p>In <span class="math-container">$(1)$<... |
176,260 | <blockquote>
<p>Let $\left\{ f_{n}\right\} $ denote the set of functions on
$[0,\infty) $ given by $f_{1}\left(x\right)=\sqrt{x} $ and
$f_{n+1}\left(x\right)=\sqrt{x+f_{n}\left(x\right)} $ for $n\ge1 $.
Prove that this sequence is convergent and find the limit function.</p>
</blockquote>
<p>We can easily show ... | Did | 6,179 | <p><strong>Hints:</strong></p>
<ul>
<li>For every $x\gt0$, the function $u_x:t\mapsto \sqrt{x+t}$ is continuous hence every convergent sequence defined by $x_{n+1}=u_x(x_n)$ for every $n\geqslant0$ has limit $z_x$ such that $u_x(z_x)=z_x$. Here, $z_x=\frac12(1+\sqrt{1+4x})$.</li>
<li>For every $x\gt0$, $z_x-u_x(t)=c_x... |
24,186 | <p>Consider the code below:</p>
<pre><code>s = Solve[(3 - Cos[4*x])*(Sin[x] - Cos[x]) == 2, x, InverseFunctions -> True];
Select[s[[All, 1, 2]], Element[#, Reals] &]
</code></pre>
<p>In MMA 8.0, I get </p>
<pre><code>{-\[Pi], \[Pi]/2, \[Pi]}
</code></pre>
<p>but in MMA 9.0, I get an empty set { }</p>
<p>Ass... | Michael E2 | 4,999 | <p>If the question is about converting general math-book expressions to pure functions, you could use something like</p>
<pre><code>SetAttributes[convert, HoldAll];
convert[expr_, vars_List] :=
With[{variables = Unevaluated@vars},
Block[variables,
Evaluate@(Hold[expr] /. Thread[vars -> Slot /@ Range@Length... |
1,579,521 | <p>Find the value of
<span class="math-container">$$ \iint_{\Sigma} \langle x, y^3, -z\rangle. d\vec{S} $$</span>
where <span class="math-container">$ \Sigma $</span> is the sphere <span class="math-container">$ x^2 + y^2 + z^2 = 1 $</span> oriented outward by using the divergence theorem.</p>
<p>So I calculate <span ... | Fundamental | 218,829 | <p>$$\int_{0}^{2\pi} \int_{0}^{\pi} \int_{0}^1 \left(\rho^2 \sin \phi \right)\underbrace{(\sqrt{3}\rho \sin \phi \sin \theta)^2}_{3y^2 \ \textrm{in spherical}} \ d\rho \ d\phi \ d\theta$$</p>
|
545,003 | <p>I have a proof that I am trying to prove and I am getting stuck at the inductive hypothesis. This is my theorem:</p>
<blockquote>
<p>For all real numbers $n>3$, the following is true: $n + 3 < n!$.</p>
</blockquote>
<p>I have proven true for $n = 4$, and will assume true for some arbitrary value $k$, i.e.,... | Newb | 98,587 | <p>As you are trying to solve this problem, I'll only give you a hint.</p>
<p>Inductive Step: we want to show $(n+1)+3 < (n+1)!$</p>
<p>That's equivalent to $n+4 < (n+1)\cdot n!$ by the property of the factorial.</p>
<p>We can distribute: $n+4 < (n\cdot n!) + (1\cdot n!)$</p>
<p>Can you take it from here?<... |
23,566 | <p>I love math, and I used to be very good at it. The correct answers came fast and intuitively. I never studied, and redid the demonstration live for the tests (sometimes inventing new ones). I was the one who answered the tricky questions in class (8 hours of math/week in high school)... You get the idea.</p>
<p>As ... | 0x0 | 6,335 | <p>Revisit all of your High school books. It will be easier to grasp and remind you of many stuff you forgot.</p>
|
1,022,950 | <p>I was reading linear dependence between vectors, where I come across below explanation:</p>
<hr>
<p>In a rectangular xy-coordinate system every vector in the plane can be expressed in
exactly one way as a linear combination of the standard unit vectors. For example, the
only way to express the vector (3, 2) as a l... | Jihad | 191,049 | <p><strong>Hint</strong>. Prove that $\forall \varepsilon > 0\exists N \forall n>N: \sqrt{\frac{n+1}{n}} > 1 - \varepsilon$ and $\sqrt{\frac{n+1}{n}} < 1 + \varepsilon$.</p>
|
184,682 | <p>I have difficulties with a rather trivial topological question: </p>
<p>A is a discrete subset of $\mathbb{C}$ (complex numbers) and B a compact subset of $\mathbb{C}$. Why is $A \cap B$ finite? I can see that it's true if $A \cap B$ is compact, i.e. closed and bounded, but is it obvious that $A \cap B$ is closed?<... | Cameron Buie | 28,900 | <p>I'm assuming that by "discrete set" you mean that $A$ has no accumulation points--that is (since we're in a Hausdorff space), that every point in the plane has a neighborhood containing at most one point of $A$. Thus, $A$ is vacuously closed (as it contains all of its accumulation points), so since $B$ is closed, th... |
1,184,961 | <p>I need to prove/show that $n^3 \leq 3^n$ for all natural numbers $n$ by strong induction. I have no clue where to begin!!!! :( I know how to do the beginning steps of showing that it's true for $k = 0$ and $k = 1$, etc but get suck on how to start the strong inductive step.</p>
| user103828 | 103,828 | <p>Assume true for $n=k$. Now let's try for $n=k+1$,
\begin{align*}
(k+1)^3 &=k^3+3k^2+3k+1 \\
&\leq 3^k + 2k^3 \\
&\leq 3^k+2 \cdot 3^k=3^{k+1}
\end{align*}
where the second step used $2k^3-3k^2-3k-1 =k^3+(k-1)^3-6k \geq 0$ (for $k\geq 3$) and the second and third step used the assumption that $k^3 \leq 3... |
2,664,286 | <p>I am confused between the usage of two words <em>for all</em> and <em>any</em>. Let us consider the example of the definition normal subgroups, A subgroup $H$ is said to be normal if $\forall g \in G, g^{-1}Hg = H$ but if I rephrase the definition of normal subgroup to $H$ is normal in $G$ if for any $g \in G, g^{-1... | user | 505,767 | <p><strong>HINT</strong></p>
<p>R is that point on the line such that the angles of PR and QR with the line are equal.</p>
<p>A trick is to consider the reflection $\bar P$ of $P$ (or Q) with respect to the line and then consider the intersection between $\bar P Q$ and the line. The intersection point is R.</p>
<p><... |
2,664,286 | <p>I am confused between the usage of two words <em>for all</em> and <em>any</em>. Let us consider the example of the definition normal subgroups, A subgroup $H$ is said to be normal if $\forall g \in G, g^{-1}Hg = H$ but if I rephrase the definition of normal subgroup to $H$ is normal in $G$ if for any $g \in G, g^{-1... | Peter Szilas | 408,605 | <p>Refer to Gimusi's drawing:</p>
<p>Let me use $\overline{AC}: = $length $AC.$</p>
<p>Let $A'$ be the reflection of $A$ with respect to the given line. </p>
<p>If $A',B,C$ are collinear then </p>
<p>$\overline {AC}+ \overline {CB}$ is shortest.</p>
<p>Proof:</p>
<p>Line $A'B$ intersects the given line at $C.$</p... |
2,407,820 | <p>$$\ln^q (1+x) \le \frac{q}{p} x^p \quad (x \ge 0, \; 0 < p \le q)$$</p>
<p>For $p=q$ this reduces to the familiar $\ln(1+x) \le x$. Otherwise I haven't had much success in proving it. General suggestions would be appreciated.</p>
| Ahmad | 411,780 | <p>put $p=\frac{1}{\ln x}$ and when $x > e$ then $p <1$ and we arrive at </p>
<p>$\ln^q(x+1) \leq q \ln x * x^{p}$ which is $\ln^q (x+1) \leq q \ln x *e$</p>
<p>when $q \geq 2$ we get that $\ln^q(x+1) \leq e q \ln x$</p>
<p>Because $\ln^q(x+1)\geq \ln^q x \geq e q \ln x$ divide by $\ln x$</p>
<p>We get tha... |
1,715,358 | <p>Being fascinated by the approximation $$\sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi -x) x}\qquad (0\leq x\leq\pi)$$ proposed, more than 1400 years ago by Mahabhaskariya of Bhaskara I (a seventh-century Indian mathematician) (see <a href="https://math.stackexchange.com/questions/976462/a-1-400-years-old-appr... | robjohn | 13,854 | <p><strong>A few approximations</strong></p>
<p>When making approximations, there is no legal or illegal. There are things that work better and things that don't. When making approximations that are supposed to work over a large range of values, often the plain Taylor series is not the best way to go. Instead, a polyn... |
157,992 | <p>Please, help me</p>
<p>Prove that $(1, i);(1,-i)$ are characteristic vectors of $\begin{bmatrix} a & b \\ -b & a \end{bmatrix}$</p>
<p>I've found the polynomial characteristic: $\lambda^2-2a\lambda+a^2+b^2$ and the roots are:</p>
<p>$\lambda_{1} = \frac{a+ib}{\lambda} \\
\lambda_{2} = \frac{a-ib}{\lambda... | Arturo Magidin | 742 | <p>In order to test if a <em>given</em> nonzero vector $\mathbf{v}$ is a characteristic vector (aka eigenvector) of a <em>given</em> matrix $A$, you do <strong>not</strong> need to find the eigenvalues, or characteristic polynomial of the matrix! All you have to do is compute $A\mathbf{v}$ and see if what you get is a ... |
120,260 | <p>Let $X$ be a simply connected smooth projective variety, whose Picard group is generated by the classes of the irreducible codimension 1 loci $D_1, \ldots, D_k$. Let $E_1, \ldots, E_r$ be other irreducible codimension 1 loci, and suppose that $X^0$ is the complement in $X$ of the divisors $D_i$ and $E_j$.</p>
<p>Su... | Daniel Litt | 6,950 | <p>There has indeed been exciting recent work in this area, by Bhargava and Shankar (see <a href="http://www-math.mit.edu/~poonen/papers/Exp1049.pdf">this Bourbaki expose by Poonen</a>) and also by <a href="http://www.math.harvard.edu/~gross/preprints/stable18.pdf">Bhargava and Gross</a>. Briefly, the work of Bhargava... |
2,633,720 | <blockquote>
<p>Prove by induction that $$ (k + 2)^{k + 1} \leq (k+1)^{k +2}$$ for $ k > 3 .$</p>
</blockquote>
<p>I have been trying to solve this, but I am not getting the sufficient insight. </p>
<p>For example, $(k + 2)^{k + 1} = (k +2)^k (k +2) , (k+1)^{k +2}= (k+1)^k(k +1)^2.$</p>
<p>$(k +2) < (k +1)^... | Anne Bauval | 386,889 | <p>A simpler (and a little stronger) statement is:
<span class="math-container">$$\forall n\ge3\quad(n+1)^n<n^{n+1}.$$</span>
We first check that <span class="math-container">$(3+1)^3=64<81=3^{3+1}.$</span>
Then, for the induction step, it is sufficient to prove (for all <span class="math-container">$n\ge3$</span... |
438,231 | <p>How should I state the general solution for the equation $\sin(4\phi)=\cos(2\phi)$.
The angles are $15$, $45$, $75$ and $135$ if I restrict myself within the range $[0,360]$</p>
| Empy2 | 81,790 | <p>$\sin 4\phi=2\sin 2\phi\cos 2\phi=\cos 2\phi$ so either $\cos 2\phi=0$ or $\sin 2\phi=1/2$. So $2\phi = n180^\circ+90^\circ$ or $2\phi=n360^\circ+30^\circ$ or $2\phi=n360^\circ+150^\circ$</p>
|
322,302 | <p>Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.</p>
<p><strong>Question</strong> What are the conjectures in your field prove... | skupers | 798 | <p>In <a href="https://arxiv.org/abs/1812.02448" rel="noreferrer">https://arxiv.org/abs/1812.02448</a>, Tadayuki Watanabe announced a disproof of the Smale conjecture in dimension 4. In particular, he shows that the inclusion <span class="math-container">$O(5) \hookrightarrow \mathrm{Diff}(S^4)$</span> is <em>not</em>... |
322,302 | <p>Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.</p>
<p><strong>Question</strong> What are the conjectures in your field prove... | Martin Väth | 165,275 | <p>The famous Nussbaum conjecture stated that every continuous map of a closed ball in a Banach space with a compact iterate (i.e. the iterate has relatively compact range) has a fixed point. Again Robert Cauty (see my previous post) proved it 2015 in the positive by showing that even a Lefschetz type fixed point theor... |
122,848 | <p>Is my calculation correct for this rotation around a point?</p>
<p>A point a(-19.94,392.11) is rotated -49.45 degrees, what is the new coordinates of point a?</p>
<p>My solution:</p>
<pre><code>x' = x*cos(0) - y*sin(0)
y' = x*sin(0) + y*cos(0)
x' = (-12.961) - (-298.0036)
y' = (15.15) + (254.92)
x' = 285.04
y' ... | Peđa | 15,660 | <p>Result of my calculation is slightly different than yours . </p>
<p>Let's denote :</p>
<p>$\theta=-49.45^\circ $</p>
<p>$x'=r\cos \alpha ~\text{ and }~y'=r\sin \alpha$</p>
<p>where $~r=\sqrt{x^2+y^2} =392.5$</p>
<p>Note that :</p>
<p>$\alpha=\arctan\left|\frac{y}{x}\right|+\frac{\pi}{2}-|\theta|$</p>
... |
25,100 | <p>Suppose one has a set $S$ of positive real numbers, such that the usual numerical ordering on $S$ is a well-ordering. Is it possible for $S$ to have any countable ordinal as its order type, or are the order types that can be formed in this way more restricted than that?</p>
| Pietro Majer | 6,101 | <p>To complete the picture (the obvious remaining part). If ${S\subset\mathbb R}$ is well ordered, then it is countable: indeed it has countable cofinality. Thus well-ordered subsets of <strong>R</strong> are <em>exactly</em> countable ordinals.</p>
|
3,479,883 | <p>I know that (I might be wrong):</p>
<ul>
<li>Symbol for empty or null set : {Ø} or {}</li>
<li>Null or empty set is 'subset of all sets' as well as 'empty or null set' set</li>
<li>So, { {} } is same as { Ø }</li>
</ul>
<p>I just want to know { {} } or { Ø } is an empty set or not ? And if yes then we can conclud... | MPW | 113,214 | <p>A set whose only element is the empty set is not empty (an empty set contains no element).</p>
<p>Think of sets a boxes. If you put a small empty box into a big box, the big box isn't empty anymore. It doesn't matter if the small box is empty or not. That's the beauty of the <span class="math-container">$\{\;\}$</s... |
4,182,153 | <p>Let A be a nonempty compact subset of <span class="math-container">$R$</span> (real numbers) and let B be a nonempty closed subset of R. Recall
that <span class="math-container">$\operatorname{dist}(A, B) = \inf{|x − y| : x ∈ A, y ∈ B}$</span>. Show that there exist <span class="math-container">$a ∈ A$</span> and <s... | usr25 | 944,317 | <p>Consider the closed set <span class="math-container">$S = A - B$</span>, which is <span class="math-container">$A + (-B)$</span>, if it contains <span class="math-container">$0$</span>, then <span class="math-container">$a = b$</span>, and <span class="math-container">$A \cap B \neq \emptyset$</span>.
Otherwise, con... |
3,299,661 | <p>I am familiar with all 3 of the entities I have listed in my question. I know the definitions of "reflexive", "symmetric", and "transitive". However, I am afraid I do not mechanistically understand the "flow" of how we ultimately generate equivalence classes from a particular relation that exhibits the 3 properties ... | José Carlos Santos | 446,262 | <ol>
<li>If <span class="math-container">$(6,6)\notin R_1$</span>, then <span class="math-container">$R_1$</span> would not be reflexive simply because <span class="math-container">$R_1$</span> being reflexive <em>means</em> that <span class="math-container">$(\forall n\in\{1,2,3,4,5,6\}):(n,n)\in R_1$</span>.</li>
<li... |
925,541 | <p>The exercise states: prove that the limit of the sequence $$a_{n+2}=(a_na_{n+1})^{1/2} \ where \ a_1 \ge 0, a_2 \ge 0 $$</p>
<p>is $L = (a_1a_2^2)^{1/3}$</p>
<p>The solutin says: $$Let \ b_n = \frac{a_{n+1}}{a_n},$$ then $$b_{n+1}= 1/\sqrt{b_n} \ for \ all \ n$$ wich implies that
$$b_{n+1}= b_1^{(-1/2)^n} \rightar... | mike | 75,218 | <p>Consider squaring the LHS and RHS of the following equation:</p>
<p>$$\prod_{j=2}^{n+1}b_j = \prod_{j=1}^{n}b_j^{-1/2}\tag{0}$$</p>
<p>We obtain:</p>
<p>$$\prod_{j=2}^{n+1}(b_j)^2 = (b_1)^{c(n)}, c(n)=\frac{2}{3}(-1+(-1/2)^n)\tag{1}$$</p>
<p>$$\prod_{j=1}^{n}b_j^{-1} = (b_1)^{d(n)}, d(n)=c(n)\tag{2}$$</p>
|
4,215 | <p>I suspect it is impossible to split a (any) 3d solid into two, such that each of the pieces is identical in shape (but not volume) to the original. How can I prove this?</p>
| Mariano Suárez-Álvarez | 274 | <p>It seems that Puppe and others proved that this is impossible for <em>any</em> strictly convex solid. See [B. L van den Waerden, Aufgabe Nr 51, <em>Elem. Math.</em> <strong>4</strong> (1949) <strong>18</strong>, 140]</p>
<p>The reference comes from <em>Unsolved problems in geometry</em> by Hallard T. Croft, K. J. F... |
433,403 | <ol>
<li>Let F(x,y) be the statement, “x can fool y,” where the domain consists of all of the people in the world. Translate this statement into symbolic logic.
a. Everyone can be fooled by somebody.</li>
</ol>
<p>Would it be: For every x.y in W, F(x,y) is in W?</p>
<p>I am not getting the gist of this...</p>
| Cameron Buie | 28,900 | <p>"For every $x,y$ in $W$, $F(x,y)$ is in $W$" translates to "For every person $x$ and person $y$ in the world, the statement '$x$ can fool $y$' is a person in the world." Does this statement make sense?</p>
<p>What you're trying to say is that for every person in the world, there is some person in the world who can ... |
354,250 | <p><strong>Remark:</strong> All the answers so far have been very insightful and on point but after receiving public and private feedback from other mathematicians on the MathOverflow I decided to clarify a few notions and add contextual information. 08/03/2020.</p>
<h2>Motivation:</h2>
<p>I recently had an interesting... | gmvh | 45,250 | <p>As to question 2, there are certainly plenty of non-trivial discrete models in statistical physics, such as the Ising or Potts models, or lattice gauge theories with discrete gauge groups, that require no partial derivatives (or indeed any operations of differential calculus) at all to formulate and simulate.</p>
<... |
102,738 | <p>I imported two sets data
one: </p>
<pre><code>data1={{0., 5.02512*10^-10}, {0.06668, 2.99284*10^-8}, {0.13336,
3.22116*10^-8}, {0.20004, 2.58191*10^-8}, {0.26672,
1.99125*10^-7}, {0.3334, 1.21646*10^-8}, {0.40008,
3.35916*10^-7}, {0.46676, 3.79768*10^-7}, {0.53344,
1.02102*10^-7}, {0.60012, 1.17535*10^-6}, {0.666... | B flat | 33,996 | <p>This works. Thank you for the help!</p>
<pre><code>SetOptions[InputNotebook[],
CounterAssignments -> {{"ItemNumbered", -1}}]
SetOptions[InputNotebook[],
StyleDefinitions ->
Notebook[{Cell[StyleData[StyleDefinitions -> "Default.nb"]],
Cell[StyleData["ItemNumbered"],
... |
207,865 | <p>It is known that all $B$, $C$ and $D$ are $3 \times 3$ matrices. And the eigenvalues of $B$ are $1, 2, 3$; $C$ are $4, 5, 6$; and $D$ are $7, 8, 9$. What are the eigenvalues of the $6 \times 6$ matrix
$$\begin{pmatrix}
B & C\\0 & D
\end{pmatrix}$$
where $0$ is the $3 \times 3$ matrix whose entries are all $... | Jacob | 825 | <p>By definition, an eigenvalue $\lambda$ of the block matrix $A$ satisfies</p>
<p>$$\det \begin{pmatrix} B-\lambda I & C \\ 0 & D-\lambda I \end{pmatrix} = 0.$$</p>
<p>Using a <a href="http://en.wikipedia.org/wiki/Determinant#Block_matrices">property of block matrix determinants</a>, we have</p>
<p>$$\det \... |
2,802,156 | <p>I have a function:</p>
<p>$${{\mathop{\rm F}\nolimits} _i}\left( {\bf{\xi }} \right) = \sum\limits_k^N {{\mathop{\rm D}\nolimits} \left( {\frac{1}{N}\sum\limits_j^N {{\mathop{\rm G}\nolimits} \left( {j,{\mathop{\rm I}\nolimits} \left( {j,{\bf{\xi }}} \right)} \right)} - {\mathop{\rm G}\nolimits} \left( {k,{\mathop... | Christian Sykes | 322,386 | <p>$${\rm F}^\prime_i(\xi)=\sum_k^N {\rm D}^\prime_k\left(\frac1N\sum_j^N\left({\rm G}_j({\rm I}_j(\xi))-{\rm G}_k({\rm I}_k(\xi))\right)\right)\cdot \frac1N\sum_j^N\left({\rm G}^\prime_j({\rm I}_j(\xi)){\rm I}^\prime_j(\xi)-{\rm G}^\prime_k({\rm I}_k(\xi)){\rm I}^\prime_k(\xi)\right)$$</p>
<p>Edit: This is the deriva... |
2,860,156 | <p>Let $A\in \mathbb{M}_3(\mathbb{R})$ be a symmetric matrix whose eigen-values are $1,1$ and $3$. Express $A^{-1}$ in the form $\alpha I +\beta A$, where $\alpha, \beta \in \mathbb{R}$.</p>
| Fred | 380,717 | <p>The minimal polynomial of $A$ is $p(x)=(x-1)(x-3)=x^2-4x+3$. By Cayley-Hamilton:</p>
<p>$$A^2-4A+3I=0.$$</p>
<p>This gives</p>
<p>$$A-4I+3A^{-1}=0.$$</p>
|
3,964,910 | <p>Let <span class="math-container">$E$</span> be a metric space, <span class="math-container">$(\mu_n)_{n\in\mathbb N}$</span> be a sequence of finite nonnegative measures on <span class="math-container">$\mathcal B(E)$</span> and <span class="math-container">$\mu$</span> be a probability measure on <span class="math-... | 0xbadf00d | 47,771 | <p>By the same argument as <a href="https://math.stackexchange.com/a/3964926/47771">presented by Botnakov N.</a> we can even show more:</p>
<p>Let <span class="math-container">$\mathcal M(E)$</span> denote the space of finite signed measures on <span class="math-container">$\mathcal B(E)$</span> equipped with the total... |
354,124 | <p>I was stumbled with a basic calculus question by a friend.</p>
<p>The question first asks to find unit vectors $v,w$ s.t $|u+v|$ is
maximal and $|u-w|$ is minimal where $u=(-2,5,3)$.</p>
<p>Then the question asks to find unit vectors $v,w$ s.t $u\cdot v$
is maximal and $|u\cdot w|$ is minimal.</p>
<p>It's easy to... | Heberto del Rio | 71,372 | <p>Given $u=(-2,5,3)$ consider the following function $f_\pm(v)=<u\pm v,u\pm v>$ which is the inner product of $u\pm v$ with itself. The critical point of $f_\pm$ coincide with the critical point of $|u\pm v|$ (why?).</p>
<p>Now $f_\pm(v)=<u\pm v,u\pm v>=<u,u>\pm 2<u,v>+<v,v>=|u|^2+|v|^2\... |
1,114,258 | <p>I am new to differential geometry and Riemannian geometry. </p>
<p>I have on two separate occasions (separated by 6 months) encountered exercises where I feel like I am not giving a complete answer. </p>
<p>Problem 1: </p>
<p><em>Show that the Gaussian curvature of the surface of a cylinder is zero.</em></p>
<p>... | Olórin | 187,521 | <p>This two groups are vector spaces over $\mathbf{Q}$, and as these two groups have the same cardinal, any basis (over $\mathbf{Q}$) of one of them has the same cardinal than has any basis (over $\mathbf{Q}$) of the other one. This allows you to show that these two $\mathbf{Q}$-vector spaces are isomorphic as $\mathbf... |
1,557,165 | <p>Prove that
$$\int_1^\infty\frac{e^x}{x (e^x+1)}dx$$
does not converge.</p>
<p>How can I do that? I thought about turning it into the form of $\int_b^\infty\frac{dx}{x^a}$, but I find no easy way to get rid of the $e^x$.</p>
| mathochist | 215,292 | <p>If we divide the top and bottom by $e^x$, we have</p>
<p>$$\int_{1}^{\infty}\frac{1}{x+x/e^x}$$</p>
<p>For large values of $x$, $x/e^x < x$, so $x+x/e^x < 2x$ and therefore $1/(x+x/e^x) > 1/2x$. Then the tail of $1/2x$ lies under the curve of $1/(x+x/e^x)$. Then since
$$\int_{1}^{\infty}\frac{1}{2x}$$
div... |
1,029,485 | <p>I wish to show the following statement:</p>
<p>$
\forall x,y \in \mathbb{R}
$</p>
<p>$$
(x+y)^4 \leq 8(x^4 + y^4)
$$</p>
<p>What is the scope for generalisaion?</p>
<p><strong>Edit:</strong></p>
<p>Apparently the above inequality can be shown using the Cauchy-Schwarz inequality. Could someone please elaborate,... | Dr. Sonnhard Graubner | 175,066 | <p>we have $8(x^4+y^4)-(x+y)^4=7x^4-4x^3y-6x^2y^2-4xy^3+7y^4=(7x^2+10xy+7y^2)(x-y)^2\geq 0$
this is true.</p>
|
3,451,374 | <p>Given that I have a random variable <span class="math-container">$\max\{K-X, 0\}$</span> where <span class="math-container">$k>0$</span> is a constant and <span class="math-container">$x$</span> is uniformly distributed on <span class="math-container">$[-K, K]$</span> or I guess more generally with any distributi... | Ragib Zaman | 14,657 | <p>I believe the approach you are trying to use is the Law of Total Expectation:</p>
<p><span class="math-container">$$\mathbb{E}[g(X)] = \mathbb{E}[g(X) \ | \ A] \ \mathbb{P}(A) + \mathbb{E}[g(x) \ | \ A^c] \ \mathbb{P}(A^c).$$</span> In your case, taking <span class="math-container">$g(X) = \max(K-X,0)$</span> and <... |
3,275,423 | <p>How do I see that for a <span class="math-container">$K$</span>-vector space <span class="math-container">$V$</span> the map</p>
<blockquote>
<p><span class="math-container">$\bigwedge^d(V^*) \times \bigwedge^d(V) \rightarrow K, (f_1 \wedge ... \wedge f_d, x_1 \wedge ... \wedge x_d) \mapsto det(f_i(x_i)_{i,j})$</... | dan_fulea | 550,003 | <p>I try to write an answer that should clear the definition of the map in the OP. By definition, it is a (multi)linear map. The main instrument is using "universality" when working with elements in the category of vector spaces and (multi)linear applications. (This would not fit as a comment, and it would be hard to ... |
3,371,104 | <p>How could I show <span class="math-container">$$\int_{\mathbb{R}}\dfrac{1}{\sqrt{1+t^{2}}}dt=\infty?$$</span> </p>
<p>I tried to use comparison test so that <span class="math-container">$$\dfrac{1}{\sqrt{1+t^{2}}}\geq \dfrac{C}{t},$$</span> for some <span class="math-container">$C$</span>, and we can use the fact t... | Bernard | 202,857 | <p><strong>Hint</strong>:</p>
<p>Its antiderivative is known:
<span class="math-container">$$\int\!\frac{\mathrm dt}{\sqrt{1+t^2}}=\operatorname{argsinh}t=\ln(t+\sqrt{t^2+1}).$$</span></p>
|
232,540 | <p>I'm trying to prove this conclusion but have some problems with one of the steps.</p>
<p>Assume $X_1,\ldots,X_n,\ldots$ is a sequence of Gaussian random variables, converging almost surely to $X$, prove that $X$ is Gaussian.</p>
<p>We use characteristics function here. Since $|\phi_{X_n}(t)|\leq 1$, by dominated c... | Shashi | 349,501 | <p>Although this question is old and it has a perfect answer already, I provide here a slightly different proof. A proof which mainly shows the convergence of <span class="math-container">$\mu_n$</span> in a funny way (which is the whole point of writing this). </p>
<p>Notice first that we have the existence and finit... |
397,274 | <p>Suppose you have a group isomorphism given by the first isomorphism theorem:</p>
<p><span class="math-container">$$G/\ker(\phi) \simeq \operatorname{im}(\phi)$$</span></p>
<p>What can we say about the group <span class="math-container">$\ker(\phi)\times \operatorname{im}(\phi)$</span>? In particular, when does the f... | Edoardo Lanari | 77,181 | <p>Even if it can't be applied to your example, I would like to point out that in the abelian case (more generally in any abelian category) it's equivalent to have a split exact sequence: $0 \to \ker(\phi) \to G \to Im(\phi) \to0$</p>
|
4,008,141 | <p>Given <span class="math-container">$G= \{ z \in \mathbb{C} | \exists \ n \in \mathbb{Z}^{+} \text{such that} \ z^n=1\}$</span>.
Define a map <span class="math-container">$f : G \to G $</span> by <span class="math-container">$$f(z)=z^k$$</span></p>
<p>where <span class="math-container">$k>1$</span> is fixed and... | Numbra | 743,703 | <p>The problem comes from the fact that <span class="math-container">$n$</span> is not fixed ! For <span class="math-container">$z \in G$</span>, <em>a priori</em>, the "corresponding <span class="math-container">$n$</span>" could be anything.</p>
<p>A good idea to start would be to understand what the elemen... |
4,008,141 | <p>Given <span class="math-container">$G= \{ z \in \mathbb{C} | \exists \ n \in \mathbb{Z}^{+} \text{such that} \ z^n=1\}$</span>.
Define a map <span class="math-container">$f : G \to G $</span> by <span class="math-container">$$f(z)=z^k$$</span></p>
<p>where <span class="math-container">$k>1$</span> is fixed and... | jasmine | 557,708 | <p>From Numbra answer</p>
<p>Motive : to show <span class="math-container">$f$</span> is onto</p>
<p><span class="math-container">$G = \{e^{\frac{2ir\pi}n}\;|\; n \in \mathbb N^\ast , 0\leq r < n\}$</span>.</p>
<p>Now let <span class="math-container">$f $</span> is given by <span class="math-container">$e^{\frac{2i... |
43,505 | <p>I am looking to make a physics based Mathematica project. Ideally the project would take around 12 hours, gathering any experimental data and analyse the findings.</p>
<p>I'd have full access to university physics labs. The project would be for 2nd year physics students in the end and would aim to introduce using M... | Zviovich | 1,096 | <p>Please check Bobthechemist site for some ideas.</p>
<p><a href="http://www.bobthechemist.com/">BobtheChemist's projects</a></p>
<p>Also, some other simple physics experiments done interfacing with Sensors and Arduino here.</p>
<p><a href="http://community.wolfram.com/groups/-/m/t/181641?p_p_auth=PThZ9lzq">An expe... |
1,894,199 | <p>Evaluate definite integral: $$\int_{-\pi/2}^{\pi/2} \cos \left[\frac{\pi n}{2} +\left(a \sin t+b \cos t \right) \right] dt$$</p>
<p>$n$ is an integer. $a,b$ real numbers.</p>
<p>The purpose of the integral - computing matrix elements of an electron Hamiltonian in an elliptic ring in the quantum box basis.</p>
<p>... | Elias Costa | 19,266 | <p>This is not a complete answer. But I think that might be helpful toward the desired solution. Fix $x_n=\frac{\pi n}{2}$. Use the Taylor series $\cos (x_n+h)=\lim_{m\to \infty}\sum^{m}_{k=0} \frac{(-1)^k}{(2k)!}f^{(k)}(x_n)\cdot (x_n+h)^{2k} $ for $h=h(t)=(a\sin t +b\cos t)$ in the interval $ [-|a|-|b|,|a|+|b|]$.
Se... |
160,542 | <p>I suspect the following integration to be wrong. My answer is coming out to be $3/5$, but the solution says $1$.</p>
<p>$$\int_0^1\frac{2(x+2)}{5}\,dx=\left.\frac{(x+2)^2}{5}\;\right|_0^1=1.$$</p>
<p>Please help out. Thanks.</p>
| Pedro | 23,350 | <p>The integration is obtained as follows:</p>
<p>$$\int 2\frac{x+2}{5}dx=\frac{2}{5}\int (x+2)d(x+2)=\frac{2}{5}\int udu=\frac{2}{5}\frac{u^2}{2}=\frac 1 5 (x+2)^2$$</p>
<p>Since $\frac 1 5 (x+2)^2$ is a primitive of $2\frac{x+2}{5}$ we can use FTCII, and get</p>
<p>$$\int 2\frac{x+2}{5}dx=\frac{(\color{red}{1}+2)^... |
160,542 | <p>I suspect the following integration to be wrong. My answer is coming out to be $3/5$, but the solution says $1$.</p>
<p>$$\int_0^1\frac{2(x+2)}{5}\,dx=\left.\frac{(x+2)^2}{5}\;\right|_0^1=1.$$</p>
<p>Please help out. Thanks.</p>
| Cameron Buie | 28,900 | <p>Here's another potential approach that you will likely find useful in the future (though it isn't really necessary, here), called "$u$-substitution".</p>
<p>Let's put $u=x+2$. Now, $x=0$ if and only if $u=2$, and $x=1$ if and only if $u=3$. Also, $$\frac{du}{dx}=\frac{d}{dx}[x+2]=1,$$ and if we treat $\frac{du}{dx}... |
373,906 | <p>(This question is <a href="https://math.stackexchange.com/questions/3859476">originally from Math.SE</a> where it was suggested that I ask the question here)</p>
<p>Let <span class="math-container">$G$</span> be a finite group with fewer than <span class="math-container">$p^2$</span> Sylow <span class="math-containe... | Thomas Browning | 95,685 | <p>Now that I understand things better, let me also give a direct proof (using essentially the same idea as Brodkey's theorem).</p>
<p>Let <span class="math-container">$P,Q,R$</span> be Sylow <span class="math-container">$p$</span>-subgroups of <span class="math-container">$G$</span>, let <span class="math-container">$... |
644,494 | <p><strong>Question</strong></p>
<blockquote>
<p>If for some real number $a$, $\lim_{x\to 0}\frac{\sin 2x + a\sin x}{x^3}$ exists, then the limit is equal to:</p>
</blockquote>
<p>Here what i have done</p>
<p>since it is of $0/0$ form applying L' Hospital's rule$$\implies\lim_{x\to0}\frac{\sin 2x + a \sin x}{x^3} ... | mathlove | 78,967 | <p>Since $3x^2\to 0,$ the numerator also has to go to $0$ when $x\to 0$. </p>
<p>Hence, $2+a=0\iff a=-2.$</p>
<p>So, you'll have
$$\lim_{x\to 0}\frac{2\cos{2x}-2\cos x}{3x^2}.$$</p>
<p>You can use L' Hospital's rule twice.</p>
|
644,494 | <p><strong>Question</strong></p>
<blockquote>
<p>If for some real number $a$, $\lim_{x\to 0}\frac{\sin 2x + a\sin x}{x^3}$ exists, then the limit is equal to:</p>
</blockquote>
<p>Here what i have done</p>
<p>since it is of $0/0$ form applying L' Hospital's rule$$\implies\lim_{x\to0}\frac{\sin 2x + a \sin x}{x^3} ... | Mhenni Benghorbal | 35,472 | <p>You can use the Taylor series</p>
<p>$$ \frac{\sin2x + a\sin x}{x^3} = \frac{(2x-(2x)^3/3!+\dots)+a (x-x^3/3!+\dots)}{x^3}$$</p>
<p>$$ \sim_{x\to 0} \frac{(a+2)x-(2^3+1)x^3/3!}{x^3}\dots\,. $$</p>
<p>Can you finish it?</p>
|
644,494 | <p><strong>Question</strong></p>
<blockquote>
<p>If for some real number $a$, $\lim_{x\to 0}\frac{\sin 2x + a\sin x}{x^3}$ exists, then the limit is equal to:</p>
</blockquote>
<p>Here what i have done</p>
<p>since it is of $0/0$ form applying L' Hospital's rule$$\implies\lim_{x\to0}\frac{\sin 2x + a \sin x}{x^3} ... | user44197 | 117,158 | <p>This is not an answer as others have done a great job of it. I will try and explain <em>how to think</em> about the problem. In the end each of us approach a problem differently so how we think may be quite different. So, this is just my thinking.</p>
<p>The sine function has only odd powers since it is an odd func... |
1,221,221 | <p>Suppose $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous and has a left derivative, $f^-$, everywhere in a neighborhood of $x.$ Suppose $f^-$ is continuous at $x.$ Does this imply that $f$ is differentiable at $x$?</p>
| Rolf Hoyer | 228,612 | <p>Only if it's 'conditionally' divergent in the sense that the positive terms form a divergent series, and also the negative terms form a divergent series. You would also need $a_n\to 0$, of course. In this case, you can use the same algorithm for rearrangement in order to force convergence to some (arbitrary) value... |
2,050,867 | <p>I would like to prove for all $x, y \in \mathbb{R}$ that $\dfrac{e^{x}+e^{y}}{2} \geq e^{\frac{x+y}{2}}$. My idea, is to show that $f(x,y) \ge 0$, it means that $(0,0)$ is the minimum of $f(x,y)$. So, I compute the equation: $\nabla f(x,y)=\begin{pmatrix}0 \\0 \end{pmatrix}$. I find that the solutions are $x=y$. <st... | Fred | 380,717 | <p>$0 \le (e^{x/2}-e^{y/2})^2=e^x-2e^{\frac{x+y}{2}}+e^y$</p>
|
469,947 | <blockquote>
<p>Show that the presentation $G=\langle a,b,c\mid a^2 = b^2 = c^3 = 1, ab = ba, cac^{-1} = b, cbc^{-1} =ab\rangle$ defines a group of order $12$.</p>
</blockquote>
<p>I tried to let $d=ab\Rightarrow G=\langle d,c\mid d^2 =c^3 = 1, c^2d=dcdc\rangle$. But I don't know how to find the order of the new pre... | Martin Brandenburg | 1,650 | <p><em>Direct proof.</em></p>
<p>$N:=\langle a,b \rangle$ is clearly a normal subgroup of $G$ with $G/N = \langle c : c^3 = 1 \rangle = C_3$, and $N$ is a quotient of the Klein four-group $V_4 = \langle a,b : a^2=b^2=1, ab=ba \rangle$. Therefore $|G|$ divides $12$. Now one checks that the permutations $a=(12)(34)$, $b... |
1,252,591 | <p>How many odd numbers can be formed using digits $0,4,5,7$.
I am getting answer $12$ but the actual answer is $14$. </p>
| gnasher729 | 137,175 | <p>There are 24 permutations of the digits, six each starting with 0, 4, 5 and 7. Four digit numbers don't start with a 0, leaving 3 times 6 permutations. If the first digit is 5 or 7 then we remove the two numbers ending in 4, so 3x6 - 2x2 = 14. </p>
|
288,974 | <p>Alright this maybe really funny but I want to know why is this wrong. We often come across identities which we prove by multiplying both the sides of the identity by a certain entity but why don't we multiply it by $0$. That way every identity will be proved in one single line. That is so stupid. I mean, by that way... | Nick | 60,044 | <p>Zero differs from other numbers and mathematical entities which represent some sort of value because it is the absence of value, <em>not a value itself</em>. Thus, it would be incorrect to say $a \cdot 0 = b \cdot 0$ means that $a = b$. Similarly, in the case of infinity: let's say that $a$ does equal $b$. However, ... |
1,579,170 | <p>This problem is dependent because it matters which one you choose, So i don't think we can do the multiplication thing in this one. </p>
<ul>
<li>Probability of ( non defective ) = 6/10 </li>
</ul>
<p>What does the question mean when it says all will be non-defective? is "all" the 2 randomly chosen telephone? How ... | David | 119,775 | <p>Use the Chinese Remainder Theorem as you have suggested. But the easiest way is to use it to <strong>check</strong> your answer, not to <strong>find</strong> the answer. Let's write $l={\rm lcm}(p-1,q-1)$.</p>
<p>We have
$$a^{p-1}\equiv1\pmod p\ ,$$
and since $l$ is a multiple of $p-1$,
$$a^l\equiv1\pmod p\ .$$
S... |
3,231,869 | <p>I am a little confused what this actually means: </p>
<p><span class="math-container">$e^{x+e^x}$</span></p>
<p>It is obviously not the same if I for example
<span class="math-container">$$e^{x}:= \lambda \\
e^{x+e^x} \neq \lambda^\lambda
$$</span></p>
| Dr. Sonnhard Graubner | 175,066 | <p>It is the same as <span class="math-container">$$e^x\cdot e^{e^x}$$</span></p>
|
1,612,353 | <blockquote>
<p>In how many ways out of $20$ students you can select $1$ treasurer, $1$ secretary and $3$ more representatives?</p>
</blockquote>
<p>I understand that for single selections I can multiply with the availability of the persons. Like for treasurer I can have $20$ options, for secretary then I have $19$ ... | vrugtehagel | 304,329 | <p>There are $20\choose 1$ options for the treasurer. Assuming one students can't have multiple roles, we have $19\choose 1$ options for the secretary and $18\choose 3$ options for the representatives, making the total number of possibilities $20\cdot 19\cdot 816=310080$.</p>
<p>Hope this helped!</p>
|
292,122 | <p>This question actually came out of a question. In some other post, I saw a reference and going through, found this, $n>0$.</p>
<p>Solve for n explicitly without calculator:
$$\frac{3^n}{n!}\le10^{-6}$$</p>
<p>And I appreciate hint rather than explicit solution.</p>
<p>Thank You.</p>
| mjqxxxx | 5,546 | <p>Note that, for $n=3m$, $$3^{-3m}{(3m)!}=\left[m\left(m-\frac{1}{3}\right)\left(m-\frac{2}{3}\right)\right]\cdots\left[1\cdot\frac{2}{3}\cdot\frac{1}{3}\right] <\frac{2}{9}\left(m!\right)^3.$$
So you have to go at least far enough so that
$$
\frac{2}{9}\left(m!\right)^3>10^{6},
$$
or $m! > \sqrt[3]{4500000} ... |
801,562 | <p>We consider that $R$ is a commutative ring with $1_R$.</p>
<p>Each $c \in R^*$(if we see it as a constant polynomial), divides each polynomial of $R[X]$.</p>
<p>($c \in R^*$ means that $c$ is invertible.)</p>
<p>I haven't undersotod it..Could you explain it to me?</p>
<p>Does it mean that if we have a polynomial... | drhab | 75,923 | <p>If $c\in R^*$ then polynomial $f=a_{0}+\cdots+a_{n}X^{n}$ can be written as $cg$ where
$g=c^{-1}a_{0}+\cdots+c^{-1}a_{n}X^{n}$.</p>
|
1,148,760 | <p>$\displaystyle \int x^7\cos x^4 dx$</p>
<p>I tried first by letting $x^4 = u$ and then using integration by parts by assigning f(x) to $u^\frac74$ and cos(u) to g'(x) and I end up getting after applying parts twice, the same integral on the RHS as what we are looking for. So I bring it in on the LHS and add it over... | abel | 9,252 | <p>make a substitution $u = x^4, du = 4x^3 dx.$ the
$$\int x^7 \cos x^4 \, dx = \frac14\int u\cos u\, du
= \frac14 \int u \, d(\sin u) = \frac14 \left( u\sin u -\int \sin u \,du\right)
=\frac14 \left( u\sin u + \cos u\right) + C = \frac14 \left( x^4\sin x^4 + \cos x^4\right) +C $$ </p>
|
3,142,339 | <p>Let <span class="math-container">$p$</span> be a real number. I am looking for all <span class="math-container">$(x,y)$</span> such that <span class="math-container">$\ln[e^{x}+e^{y}]=px+(1-p)y$</span>. My effort:</p>
<p>Take exponent of both sides to obtain <span class="math-container">$e^{x}+e^{y}=e^{px}e^{(1-p)y... | Eric Towers | 123,905 | <p>For future complex continued fractions...</p>
<p>For a continued fraction to converge, the sequence of convergents (values finite initial segments of the partial fraction expression) must converge to a particular complex number, that is, to a specific argument and magnitude. For the real continued fraction you men... |
1,675,329 | <p>What's the value of $\sum_{i=1}^\infty \frac{1}{i^2 i!}(= S)$?</p>
<p>I try to calculate the value by the following.</p>
<p>$$\frac{e^x - 1}{x} = \sum_{i=1}^\infty \frac{x^{i-1}}{i!}.$$
Taking the integral gives
$$ \int_{0}^x \frac{e^t-1}{t}dt = \sum_{i=1}^\infty \frac{x^{i}}{i i!}. $$</p>
<p>In the same, we gets... | TOM | 118,685 | <p>By A.S.'s comment, we gets
$$\int_{s=0}^x \frac{1}{s} \int_{t=0}^s \frac{e^t-1}{t}dt ds = \int_{t=0}^x \frac{e^t-1}{t}\int_{s=t}^x \frac{1}{s}ds dt = \int_{0}^x \frac{(e^t-1) (\log{x} - \log{t})}{t}dt.$$</p>
<p>So, we holds
$$S = - \int_{0}^1 \frac{(e^t-1) \log{t}}{t}dt = - \int_{- \infty}^0 (e^{e^u}-1) u du.$$</... |
934,660 | <p>Prove that for $ n \geq 2$, n has at least one prime factor.</p>
<p>I'm trying to use induction. For n = 2, 2 = 1 x 2. For n > 2, n = n x 1, where 1 is a prime factor. Is this sufficient to prove the result? I feel like I may be mistaken here.</p>
| Sheheryar Zaidi | 131,709 | <p>Inductive case: Assume $n$ has prime factors: It is either a prime, then it's got a prime factor (itself), and then $n+1$ is even and has 2 as a prime factor. If $n$ isn't prime, then the FTA says it has a unique prime factorization and $n+1$ is either prime or FTA says it has a prime factorization. </p>
<p>Inducti... |
4,491,251 | <p>Per the question title, what's the easiest way to evaluate the following?
<span class="math-container">$$\int_0^{\pi/6}\sec x\,dx$$</span></p>
<p>You can do something like computing the derivatives of <span class="math-container">$\sec x$</span> and <span class="math-container">$\tan x$</span>, adding them up, compu... | Bob Dobbs | 221,315 | <p>Let's make the famous <span class="math-container">$z=\tan(\frac{x}{2})$</span> substitution. Then <span class="math-container">$\sec x=\frac{1+z^2}{1-z^2}$</span> and <span class="math-container">$dx=\frac{2dz}{1+z^2}$</span>. Knowing that <span class="math-container">$\tan(\frac{\pi}{12})=2-\sqrt{3}$</span>, we co... |
4,491,251 | <p>Per the question title, what's the easiest way to evaluate the following?
<span class="math-container">$$\int_0^{\pi/6}\sec x\,dx$$</span></p>
<p>You can do something like computing the derivatives of <span class="math-container">$\sec x$</span> and <span class="math-container">$\tan x$</span>, adding them up, compu... | Quanto | 686,284 | <p><span class="math-container">$$\int \sec x\,dx= \int \frac{\sec^2x}{\sqrt{1+\tan^2x}}dx=\sinh^{-1}(\tan x)+C
$$</span></p>
|
1,249,707 | <blockquote>
<p>Assume <span class="math-container">$V$</span> to be a finite dimensional vector space. Define the algebraic multiplicity <span class="math-container">$am(\lambda)$</span> of an eigenvalue <span class="math-container">$\lambda$</span> of a linear operator <span class="math-container">$T:V\to V$</span> a... | JR2 | 1,066,996 | <p>I would like to complement Marc van Leeuwen's answer. I am going to use some results from the PDF that Marc linked (<a href="https://www.maa.org/sites/default/files/pdf/awards/Axler-Ford-1996.pdf" rel="nofollow noreferrer">Axler's paper</a>). First, we know that <span class="math-container">$V$</span> can be written... |
173,387 | <p>How can I indent properly long code in <em>Mathematica</em>?
Are there some best practices?</p>
| Fraccalo | 40,354 | <p>As already said in other answers, this is very subjective, but here a tip I find very useful for coding plots: I put every command on a different line, and I use the comma separator at the beginning of the line.
This is quite handy for commenting parts of the code, to enable/disable some plot options quickly (i.e. j... |
4,280,328 | <p>I think the substitution <span class="math-container">$x=\xi+\eta,$</span> <span class="math-container">$y=\xi-\eta$</span> can be done. Then the equation takes the form <span class="math-container">$$ \begin{gathered} 38(\xi^{2}+\eta^{2})=221+33(\xi^{2}-\eta^{2}) \\ 5 \xi^{2}+71 \eta^{2}=221 \end{gathered} $$</span... | sirous | 346,566 | <p>A simpler approach:</p>
<p><span class="math-container">$19y^2-(33x)y+19x^2-221=0$</span></p>
<p>we solve this for y:</p>
<p><span class="math-container">$\Delta=(33x)^2-4\times 19\times (19x^2-221)$</span></p>
<p>Or:</p>
<p><span class="math-container">$\Delta=16796-355\geq 0\Rightarrow x^2<47.31$</span></p>
<p>... |
293,047 | <p>When I am reading through higher Set Theory books I am frequently met with statements such as '$V$ is a model of ZFC' or '$L$ is a model of ZFC' where $V$ is the Von Neumann Universe, and $L$ the Constructible Universe. For instance, in Jech's 'Set Theory' pg 176, in order to prove the consistency of the Axiom of Ch... | Joel David Hamkins | 1,946 | <p>What is shown in the cases you mention is not that the model is a model of ZFC, made as a single statement, but rather the <em>scheme</em> of statements that the model satisfies every individual axiom of ZFC, as a separate statement for each axiom. </p>
<p>The difference is between asserting "$L$ is a model of ZFC"... |
1,480,671 | <p>How to prove $\int_{0}^{\infty}{h(t)\mathbb{E}(I(X>t))dt}=\mathbb{E}(\int_{0}^{\infty}{h(t)I(X>t)dt})$.
Can I treat $h(t)$ as a constant respect to $X$? Then, directly get the result?</p>
<p>The point is I do not understand what $\mathbb{E}(\int_{0}^{\infty}{h(t)I(X>t)dt})$ is.</p>
| John Dawkins | 189,130 | <p>The integral $\int_0^\infty h(t)I(X>t)\,dt$ is a random variable, call it $Y$. The role of the indicator random variable $I(X>t)$ is to restrict the $t$-integration to the (random) interval $(0,X)$. In other words,
$$
Y(\omega) =\int_0^{X(\omega)} h(t)\,dt,
$$
for each sample point $\omega$ in the sample space... |
1,894,867 | <p>Let $n=3^{1000}+1$. Is n prime?</p>
<p>My working so far:</p>
<p>$n=3^{1000}+1 \cong 1 \mod 3$</p>
<p>I notice that n is of form; $n=3^n+1$</p>
<p>Seeking advice tips, and methods on progressing this.</p>
| Mythomorphic | 152,277 | <p>Taking binomial expansion,</p>
<p>\begin{align}
3^{1000}+1&=(2+1)^{1000}+1\\
&=1+\sum_{k=0}^{1000}{1000\choose k}2^k1^{1000-k}\\
&=1+{1000\choose 0}+\sum_{k=1}^{1000}{1000\choose k}2^k\\
&=2\left[1+\sum_{k=1}^{1000}{1000\choose k}2^{k-1}\right]
\end{align}</p>
<p>So $3^{1000}+1$ is composite.</p>
|
1,102,928 | <p>Let $\mathcal{H}$ be a Hilbert space. I am trying to show that every self-adjoint idempotent continuous linear transformation is the orthogonal projection onto some closed subspace of $\mathcal{H}$. If $P$ is such an operator, the obvious thing is to consider $S=\{Px:x\in\mathcal{H}\}$. However, I'm having trouble s... | copper.hat | 27,978 | <p>A convenient way to check for closure of subspaces is to try to write the subspace as the kernel of some continuous operator.</p>
<p>Note that $(I-P)x = 0$ <strong>iff</strong> $Px=x$.</p>
<p>Note that $x \in S$ <strong>iff</strong> $x = Py$ for some $y$ <strong>iff</strong> $Px = x$, and so
$S = \ker (I-P)$. Henc... |
1,375,365 | <p>Find all polynomials for which </p>
<p>What I have done so far:
for $x=8$ we get $p(8)=0$
for $x=1$ we get $p(2)=0$</p>
<p>So there exists a polynomial $p(x) = (x-2)(x-8)q(x)$</p>
<p>This is where I get stuck. How do I continue?</p>
<p><strong>UPDATE</strong></p>
<p>After substituting and simplifying I get
$(x-... | drhab | 75,923 | <p>The route you take is fruitful.</p>
<p>$p\left(x\right)=\left(x-2\right)\left(x-8\right)q\left(x\right)$
leads to:</p>
<p>$$\left(x-4\right)q\left(2x\right)=2\left(x-2\right)q\left(x\right)$$</p>
<p>Then $4$ must be a root of $q$, so $q\left(x\right)=\left(x-4\right)r\left(x\right)$ leading to:</p>
<p>$$r\left(2... |
1,375,365 | <p>Find all polynomials for which </p>
<p>What I have done so far:
for $x=8$ we get $p(8)=0$
for $x=1$ we get $p(2)=0$</p>
<p>So there exists a polynomial $p(x) = (x-2)(x-8)q(x)$</p>
<p>This is where I get stuck. How do I continue?</p>
<p><strong>UPDATE</strong></p>
<p>After substituting and simplifying I get
$(x-... | Eric Towers | 123,905 | <p>The following is essentially @drhab's solution, but uses only one idea repeatedly.</p>
<p>From $$ (x-8)p(2x) = 8(x-1)p(x) $$ we see $x-8$ divides $p(x)$. Let $p(x) = (x-8)p_1(x)$ and substitute, yielding $$ 2(x-8)(x-4)p_1(2x) = 8(x-1)(x-8)p_1(x) $$
From this we see $x-4$ divides $p_1(x)$. Let $p_1(x) = (x-4)p_2(x... |
444,486 | <p>I am teaching myself real analysis, and in this particular set of lecture notes, the <a href="http://www.math.louisville.edu/~lee/RealAnalysis/IntroRealAnal-ch01.pdf" rel="nofollow">introductory chapter on set theory</a> when explaining that not all sets are countable, states as follows:</p>
<blockquote>
<p>If $S... | jwg | 64,062 | <p>To augment Kendra Lynne's answer, what does it mean to say that signal analysis in $\mathbb{R}^2$ isn't as 'clean' as in $\mathbb{C}$?</p>
<p>Fourier series are the decomposition of periodic functions into an infinite sum of 'modes' or single-frequency signals. If a function defined on $\mathbb{R}$ is periodic, say... |
444,486 | <p>I am teaching myself real analysis, and in this particular set of lecture notes, the <a href="http://www.math.louisville.edu/~lee/RealAnalysis/IntroRealAnal-ch01.pdf" rel="nofollow">introductory chapter on set theory</a> when explaining that not all sets are countable, states as follows:</p>
<blockquote>
<p>If $S... | G Cab | 317,234 | <p>Electrical engineers are much entangled in the complex numbers field just because a fundamental circuit block, like it is a RC, "works perfectly" with the complex numbers: its <a href="https://en.wikipedia.org/wiki/Electrical_impedance" rel="nofollow noreferrer">impedance</a> looks to be "naturally&qu... |
654,408 | <p>I know that the volume form on $S^1$ is $\omega= ydx-xdy$. But how I can derive that? The only things that I know are the definition of differential q-form, and the fact that the vector field $v= y \frac{\partial}{\partial x}-x\frac{\partial}{\partial y}$ never vanishes on $S^1$.</p>
| Martín-Blas Pérez Pinilla | 98,199 | <p>See the proof of Proposition 12.6 in <a href="http://www.math.toronto.edu/mat1300/orientation.11.pdf" rel="nofollow">http://www.math.toronto.edu/mat1300/orientation.11.pdf</a>.</p>
<p>EDIT: Wikipedia gives the following reference for the deduction of the generalization of your formula: Flanders, Harley (1989). Diff... |
2,038,520 | <p>I know that the series b. converges as $\sum \frac{1}{n^p}$ converges for $p>1$, So a. also converges. I want to know the sum.</p>
<blockquote>
<blockquote>
<p>a.$1+\frac{1}{9}+\frac{1}{25}+\frac{1}{49}+.....$</p>
<p>$b.1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+.....$</p>
</blockquote>
</blockquote... | Ethan Alwaise | 221,420 | <p>The Riemann zeta function is defined as
$$\zeta(s) = \sum_{n=1}^{\infty}\frac{1}{n^s}.$$
The value of $\zeta(2)$ is known to be $\frac{\pi^2}{6}$. Thus
$$\sum_{n=0}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}.$$
The series in a. in your post can be written as
$$\sum_{n=1}^{\infty}\frac{1}{n^2} - \sum_{n=1}^{\infty}\fra... |
4,107,920 | <p>the question tells me that <span class="math-container">$P(A|B)>P(A)$</span> and needs me to prove: <Br></p>
<ol>
<li><span class="math-container">$P(B|A)>P(B)$</span> <br></li>
<li><span class="math-container">$P(B^c|A)<P(B^c)$</span></li>
</ol>
<p>In general all I want to ask is do I need to care that <sp... | Kavi Rama Murthy | 142,385 | <p>If <span class="math-container">$P(A)=0$</span> then the hypothesis becomes <span class="math-container">$0 >0$</span> which is false. So we must have <span class="math-container">$P(A) >0$</span> and 1) follows from the hyptohesis by your argument.</p>
|
189,069 | <p>The Survival Probability for a walker starting at the origin is defined as the probability that the walker stays positive through n steps. Thanks to the Sparre-Andersen Theorem I know this PDF is given by</p>
<pre><code>Plot[Binomial[2 n, n]*2^(-2 n), {n, 0, 100}]
</code></pre>
<p>However, I want to validate this ... | Mr.Wizard | 121 | <p>Something seems odd to me about your code. You are summing twice, once with <a href="https://reference.wolfram.com/language/ref/Accumulate.html" rel="noreferrer"><code>Accumulate</code></a> and once with <a href="https://reference.wolfram.com/language/ref/FoldList.html" rel="noreferrer"><code>FoldList</code></a>. ... |
1,121,845 | <p>let $G$ be a multiplicative group of non-zero complex analysis.consider the group homomorphism $\phi:G\rightarrow G$ defined by $\phi(z)=z^4$.</p>
<p>1.Identify kernel of $\phi=H$.</p>
<p>2.Identify $G/H$</p>
<p>My try:</p>
<p>let $z\in \ker \phi$ then $\phi(z)=1\implies z^4=1$
let $z=re^{i\theta}\implies r^4... | Aaron Maroja | 143,413 | <p>Hint: </p>
<ol>
<li><p>If $z \in \ker \phi$ then $\phi (z) = 1$. Think of the fourth root of unit. </p></li>
<li><p>Use the <a href="http://en.wikipedia.org/wiki/Isomorphism_theorem" rel="nofollow">Isomorphism Theorem</a>.</p></li>
</ol>
<p>If $\xi$ is a n-th root of unit and $z ^n = a$ then $a\xi$ is a root of $... |
2,137,591 | <blockquote>
<p>$$\int \frac{1}{x+x\log x}\,dx$$</p>
</blockquote>
<p>I couldn't use any of the integration techniques to solve this, any help will be appreciated!</p>
| Chinny84 | 92,628 | <p>$$
\int \frac{1}{x}\frac{1}{1+\log x}dx
$$
let $1+ \log x = u\implies du =\frac{1}{x}\frac{1}{\ln 10}dx$ </p>
<p>then we have
$$
\int \ln 10\frac{1}{u}du = \ln 10 \ln u + C
$$</p>
|
1,456,224 | <p>I've been asked to compute the Euler-Lagrange equation and second variation of the functional $$I[y]=\int_{a}^{b}(y'^2+y^4)dx$$
with boundary conditions $y(a)=\alpha$, $y(b)=\beta$. It's easy to see that $$I[y+\delta y]=I[y]+\int_{a}^{b}\delta y(4y^{3}-2y'') dx+\int_{a}^{b}(6y^{2}\delta y^{2}+\delta y'^{2})dx$$
So t... | Qmechanic | 11,127 | <p>OP has essentially already proven that there is only a trivial solution $y\equiv 0$. See also the answer by John Ma. Now OP is pondering <em>Why?</em> Perhaps he would appreciate a bit of physics intuition: The <a href="http://en.wikipedia.org/wiki/Lagrangian_mechanics" rel="nofollow">Lagrangian</a> $L=T-V$ describe... |
51,509 | <p>Here is a problem due to Feynman. If you take 1 divided by 243 you get 0.004115226337 .... It goes a little cockeyed after 559 when you're carrying out the decimal expansion, but it soon straightens itself out and repreats itself nicely. Now I want to see how many times it repeats itself. Does it do this indefinitel... | Dr. belisarius | 193 | <pre><code>RealDigits[1/243]
(*
{{{4, 1, 1, 5, 2, 2, 6, 3, 3, 7, 4, 4, 8, 5, 5, 9, 6, 7, 0, 7, 8, 1, 8, 9, 3, 0, 0}}, -2}
*)
</code></pre>
|
51,509 | <p>Here is a problem due to Feynman. If you take 1 divided by 243 you get 0.004115226337 .... It goes a little cockeyed after 559 when you're carrying out the decimal expansion, but it soon straightens itself out and repreats itself nicely. Now I want to see how many times it repeats itself. Does it do this indefinitel... | srgntoptics | 63,808 | <p>When you examine the repetitions over a larger scale, another interesting repetition shows up.</p>
<p>using the same code as eldo:</p>
<pre><code>Count[#, Max@#] &[ StringLength /@ Rest@StringSplit[ToString@N[1/243, 10^2], "00"]]
</code></pre>
<p>with <code>10^2</code> digits all the way to <code>10^9<... |
2,027,337 | <p>My homework sets up the problem accordingly:</p>
<blockquote>
<p>An object moves horizontally in one dimension with a velocity given by
v(t) = $8\cos\left(\frac{\pi \cdot t}{6}\right)$ m/s.</p>
<p>Find the The position of the object is given by s(t) =
$s\left(t\right)=\int _0^t\:v\left(y\right)\:dy\:$ ... | A.D. | 294,708 | <p>You need find the function position $s(t)$, how you know the functon velocity $v(t)$ your problem is equal to find indefined integral $\int v(t)dt$. Let be $g(t) = \frac{\pi - t}{ 6}$, this implies that $\frac{dg}{dt} = -1/6$ and therefore
$$ v(t) = -48 g'\cos{g}$$
this implies that </p>
<p>$$\int v(t) dt = \int 4... |
2,027,337 | <p>My homework sets up the problem accordingly:</p>
<blockquote>
<p>An object moves horizontally in one dimension with a velocity given by
v(t) = $8\cos\left(\frac{\pi \cdot t}{6}\right)$ m/s.</p>
<p>Find the The position of the object is given by s(t) =
$s\left(t\right)=\int _0^t\:v\left(y\right)\:dy\:$ ... | MPW | 113,214 | <p>To evaluate $$\int_a^bk\cos cx\; dx$$
Put $$u=cx$$
$$du = c\; dx$$
so
$$dx = \frac1c (c\; dx) = \frac1c \; du$$
$$x=a \iff u = ca$$
$$x=b \iff u = cb$$
and
$$\int_a^b k\cos cx\; dx = \frac kc\int_{ca}^{cb}\cos u\; du
= \left[\frac kc\sin u\right]_{ca}^{cb}= \frac kc(\sin cb - \sin ca)$$
In your case, $a=0$, $b=t$, $... |
3,888,365 | <p>I have been trying to understand this limit:</p>
<p><span class="math-container">$$\lim_{x \to 0}\frac{tan(x)-sin(x)}{x^2}$$</span></p>
<p>When aplying the l'Hopital rule I arrive to the limit being <span class="math-container">$0$</span> but when doing things organically I get an indetermination:</p>
<p><span class... | Somos | 438,089 | <p>In the numerator, since both <span class="math-container">$\tan(x)$</span> and <span class="math-container">$\sin(x)$</span> are odd functions, the difference is also an odd function. The denominator <span class="math-container">$x^2$</span> is an even function. The quotient is an odd function. If the limit as <span... |
2,384,422 | <p>I'm really stuck on how to go about solving the following first order ODE; I've got little idea on how to approach it, and I'd really appreciate if someone could give me some hints and/or working for a solution so I can have a reference point on how to approach these sorts of problems.</p>
<p>The following is one o... | Frieder Jäckel | 440,045 | <p>I always like to think of these type of ODE's in terms of the product rule.
\begin{equation}x=y'e^{\sin(x)}+y\cos(x)e^{\sin(x)}=\left(ye^{\sin(x)}\right)'
\end{equation}
So integrating both sides and dividing by $e^{\sin(x)}$ yields\begin{equation}y=e^{-\sin(x)}\left(\frac{1}{2}x^2+c\right).
\end{equation}</p>
|
2,503,306 | <p>Suppose $g{^n}$=e. Show the order of $g$ divides $n$.</p>
<p>Would I use Eulers Theorem???;</p>
<p>$a{^{\phi p}}$ $\equiv1 \pmod p$</p>
<p>$a{^{p-1}}\equiv1 \pmod p$</p>
<p>$a{^p}\equiv a\pmod p$</p>
<p>So then I would have </p>
<p>$g{^n}\equiv g\pmod n$</p>
<p>then I think you use the $\gcd$, which states $\... | José Carlos Santos | 446,262 | <p>Let $o$ be the order of $g$. Then $n$ can be written as $oq+r$, with $r\in\{0,1,\ldots,o-1\}$. Therefore\begin{align}e&=g^n\\&=g^{oq+r}\\&=(g^o)^q.g^r\\&=g^r.\end{align} Since $g^r=1$, since $o$ is the order of $g$ and since $r<o$, $r$ can only be equal to $0$. And this means that $o\mid n$.</p>
|
203,505 | <p>Let <span class="math-container">$P(x)$</span> be a non-constant polynomial with real coefficients.</p>
<p>Can <a href="http://en.wikipedia.org/wiki/Natural_density" rel="noreferrer">natural density</a> of</p>
<p><span class="math-container">$$\{n\ |\ \lfloor P(n)\rfloor \ \text{is prime.}\}$$</span></p>
<p>be posit... | Terry Tao | 766 | <p>No. There are two cases. Firstly, suppose that one of the non-constant coefficients of $P$ is irrational. Then, by the Weyl equidistribution theorem, $\lfloor P(n) \rfloor$ is equidistributed mod $W$ for any modulus $W$, which already limits the natural density of the prime-producing $n$ to be at most $\phi(W)/W$... |
1,267,395 | <p>Julie is required to pay a 2 percent tax on all income over 3,000. She also has to pay 2.5 percent on all income over 20,000. She earned more than 20,000 and paid 992.50 what was her total income</p>
| DeepSea | 101,504 | <p>Let $x$ be her total income, then we have: $0.02(x-3,000) + 0.025(x-20,000) = 992.5$ Can you solve this linear equation?</p>
|
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