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 |
|---|---|---|---|---|
2,981,745 | <p>Wolfram Alpha shows that <span class="math-container">$$\int_{0}^{2\pi}x^2\ln (1-\cos x)dx = -\frac{8}{3} \pi (\pi^2 \ln(2) + 3 \zeta(3))$$</span>
I tried to use the Fourier series
<span class="math-container">$$\ln (1-\cos x)=-\sum_{n=1}^{\infty} \frac{\cos^nx}{n}.$$</span>
I am not sure how to continue from this ... | Larry | 587,091 | <p>Thanks mrtaurho, I am able to finish it.
<span class="math-container">$$\begin{align}
\int_{0}^{2\pi}\pi^2\ln(1-\cos x)dx
&=-\frac{8\pi^3}3\ln(2)-\sum_{n=1}^{\infty}\frac{16}n\int_0^{\pi}x^2\cos(2nx)~dx\\
&=-\frac{8\pi^3}3\ln(2)-\sum_{n=1}^{\infty}\frac{16}n\left[-x^2\frac{\sin(2nx)}{2n}+\frac{2x\cos(2nx)}{4... |
666,461 | <p>The function $f(x)=x+\log x$ has only one root on $(0,\infty)$ which is in $(0,1)$.</p>
<p>Using the Intermediate value theorem: $f$ is continuous on $(0,\infty)$ and $f(0)=0+\log(0)=-\infty<0$ and $f(1)=1+\log(1)=1>0$. So there exists an $x$ such $f(x)=0$.</p>
<p>But how to show that this $x$ is the only ro... | Mhenni Benghorbal | 35,472 | <p><strong>Hints:</strong></p>
<p>1) $\lim_{x\to 0^{+}}f(x)=-\infty .$</p>
<p>2) $f(1)=1$</p>
<p>3) $f(x)$ is increasing.</p>
<p>Can you conclude?</p>
|
2,930,413 | <p>The problem is as shown. I tried using gradient and Hessian but can not make any conclusions from them. Any ideas?</p>
<p><span class="math-container">$$\max x_1^{a_1}x_2^{a_2}\cdots x_n^{a_n}$$</span></p>
<p>subject to</p>
<p><span class="math-container">$$\sum_{i=1}^nx_i=1,\quad x_i\geq 0,\quad i=1,2,\ldots,n,$... | David G. Stork | 210,401 | <p><span class="math-container">$$x(t) = A \cos (2 \pi \omega t + \phi)$$</span></p>
<p>where <span class="math-container">$\omega = 1$</span>.</p>
|
2,930,413 | <p>The problem is as shown. I tried using gradient and Hessian but can not make any conclusions from them. Any ideas?</p>
<p><span class="math-container">$$\max x_1^{a_1}x_2^{a_2}\cdots x_n^{a_n}$$</span></p>
<p>subject to</p>
<p><span class="math-container">$$\sum_{i=1}^nx_i=1,\quad x_i\geq 0,\quad i=1,2,\ldots,n,$... | user0102 | 322,814 | <p><strong>HINT</strong></p>
<p>Suppose <span class="math-container">$x(t) = e^{kt}$</span>, where <span class="math-container">$k\in\mathbb{C}$</span>:
<span class="math-container">\begin{align*}
x^{\prime\prime} + x = 0 \Longleftrightarrow k^{2}e^{kt} + e^{kt} = 0 \Longleftrightarrow k^{2} + 1 = 0 \Longleftrightarro... |
743,473 | <p>A long Weierstrass equation is an equation of the form
$$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$
Why are the coefficients named $a_1, a_2, a_3, a_4$ and $a_6$ in this manner, corresponding to $xy, x^2, y, x$ and $1$ respectively? Why is $a_5$ absent?</p>
| drhab | 75,923 | <p>1) $$P\left(M\right)=P\left(M|L\right)P\left(L\right)+P\left(M|L^{c}\right)P\left(L^{c}\right)$$ </p>
<p>2) $$P\left(L\mid M^{c}\right)=\frac{P\left(L\cap M^{c}\right)}{P\left(M^{c}\right)}=\frac{P\left(L\cap M^{c}\right)}{P\left(L\right)}\frac{P\left(L\right)}{P\left(M^{c}\right)}=P\left(M^{c}\mid L\right)\frac{P\... |
283,824 | <p>Let $x$ be a random vector uniformly distributed on the unit sphere $\mathbb{S}^{n-1}$. Let $V$ be a linear subspace of dimension $k$ and let $P_V(x)$ be the orthogonal projection of $x$ onto $V$.
I have seen quoted in the literature that
\begin{align}
\mathbb{P}[|\left\| P_V(x)\right\|_2 - \sqrt{k/n} | \le \epsilo... | Iosif Pinelis | 36,721 | <p>In general, under regularity conditions such as compactness, one can use standard minimax duality argument ($\min_y\max_x F(x,y)=\max_x\min_y F(x,y)$ -- for bilinear or, more generally, concave-convex functions $F$) to quickly show that
\begin{multline}
W(\mu_1, \mu_2, \dots, \mu_m):=\inf_\gamma\int c d\gamma \\
... |
4,138,292 | <p>Can anybody help me understand why these terms are always reciprocals? (theta <= 45°)</p>
<p><span class="math-container">$$ x = \frac{1}{\cos \theta} + \tan{\theta} $$</span>
<span class="math-container">$$ \frac{1}{x} = \frac{1}{\cos \theta} - \tan{\theta} $$</span></p>
<p>I understand that if we multiply the... | bjcolby15 | 122,251 | <p>Proof:</p>
<p>If <span class="math-container">$$x = \sec \theta + \tan \theta \ (\text {here} \sec \theta = \dfrac {1}{\cos \theta})$$</span></p>
<p>Then <span class="math-container">$$ \dfrac {1}{x} = \dfrac {1}{\sec \theta+\tan \theta}$$</span></p>
<p>Rationalizing the denominator by multiplying by <span class="ma... |
881,141 | <p>Let $A$ and $B$ be two covariance matrices such that $AB=BA$. Is $AB$ a covariance matrix?</p>
<p>A covariance matrix must be symmetric and positive semi definite. The symmetry of $AB$ can be proved as follows:
$$(AB)^T = B^TA^T = BA = AB$$</p>
<p>The question is, how to prove or disprove the positive semi definit... | Quang Hoang | 91,708 | <p>Two commuting matrices can be diagonalized by the same matrix. The positive semi definite follows immediately.</p>
|
881,141 | <p>Let $A$ and $B$ be two covariance matrices such that $AB=BA$. Is $AB$ a covariance matrix?</p>
<p>A covariance matrix must be symmetric and positive semi definite. The symmetry of $AB$ can be proved as follows:
$$(AB)^T = B^TA^T = BA = AB$$</p>
<p>The question is, how to prove or disprove the positive semi definit... | Qaswed | 333,427 | <p>Your question about "the positive semi definitive character of $AB$" can be answered as follows: $A\cdot B$ is positive semi definite if and only if $A \cdot B$ is normal (i.e. $(A\cdot B)^T\cdot (A\cdot B) = (A\cdot B)\cdot (A \cdot B)^T$). Reference: Meenakshi, A. and C. Rajian (1999). <em>On a product of positiv... |
228,036 | <p>I quote from the <a href="http://en.wikipedia.org/wiki/Von_Neumann_cardinal_assignment" rel="nofollow">Wikipedia article</a>:</p>
<p>"So (assuming the axiom of choice) we identify $\omega_\alpha$ with $\aleph_\alpha$, except that the notation $\aleph_\alpha$ is used for writing cardinals, and $\omega_\alpha$ for w... | Asaf Karagila | 622 | <p>The $\aleph$ numbers are <em>defined</em> as the initial ordinals, without any appeal to the axiom of choice. This is done by a transfinite recursion over the ordinals. $\aleph_\alpha$ is therefore the ordinal $\omega_\alpha$, which is the unique initial ordinal such that the set of all initial ordinals strictly sma... |
2,441,630 | <p>The operator given is the right-shift operator $T$ on $l^2$. We show that $\lambda=1$ is in the residual spectrum. Therefore we show that $(I-T)$ is injective but fails to have a dense range. While injectivity is clear, I fail to understand why the following shows that the range is not dense:</p>
<p>Let $y=(I-T)x$.... | John Hughes | 114,036 | <p>Not a complete answer, but at least some direction: </p>
<ol>
<li>I don't have an answer, although the fact that $T$ preserves orthogonality probably comes into it somewhere. </li>
</ol>
<p>But assuming this part to have been addressed, we have:</p>
<ol start="2">
<li>It's not dense because, for instance, the ope... |
1,946,438 | <p>I solved the equation $e^{e^z}=1$ and it seemed to easy so I suspect I must be missing something.</p>
<blockquote>
<p>Would someone please check my answer?</p>
</blockquote>
<p>My original answer:</p>
<p>$e^{e^z}=1$ if and only if $e^z = 2\pi i k$ for $k\in \mathbb Z$ if and only if $z=\ln(2\pi i k)$ for $k\in ... | Community | -1 | <p>Since $1$ can be written $1=e^0$ it follows that a first solution is
$$e^z=i2\pi k,\qquad (k \in \mathbb{Z})$$</p>
<p>If $k=0$ there are no solutions since $e^z$ is never zero.</p>
<p>If $k>0$ write $i2 \pi k$ in exponential polar form and you should find that
$$|i2 \pi k|=2 \pi |k| = 2 \pi k, \qquad (\text{sin... |
4,151 | <p>Since it's currently summer break, and I've a bit more time than normal, I've been organizing my old notes. I seem to have an almost unwieldy amount of old handouts and tests from classes previously taught. I'm hesitant to get rid of these, as they may provide useful for some future course. Because I adjunct at a fe... | JPBurke | 759 | <p>While my data organizing problems are usually more related to research, they still often involve notes, papers, and sheets of data collection interview items that are much like handouts.</p>
<p>One solution that I have used (although I am still perfecting it) is to scan everything and get rid of the paper copies.</... |
57,769 | <p>Consider a finitely axiomatized theory $T$ with axioms $\phi_1,...,\phi_n$ over a first-order language with relation symbols $R_1,...,R_k$ of arities $\alpha_1,...,\alpha_k$. Consider the atomic formulas written in the form $(x_1,...,x_{\alpha_j})\ \varepsilon R_j$.</p>
<p>Translate this theory into a (finite) set-... | Robert Israel | 8,508 | <p>In dealing with gravitational waves, yes. A quick Google search came up with lots of hits, e.g. <a href="http://arxiv.org/ftp/gr-qc/papers/0701/0701008.pdf" rel="nofollow">http://arxiv.org/ftp/gr-qc/papers/0701/0701008.pdf</a></p>
|
118,232 | <p>For example I have </p>
<pre><code>square = Graphics[Polygon[{{0, 0} ,{0, 1}, {1, 1}, {1, 0}}]]
</code></pre>
<p>What functions can I apply to <code>sqaure</code> to extract the coordinates of the polygon? It is necessary to do this kind of extraction when I have a graphics object as an argument of a function, and... | bill s | 1,783 | <p>I'm afraid that you have calculated the stability incorrectly. Here is the Jacobian of your system:</p>
<pre><code>a = 2; b = 1; c = 2.5; d = 1.2; k = 2.8;
jac[x_, y_] := {{D[a x*(1 - x/k) - b x*y, x], D[a x*(1 - x/k) - b x*y, y]},
{D[-c y + d x*y, x], D[-c y + d x*y, y]}};
</code></pre>
<p>At the ... |
1,487,966 | <p>I have been looking at stereographic projections in books, online but they all seem...I don't know how else to put this, but very pedantic yet skipping the details of calculations.</p>
<p>Say, I have a problem here which asks;</p>
<blockquote>
<p>Let <span class="math-container">$n \geq 1$</span> and put <span class... | mr_e_man | 472,818 | <p>For the function $g$, we need to find a point $x=g(y)$ on the unit sphere $x^2=1$, which is also on the line between $y$ and the pole $p$. (Also $p^2=1$.) So there should be some scalar $k$ such that</p>
<p>$$x = p + k(y-p)$$</p>
<p>$$x^2 = p^2 + 2k\,p\cdot(y-p) + k^2(y-p)^2$$</p>
<p>Cancel $x^2=p^2$, and $k\neq0... |
8,023 | <p>I'm looking for an easily-checked, local condition on an $n$-dimensional Riemannian manifold to determine whether small neighborhoods are isometric to neighborhoods in $\mathbb R^n$. For example, for $n=1$, all Riemannian manifolds are modeled on $\mathbb R$. When $n=2$, I believe that it suffices for the scalar c... | Deane Yang | 613 | <p>If the Riemannian metric is twice differentiable in some co-ordinate system, then this holds in any dimension if and only if the Riemann curvature tensor vanishes identically.</p>
<p>In dimension 2, it suffices for the scalar curvature to vanish.
In dimension 3, it suffices for the Ricci curvature to vanish.
In hig... |
8,023 | <p>I'm looking for an easily-checked, local condition on an $n$-dimensional Riemannian manifold to determine whether small neighborhoods are isometric to neighborhoods in $\mathbb R^n$. For example, for $n=1$, all Riemannian manifolds are modeled on $\mathbb R$. When $n=2$, I believe that it suffices for the scalar c... | José Figueroa-O'Farrill | 394 | <p>Deane already answered the question. I just want to add that knowing the existence of local flat coordinates (by the vanishing of the curvature) and actually <em>finding</em> the flat coordinates are two very different things. I've had "fun" in the past finding explicit flat coordinates for flat metrics and it can... |
3,869,237 | <p>I know this is quite weird or it does not make much sense, but I was wondering, does <span class="math-container">$\int e^{dx}$</span> has any meaning or whether it makes sense at all? If it does means something, can it be integrated and what is the result?</p>
| Community | -1 | <p>The differential in an integral is essentially a symbolic way to show on which variable you integrate and is not to be taken as a factor. With this in mind,</p>
<p><span class="math-container">$$\int e^{dx}$$</span> is just syntactically unparsable, in the same way as</p>
<p><span class="math-container">$$\sin(x=(+$... |
3,421,858 | <p><span class="math-container">$\sqrt{2}$</span> is irrational using proof by contradiction.</p>
<p>say <span class="math-container">$\sqrt{2}$</span> = <span class="math-container">$\frac{a}{b}$</span> where <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are positive integers. </... | Noah Schweber | 28,111 | <p>This is a bit oddly phrased. I think it would be better to write it as follows:</p>
<ul>
<li><p>Suppose <span class="math-container">$\sqrt{2}$</span> is rational. Let <span class="math-container">$b$</span> be the smallest positive integer such that <span class="math-container">$b\sqrt{2}$</span> is a positive int... |
3,730,083 | <p>If <span class="math-container">$a,b>0$</span> and <span class="math-container">$Q=\{x_1, x_2, x_3,..., x_a\}$</span> a subset of the natural numbers <span class="math-container">$1, 2, 3,..., b$</span> such that, for <span class="math-container">$x_i+x_j<b+1$</span> with <span class="math-container">$1 ≤ i ≤ ... | user | 293,846 | <p>Yes there is. The number of vertices is <span class="math-container">$2^n$</span> and the number of edges is <span class="math-container">$n2^{n-1}$</span>. Generally the number of <span class="math-container">$d$</span>- dimensional elements in <span class="math-container">$n$</span>- dimensional cube is
<span clas... |
178,823 | <p>How would I prove the following trig identity? </p>
<blockquote>
<p><span class="math-container">$$\frac{ \cos (A+B)}{ \cos A-\cos B}=-\cot \frac{A-B}{2} \cot \frac{A+B}{2} $$</span></p>
</blockquote>
<p>My work thus far has been:
<span class="math-container">$$\dfrac{2\cos\dfrac{A+B}{2} \cos\dfrac{A-B}{2}}{-2\s... | Lance Helsten | 26,902 | <p>A couple of hints: use the $\cot$ identity on the RHS to put in terms of $\sin$ and $\cos$, use a product to sum identity, and pay attention to odd and even functions. This should simplify it to an easier expression.</p>
|
428,530 | <p>Let $\Omega := [0, 1] \times [0,\pi]$. We are searching for a function $u$ on $\Omega$ s.t.
$$
\Delta u =0
$$
$$
u(x,0) = f_0(x), \quad u(x,1) = f_1(x), \quad u(0,y) = u(\pi,y) = 0
$$ with
$$
f_0(x) = \sum_{k=1}^\infty A_k \sin kx \quad, f_1(x) = \sum_{k=1}^\infty B_k \sin kx
$$
If I use seperation of variables, ... | Community | -1 | <p>After some time, work and help by Avitus I finally got it :</p>
<p>Assume <span class="math-container">$u(x,y) = A(x)B(y)$</span> with <span class="math-container">$A,B \neq 0$</span> on <span class="math-container">$\Omega$</span>. This yields to
<span class="math-container">$$
A''(x)+\lambda A(x) = 0 , \quad B''(y... |
237,142 | <p>I am having a problem with the final question of this exercise.</p>
<p>Show that $e$ is irrational (I did that). Then find the first $5$ digits in a decimal expansion of $e$ ($2.71828$).</p>
<p>Can you approximate $e$ by a rational number with error $< 10^{-1000}$ ? </p>
<p>Thank you in advance</p>
| lhf | 589 | <p>In the standard proof that $e$ is irrational, one first proves that
$$
0 < e -s_n < \frac1{n!n}
\qquad\mbox{where}\qquad
s_n = \sum_{k=0}^n \frac1{k!}
$$
So you only need to find $n$ such that $\frac1{n!n}< 10^{-1000}$ or $n!>10^{1000}$. You can use Stirling's approximation for that I guess. <a href="htt... |
2,130,658 | <p>How would I go about proving this mathematically? Having looked at a proof for a similar question I think it requires proof by induction. </p>
<p>It seems obvious that it would be even by thinking about the first few cases. As for $n=0$ there will be no horizontal dominoes which is even, and for $n=1$ there can onl... | fleablood | 280,126 | <p>Ah, you mean the number of horizontal dominos in <em>BOTH</em> rows combined is even!</p>
<p>Consider each square in the grid. Either it is occupied by a vertical or horizontal domino. If it is occupied by a vertical domino, than so is the square in the grid directly below or directly above. If it is occupied by... |
1,635,136 | <p>In Kelley's "General topology" (in the "Appendix") the <em>full classes</em> $X$ are defined as those with the property
$$
\forall A\in X\quad A\subseteq X.
$$
In the Russian translation it is added that this is equivalent to the property that the relation $\in$ is transitive on $X$:
$$
\forall A,B,C\in X\quad A\in ... | hmakholm left over Monica | 14,366 | <p>The first property (the one you call "full class") is more commonly known as a <strong>transitive</strong> class/set in set theory.</p>
<p>It is not entirely uncommon to be confused by this name and get the misconception that this is the same as "a class on which the $\in$ relation is transitive" -- and apparently ... |
1,635,136 | <p>In Kelley's "General topology" (in the "Appendix") the <em>full classes</em> $X$ are defined as those with the property
$$
\forall A\in X\quad A\subseteq X.
$$
In the Russian translation it is added that this is equivalent to the property that the relation $\in$ is transitive on $X$:
$$
\forall A,B,C\in X\quad A\in ... | Pedro Sánchez Terraf | 212,120 | <p>I already posted one counterexample in the comment section. The one for the other direction is the following. </p>
<p>Intuitively, the class $V$ of all sets is transitive but it contains non transitive sets. So it is enough to get a “smaller” counterexample. For instance, $$
V_3=\mathcal{P}^3(\emptyset)=
\{0,\{0\},... |
2,965,082 | <blockquote>
<p>Suppose that <span class="math-container">$(X,\ d)$</span> and <span class="math-container">$(Y,\ \rho)$</span> are metric spaces, that
<span class="math-container">$f_n:X\to Y$</span> is continuous for each <span class="math-container">$n$</span>, and that <span class="math-container">$(f_n)$</span... | Alex Ortiz | 305,215 | <p>If we are more precise, we may see where the error is. I will use <span class="math-container">$|y-x|$</span> to mean <span class="math-container">$d(y,x)$</span> since I think it makes it clearer than working with <span class="math-container">$d$</span> and <span class="math-container">$\rho$</span>. It won't chang... |
1,707,132 | <blockquote>
<p>Let $X$ be a contractible space (i.e., the identity map is homotopic to the constant map). Show that $X$ is simply connected.</p>
</blockquote>
<p>Let $F$ be the homotopy between $\mathrm{id}_X$ and $x_0$, that is $F:X\times [0,1]\to X$ is a continuous map such that
$$ F(x,0)=x,\quad F(x,1)=x_0$$
fo... | Faraad Armwood | 317,914 | <p>Since $X$ is contractible there exists a homotopy $h=h_t: id_X \to x_0$ where $x_0 \in X$. Therefore, if we take any loop $\alpha: I \to X$ then $h|_{\alpha}$ takes $\alpha$ to $x_0$. Can you show $X$ is path-connected? You know $h_t$ takes every point in $X$ to $x_0$ therefore given any $x,y \in X$ then $h_{2t}(x) ... |
1,707,132 | <blockquote>
<p>Let $X$ be a contractible space (i.e., the identity map is homotopic to the constant map). Show that $X$ is simply connected.</p>
</blockquote>
<p>Let $F$ be the homotopy between $\mathrm{id}_X$ and $x_0$, that is $F:X\times [0,1]\to X$ is a continuous map such that
$$ F(x,0)=x,\quad F(x,1)=x_0$$
fo... | Ivin Babu | 704,464 | <p>Given that X is contractible.<br />
Hence there exists a function <span class="math-container">$α:X \to X$</span>, <span class="math-container">$α(x) =x_0 $</span> which is homotopic to the identity map on <span class="math-container">$X$</span>.<br />
Hence the subspace {<span class="math-container">$x_0$</span>} i... |
2,666,568 | <p>I have a dynamical system: $\dot{\mathbf x}$= A$\mathbf x$ with $\mathbf x$=
$\bigl( \begin{smallmatrix} x \\ y\end{smallmatrix} \bigr)$ and A =
$\bigl( \begin{smallmatrix} 3 & 0 \\ \beta & 3 \end{smallmatrix} \bigr). \beta$ real, time-independent.</p>
<p>I calculated the eigenvalue $\lambda$ = 3 with the ... | Will Jagy | 10,400 | <p>Although it does not really matter, it is traditional (well, in the U.S.) to put the $1$ in the Jordan form above the diagonal. I have been noticing students lately getting to the Jordan form but failing to write things in the reverse( and actually useful) order.
$$
\left(
\begin{array}{rr}
0 & \frac{1}{\beta} \... |
394,321 | <p>I'm studying the asymptotic behavior <span class="math-container">$(n \rightarrow \infty)$</span> of the following formula, where <span class="math-container">$k$</span> is a given constant.
<span class="math-container">$$ \frac{1}{n^{k(k+1)/(2n)}(2kn−k(1+k) \ln n)^2}$$</span></p>
<p>I'm trying to do a series expans... | mrf | 19,440 | <p>It depends on whether you view the expression as a function of $x$ or as a function of $y$. If you view it as a function of $x$, the decomposition is already done.</p>
<p>If you view it as a function of $y$, make the usual ansatz:</p>
<p>$$\frac{y}{(x-y)(y-1)} = \frac{A}{x-y} + \frac{B}{y-1}.$$</p>
<p>$A$ and $B$... |
394,321 | <p>I'm studying the asymptotic behavior <span class="math-container">$(n \rightarrow \infty)$</span> of the following formula, where <span class="math-container">$k$</span> is a given constant.
<span class="math-container">$$ \frac{1}{n^{k(k+1)/(2n)}(2kn−k(1+k) \ln n)^2}$$</span></p>
<p>I'm trying to do a series expans... | Community | -1 | <p>$$\text{Plot3D}\left[\left\{\frac{y^4}{(x-1) (y-1)}-\frac{y \left(x^2 y^2+x^2 y+x^2+x y^2+x y+y^2\right)}{x^3},\frac{y}{(y-1) (x-y)},x \left(y+\frac{1}{y}+1\right)+y+1\right\},\{x,-2,2\},\{y,-2,2\},\text{PlotLegends}\to \text{Automatic}\right]$$
use this
$$\frac{y^4}{(x-1) (y-1)}-\frac{y \left(x^2 y^2+x^2 y+x^2+x y^... |
2,965,459 | <p>Some curves defined by polynomial equations are disconnected over reals but not over complexes, e.g., <span class="math-container">$x y - 1 = 0$</span>. How can we convince someone with background only on equations over reals that the curve drawn by above equation is connected over complexes? Is a plot or something... | Community | -1 | <p>You can connect any two complex points <span class="math-container">$u, v$</span> with a spiral</p>
<p><span class="math-container">$$u^{1-t}v^t$$</span> where <span class="math-container">$t$</span> runs from <span class="math-container">$0$</span> to <span class="math-container">$1$</span>.</p>
<p>More precisely... |
1,722,287 | <p>So far I know that when matrices A and B are multiplied, with B on the right, the result, AB, is a linear combination of the columns of A, but I'm not sure what to do with this. </p>
| Bernard | 202,857 | <p>Consider the linear maps associated to $A$ and $B$ in canonical bases, respectively ($K$ denotes the base field):
\begin{align*}
f\colon K^p\to K^q\\
g\colon K^n\to K^p
\end{align*}
$\DeclareMathOperator\rk{rank}\DeclareMathOperator\img{Im}$The rank of a matrix is the dimension of the image of the associated linear... |
1,007,399 | <p>I came across following problem</p>
<blockquote>
<p>Evaluate $$\int\frac{1}{1+x^6} \,dx$$</p>
</blockquote>
<p>When I asked my teacher for hint he said first evaluate</p>
<blockquote>
<p>$$\int\frac{1}{1+x^4} \,dx$$</p>
</blockquote>
<p>I've tried to factorize $1+x^6$ as</p>
<p>$$1+x^6=(x^2 + 1)(x^4 - x^2 +... | MathArt | 319,307 | <p>By partial fraction expansion, <span class="math-container">$$I=\int{1\over x^6+1}dx=\int{1\over f(x)}={1\over f'(x_1)}\int{dx\over x-x_1}+{1\over f'(x_2)}\int{dx\over x-x_2}\ldots+{1\over f'(x_6)}\int{dx\over x-x_6}=\sum_{k=0}^6\ln(x-x_k)^{1\over f'(x_k)}$$</span> where <span class="math-container">$x_k$</span> are... |
1,007,399 | <p>I came across following problem</p>
<blockquote>
<p>Evaluate $$\int\frac{1}{1+x^6} \,dx$$</p>
</blockquote>
<p>When I asked my teacher for hint he said first evaluate</p>
<blockquote>
<p>$$\int\frac{1}{1+x^4} \,dx$$</p>
</blockquote>
<p>I've tried to factorize $1+x^6$ as</p>
<p>$$1+x^6=(x^2 + 1)(x^4 - x^2 +... | Community | -1 | <p><strong>Hint:</strong></p>
<p><span class="math-container">$$1+x^6$$</span> factors with the sixth roots of minus one, <span class="math-container">$\pm i$</span> and <span class="math-container">$\dfrac{\pm\sqrt3\pm i}2$</span> and by grouping the conjugate roots, we obtain a real factorization:</p>
<p><span class=... |
3,616,969 | <p>How to prove that if <span class="math-container">$n$</span> a natural number then <span class="math-container">$n^2$</span> never ends with the digits 2,3,7,8</p>
| Daniel | 530,757 | <p>Every natural number can be represented as :<span class="math-container">$$n=10k+r$$</span>where <span class="math-container">$k$</span> is an integer and <span class="math-container">$r$</span> is the remainder <span class="math-container">$(0,1,...,9)$</span> , now look at <span class="math-container">$$(10k+r)^{... |
3,616,969 | <p>How to prove that if <span class="math-container">$n$</span> a natural number then <span class="math-container">$n^2$</span> never ends with the digits 2,3,7,8</p>
| Mostafa Ayaz | 518,023 | <p>They do not end with those numbers because they end with <span class="math-container">$$0^2\equiv0\\1^2\equiv1\\2^2\equiv4\\3^2\equiv9\\4^2\equiv6\\5^2\equiv5\\6^2\equiv6\\7^2\equiv9\\8^2\equiv4\\9^2\equiv1$$</span>hence <span class="math-container">$\{0,1,4,5,6,9\}$</span></p>
|
3,616,969 | <p>How to prove that if <span class="math-container">$n$</span> a natural number then <span class="math-container">$n^2$</span> never ends with the digits 2,3,7,8</p>
| Community | -1 | <p>Assume that <span class="math-container">$n$</span> end with <span class="math-container">$0$</span> then <span class="math-container">$n^2$</span> end with <span class="math-container">$0$</span> too.
Now assume that end with <span class="math-container">$1$</span> then <span class="math-container">$n^2$</span> end... |
85,470 | <p>We decided to do secret Santa in our office. And this brought up a whole heap of problems that nobody could think of solutions for - bear with me here.. this is an important problem.</p>
<p>We have 4 people in our office - each with a partner that will be at our Christmas meal.</p>
<p>Steve,
Christine,
Mark,
Mary,... | Dimitar Slavchev | 20,048 | <p>1) You could have a third party party handle the distribution of cards in the hat so that every draw will be valid.</p>
<p>And after each draw the third party will remove invalid cards and put valid ones for the next draw. That should happen without any of the participants knowledge of how much cards are placed and... |
842,365 | <blockquote>
<p>Show that a field <span class="math-container">$\mathbb{F}$</span> is finite if and only if its multiplicative group <span class="math-container">$\mathbb{F}^{\times}$</span> is finitely generated.</p>
</blockquote>
<p>The "<span class="math-container">$\Rightarrow$</span>" implication is obvious, bu... | Martin Argerami | 22,857 | <p>Even if you had only two elements per "axis", the number of points in your array is uncountable. That's precisely what Cantor's diagonal argument shows.</p>
<p>Technically, what I'm saying is that the Cartesian product of countably many copies of $\{0,1\} $ is uncountable. </p>
|
2,952,556 | <p>I tried to to solve this but what I found is </p>
<pre><code>A is not necessary for B
</code></pre>
<blockquote>
<p>I could be wrong</p>
</blockquote>
<pre><code>= not(A is necessary for B)
= not(not(B) or A)
= not(A) and B
</code></pre>
<p>but it doesn't make sense.
Let's take an example:</p>
<pre><code>A = ... | Vera | 169,789 | <p>"<span class="math-container">$A$</span> necessary for <span class="math-container">$B$</span>" is actually "<span class="math-container">$B$</span> implies <span class="math-container">$A$</span>"</p>
<p>So "<span class="math-container">$A$</span> not necessary for <span class="math-container">$B$</span>" is actua... |
2,952,556 | <p>I tried to to solve this but what I found is </p>
<pre><code>A is not necessary for B
</code></pre>
<blockquote>
<p>I could be wrong</p>
</blockquote>
<pre><code>= not(A is necessary for B)
= not(not(B) or A)
= not(A) and B
</code></pre>
<p>but it doesn't make sense.
Let's take an example:</p>
<pre><code>A = ... | Graham Kemp | 135,106 | <p>When you interpret "is necessary" as "is implied by" then indeed we have "<span class="math-container">$A$</span> is not neccessary for <span class="math-container">$B$</span>" exactly where "not <span class="math-container">$A$</span> yet <span class="math-container">$B$</span>".</p>
<p>Check the truth table: <spa... |
3,510,233 | <blockquote>
<p>If <span class="math-container">$\sin\left(\operatorname{cot^{-1}}(x + 1)\right) = \cos\left(\tan^{-1}x\right)$</span>, then find the value of <span class="math-container">$x$</span>.</p>
</blockquote>
<p>Please solve this question by using <span class="math-container">$\cos\left(\dfrac\pi2 - \theta\... | Math1000 | 38,584 | <p>The rate at which the wasp population is growing is given by
<span class="math-container">$$
P'(t) = \frac{25000000 e^{-\frac{t}{2}}}{\left(1000 e^{-\frac{t}{2}}+1\right)^2}.
$$</span>
We want to find the value of <span class="math-container">$t$</span> that maximizes this function. Since <span class="math-container... |
592,560 | <p>Let G be an abelian group. Show that, if G is not cyclic, then for all $x\in G$, there is a divisor $d$ of $n = |G|$ which is strictly smaller than n satisfying $x^d=1$. </p>
<p>I'm guessing that this is a consequence of Lagrange's Theorem. We can have that G is a disjoint union of left cosets that all have the sam... | lhf | 589 | <ul>
<li><p>Consider $E=\{ e \in \mathbb Z : x^e =1 \mbox{ for all } x\in G \} $, the set of <em>exponents</em> of $G$.<br>
Then $E$ is a subgroup of $\mathbb Z$ and so $E=m\mathbb Z$.</p></li>
<li><p>By Lagrange's Theorem, $n\in E$ and so $m$ divides $n$.</p></li>
<li><p>$m$ is the lcm of the orders of all elements of... |
1,557,688 | <p>I want to show that the two metrics are equivalent. </p>
<p>Suppose we have a metric space $X \times Y$. Two metrics are defined as:</p>
<p>$d_{X \times Y}((x, y), (x', y')) := \max\{d_X(x, x'), d_Y(y, y')\}$</p>
<p>$d'_{X \times Y}((x, y), (x', y')) := d_X(x, x')+d_Y(y, y')$</p>
<p>Here is my attempt at proof:<... | Asker | 201,024 | <p>According to your book, if you have $N$ different things that can be picked, $K$ is the number of things from those $N$ things which would be considered "successes". $X$ is then the number of successes from those $K$ successes <em>that are actually picked.</em> </p>
<p>The number of trials is not $K$. The number of... |
3,756,436 | <p>Recently I was doing a physics problem and I ended up with this quadratic in the middle of the steps:</p>
<p><span class="math-container">$$ 0= X \tan \theta - \frac{g}{2} \frac{ X^2 \sec^2 \theta }{ (110)^2 } - 105$$</span></p>
<p>I want to find <span class="math-container">$0 < \theta < \frac{\pi}2$</span> ... | Community | -1 | <p>This is a similar but simple approach which gives the same result. Seeing that <span class="math-container">$y=\tan(\theta)$</span> can take any positive value, we go for maximizing <span class="math-container">$x$</span> and get <span class="math-container">$\max(x) = 1123$</span>.</p>
<p>We have: <span class="math... |
3,756,436 | <p>Recently I was doing a physics problem and I ended up with this quadratic in the middle of the steps:</p>
<p><span class="math-container">$$ 0= X \tan \theta - \frac{g}{2} \frac{ X^2 \sec^2 \theta }{ (110)^2 } - 105$$</span></p>
<p>I want to find <span class="math-container">$0 < \theta < \frac{\pi}2$</span> ... | River Li | 584,414 | <p>By letting <span class="math-container">$u = \tan \theta \in (0, \infty)$</span>, the equation is written as
<span class="math-container">$$\frac{g}{2} \frac{ X^2 (1+u^2) }{ (110)^2 } - X u + 105 = 0.$$</span>
The equation has real roots if and only if its discriminant is non-negative, that is,
<span class="math-co... |
1,276,957 | <p>These are the provided notes:</p>
<blockquote>
<p><img src="https://i.stack.imgur.com/NesWm.png" alt="Blockquote"></p>
</blockquote>
<p>These are the provided questions:</p>
<blockquote>
<p><img src="https://i.stack.imgur.com/t0Ta7.png" alt="Blockquote"></p>
</blockquote>
<p>I do not understand when I should... | danimal | 202,026 | <p>The point is, I think here, that the sine, cosine and tangent functions are all many-to-one, that is there are an infinite number of different values of $\theta$ that will give the <em>same</em> value of $\sin\theta$ (and the same for $\cos\theta$ and $\tan\theta$). If you consider the graph of, for example, $y=\sin... |
1,276,957 | <p>These are the provided notes:</p>
<blockquote>
<p><img src="https://i.stack.imgur.com/NesWm.png" alt="Blockquote"></p>
</blockquote>
<p>These are the provided questions:</p>
<blockquote>
<p><img src="https://i.stack.imgur.com/t0Ta7.png" alt="Blockquote"></p>
</blockquote>
<p>I do not understand when I should... | Stephen P | 233,086 | <p>If we are just looking at acute and obtuse angles</p>
<p>$\sin (180 - \theta ) = \sin \theta $ tells you that sin is positive for both acute and obtuse angles. Try it, type in to your calculator any angle from 0 to 180 and you get a positive numbers.</p>
<p>The cos and tan expressions tell you that for obtuse angl... |
1,276,957 | <p>These are the provided notes:</p>
<blockquote>
<p><img src="https://i.stack.imgur.com/NesWm.png" alt="Blockquote"></p>
</blockquote>
<p>These are the provided questions:</p>
<blockquote>
<p><img src="https://i.stack.imgur.com/t0Ta7.png" alt="Blockquote"></p>
</blockquote>
<p>I do not understand when I should... | Fax | 239,640 | <p>The question asks what the angles are if they are obtuse, i.e. between 90 and 180 degrees. The notes provide the conversions, e.g. if you get sin(30) = 0.5 then you convert it to an obtuse angle through sin(30) = sin(180 - 30) from the first note.</p>
|
1,276,957 | <p>These are the provided notes:</p>
<blockquote>
<p><img src="https://i.stack.imgur.com/NesWm.png" alt="Blockquote"></p>
</blockquote>
<p>These are the provided questions:</p>
<blockquote>
<p><img src="https://i.stack.imgur.com/t0Ta7.png" alt="Blockquote"></p>
</blockquote>
<p>I do not understand when I should... | John Joy | 140,156 | <p>Try Googling the following terms.</p>
<ul>
<li>unit circle</li>
<li>standard position</li>
<li>terminal size</li>
<li>initial side</li>
<li>reference angle</li>
</ul>
<p>Draw a diagram, and label the point where the terminal side intersects the unit circle as $A(\sin\theta, \cos\theta)$. Plot also point $B(\sin(18... |
754,888 | <p>The letters that can be used are A, I, L, S, T. </p>
<p>The word must start and end with a consonant. Exactly two vowels must be used. The vowels can't be adjacent.</p>
| fgp | 42,986 | <p>If $a$ has prime factorization $$
a = \prod_{k=1}^n p_k^{e_k} \text{ where } e_k \in \mathbb{N}, p_k \text{ prime}
$$
then $$
\sqrt[n]{a} \in \mathbb{N} \text{ exactly if $n \mid e_k$ for all $k$, meaning if } n \mid \textrm{gcd}(e_1,\ldots,d_k).
$$
The largest such $n$ is thus $\textrm{gcd}(e_1,\ldots,d_k)$.</... |
167,446 | <p>Let $p$ be a prime number, $C_p$-cyclic group of order $p$, and $G$ an elementary p-group of order $p^n$. Let us denote by Cext$(G,C_p)$ the group of all central extensions of $C_p$ by $G$. Is the number of non isomorphic groups in Cext$(G,C_p)$ known as a function of $n$? </p>
| Lenny Krop | 50,922 | <p>I'd like to thank everyone who has responded to my query.
The question was whether $I(n,p)$= # of non isomorphic groups in $\mathrm{Cext}(G,C_p)$ is known. My impression so far is that it is not.
Regarding discussion, first $\mathrm{Cext}(G,A)$, with $A$ abelian stands for $\mathrm{Opext}(G,A,\text{triv})$, hence b... |
269,178 | <p>i would like to know where i could find a plot of</p>
<p>$$ J_{ia}(2\pi i)$$ (1)</p>
<p>using Quantum mechanics i have conjectured that if $ a= \frac{x}{2} $ and $ i= \sqrt{-1} $ then </p>
<p>$$ J_{it}(2\pi i)\approx0=\zeta (1/2+2it)$$ at least for big $ t \to \infty $ (2)</p>
<p>however i do not know how to che... | GEdgar | 442 | <p>Here is what I get in Maple (first real part, then imaginary part)</p>
<p><img src="https://i.stack.imgur.com/SwjTu.jpg" alt="Bessel"></p>
<p>But here is your zeta</p>
<p><img src="https://i.stack.imgur.com/FfXZ2.jpg" alt="zeta"></p>
|
3,541,910 | <p>Let </p>
<p><span class="math-container">$$a_n = \frac{1}{1^3\cdot 1}+\frac{1}{1^3\cdot 2+2^3\cdot 1} + \cdots +\frac{1}{1^3\cdot n+2^3\cdot (n-1)+\cdots+n^3\cdot 1}$$</span></p>
<p>Does this sequence converge to a simple number?</p>
<p>My thought was to compute each denominator:</p>
<p><span class="math-contain... | Community | -1 | <p>Your sum is:
<span class="math-container">$$\sum_{k=1}^{n}\frac{1}{\left(1^{3}\cdot k\right)+\left(2^{3}\cdot\left(k-1\right)\right)+...+\left(k^{3}\cdot\left(1\right)\right)}=\sum_{k=1}^{n}\frac{1}{\color{red}{\sum_{m=1}^{k}m^{3}\cdot\left(k+1-m\right)}}$$</span>
For the red part we have:
<span class="math-containe... |
3,541,910 | <p>Let </p>
<p><span class="math-container">$$a_n = \frac{1}{1^3\cdot 1}+\frac{1}{1^3\cdot 2+2^3\cdot 1} + \cdots +\frac{1}{1^3\cdot n+2^3\cdot (n-1)+\cdots+n^3\cdot 1}$$</span></p>
<p>Does this sequence converge to a simple number?</p>
<p>My thought was to compute each denominator:</p>
<p><span class="math-contain... | Gary | 83,800 | <p>Continuing in a different way, you can express the limit in terms of the digamma function as follows:
<span class="math-container">$$
\mathop {\lim }\limits_{n \to + \infty } 30\left[ {\sum\limits_{k = 1}^n {\frac{1}{k}} + \sum\limits_{k = 1}^n {\frac{1}{{k + 1}}} + \sum\limits_{k = 1}^n {\frac{1}{{k + 2}}} - \f... |
201,576 | <p>Struggling with something basic. Suppose when defining f[x_] the outcome depends on Sign[x]. When calling this function, how do I tell Mathematica the sign of the argument?
My attempt:</p>
<pre><code>f[x_]=x Sign[x];
Assuming[a>0, f[a]]
</code></pre>
<p>The output I get is </p>
<pre><code>a Sign[a]
</code></p... | Bill | 18,890 | <pre><code>f[x_]=x Sign[x];
Assuming[a>0,FunctionExpand[f[a]]]
</code></pre>
<p>returns a</p>
<pre><code>Assuming[a>0,Simplify[f[a]]]
</code></pre>
<p>returns a</p>
|
19,253 | <p><strong>Bug introduced in 7.0.1 or earlier and fixed in 10.0.2 or earlier</strong></p>
<hr>
<p>I have access to two versions of Mathematica, version 7.0.1 on Linux and version 8 on Windows. When I try the following two lines on version 7, the kernel quits when it tries to plot. In version 8 it plots just fine. ... | Alexey Popkov | 280 | <p>I also had such problem in version 7 when the data grid is too regular. I think it is a limitation of the triangulation algorithm used in v.7. To avoid this you could try to perturbate original grid a bit by adding a small random noise to the {x ,y} values: <a href="https://groups.google.com/d/msg/comp.soft-sys.math... |
19,253 | <p><strong>Bug introduced in 7.0.1 or earlier and fixed in 10.0.2 or earlier</strong></p>
<hr>
<p>I have access to two versions of Mathematica, version 7.0.1 on Linux and version 8 on Windows. When I try the following two lines on version 7, the kernel quits when it tries to plot. In version 8 it plots just fine. ... | Simon Woods | 862 | <p>You can use the undocumented <code>Method</code> option <code>"DelaunayDomainScaling"</code> to deal with the problem. I don't know any details but I assume it works by rescaling the data inside the triangulation algorithm:</p>
<pre><code>ListContourPlot[temp, Method -> {"DelaunayDomainScaling" -> True}, Plot... |
114,754 | <p>I have several questions concerning the proof. I don't think I quite understand the details and motivation of the proof. Here is the proof given by our professor.</p>
<p>The space of polynomials $F[x]$ is not finite-dimensional.</p>
<p><em>Proof</em>. Suppose
$$F[x] = \operatorname{span}\{f_1,f_2,\dots,f_n\}$$</p>... | Brian M. Scott | 12,042 | <p>To say that a polynomial $p(x)$ is <em>identically zero</em> is to say that $p(x)=0$ for <strong>all</strong> values of $x$. Here the coefficients $a_1,\dots,a_n$ were chosen so that $$x^N=a_1f_1(x)+a_2f_2(x)+\dots+a_nf_n(x)$$ for all values of $x$, and $G(x)$ was defined by $$G(x)=x^N-a_1f_1(x)-a_2f_2(x)-\dots-a_nf... |
3,345,063 | <p><a href="https://i.stack.imgur.com/T3Ue9.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/T3Ue9.png" alt="Given a triangle whose apex angle is \theta" /></a></p>
<p><strong>Given a triangle with two circles and apex angle equals <span class="math-container">$\theta$</span>.</strong></p>
<p><em><str... | MachineLearner | 647,466 | <p>Use the lower picture and note that the red lines are perpendicular to the outer edges.</p>
<p>Using this knowledge we can set up the following system.</p>
<p><span class="math-container">$$\sin\dfrac{\theta}{2} = \dfrac{r}{r+s}$$</span>
<span class="math-container">$$\sin\dfrac{\theta}{2} = \dfrac{R}{R+2r+s}$$</spa... |
1,855,650 | <p>Need to solve:</p>
<p>$$2^x+2^{-x} = 2$$</p>
<p>I can't use substitution in this case. Which is the best approach?</p>
<p>Event in this form I do not have any clue:</p>
<p>$$2^x+\frac{1}{2^x} = 2$$</p>
| Aakash Kumar | 346,279 | <p>$$2^x =y $$
$$y+ \frac{1}{y} =2$$
Using AM-GM INEQUALITY
$$\frac{({y+ \frac{1}{y}})}{2} \ge \sqrt {y.\frac{1}{y}}$$</p>
<p>$$y+ \frac{1}{y}\ge 2$$</p>
|
1,855,650 | <p>Need to solve:</p>
<p>$$2^x+2^{-x} = 2$$</p>
<p>I can't use substitution in this case. Which is the best approach?</p>
<p>Event in this form I do not have any clue:</p>
<p>$$2^x+\frac{1}{2^x} = 2$$</p>
| Community | -1 | <p><strong>A contrived solution</strong>:</p>
<p>Write</p>
<p>$$\frac{e^{x\ln2}+e^{-x\ln2}}2=1=\cosh(x\ln2),$$</p>
<p>then</p>
<p>$$x\ln2=\cosh^{-1}1=0.$$</p>
<hr>
<p><strong>Another one</strong>:</p>
<p>By inspection, $x=0$ is a solution.</p>
<p>The derivative of the LHS is</p>
<p>$$2^x\ln2-2^{-x}\ln2,$$ whic... |
161,029 | <p>I have not seen a problem like this so I have no idea what to do.</p>
<p>Find an equation of the tangent to the curve at the given point by two methods, without elimiating parameter and with.</p>
<p>$$x = 1 + \ln t,\;\; y = t^2 + 2;\;\; (1, 3)$$</p>
<p>I know that $$\dfrac{dy}{dx} = \dfrac{\; 2t\; }{\dfrac{1}{t}}... | Américo Tavares | 752 | <ol>
<li><p><strong>Eliminating the parameter</strong> $t$. The given system of two parametric equations $$\begin{eqnarray*}\left\{
\begin{array}{c}
x=1+\ln t \\
y=t^{2}+2
\end{array}
\right. \end{eqnarray*} \tag{A}$$ is equivalent successively to
$$\begin{eqnarray*}
\left\{
\begin{array}{c}
x-1=\ln t \\
y=t^{2}+2
... |
192,883 | <p>Can anyone please give an example of why the following definition of $\displaystyle{\lim_{x \to a} f(x) =L}$ is NOT correct?:</p>
<p>$\forall$ $\delta >0$ $\exists$ $\epsilon>0$ such that if $0<|x-a|<\delta$ then $|f(x)-L|<\epsilon$</p>
<p>I've been trying to solve this for a while, and I think it w... | Fly by Night | 38,495 | <p>Let's consider a counter example. Let's use your definition to prove that the limit of $x$, as $x$ tends towards 1, is 2. For all $\delta > 0$, we claim that there exists $\varepsilon_{\delta} > 0$ such that: </p>
<p>If $|x - 1| < \delta$ then $|x-2| <\varepsilon_{\delta}$. </p>
<p>We could define $\va... |
3,900,962 | <p>In Valter Moretti's "<a href="https://rads.stackoverflow.com/amzn/click/com/8847028345" rel="nofollow noreferrer" rel="nofollow noreferrer">Spectral Theory and Quantum Mechanics</a>," Remark 4.2 (1), he claims:</p>
<blockquote>
<p>Compactness is hereditary, in the sense that it is passed on to induced topo... | QuantumSpace | 661,543 | <p>False. <span class="math-container">$X=K=[0,1]$</span> is compact but <span class="math-container">$K \cap A=A=(0,1)$</span> is non-compact.</p>
<p>It is true that compactness is inherited by closed subsets. Indeed, let <span class="math-container">$A\subseteq X$</span> be a closed set. If <span class="math-containe... |
3,900,962 | <p>In Valter Moretti's "<a href="https://rads.stackoverflow.com/amzn/click/com/8847028345" rel="nofollow noreferrer" rel="nofollow noreferrer">Spectral Theory and Quantum Mechanics</a>," Remark 4.2 (1), he claims:</p>
<blockquote>
<p>Compactness is hereditary, in the sense that it is passed on to induced topo... | Anatoliy Lotkov | 422,463 | <p>The author probably means that compactness is inherent property of subspace: you don't need the space in which it is contained to deduce its compactness.</p>
<p>In more rigorous terms. Let <span class="math-container">$ (X, \, \tau_X)$</span> be a topological space. Consider its compact subspace <span class="math-co... |
137,434 | <p>I would like to draw the solution of the following equation in a log-log plot of $\lambda$ against $M$.</p>
<p>$$10^{-10} = \frac{\lambda^{2}}{(4\pi)^{2}}\int_{0}^{1}dz\frac{(1-x)^{3}}{(1-x)^{2}+zM^{2}}$$</p>
<p>Here is my code:</p>
<pre><code>table = Table[{Exp[M], λ /.
NSolve[((λ^2)/(4 *Pi)^(2)) *
... | David G. Stork | 9,735 | <p>Here is your graph, once you delete the non-converged values:</p>
<pre><code>ListLogLogPlot[table, AxesOrigin -> {1,1}]
</code></pre>
<p><a href="https://i.stack.imgur.com/LXLwL.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/LXLwL.png" alt="enter image description here"></a></p>
|
137,434 | <p>I would like to draw the solution of the following equation in a log-log plot of $\lambda$ against $M$.</p>
<p>$$10^{-10} = \frac{\lambda^{2}}{(4\pi)^{2}}\int_{0}^{1}dz\frac{(1-x)^{3}}{(1-x)^{2}+zM^{2}}$$</p>
<p>Here is my code:</p>
<pre><code>table = Table[{Exp[M], λ /.
NSolve[((λ^2)/(4 *Pi)^(2)) *
... | Dr. Wolfgang Hintze | 16,361 | <p>Why not make things easier and solve the integral exactly?</p>
<p>After making the integration variable consistently $\text{x}$ and changing it to $\text{x}\to 1-\text{x}$, the integral can be calculated explicitly. (In order to assist Mathematica in solving the integral we also assume temporarily that $\text{M}<... |
440,242 | <p>I'm pretty sure almost all mathematicians have been in a situation where they found an interesting problem; they thought of many different ideas to tackle the problem, but in all of these ideas, there was something missing- either the "middle" part of the argument or the "end" part of the argumen... | Hollis Williams | 119,114 | <p>The only generic advice that can be offered here is that deciding what to do in this situation is quite a difficult skill, but one that you are expected to develop as a PhD student. You ultimately might have to think of the “cost-benefit” analysis.</p>
<p>Is the time and energy which you would expend continuing to w... |
440,242 | <p>I'm pretty sure almost all mathematicians have been in a situation where they found an interesting problem; they thought of many different ideas to tackle the problem, but in all of these ideas, there was something missing- either the "middle" part of the argument or the "end" part of the argumen... | Geordie Williamson | 919 | <p>One stochastic algorithm that sometimes works for me:</p>
<ol>
<li><p>Take Pólya's "How to solve it" from bookshelf.</p>
</li>
<li><p>Open at random page, and read for a few pages.</p>
</li>
<li><p>Now attempt problem again.</p>
</li>
</ol>
|
440,242 | <p>I'm pretty sure almost all mathematicians have been in a situation where they found an interesting problem; they thought of many different ideas to tackle the problem, but in all of these ideas, there was something missing- either the "middle" part of the argument or the "end" part of the argumen... | Gabe K | 125,275 | <p>In my experience, one key aspect to avoid getting stuck is the choice of problem in the first place. With a good project, there are many smaller questions or computations to be done when you are stuck at a particularly difficult point with the main project. Doing these tasks might not solve your particular issue, bu... |
1,097,134 | <p>this is something that came up when working with one of my students today and it has been bothering me since. It is more of a maths question than a pedagogical question so i figured i would ask here instead of MESE.</p>
<p>Why is $\sqrt{-1} = i$ and not $\sqrt{-1}=\pm i$?</p>
<p>With positive numbers the square r... | Milo Brandt | 174,927 | <p>Well, if you are considering that $y=\sqrt{x}$ is the <em>relation</em> $y^2=x$, then, yes, $\pm i$ are both solutions to $\sqrt{-1}$. However, this is not usually how square roots are defined. Typically we say:
$$\sqrt{1}=1$$
Not plus or minus $1$ - just $1$. This means that $\sqrt{x}$ is a "right inverse" of $x^2$... |
1,097,134 | <p>this is something that came up when working with one of my students today and it has been bothering me since. It is more of a maths question than a pedagogical question so i figured i would ask here instead of MESE.</p>
<p>Why is $\sqrt{-1} = i$ and not $\sqrt{-1}=\pm i$?</p>
<p>With positive numbers the square r... | Andrew D. Hwang | 86,418 | <p>If $x$ is a non-negative real number, there's an unambiguous interpretation for the expression $\sqrt{x}$, namely, the <em>non-negative</em> square root of $x$. (The $\pm$ signs aren't "part of the square root function", which is why they have to be included explicitly when "solving an equation by taking square root... |
1,680,269 | <p>Here $\mathbb{Z}_{n}^{*}$ means $\mathbb{Z}_{n}-{[0]_{n}}$</p>
<p>My attempt:</p>
<p>$(\leftarrow )$</p>
<p>$p$ is a prime, then, for every $[x]_{n},[y]_{n},[z]_{n}$ $\in (\mathbb{Z}_{n}^{*},.)$ are verified the following:</p>
<p>1) $[x]_{n}.([y]_{n}.[z]_{n}) = ([x]_{n}.[y]_{n}).[z]_{n}$, since from the operatio... | Community | -1 | <p>By definition of infimum there are sequences $(x_n)_{n\in\mathbb{N}}\subseteq K, (y_n)_{n\in\mathbb{N}}\subseteq L$ with $\lim_{n\to\infty}|x_n-y_n|=d$. From compactness we have convergent subsequences $x_{n_k}, y_{n_k}$ (it's possible to take the same indices $n_k$!), so there are $x_0\in K, y_0\in L$ with $\lim_{k... |
1,298,971 | <p>Any help on this problem is greatly appreciated! I'm completely stuck</p>
<p>School board officials are debating whether to require all high school seniors to take a proficiency exam before graduating. A student passing all three parts (mathematics, language skills, and general knowledge) would be awarded a diploma... | André Nicolas | 6,312 | <p>Consider a square such that the distance from the centre to any side is $r$. Then the area of the square is $4r^2$, and the perimeter of the square is $8r$, which is the derivative of $4r^2$. </p>
<p>So your circle rule works for the square, if we use the right parameter to describe its size.</p>
<p><strong>Explor... |
1,869,119 | <p>Show that the Monotone Convergence Theorem may not hold for decreasing sequences of functions.</p>
<p>Suppose $\left\{f_{n}\right\}$ is a sequence of nonnegative decreasing functions converging to $f$ pointwise. I know that if $f_{1}$ is finite,we can construct the sequence say $\left\{f_{1}-f_{n}\right\}$ which is... | Aweygan | 234,668 | <p>Define the sequence $\{f_n\}$ on $[0,\infty)$ by letting $f_n(0)=0$ for all $n$, and
$$ f_n(x)=\frac{1}{nx}\qquad (x\in(0,\infty). $$
Then $f_n$ is monotonically decreasing, and converges pointwise to $0$ on $[0,\infty)$. However,
$$\int_{[0,\infty)}f_n\ dm=\infty,$$
while
$$\int_{[0,\infty)}\lim_{n\to\infty}f_n\ d... |
2,231,949 | <p>To find the minimal polynomial of $i\sqrt{-1+2\sqrt{3}}$, I need to prove that
$x^4-2x^2-11$ is irreducible over $\Bbb Q$. And I am stuck. Could someone please help? Thanks so much!</p>
| Alex Macedo | 400,433 | <p>If you have a candidate $f$ for the minimal polynomial of an algebraic element $\alpha$ over $\mathbf Q$ but you don't know if $f$ is really irreducible and want to avoid trying possible factorizations, you might want to compute the degree $$[\mathbf Q(\alpha): \mathbf Q]$$ and check if it agrees with the degree of ... |
720,969 | <p>While studying real analysis, I got confused on the following issue.</p>
<p>Suppose we construct real numbers as equivalence classes of cauchy sequences. Let $x = (a_n)$ and $y= (b_n)$ be two cauchy sequences, representing real numbers $x$ and $y$.</p>
<p>Addition operation $x+y$ is defined as $x+y = (a_n + b_n)$... | Christian Blatter | 1,303 | <p>You have to differentiate typographically between (a) sequences and (b) equivalence classes of sequences, i.e., real numbers.</p>
<p>Write $x$ for the Cauchy sequence $(x_n)_{n\geq1}$ and $[x]$ for the equivalence class represented by $x$.</p>
<p>Since addition of real numbers is described in terms of representant... |
124,662 | <p>Is topology on $\mathbb{R}/\mathbb{Z}$ compact? If it is, how to prove it? </p>
<p>$\mathbb{R}/\mathbb{Z}$ denotes the set of equivalence classes of the set of real numbers, two real numbers being equivalent if and only if their difference is an integer.</p>
| Rudy the Reindeer | 5,798 | <p>Or use that $S^1$ is compact and try to write down a homeomorphism between $\mathbb R/ \mathbb Z$ and $S^1$.</p>
|
92,660 | <p>Let $X$ be a nonsingular projective variety over $\mathbb{C}$, and let $\widetilde{X}$ be the blow-up of X at a point $p\in X$.
What relationships exist between the degrees of the Chern classes of $X$ (i.e. of the tangent bundle of $X$) and the degrees of the Chern classes of $\widetilde{X}$?</p>
<p>Thanks.</p>
| Georges Elencwajg | 450 | <p>For the first Chern class you get the simple formula<br>
$$c_1(\tilde X)=p^*c_1(X)- (n-1)E$$
where $p:\tilde X \to X$ is the projection and $E$ the exceptional divisor. </p>
<p>In general the formula is more complicated and I'll refer you to Fulton's <em>Intersection Theory</em>, where the formula you require i... |
1,550,841 | <p>$\int e^{2\theta}\ \sin 3\theta\ d\theta$</p>
<p>After Integrating by parts a second time, It seems that the problem will repeat for ever. Am I doing something wrong. I would love for someone to show me using the method I am using in a clean and clear fashion. Thanks. <a href="https://i.stack.imgur.com/FipiE.jpg" r... | Ian | 83,396 | <p>When you do it with integration by parts, you have to go in the "same direction" both times. For instance, if you initially differentiate $e^{2 \theta}$, then you need to differentiate $e^{2 \theta}$ again; if you integrate it, you will wind up back where you started. If you do this, you should find something of the... |
1,550,841 | <p>$\int e^{2\theta}\ \sin 3\theta\ d\theta$</p>
<p>After Integrating by parts a second time, It seems that the problem will repeat for ever. Am I doing something wrong. I would love for someone to show me using the method I am using in a clean and clear fashion. Thanks. <a href="https://i.stack.imgur.com/FipiE.jpg" r... | Yes | 155,328 | <p>Here is a plain answer for your reference:</p>
<p>Let $\simeq$ denote the equality sign up to a constant.
We have
$$
\int e^{2x}\sin 3x dx = \int \sin 3x de^{2x}\frac{1}{2} \simeq \frac{1}{2}[ e^{2x}\sin 3x - 3 \int e^{2x} \cos 3x dx ]\\ = \frac{1}{2}e^{2x}\sin 3x - \frac{3}{2} \int e^{2x}\cos 3x dx;
$$
we have
$$... |
143,655 | <p>According to <a href="http://en.wikipedia.org/wiki/Lipschitz_continuity#Properties" rel="nofollow noreferrer">wikipedia</a> a function <span class="math-container">$f\colon \mathbb{R}^n\to\mathbb{R}^n$</span> that is continuously differentiable, is also locally Lipschitz.</p>
<p>I there someone who knows a good refe... | copper.hat | 27,978 | <p>The proof on $\mathbb{R}^n$ is fairly straightforward.</p>
<p>Choose some ball $B(\hat{x},\epsilon)$. The closure is compact, so the derivative $\frac{\partial f}{\partial x}$ is bounded by some $L$ on the ball. Now suppose $x,y \in B(\hat{x},\epsilon)$, then using Taylor's formula, we have:
$$f(x)-f(y) = \int_o^1 ... |
4,549,300 | <p>Matrix C of size n<span class="math-container">$\times$</span>n is symmetric . Zero is a simple eigenvalue of C. The associated eigenvector is q. For <span class="math-container">$\epsilon$</span>>0, the equation <span class="math-container">$Cx+\epsilon x=d$</span> in x, where x and d are n-dimensional Column v... | Ted Shifrin | 71,348 | <p>I assume here that <span class="math-container">$C$</span> is a <em>real</em> symmetric matrix.</p>
<p><strong>HINT</strong>: We assume <span class="math-container">$q$</span> is a unit vector. Apply the Spectral Theorem to write
<span class="math-container">$$C+\epsilon I = \epsilon qq^\top + \sum_{i=1}^{n-1} (\lam... |
34,557 | <p>I just came across a spam answer which is extremely vulgar (sexual). I flagged it for moderator attention, and then it occurred to me that I could edit it and erase its contents by blanking it till a moderator gets to look at it. Is this an acceptable thing to do?</p>
| KReiser | 21,412 | <p>From the comments: No. Instead, flag it as <em>rude or abusive</em> and move on.</p>
|
107,399 | <p>Let's say we have a set a\of associations:</p>
<pre><code>dataset = {
<|"type" -> "a", "subtype" -> "I", "value" -> 1|>,
<|"type" -> "a", "subtype" -> "II", "value" -> 2|>,
<|"type" -> "b", "subtype" -> "I", "value" -> 1|>,
<|"type" -> "b", "subtype" -> ... | Leonid Shifrin | 81 | <p>I have posted code doing a very similar thing <a href="https://mathematica.stackexchange.com/a/54493/81">here</a> - the functions <code>pushUp</code> and <code>pushUpNested</code>. That code was more general, since there I provided a declarative interface to group by values <em>or their parts</em>. To do what you ne... |
2,152,914 | <p>Say I have a random function $f: \mathbb{Z}_N \rightarrow \mathbb{Z}_N$. (by random I mean for each possible input we choose an output in an uniform distribution from $\mathbb{Z}_N$) <br/>
So I know that for all $x,y \in \mathbb{Z}_N$ I have $Pr[f(x)=y]={1 \over |\mathbb{Z}_N|}$. <br/>
Now I look at $f'(x):=f(f(x))... | Peter | 82,961 | <p>Yes, because the value of $f(x)$ is irrelevant for the determination of $f(f(x))$</p>
|
2,152,914 | <p>Say I have a random function $f: \mathbb{Z}_N \rightarrow \mathbb{Z}_N$. (by random I mean for each possible input we choose an output in an uniform distribution from $\mathbb{Z}_N$) <br/>
So I know that for all $x,y \in \mathbb{Z}_N$ I have $Pr[f(x)=y]={1 \over |\mathbb{Z}_N|}$. <br/>
Now I look at $f'(x):=f(f(x))... | Empy2 | 81,790 | <p>Here is one difference:
$f(x)$ probably has around $N(1-1/e)$ different values. The chance a value never appears is $(1-1/N)^N\approx 1/e$</p>
<p>$f(f(x))$ probably has around $N(1-1/e)^2$ different values.</p>
<p>EDIT:
Let $M=N/e$ be the number of 'orphans' that are not a value of $f(x)$.<br>
Let $P=N-M$ be t... |
97,393 | <p>The polynomial</p>
<p>$F(x) = x^5-9x^4+24x^3-24x^2+23x-15$</p>
<p>has roots $x=1$ and $x=j$. Calculate all the roots of the polynomial.</p>
<p>I was told I had to use radicals or similar to solve this but after reading up on it I'm still confused about how to solve it.</p>
| John R Ramsden | 18,548 | <p>Knowing that one root is $x = 1$ means $F(x)$ has a factor $x - 1$. So you can either obtain the complementary factor by long division, or note that:</p>
<p>$\begin{align}F(x) &= x^4(x - 1) + 24x^2(x - 1) - 8x^4 + 23x - 15
\\ &= x^4(x - 1) + 24x^2(x - 1) - 8x(x^3 - 1) + 15(x - 1)\end{align}$</p>
<p>so ... |
3,172,485 | <p>Consider the familiar trigonometric identity: <span class="math-container">$\cos^3(x) = \frac{3}{4} \cos(x) + \frac{1}{4} \cos(3x)$</span></p>
<p>Show that the identity above can be interpreted as Fourier series expansion.</p>
<p>so we know that cos is periodic between <span class="math-container">$\pi$</span> and... | PrincessEev | 597,568 | <p>Recall: from a familiar trigonometric identity,</p>
<p><span class="math-container">$$\cos^3(x) = \cos(x)\color{blue}{\cos^2(x)} = \cos(x)\color{blue}{(1 - \sin^2(x))}$$</span></p>
<p>Thus,</p>
<p><span class="math-container">$$\int \cos^3(x)dx = \int (1-\sin^2(x))\cos(x)dx$$</span></p>
<p>Make the <span class="... |
255,164 | <p>$\newcommand{\al}{\alpha}$
$\newcommand{\euc}{\mathcal{e}}$
$\newcommand{\Cof}{\operatorname{Cof}}$
$\newcommand{\Det}{\operatorname{Det}}$</p>
<p>Let $M,N$ be smooth $n$-dimensional Riemannian manifolds (perhaps with smooth boundary), and let $\, f:M \to N$ be a smooth <strong>immersion</strong>. Let $\Omega^k(M,... | Igor Khavkine | 2,622 | <p>Suppose that the vector field $v$ is a non-vanishing covariantly constant on $N$, then its pullback $f^*v$ is a constant section of $f^*TN \to M$ with respect to the pulled back connection. Then, for any $(n-1)$-form $\alpha$ and $(n-2)$-form $\beta$ on $M$, we have $d_{n-2} (\beta \otimes f^*v) = (d\beta) \otimes f... |
2,484 | <p>It's been quite a while since I was tutoring a high school student and even longer since not a gifted one.</p>
<p>However, this time, something was amiss. I have asked him to show me how he does some exercise, and then another and the only thing I wanted to do was to shout:</p>
<blockquote>
<p><strong>You are do... | JvR | 1,193 | <p><em>(This is a very, very long answer because I want to highlight both the likely mindset of such a pupil, and possible approaches to win them over. Also, please check <a href="https://matheducators.stackexchange.com/questions/895/student-poisoned-experience-with-math">Student Poisoned Experience with Math</a> in c... |
405,953 | <p>Let</p>
<ul>
<li><span class="math-container">$E$</span> be the usual sobolev space <span class="math-container">$H^{1}_{0}(\Omega)$</span> on a smoothly bounded domain <span class="math-container">$\Omega$</span>,</li>
<li><span class="math-container">$E_{k}$</span> be its subspace spanned by the first <span class=... | user378654 | 378,654 | <p>This does not answer the question asked (see the other answer for a good counterexample) and I don't know if it's relevant to the paper. However, if <span class="math-container">$u$</span> is sufficiently far from <span class="math-container">$\varphi_1$</span>, the first eigenfucntion, you do get a positive answer.... |
8,816 | <p>What is the result of multiplying several (or perhaps an infinite number) of binomial distributions together?</p>
<p>To clarify, an example.</p>
<p>Suppose that a bunch of people are playing a game with k (to start) weighted coins, such that heads appears with probability p < 1. When the players play a round, t... | Mikhail Bondarko | 2,191 | <p>If A is an elliptic curve then G (in your notation) is finite. Yet it seems that for a square of an elliptic curve you get something infinite and very far from being algebraic. If $A=B\times B$, $B$ is a 'general' elliptic curve, then it seems that $G(A)=GL_2(\Bbb Z)$; this is a 'large' discrete group. Another examp... |
3,632,576 | <p>Considering that input <span class="math-container">$x$</span> is a scalar, the data generation process works as follows:</p>
<ul>
<li>First, a target t is sampled from {0, 1} with equal probability.</li>
<li>If t = 0, x is sampled from a uniform distribution over the interval
[0, 1]. </li>
<li>If t = 1, x is sampl... | Sam | 584,704 | <p>It's simpler if you keep the absolute values and use the same idea: <span class="math-container">$$\left|\frac{1}{x}\int_0^x f(t)dt - a\right| $$</span> <span class="math-container">$$= \left|\frac{1}{x}\int_0^N f(t)dt + \frac{1}{x}\int_N^xf(t)dt - a\right|$$</span>
<span class="math-container">$$\le\left|\frac{1}{... |
186,878 | <p>Is there a way to find the geoposition of a given distance from start in a <code>GeoPath</code>? I want to mark equidistant positions along a track, for example, a mark every 500 km along the path given by</p>
<pre><code>path=GeoGraphics[
GeoPath[{
Entity["City", {"Boston", "Massachusetts", "UnitedStates"}],
... | Greg Hurst | 4,346 | <p>Here's my attempt to parametrize the path by arclength, where here arclength is <code>GeoLength</code>.</p>
<p>First I build up a function that can be used on many values:</p>
<pre><code>ParametrizeGeoPath[g_GeoPath, t_] := ParametrizeGeoPath[g][t]
ParametrizeGeoPath[GeoPath[locs_, args___]] :=
Block[{line, nod... |
359,212 | <p>I mean, $\Bbb Z_p$ is an instance of $\Bbb F_p$, I wonder if there are other ways to construct a field with characteristic $p$?
Thanks a lot!</p>
| Ittay Weiss | 30,953 | <p>Just to supplement the other answers: As stated in the other answers, for every prime power $p^r$, $r>0$, there is a unique (up to isomorphism) field with $p^r$ elements. There are also infinite fields of characteristic $p$, for instance if $F$ is any field of characteristic $p$ (e.g., $\mathbb Z_p$), the field $... |
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