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# Leon Green’s theorem

The fundamental objects of study in higher-order Fourier analysis are nilmanifolds, or in other words spaces given as a quotient ${G/\Gamma}$ of a connected nilpotent Lie group ${G}$ by a discrete cocompact subgroup ${\Gamma}$. Starting with Furstenberg’s work on Szemeredi’s theorem and the multiple recurrence theorem, work by Host and Kra, Green and Tao, and several others has gradually established that nilmanifolds control higher-order linear configurations in the same way that the circle, as in the Hardy-Littlewood circle method, controls first-order linear configurations.

Of basic importantance in the study of nilmanifolds is equidistribution: one needs to know when the sequence ${g^n x}$ equidistributes and when it is trapped inside a subnilmanifold. It turns out that this problem was already studied by Leon Green in the 60s. To describe the theorem note first that the abelianisation map ${G\rightarrow G/G_2}$ induces a map from ${G/\Gamma}$ to a torus ${G/(G_2\Gamma)}$ which respects the action of ${G}$, and recall that equidistribution on tori is well understood by Weyl’s criterion. Leon Green’s beautiful theorem then states that ${g^n x}$ equidistributes in the nilmanifold if and only if its image in the torus ${G/(G_2\Gamma)}$ equidistributes.

Today at our miniseminar, Aled Walker showed us Parry’s nice proof of this theorem, which is more elementary than Green’s original proof. During the talk there was some discussion about the importantance of various hypotheses such as “simply connected” and “Lie”. It turns out that the proof works rather generally for connected locally compact nilpotent groups, so I thought I would record the proof here with minimal hypotheses. The meat of the argument is exactly as in Aled’s talk and, presumably, Parry’s paper.

Let ${G}$ be an arbitrary locally compact connected nilpotent group, say with lower central series

$\displaystyle G=G_1\geq G_2 \geq\cdots\geq G_s\geq G_{s+1}=1,$

and let ${\Gamma\leq G}$ be a closed cocompact subgroup. Under these conditions the Haar measure ${\mu_G}$ of ${G}$ induces a ${G}$-invariant probability measure ${\mu_{G/\Gamma}}$ on ${G/\Gamma}$. We say that ${x_n\in G/\Gamma}$ is equidistributed if for every ${f\in C(G/\Gamma)}$ we have

$\displaystyle \frac{1}{N}\sum_{n=0}^{N-1} f(x_n) \rightarrow \int f \,d\mu_{G/\Gamma}.$

We fix our attention on the sequence

$\displaystyle x_n = g^n x$

for some ${g\in G}$ and ${x\in G/\Gamma}$. As before we have an abelianisation map

$\displaystyle \pi: G/\Gamma\rightarrow G/G_2\Gamma$

from the ${G}$-space ${G/\Gamma}$ to the compact abelian group ${G/G_2\Gamma}$. We define equidistribution on ${G/G_2\Gamma}$ similarly. The theorem is then the following.

Theorem 1 (Leon Green’s theorem) For ${g\in G}$ and ${x\in G/\Gamma}$ the following are equivalent.

1. The sequence ${g^n x}$ is equidistributed in ${G/\Gamma}$.
2. The sequence ${\pi(g^n x)}$ is equidistributed in ${G/G_2\Gamma}$.
3. The orbit of ${\pi(g)}$ is dense in ${G/G_2\Gamma}$.
4. ${\chi(\pi(g))\neq 0}$ for every nontrivial character ${\chi:G/G_2\Gamma\rightarrow\mathbf{R}/\mathbf{Z}}$.

Item 1 above trivially implies every other item. The implication 4${\implies}$3 (a generalised Kronecker theorem) follows by pulling back any nontrivial character of ${(G/G_2\Gamma)/\overline{\langle\pi(g)\rangle}}$. The implication 3${\implies}$2 (a generalised Weyl theorem) follows from the observation that every weak* limit point of the sequence of measures

$\displaystyle \frac{1}{N}\sum_{n=0}^{N-1} \delta_{\pi(g^n x)}$

]must be shift-invariant and thus equal to the Haar measure. So the interesting content of the theorem is 2${\implies}$1.

A word about the relation to ergodicity: By the ergodic theorem the left shift ${\tau_g:x\mapsto gx}$ is ergodic if and only if for almost every ${x}$ the sequence ${g^n x}$ equidistributes; on the other hand ${\tau_g}$ is uniquely ergodic, i.e., the only ${\tau_g}$-invariant measure is the given one, if and only if for every ${x}$ the sequence ${g^n x}$ equidistributes. Thus to prove the theorem above we must not only prove that ${\tau_g}$ is ergodic but that it is uniquely ergodic. Fortunately one can prove these two properties are equivalent in this case.

Lemma 2 If ${\tau_g:G/\Gamma\rightarrow G/\Gamma}$ is ergodic then it’s uniquely ergodic.

The following proof is due to Furstenberg.

Proof: By the ergodic theorem the set ${A}$ of ${\mu_{G/\Gamma}}$-generic points, in other words points ${x}$ for which

$\displaystyle \frac{1}{N}\sum_{n=0}^{N-1} f(g^n x) \rightarrow \int f \,d\mu_{G/\Gamma}$

for every ${f\in C(G/\Gamma)}$, has ${\mu_{G/\Gamma}}$-measure ${1}$, and clearly if ${x\in A}$ and ${c\in G_s}$ then ${xc\in A}$, so ${A = p^{-1}(p(A))}$, where ${p}$ is the projection of ${G/\Gamma}$ onto ${G/G_s\Gamma}$.

Now let ${\mu'}$ be any ${\tau_g}$-invariant ergodic measure. By induction we may assume that ${\tau_g:G/G_s\Gamma\rightarrow G/G_s\Gamma}$ is uniquely ergodic, so we must have ${p_*\mu' = p_*\mu_{G/\Gamma}}$, so

$\displaystyle \mu'(A) = p_*\mu'(p(A)) = p_*\mu_{G/\Gamma}(p(A)) = \mu_{G/\Gamma}(A) = 1.$

But by the ergodic theorem the set of ${\mu'}$-generic points must also have ${\mu'}$-measure ${1}$, so there must be some point which is both ${\mu_{G/\Gamma}}$– and ${\mu'}$-generic, and this implies that ${\mu'=\mu_{G/\Gamma}}$. $\Box$

We need one more preliminary lemma about topological groups before we really get started on the proof.

Lemma 3 If ${H}$ and ${K}$ are connected subgroups of some ambient topological group then ${[H,K]}$ is also connected.

Proof: Since ${(h,k)\mapsto [h,k]=h^{-1}k^{-1}hk}$ is continuous certainly ${C = \{[h,k]:h\in H,k\in K\}}$ is connected, so ${C^n = CC\cdots C}$ is also connected, so because ${1\in C^n}$ for all ${n}$ we see that ${[H,K]=\bigcup_{n=1}^\infty C^n}$ is connected. $\Box$

Thus if ${G}$ is connected then every term ${G_1,G_2,G_3,\dots}$ in the lower central series of ${G}$ is connected.

We can now prove Theorem 1. As noted it suffices to prove that ${\tau_g}$ acts ergodically on ${G/\Gamma}$ whenever it acts ergodically on ${G/G_2\Gamma}$. By induction we may assume that ${\tau_g}$ acts ergodically on ${G/G_s\Gamma}$. So suppose that ${f\in L^2(G/\Gamma)}$ is ${\tau_g}$-invariant. By decomposing ${L^2(G/\Gamma)}$ as a ${\overline{G_s\Gamma}/\Gamma}$-space we may assume that ${f}$ obeys

$\displaystyle f(cx)=\gamma(c)f(x)\quad(c\in G_s, x\in G/\Gamma)$

for some character ${\gamma:G_s\Gamma/\Gamma\rightarrow S^1}$. In particular ${|f|}$ is both ${G_s}$-invariant and ${\tau_g}$-invariant, so it factors through a ${\tau_g}$-invariant function ${G/G_s\Gamma\rightarrow\mathbf{R}}$, so it must be constant, say ${1}$. Moreover for every ${b\in G_{s-1}}$ the function

$\displaystyle \Delta_bf(x) = f(bx)\overline{f(x)}$

is ${G_s}$-invariant, and also a ${\tau_g}$ eigenvector:

$\displaystyle \Delta_bf(gx) = \gamma([b,g])\Delta_bf(x).$

By integrating this equation we find that either ${\gamma([b,g])=1}$, so ${\Delta_bf}$ is constant, or ${\int\Delta_bf \,d\mu_{G/\Gamma}= 0}$, so either way we have

$\displaystyle \int \Delta_bf\,d\mu_{G/\Gamma}\in \{0\}\cup S^1.$

But since ${\int\Delta_bf\,d\mu_{G/\Gamma}}$ is a continuous function of ${b}$ and equal to ${1}$ when ${b=1}$ we must have ${\gamma([b,g])=1}$ for all sufficiently small ${b}$, and thus for all ${b}$ by connectedness of ${G_{s-1}}$ and the identity

$\displaystyle [b_1b_2,g]=[b_1,g][b_2,g].$

Thus setting ${\gamma(b)=\Delta_bf}$ extends ${\gamma}$ to a homomorphism ${G_{s-1}\rightarrow S^1}$. In fact we can extend ${\gamma}$ still further to a function ${G\rightarrow D_1}$, where ${D_1}$ is the unit disc in ${\mathbf{C}}$, by setting

$\displaystyle \gamma(a) = \int \Delta_af\,d\mu_{G/\Gamma}.$

Now if ${a\in G}$ and ${b\in G_{s-1}}$ then

$\displaystyle \gamma(ba) = \int f(bax) \overline{f(x)}\,d\mu_{G/\Gamma} = \int \gamma(b)f(ax)\overline{f(x)}\,d\mu_{G/\Gamma}=\gamma(b)\gamma(a),$

and

$\displaystyle \gamma(ab) = \int f(abx)\overline{f(x)}\,d\mu_{G/\Gamma} = \int f(ax) \overline{f(b^{-1}x)}\,d\mu_{G/\Gamma} = \int f(ax) \overline{\gamma(b^{-1})}\overline{f(x)}\,d\mu_{G/\Gamma} = \gamma(b)\gamma(a),$

so

$\displaystyle \gamma(b)\gamma(a)=\gamma(ab)=\gamma(ba[a,b]) = \gamma(ba)\gamma([a,b])=\gamma(b)\gamma(a)\gamma([a,b]).$

Since ${|\gamma(b)|=1}$ we can cancel ${\gamma(b)}$, so

$\displaystyle \gamma(a)(\gamma([a,b])-1) = 0.$

Finally observe that ${\gamma(a)}$ is a continuous function of ${a}$, and ${\gamma(1)=1}$, so we must have ${\gamma([a,b])=1}$ for all sufficiently small ${a}$, and thus by connectedness of ${G}$ and the identity

$\displaystyle [a_1a_2,b]=[a_1,b][a_2,b]$

we must have ${\gamma([a,b])=1}$ identically. But this implies that ${\gamma}$ vanishes on all ${s}$-term commutators and thus on all of ${G_s}$, so in fact ${f}$ factors through ${G/G_s\Gamma}$, so it must be constant. This finishes the proof.

A remark is in order about the possibility that some of the groups ${G_i}$ and ${G_i\Gamma}$ are not closed. This should not matter. One could either read the above proof as it is written, noting carefully that I never said groups should be Hausdorff, or, what’s similar, instead modify it so that whenever you quotient by a group ${H}$ you instead quotient by the group ${\overline{H}}$.

Embarrassingly, it’s difficult to come up with a non-Lie group to which this generalised Leon Green’s theorem applies. It seems that many natural candidates have the property that ${G}$ is not connected but ${G/\Gamma}$ is: for example consider

$\displaystyle \left(\begin{array}{ccc}1&\mathbf{R}\times\mathbf{Q}_2&\mathbf{R}\times\mathbf{Q}_2\\0&1&\mathbf{R}\times\mathbf{Q}_2\\0&0&1\end{array}\right)/\left(\begin{array}{ccc}1&\mathbf{Z}[1/2]&\mathbf{Z}[1/2]\\0&1&\mathbf{Z}[1/2]\\0&0&1\end{array}\right).$

So it would be interesting to know whether the theorem extends to such a case. Or perhaps there are no interesting non-Lie groups for this theorem, which would be a bit of a let down.

# The idempotent theorem

Let ${G}$ be a locally compact abelian group and let ${M(G)}$ be the Banach algebra of regular complex Borel measures on ${G}$. Given ${\mu\in M(G)}$ its Fourier transform

$\displaystyle \hat{\mu}(\gamma) = \int \overline{\gamma}\,d\mu,$

is a continuous function defined on the Pontryagin dual ${\hat{G}}$ of ${G}$. If the measure ${\mu}$ is “nice” in some way then we expect some amount of regularity from the function ${\hat{\mu}}$. For instance if ${\mu}$ is actually an element of the subspace ${L^1(G)\subset M(G)}$ of measures absolutely continuous with respect to the Haar measure of ${G}$ then the Riemann-Lebesgue lemma asserts ${\hat{\mu}\in C_0(\hat{G})}$.

The idempotent theorem of Cohen, Helson, and Rudin describes the structure of measures ${\mu}$ whose Fourier transform ${\hat{\mu}}$ takes a discrete set of values, or equivalently, since ${\|\hat{\mu}\|_\infty\leq\|\mu\|}$, a finite set of values. To describe the theorem, note that we can define ${P(\mu)}$ for any polynomial ${P}$ by taking appropriate linear combinations of convolution powers of ${\mu}$, and moreover we have the relation ${\widehat{P(\mu)} = P(\hat{\mu})}$, where on the right hand side we apply ${P}$ pointwise. Thus if ${\hat{\mu}}$ takes only the values ${a_1,\dots,a_n}$ then by setting

$\displaystyle P_i(x) = \prod_{j\neq i} (x-a_j)/(a_i-a_j)$

we obtain a decomposition ${\mu = a_1\mu_1 + \cdots + a_n\mu_n}$ of ${\mu}$ into a linear combination of measures ${\mu_i=P_i(\mu)}$ whose Fourier transforms ${\hat{\mu_i} = P_i(\hat{\mu})}$ take only values ${0}$ and ${1}$. Such measures are called idempotent, because they are equivalently defined by ${\mu\ast\mu=\mu}$. By the argument just given it suffices to characterise idempotent measures: this explains the name of the theorem.

The most obvious example of an idempotent measure is the Haar measure ${m_H}$ of a compact subgroup ${H\leq G}$. Moreover we can multiply any idempotent measure ${\mu}$ by a character ${\gamma\in\hat{G}}$ to obtain a measure ${\gamma\mu}$ defined by

$\displaystyle \int f \,d(\gamma\mu) = \int f\gamma\,d\mu.$

This measure ${\gamma\mu}$ will again be idempotent, as

$\displaystyle \begin{array}{rcl} \int f\,d(\gamma \mu\ast\gamma \mu) &=& \int\int f(x+y)\gamma(x)\gamma(y)\,d\mu(x)d\mu(y) \\ &=& \int\int f(x+y) \gamma(x+y)\,d\mu(x)d\mu(y) \\ &=& \int f\gamma\,d\mu. \end{array}$

If we add or subtract two idempotent measures then though we may not have again an idempotent measure we certainly have a measure whose Fourier transform takes integer values. On reflection, it feels more natural in the setting of harmonic analysis to require that ${\hat{\mu}}$ takes values in a certain discrete subgroup than to require that it take values in ${\{0,1\}}$, so we relax our restriction so. The idempotent theorem states that we have already accounted for all those ${\mu}$ such that ${\hat{\mu}}$ is integer-valued.

Theorem 1 (The idempotent theorem) For every ${\mu\in M(G)}$ such that ${\hat{\mu}}$ is integer-valued there is a finite collection of compact subgroups ${G_1,\dots,G_k\leq G}$, characters ${\gamma_1,\dots,\gamma_k\in\hat{G}}$, and integers ${n_1,\dots,n_k\in\mathbf{Z}}$ such that

$\displaystyle \mu = n_1\gamma_1 m_{G_1} + \cdots + n_k\gamma_k m_{G_k}.$

As a consequence we deduce a structure theorem for ${\mu}$ with ${\hat{\mu}}$ taking finitely many values, as we originally wanted: for every such ${\mu}$ there is a finite collection of compact subgroups ${G_1,\dots,G_k\leq G}$, characters ${\gamma_1,\dots,\gamma_k\in\hat{G}}$, and complex numbers ${a_1,\dots,a_k\in\mathbf{C}}$ such that

$\displaystyle \mu = a_1\gamma_1 m_{G_1} + \cdots + a_k \gamma_k m_{G_k}.$

The theorem was first proved in the case of ${G=\mathbf{R}/\mathbf{Z}}$ by Helson in 1953: in this case the theorem states simply that if ${\hat{\mu}}$ is integer-valued then it differs from some periodic function in finitely many places. In 1959 Rudin gave the theorem its present form and proved it for ${(\mathbf{R}/\mathbf{Z})^d}$. Finally in 1960 Cohen proved the general case, in the same paper in which he made the first substantial progress on the Littlewood problem. The proof was subsequently simplified a good deal, particularly by Amemiya and Ito in 1964. We reproduce their proof here.

First note that if ${\hat{\mu}}$ is integer-valued then ${\mu}$ is supported on a compact subgroup. Indeed by inner regularity there is a compact set ${K}$ such that ${|\mu|(K^c)<0.1}$, the set ${U}$ of all ${\gamma\in\hat{G}}$ such that ${|1-\gamma|<0.1/\|\mu\|}$ on ${K}$ is then open, and if ${\gamma\in U}$ then

$\displaystyle \|\gamma\mu-\mu\| = \int_G |\gamma-1|\,d|\mu| \leq \int_K + \int_{K^c} < 0.1 + 0.1 < 1.$

But if ${\gamma\mu\neq\mu}$ then

$\displaystyle \|\gamma\mu-\mu\|\geq \|\widehat{\gamma\mu}-\hat{\mu}\|_\infty \geq 1,$

so ${\gamma\mu=\mu}$ for all ${\gamma\in U}$. Thus ${\Gamma=\{\gamma\in\hat{G}: \gamma\mu=\mu\}}$ is an open subgroup of ${\hat{G}}$, so by Pontryagin duality its preannihilator ${\Gamma^\perp = \{g\in G: \gamma(g)=1 \text{ for all }\gamma\in\Gamma\}}$ is a compact subgroup of ${G}$. Clearly ${\mu}$ is supported on ${\Gamma^\perp}$. Thus from now on we assume ${G}$ is compact.

Fix a measure ${\mu\in M(G)}$ and let ${A=\{\gamma\mu: \gamma\in\hat{G}\}}$.

Lemma 2 If ${\nu}$ is a weak* limit point of ${A}$ then ${\|\nu\|<\|\mu\|}$.

Proof: Fix ${\epsilon>0}$ and suppose we could find ${f\in C(G)}$ such that ${\|f\|_\infty\leq 1}$ and ${\int f\,d\nu > (1-\epsilon)\|\mu\|}$. Let ${\gamma\mu}$ be close enough to ${\nu}$ that ${\Re\int f\gamma\,d\mu > (1-\epsilon)\|\mu\|}$. Write ${\mu = \theta|\mu|}$ and ${f\gamma\theta = g + ih}$. Then if ${Z}$ is the complex number ${Z = \int (g+i|h|)\,d|\mu|}$, then ${|Z|\leq\|\mu\|}$ and

$\displaystyle \Re Z = \int g \,d|\mu| = \Re\int f\gamma\,d\mu > (1-\epsilon)\|\mu\|,$

so we must have

$\displaystyle \Im Z = \int |h|\,d|\mu| \leq (1-(1-\epsilon)^2)^{1/2}\|\mu\| \leq 2\epsilon^{1/2}\|\mu\|.$

Thus also

$\displaystyle \int |1 - f\gamma\theta| \,d|\mu| \leq \int |1 - g|\,d|\mu| + \int |h|\,d|\mu| \leq 3\epsilon^{1/2}\|\mu\|.$

But if this holds for both ${\gamma_1\mu}$ and ${\gamma_2\mu}$, say with ${\gamma_1\mu\neq\gamma_2\mu}$, then we have

$\displaystyle 1\leq \|\gamma_1\mu-\gamma_2\mu\| \leq \int |\gamma_1 - f\gamma_1\gamma_2\theta|\,d|\mu| + \int |\gamma_2 - f\gamma_1\gamma_2\theta|\,d|\mu| \leq 6\epsilon^{1/2}\|\mu\|,$

so ${\epsilon \geq 1/(36\|\mu\|^2)}$, so

$\displaystyle \|\nu\| \leq \|\mu\| - \frac{1}{36\|\mu\|}.$

This proves the lemma. $\Box$

Lemma 3 If ${\nu}$ is a weak* limit point of ${A}$ then ${\nu}$ is singular with respect to the Haar measure ${m_G}$ of ${G}$.

Proof: By the Radon-Nikodym theorem we have a decomposition ${\mu = f m_G + \mu_s}$ for some ${f\in L^1(G)}$ and some ${\mu_s}$ singular with respect to ${m_G}$. By the Riemann-Lebesgue lemma then ${\nu}$ is a limit point of ${\{\gamma\mu_s:\gamma\in\hat{G}\}}$. Thus for any open set ${U}$ and ${f\in C(G)}$ such that ${\|f\|_\infty\leq 1}$ and ${f=0}$ outside of ${U}$ we have

$\displaystyle \left|\int f\,d\nu\right| \leq \sup_\gamma \left|\int f\gamma \,d\mu_s\right| \leq |\mu_s|(U),$

so ${|\nu|(U)\leq |\mu_s|(U)}$. This inequality extends to Borel sets in the usual way, so ${\nu}$ is singular. $\Box$

The theorem follows relatively painlessly from the two lemmas. Fix ${\mu\in M(G)}$ with ${\hat{\mu}}$ integer-valued and let ${A = \{\gamma\mu: \int\gamma\,d\mu\neq 0\}}$. Then ${\overline{A}}$ is weak* compact, so because ${\|\cdot\|}$ is lower semicontinuous in the weak* topology there is some ${\nu\in\overline{A}}$ of minimal norm. Since ${\int d\nu}$ is an integer different from ${0}$ we must have ${\nu\neq 0}$. Thus by Lemma~2 the set ${\{\gamma\nu: \int\gamma\,d\nu\neq 0\}}$ is finite. But this implies that

$\displaystyle \nu = (n_1 \gamma_1 + \cdots + n_k \gamma_k) m_H \ \ \ \ \ (1)$

for some ${n_1,\dots,n_k\in\mathbf{Z}}$, ${\gamma_1,\dots,\gamma_k\in\hat{G}}$, and ${H=\{\gamma:\gamma\nu=\nu\}^\perp}$ the support group of ${\nu}$. In particular ${\nu}$ is absolutely continuous with respect to ${m_H}$, so because ${\nu|_H}$ is in the weak* closure of ${\{\gamma\mu|_H:\gamma\in\hat{G}\}}$ we deduce from Lemma 2 that ${\nu|_H = \gamma\mu|_H}$ for some ${\gamma}$. Thus ${\mu|_H}$ is a nonzero measure of the form (1) and we have an obvious mutually singular decomposition

$\displaystyle \mu = \mu|_H + (\mu-\mu|_H).$

Since ${\|\mu-\mu|_H\| = \|\mu\| - \|\mu|_H\|\leq\|\mu\|-1}$ the theorem follows by induction.


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# Simpson's Rule to find the volume of f(x) rotated about the x and y axis.

1. Jun 21, 2012

### arishorts

1. The problem statement, all variables and given/known data

Answers in the back of the book

2. Relevant equations

Simpson's Rule: (dx/3)* sum of(sequence of coefficients {1,4,2....2,4,1}*sequence of function values{f(0), f(1), f(2).....f(n-2),f(n-1), f(n)})

Volume using Shells: 2π ∫ (radius)(height) dx

3. The attempt at a solution

I found the area (≈) under the curve using Simpson's law, how do i rotate it without a given f(x)? The book doesn't ask for the specific method (either shells or cross-sections), but I'd like to understand how to do both. Please help.

2. Jun 21, 2012

### LCKurtz

The integral for a volume of revolution of $f(x)$ between $a$ and $b$ is$$V=\pi\int_a^b f^2(x)dx$$Do Simpson's rule on that, not on the area integral.

3. Jun 21, 2012

### arishorts

we're not given f(x) though. We're only given the values of f(x)

4. Jun 21, 2012

### LCKurtz

So you can figure out the values of $\pi f^2(x)$ and do Simpsons rule just like you did for the area.

5. May 26, 2013

### z37002

Great question. Would someone please give more details as to combining function and s rule?

format: plain


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# Converting ConditionalExpression to Piecewise

I have Solve[] return a list like {{F->ConditionalExpression[EXPR1, 0<z<a1]},{F->ConditionalExpression[EXPR2, a1<z<a2]},...} and so on -- EXPR* and a1,a2,... are somewhat complicated expressions, containing things like Root[]:

EQ = F^4 (138880 z + 318688 F z + 98415 F^10 z + 6561 F^9 (5 + 4 z) +
405 F^5 (625 + 1136 z) + 243 F^6 (-2500 + 5077 z)) ==
1024 z + 11008 F z + 40192 F^2 z + 33536 F^3 z + 46944 F^6 z +
709776 F^7 z + 368388 F^8 z + 4374 F^12 (50 + 43 z) +
4374 F^11 (-125 + 202 z);

ftz = Solve[EQ && 0 < F < 1 && 0 < z < 1, F];

e.g. the first conditional:

In[100]:= N[(F /. ftz[[1]])]

Out[100]= ConditionalExpression[
Root[-1024 z - 11008 z #1 - 40192 z #1^2 - 33536 z #1^3 +
138880 z #1^4 + 318688 z #1^5 - 46944 z #1^6 - 709776 z #1^7 -
368388 z #1^8 + (253125 + 460080 z) #1^9 + (-607500 +
1233711 z) #1^10 + (546750 - 883548 z) #1^11 + (-218700 -
188082 z) #1^12 + (32805 + 26244 z) #1^13 + 98415 z #1^14 &,
2], 0. < z < 0.674139]

I need to (numerically) integrate this function for given parameter values, which Mathematica refuses to do. However, if I convert the expression into a Piecewise function: for example as

ff = Piecewise[{{ftz[[1]][[1]][[2]][[1]], ftz[[1]][[1]][[2]][[2]]},
{ftz[[2]][[1]][[2]][[1]], ftz[[2]][[1]][[2]][[2]]},
{0, z == 0},
{1, z == 1}}];

it easily computes the integral. Unfortunately, the number of intervals and the breakpoints are parameter dependent, and this must be done automatically as the function is called inside another function.

Does anyone have any suggestions on how to convert such a list of conditionals into a Piecewise function, or how to otherwise compute the integral? Thanks!

-
This is an interesting question, however it is not clear what is the function, what are parameters, what is your expected Piecewise form etc. More details are needed to give a constructive answer. ConditionalExpression is new in M8 and I suspect it is not quite seamlessly integrated with the rest of the system. –  Artes Jun 15 '12 at 17:14
Thanks, I added an example equation. Unfortunately I do not have an earlier version of Mathematica installed to see what Solve returns there :( –  laxxy Jun 15 '12 at 18:04
By the way, when multiple indexing, you can use a single call to Part. It'll be more efficient and neater. x[[1]][[2]]==x[[1, 2]] –  Rojo Jun 16 '12 at 15:27

That conversion is straightforward, in this case. It is a one-liner

Piecewise[List @@@ Flatten[N@ftz][[All, 2]] ]

Now let me explain. As you have noticed, Solve returns solutions of the form:

{ {_ -> _}, {_ -> _}, ...}

which is what Flatten reduces to

{_ -> _, _ -> _, ...}

Then, I take only the right hand side of each Rule via Flatten[...][[All,2]] leaving us with something of the form

{ConditionalExpression[...], ...}

So, we need to replace ConditionalExpression with List to make it palatable to Piecewise, and the it is done using the shorthand form of Apply at level 1, @@@. Finally, we feed the result into Piecewise and get what you want.

As pointed out in the comments, the simpler form

Piecewise[List @@@ Last @@@ N@ftz]

works just as well.

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Thanks, this works! :) –  laxxy Jun 15 '12 at 21:56
(+1) could also apply Apply twice: Piecewise[List @@@ Last @@@ N@ftz] or List @@@ Last @@@ N@ftz // Piecewise :) –  kguler Jun 15 '12 at 21:56
@kguler updated my answer with that form. –  rcollyer Jun 15 '12 at 21:58

It may not always be appropriate to convert a ConditionalExpression into Piecewise, because the two are not equivalent, and you may be changing the meaning. This is because Piecewise IMPOSES a default value of 0 if the conditions are not satisfied, whereas ConditionalExpression leaves the expression as undefined.

For example, given:

In[1]:=  expr = ConditionalExpression[1, x > 3]

In[2]:=  Simplify[expr, x < -2]
Out[2]=  Undefined

whereas:

In[3]:=  Simplify[Piecewise[{{1, x > 3}}],  x < -2]
Out[3]=  0

But, if you do want to impose the swap, say given:

In[4]:=  sol = Integrate[x^n, {x, 0, 1}]
Out[4]=  ConditionalExpression[1/(1 + n), Re[n] > -1]

... then an easy way to do it is:

In[5]:=  sol /. ConditionalExpression[xx_, yy_] :> Piecewise[{{xx, yy}}]

-


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# Thread: A partial order set with a non-unique maximal element

1. ## A partial order set with a non-unique maximal element

Find an example of a partial order set with a maximal element that is not unique.

My question here is, since a maximal element is define as:

m is a max of a set X if $m \leq x$, then x = m.

Not unique means that there are at least two, but how can I find two such elements? If we have two different max, say m and n, then either $m \leq n$ or $n \leq m$ since $m \neq n$, but then by definition of a maximal element, m is n...

Thank you!

2. let $x\le y$ if $x^2\le y^2$. And to this on the interval $[-a,a]$ for $a>0$.

Find an example of a partial order set with a maximal element that is not unique.
Remember this is a partial order not a total ordering.
Consider $\{1,2,3,4,5,6,7,8,9\}$ ordered by divides, $
x \prec y \Leftrightarrow x|y$
.
If you want something a little more concrete use sets. Supose A= {a}, B= {a, b}, C= {a, c}, D= {a, b, c} with " $\le$" defined by "inclusion": [tex]X\le Y[tex] if and only if X is a subset of Y. That is partially ordered because neither B nor C is a subset of the other. $B\le C$ is false because B is not a subset of C. $C\le B$ is not true because C is not a subset of B. Clearly B= C is not true. So this is a partially ordered set in which D is the maximum (and so "maximal").


text:
# zbMATH — the first resource for mathematics

The Heegaard genus of amalgamated 3-manifolds. (English) Zbl 1081.57018
Genera of closed $$3$$-manifolds obtained from two $$3$$-manifolds by glueing along their boundaries are studied. Let $$g(X)$$ denote the genus of $$X$$. Let $$M_1, M_2$$ be simple $$3$$-manifolds. Suppose that there is a homeomorphism $$h_i : \partial M_i \rightarrow S$$ for $$i=1, 2$$, where $$S$$ is connected and of genus at least two. Let $$\psi : S \rightarrow S$$ be a pseudo-Anosov homeomorphism. The main theorem states that $$g(M_1 \cup_{\Psi} M_2) = g(M_1) + g(M_2) - g(S)$$, if $$\Psi = h_2^{-1} \psi^n h_1$$ and $$| n|$$ is sufficiently large. Furthermore, any minimal genus Heegaard splitting for $$M_1 \cup_{\Psi} M_2$$ is obtained from splittings of $$M_1$$ and $$M_2$$ by amalgamation and is weakly reducible. Amalgamation was defined by J. Schultens [Proc. London Math. Soc. (3) 67, 425–448 (1993; Zbl 0789.57012)]. This holds even for $$\Psi=h_2^{-1} \phi h_1$$ with $$\phi$$ of sufficiently large distance. For general glueing homeomorphisms, certain inequalities have been discovered by K. Johannson [Topology and Combinatorics of $$3$$-Manifolds, Springer-Verlag (Berlin) (1995; Zbl 0820.57001)] and J. Schultens [Heegaard genus formula for Haken manifolds, preprint]. Genera of non-minimal Heegaard splittings, and that of $$M/\sim$$ obtained by glueing two components of $$\partial M$$ are also studied. We summarise the proof of the main theorem. $$M_1 \bigcup_{\Psi} M_2$$ has a hyperbolic structure by the geometrisation theorem. There are an arbitrarily large number of parallel copies of $$S$$ in $$M_1 \cup_{\Psi} M_2$$ such that any two adjacent ones have distance at least one from each other, provided that $$| n|$$ is sufficiently large. Untelescoping a minimal genus Heegaard splitting $$F$$, we construct a generalised Heegaard splitting. The positive splitting surfaces $$F_+$$ may be isotoped in a small neighbourhood of a minimal surface. Then we can find a copy of $$S$$ disjoint from $$F_+$$. It is parallel to some component of the generalised splitting surfaces, and hence $$F$$ is an amalgamation of Heegaard splittings of $$M_1$$ and $$M_2$$.

##### MSC:
 57N10 Topology of general $$3$$-manifolds (MSC2010)
##### Keywords:
Haken manifold; minimal surfaces; hyperbolic geometry
Full Text:
##### References:
 [4] Lackenby M.: Heegaard splittings, the virtually Haken conjecture and Property ($$\tau$$), Preprint. · Zbl 1110.57015 [5] Morgan, J.: On Thurstons uniformization theorem for three-dimensional manifolds, In: The Smith Conjecture, Pure Appl. Math. 112, Academic Press, New York, 1979, pp. 37– 125 [6] Pitts, J. and Rubinstein, J. H.: Existence of minimal surfaces of bounded topological type in three-manifolds. In: Miniconference on Geometry and Partial Di.erential Equations (Canberra, 1985), pp. 163–176 [7] Rubinstein, J. H.: Minimal surfaces in geometric 3-manifolds, Preprint · Zbl 1119.53042 [8] Scharlemann, M.: Heegaard splittings of compact 3-manifolds, Handbook of Geometric Topology, Elsevier, Amsterdam, 2002, pp. 921-953 · Zbl 0985.57005 [9] Scharlemann M. and Thompson, A.: Thin position for 3-manifolds, In: Geometric Topology (Haifa, 1992), Contemp. Math. 164, Amer. Math. Soc., Providence, 1992, pp. 231-238. · Zbl 0818.57013 [12] Schultens, J.: Heegaard genus formula for Haken manifolds, Preprint. · Zbl 1099.57019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
# Changeset 787:c2230649a493 in lemon-1.2

Ignore:
Timestamp:
11/15/09 19:57:02 (10 years ago)
Branch:
default
Phase:
public
Message:

Various doc improvements (#331)

• Add notes to the graph classes about the time of item counting.
• Clarify the doc for run() in BFS and DFS.
• Other improvements.
Location:
lemon
Files:
11 edited

Unmodified
Removed

• ## lemon/bfs.h

 r717 ///Runs the algorithm to visit all nodes in the digraph. ///This method runs the %BFS algorithm in order to ///compute the shortest path to each node. /// ///The algorithm computes ///- the shortest path tree (forest), ///- the distance of each node from the root(s). ///This method runs the %BFS algorithm in order to visit all nodes ///in the digraph. /// ///\note b.run(s) is just a shortcut of the following code. ///Runs BFS algorithm to visit all nodes in the digraph. ///This method runs BFS algorithm in order to compute ///the shortest path to each node. ///This method runs BFS algorithm in order to visit all nodes ///in the digraph. void run() { /// \brief Runs the algorithm to visit all nodes in the digraph. /// /// This method runs the %BFS algorithm in order to /// compute the shortest path to each node. /// /// The algorithm computes /// - the shortest path tree (forest), /// - the distance of each node from the root(s). /// This method runs the %BFS algorithm in order to visit all nodes /// in the digraph. /// /// \note b.run(s) is just a shortcut of the following code.
• ## lemon/dfs.h

 r717 ///Runs the algorithm to visit all nodes in the digraph. ///This method runs the %DFS algorithm in order to compute the ///%DFS path to each node. /// ///The algorithm computes ///- the %DFS tree (forest), ///- the distance of each node from the root(s) in the %DFS tree. ///This method runs the %DFS algorithm in order to visit all nodes ///in the digraph. /// ///\note d.run() is just a shortcut of the following code. ///Runs DFS algorithm to visit all nodes in the digraph. ///This method runs DFS algorithm in order to compute ///the DFS path to each node. ///This method runs DFS algorithm in order to visit all nodes ///in the digraph. void run() { /// \brief Runs the algorithm to visit all nodes in the digraph. /// This method runs the %DFS algorithm in order to /// compute the %DFS path to each node. /// /// The algorithm computes /// - the %DFS tree (forest), /// - the distance of each node from the root(s) in the %DFS tree. /// This method runs the %DFS algorithm in order to visit all nodes /// in the digraph. /// /// \note d.run() is just a shortcut of the following code.
• ## lemon/dijkstra.h

 r717 ///The type of the arc lengths. typedef typename TR::LengthMap::Value Value; typedef typename TR::Value Value; ///The type of the map that stores the arc lengths. typedef typename TR::LengthMap LengthMap;
• ## lemon/edge_set.h

 r778 /// all arcs incident to the given node is erased from the arc set. /// /// This class fully conforms to the \ref concepts::Digraph /// "Digraph" concept. /// It provides only linear time counting for nodes and arcs. /// /// \param GR The type of the graph which shares its node set with /// this class. Its interface must conform to the /// \ref concepts::Digraph "Digraph" or \ref concepts::Graph "Graph" /// concept. /// /// This class fully conforms to the \ref concepts::Digraph /// "Digraph" concept. template class ListArcSet : public ArcSetExtender > { /// incident to the given node is erased from the arc set. /// /// This class fully conforms to the \ref concepts::Graph "Graph" /// concept. /// It provides only linear time counting for nodes, edges and arcs. /// /// \param GR The type of the graph which shares its node set /// with this class. Its interface must conform to the /// \ref concepts::Digraph "Digraph" or \ref concepts::Graph "Graph" /// concept. /// /// This class fully conforms to the \ref concepts::Graph "Graph" /// concept. template /// arcs. Therefore the arcs cannot be erased from the arc sets. /// /// This class fully conforms to the \ref concepts::Digraph "Digraph" /// concept. /// It provides only linear time counting for nodes and arcs. /// /// \warning If a node is erased from the underlying graph and this /// node is the source or target of one arc in the arc set, then /// the arc set is invalidated, and it cannot be used anymore. The /// validity can be checked with the \c valid() member function. /// /// This class fully conforms to the \ref concepts::Digraph /// "Digraph" concept. template class SmartArcSet : public ArcSetExtender > { /// edges cannot be erased from the edge sets. /// /// This class fully conforms to the \ref concepts::Graph "Graph" /// concept. /// It provides only linear time counting for nodes, edges and arcs. /// /// \warning If a node is erased from the underlying graph and this /// node is incident to one edge in the edge set, then the edge set /// is invalidated, and it cannot be used anymore. The validity can /// be checked with the \c valid() member function. /// /// This class fully conforms to the \ref concepts::Graph /// "Graph" concept. template class SmartEdgeSet : public EdgeSetExtender > {
• ## lemon/full_graph.h

 r780 /// only in the concept class. /// /// This class provides constant time counting for nodes and arcs. /// /// \note FullDigraph and FullGraph classes are very similar, /// but there are two differences. While this class conforms only /// completely static, the nodes can be indexed with integers from /// the range [0..nodeNum()-1]. /// The index of a node is the same as its ID. /// \sa index() Node operator()(int ix) const { return Parent::operator()(ix); } /// completely static, the nodes can be indexed with integers from /// the range [0..nodeNum()-1]. /// The index of a node is the same as its ID. /// \sa operator()() static int index(const Node& node) { return Parent::index(node); } /// only in the concept class. /// /// This class provides constant time counting for nodes, edges and arcs. /// /// \note FullDigraph and FullGraph classes are very similar, /// but there are two differences. While FullDigraph /// completely static, the nodes can be indexed with integers from /// the range [0..nodeNum()-1]. /// The index of a node is the same as its ID. /// \sa index() Node operator()(int ix) const { return Parent::operator()(ix); } /// completely static, the nodes can be indexed with integers from /// the range [0..nodeNum()-1]. /// The index of a node is the same as its ID. /// \sa operator()() static int index(const Node& node) { return Parent::index(node); }
• ## lemon/grid_graph.h

 r735 /// Most of its member functions and nested classes are documented /// only in the concept class. /// /// This class provides constant time counting for nodes, edges and arcs. class GridGraph : public ExtendedGridGraphBase { typedef ExtendedGridGraphBase Parent;
• ## lemon/hypercube_graph.h

 r780 /// Most of its member functions and nested classes are documented /// only in the concept class. /// /// This class provides constant time counting for nodes, edges and arcs. /// /// \note The type of the indices is chosen to \c int for efficiency
• ## lemon/list_graph.h

 r741 ///only in the concept class. /// ///This class provides only linear time counting for nodes and arcs. /// ///\sa concepts::Digraph ///\sa ListGraph ///\brief Erase a node from the digraph. /// ///This function erases the given node from the digraph. ///This function erases the given node along with its outgoing and ///incoming arcs from the digraph. /// ///\note All iterators referencing the removed node or the connected ///arcs are invalidated, of course. void erase(Node n) { Parent::erase(n); } /// ///This function erases the given arc from the digraph. /// ///\note All iterators referencing the removed arc are invalidated, ///of course. void erase(Arc a) { Parent::erase(a); } ///This function erases all nodes and arcs from the digraph. /// ///\note All iterators of the digraph are invalidated, of course. void clear() { Parent::clear(); ///only in the concept class. /// ///This class provides only linear time counting for nodes, edges and arcs. /// ///\sa concepts::Graph ///\sa ListDigraph ///\brief Erase a node from the graph. /// /// This function erases the given node from the graph. /// This function erases the given node along with its incident arcs /// from the graph. /// /// \note All iterators referencing the removed node or the incident /// edges are invalidated, of course. void erase(Node n) { Parent::erase(n); } /// /// This function erases the given edge from the graph. /// /// \note All iterators referencing the removed edge are invalidated, /// of course. void erase(Edge e) { Parent::erase(e); } /// Node validity check ///This function erases all nodes and arcs from the graph. /// ///\note All iterators of the graph are invalidated, of course. void clear() { Parent::clear();
• ## lemon/smart_graph.h

 r780 ///only in the concept class. /// ///This class provides constant time counting for nodes and arcs. /// ///\sa concepts::Digraph ///\sa SmartGraph /// only in the concept class. /// /// This class provides constant time counting for nodes, edges and arcs. /// /// \sa concepts::Graph /// \sa SmartDigraph
• ## lemon/static_graph.h

 r779 /// only in the concept class. /// /// This class provides constant time counting for nodes and arcs. /// /// \sa concepts::Digraph class StaticDigraph : public ExtendedStaticDigraphBase {
Note: See TracChangeset for help on using the changeset viewer.


text:
## mathsux4real Group Title Please help :0 Find the term containing x^42 of (x^3-3)^16 2 years ago 2 years ago

1. satellite73 Group Title

well first we need $$\frac{42}{3}=14$$ to know which term we want

2. satellite73 Group Title

so it is either the 14th term or the second term depending on how you are counting. makes no difference. the term looks like $\dbinom{16}{2}(x^3)^{14}(-3)^2$

3. satellite73 Group Title

$\dbinom{16}{2}=\frac{16\times 15}{2}=8\times 15=120$ and $$(-3)^2=9$$ multiply to get your answer


text:
info:user_guide

# COR@L User Guide

This page contains instructions and guidelines for COR@L users for accessing the resources available in the laboratory. These instructions are far from comprehensive but represent a basic guide for getting started. We hope you find these guidelines to be helpful on your path to becoming an expert on the tools you need in your research. Feel free to contact Suresh Bolusani if you have any questions or suggestions for additions to this page.

## Getting Started with Linux

If you are new to Linux, then you may find the following links to be useful.

## Logging into COR@L

Every user with a COR@L account can access the machines listed here using her/his username and password. However, if you are outside Lehigh’s network, then you need to log into shark.ie.lehigh.edu or coral.ie.lehigh.edu and then ssh into another machine on the list. Please keep in mind that shark is the file server and coral is the web server. Therefore, these two machines should not be used for big and/or long running jobs. If you have such a job, then please use another computer in the list.

#### Linux users

If you are a Linux user, then logging into COR@L is straightforward. The following command uses ssh, which is typically available on standard Linux distributions:

ssh username@machinename.ie.lehigh.edu

Your COR@L username and password are the same for all machines. COR@L uses NIS to manage passwords, so passwords can be at most 8 characters long. Any extra characters will be ignored. If you wish to change your password, then do not use the passwd command. Use yppasswd instead.

#### Windows users

If you are using Windows and want to sign into a COR@L machine, then you need to install an ssh client first. There are several free software options. You can check Wikipedia for more information about your choices. A common free ssh client is putty, which can be downloaded here. This ssh client is easy to use and it is well documented here.

Back

## Compiling Programs

Eclipse is available on all machines. Use gcc to compile C programs, g++ to compile C++ programs, and f77 to compile Fortran programs. gdb may be used for debugging. valgrind may be used to check for memory leaks. [Purify/Quantify] are also available for checking memory leaks and for profiling.

## Optimization Software

Below is a list of the tools available for supporting optimization research at COR@L.

• COIN-OR
• CPLEX
• EXPRESS
• GLPK
• GUROBI
• MOSEK

• AMPL
• GAMS
• PULP

Back

## Coin-all in mac

This is a description of the steps followed to install coin-or on a OS X Mountain Lion using macports:

1. Install macports, there is a nice installation guide in this link
2. Install gcc47: To do this open a terminal and type “sudo port install gcc47″.
3. Install subversion: “sudo port install subversion”.
5. Move to directory coin-all and do “mkdir buildmac”
6. Download all the Third Party sources. Got to the ThirdParty folder and run the get script for: ASL, Blas, Lapack, Mumps, Metis, Glpk
7. Move to the directory buildmac
8. Run the configure script (This command is taken from link):
../configure –disable-shared F77=gfortran-mp-4.7 FFLAGS=”-fexceptions -m64 -fbackslash” CFLAGS=”-fno-common -no-cpp-precomp -fexceptions -m64″ CXXFLAGS=”-fno-common -no-cpp-precomp -fexceptions -m64″
9. make tests (the DyLP test fails, rename the folder DyLP/test to DyLP/test.back if you want to run the other tests)
10. make install, remember that you need to use the option -prefix=/path/to/install if you want to specify a particular path to install the binaries

## Using CONDOR on COR@L

Also, check the main thread: CONDOR

• To submit a job to the condor pool you need a condor.sub file similar to this one:
 ################################################################################# # This line is important to get the license variables of mosek, cplex and others # This line is needed only once in the file getenv = TRUE # This is the standard universe, this should work fine with most of the tasks # This line is needed only once in the file Universe = vanilla # This is where you specify the executable you want to use # Advice: absolute paths will ensure that you are using the file you want Executable = mosek # This is where you put the arguments for the executable arguments = -into r12c15k5i10.sol r12c15k5i10.mps # This tells condor where to place the outputs it can generate, they are really useful to identify # if there are problems with the job you are trying to run Output = r12c15k5i10.out Error = r12c15k5i10.err Log = r12c15k5i10.log # This instruct condor to put the job in the queue queue # You can put as many jobs as you need repeating the same lines with different information # Notice that you don’t need to specify the executable again, unless you want to use a different # file

arguments = -into r12c15k5i15.sol r12c15k5i15.mps Output = r12c15k5i10.out Error = r12c15k5i10.err Log = r12c15k5i10.log queue #######################################################################################################

• Once you have your condor file you can send it to the pool with the command “condor_submit condor.sub”
• To check the status of the job you run “condor_q -global” and your username will appear under the column owner for the jobs you are currently running
• To kill a job you use the command “condor_rm ID”, where ID is the number in the ID column of the output of condor_q associated with the job you want to remove. You can remove only those jobs you are the owner.


text:
Alice has two secret numbers, a and b. She publishes c1=E(a), c2=E(b) and c3=E(a+b). Is there an encryption system E such that anyone would be able to prove that the c3 as published by Alice is actually E(a+b)?

EDIT: Modified the question to use "anyone" instead of "Bob".

Note that this is different from homomorphic encryption. In case of homomorphic encryption Bob is able to compute E(a+b) himself, but he is not necessarily able to prove that E(a+b) as computed by Alice is correct. What's needed here is a system where Bob is not necessarily able to compute E(a+b) but is able to prove Alice's result.

• Btw, there's a system that can be used this way outlined in the accepted answer here: crypto.stackexchange.com/questions/6732/… however it requires revealing r3 to Bob. Can we do better? – Martin Sustrik Oct 21 '13 at 8:44
• – user991 Oct 21 '13 at 9:39
• How would you approach the problem via ZKP? Sorry, I am not a cryptographer, so I'm just trying to get my head around this. – Martin Sustrik Oct 21 '13 at 10:27
• You may specify more precisely why you are requiring this feature. If you have $c_1$ and $c_2$ in an authentic fashion, then anyone can recompute $c_3'$ and compare it to the obtained $c_3$ (this operation is deterministic in homomorphic encryption schemes). You may also look here: crypto.stackexchange.com/questions/10048/… – DrLecter Oct 21 '13 at 11:08
• Imagine a game with cards. Each participant has secret information (the cards in the hand) that he doesn't want to disclose to other participants. He can also perform an operation (say exchange one of the cards) but he doesn't want to disclose the result of the operation, i.e. the resulting set of cards in the hand should remain secret. However, other participants want to ensure that he played by rules, i.e. he exchanged exacly one card. In general, the goal is to ensure that every participant plays by rules without disclosing their secret. – Martin Sustrik Oct 21 '13 at 11:30

One example of such a system is ElGamal over a group $G$ generated by $g$. The public key is $y$ and an encryption $(x,w)$ of a message $m$ is of the form $(g^r, y^r m)$ for some integer $r$.
You have three ciphertexts $(x_1, w_1)$, $(x_2, w_2)$ and $(x_3, w_3)$ which are encryptions of $m_1$, $m_2$ and $m_1m_2$, respectively. The random integers used is $r_1$, $r_2$ and $r_3$, respectively. Note that $r_3$ isn't $r_1+r_2$, it is a random number. You want to produce a proof that the encryptions are correct.
The idea is to observe that the ciphertext $(x_1x_2/x_3, w_1 w_2 / w_3)$ is an encryption of $1$. As such, you only need to prove that it is an encryption of $1$. This amounts to proving that the logarithm of $x_1x_2/x_3$ to the base $g$ equals the logarithm of $w_1w_2/w_3$ to the base $y$, which is done using a standard proof of equality of discrete logarithms.
• Didn't @martin-sustrik said that he didn't want to use any homomorphic cryptosystem? In your solution you're assuming that anyone has access to $c_1=(x_1,w_1)$ and $c_2=(x_2,w_2)$, and if that was the case, then anyone could also compute $E(a+b)$. – LRM Oct 23 '13 at 10:44
• Yes, but still you're using a homomorphic cryptosystem, otherwise $\frac{c_1 c_2}{c_3}$ wouldn't give you the encryption of $1$, isn't it? – LRM Oct 24 '13 at 8:08


text:
## anonymous 4 years ago 33 1/3% of 240

1. anonymous

(33 1/3)% of 240 = 80

2. anonymous

(33 1/3)% of 240 = 80

3. phi

rewrite 33 1/3 % as $\frac{100}{3} \% = \frac{100}{3 \cdot 100} = \frac{1}{3}$


text:
Is there any guarantee of maximum distance to next prime?

When watching the Numberphile video about Highly Composite Number, I spotted something that aroused some of my doubts. One of properties suggested by Ramanujan was that the highly composite number's powers of prime factors are in order of decreasing order, with the highest prime factor almost always (with exactly 2 exceptions: 4 and 36.) appearing with power of 1.

It seems to me this assertion hinges upon the next prime after the last being lower than the square of the previous one. While π(N) shows the average distance between the consecutive primes would be significantly lower than between the prime and it's square, as I understand it's more of a probabilistic thing, and while very unlikely, it's not guaranteed next prime will be found within pretty much any finite distance of the prior one. So is this property of highly composite numbers just a conjecture based on dwindling probability of such a gap between primes ever appearing, or is there some solid proof to it?

• en.m.wikipedia.org/wiki/Bertrand%27s_postulate Aug 30 '20 at 0:03
• On the opposite end of the spectrum, there are very small prime gaps. According to en.wikipedia.org/wiki/Prime_gap#Upper_bounds there are infinitely many prime gaps with a gap less than 247. Aug 30 '20 at 0:32
• @Houston: It is suspected there's an infinite number of twin primes, so with a gap of 2.
– SF.
Aug 30 '20 at 0:51

We can do even better than finding a prime smaller than the square of the previous: Bertrand's postulate, or the Bertrand-Chebyshev theorem, tells us that there is always a prime between $$n$$ and $$2n$$.

As a side note, the distribution of the primes is not at all "a probabilistic thing". Every statement about prime numbers, in specific ranges, or based on variables, etc. is either true or false, though it may be very difficult or impossible to prove or disprove some such statements. The prime number theorem and many similar results tell us some things which are always true as patterns in the limit when we take averages of large ranges. Since this is similar to some patterns in random variables which are almost always true when we take averages of many samples, some of the same techniques for dealing with random variables can be applied in much the same ways to properties of the prime numbers. But these arguments can only tell us about average properties for large numbers, and don't always help attempting to show something always or never happens, or applies in any way to small numbers.

• You can do even better. It follows from the prime number theorem that for every $\epsilon$ there is a prime between $n$ and $(1 + \epsilon)n$ for all sufficiently large $n$. Aug 30 '20 at 14:46
• Could you elaborate on or make more formal your sentence "Every statement about prime numbers, in specific ranges, or based on variables, etc. is either true or false, though it may be very difficult or impossible to prove or disprove some such statements."? I guess I know what you want to express, but provability/independence is already a non-trivial topic on its own that it's best to express terms thereof clearly. Aug 30 '20 at 15:02
• @EthanBolker I don't think that helps with the question here, of the distance from a particular $n$ to the next prime. Aug 31 '20 at 23:27
• @aschepler Maybe not. But it may be worth pointing out that for large $n$ you can do a lot better than Bertrand. Aug 31 '20 at 23:31
• @ComFreek Maybe that was a bit bold. The "or impossible" is a reference to Gödel's theorems and related results, yes. But "is either true or false" gets into a murky mix of theory modeling and philosophy. It assumes the Peano axioms describe something specific, just incompletely. But maybe that's from a misleading intuition, that I think I understand natural numbers and their common operations as obvious enough that there's exactly one thing (model) they really are about. Aug 31 '20 at 23:36

If I understand your question, you are asking if there is a bound on the gaps between primes or if at some point in the primes there could be a huge gap that is way out of the ordinary.

Well, there is a bound from this wikipedia page that gives the following bounds on the nth prime number (for n > 6):

$$\ln(n) + \ln(\ln(n)) - 1 < \frac{p_n}{n} < \ln(n) + \ln(\ln(n))$$

Now let's assume we have $$p_n$$ and $$p_{n+1}$$ where $$p_n$$ is the smallest possible and $$p_{n+1}$$ is the largest possible. Then the difference would be:

$$g_n < (n+1)(\ln(\ln(n+1)) + \ln(n+1)) - n(\ln(\ln(n)) + \ln(n) - 1)$$

$$g_n$$ is the gap between the nth and the n+1 th prime. If you are willing to be a little less exact you can assume that $$\ln(n+1)=\ln(n)$$ because as $$n \to \infty$$, $$\ln(n+1)\to\ln(n)$$, you get this result:

$$g_n ≲ \ln(\ln(n)) + \ln(n) + n$$

So it seems that it would be impossible for $$p_n > p_{n-1}^2$$. There is also something I heard on numberphile which is that there is always a prime between $$x$$ and $$2x$$ for any $$x$$.


text:
# I Property of Jacobian Determinant

#### Gear300

We can denote the jacobian of a vector map $\pmb{g}(\pmb{x})$ by $\nabla \pmb{g}$, and we can denote its determinant by $D \pmb{g}$. We were asked to prove that

$\sum_j \frac{\partial ~ {cof}(D \pmb{g})_{ij}}{\partial x_j} = 0$

generally holds so long as the $g_i$ are suitably differentiable not too long ago. Looking around, this property is mentioned in passing in papers here and there. Does this result go by a headword that I can look up. Like a theorem name of some sort? It is rumored to be an essential property for things like degree theory and the brouwer fixed point theorem.

Last edited:
Related Linear and Abstract Algebra News on Phys.org

#### mathwonk

Homework Helper
i can't really understand your notation, I assume cof stands for cofactor, but since to me a determinant is a number, hence does not have a cofactor, rather a matrix does. I have never seen this inequality in my study of either degree theory or brouwer's fix point theorem, and do not really know what it is saying, but just off the top of my head it looks as if it might follow from the equality of mixed 2nd partials. you might try working it out for n = 2 or 3.

well here is a discussion that gives a reason for it: see p. 2: he calls the general formula he is deriving Jacobi's formula, but this is just a step in the derivation.

also see page 6 in these notes:

https://www.gotohaggstrom.com/Jacobi’s formula for the derivative of a determinant.pdf

Last edited:

#### Gear300

You...are absolutely correct, my friend. I meant
$\sum_j \frac{ \partial ~ cof( \nabla \pmb{g} ) }{ \partial x_j } = 0 ~.$
But that aside, the above statement is true. I had already proved it, so I was inquiring its significance. For example, I've seen it in the second page (pg 17) of the following link, The Jacobian Determinant Revisited. Although, I haven't searched around much thereafter. I am familiar with the content you've referred to above, so I think my inquiry is different. Funny thing is that I haven't bothered reading the paper I referenced. Perhaps I should give it a swirl.

Last edited:

"Property of Jacobian Determinant"

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text:
Clocks Qns on relativity

1. Jan 15, 2007

Delzac

1. The problem statement, all variables and given/known data

At what speed does a clock have to move in order to run at a rate which is one half that of a clock at rest?

2. Relevant equations

$$\Delta T = \gamma \Delta T_0$$

3. The attempt at a solution

For clocks to run at half it's spd,

$$\Delta T = 2 \Delta T_0$$

$$\gamma = 2$$

which works out to v = 0.866c

Is this correct?

Qns 2 :
i saw in the physic formulary list this :

Clock desynchronization: Moving clocks, synchronized in their rest frame but separated by a distance D along their direction of motion, are not synchronized in the stationary frame; The front clock lags the rear clock by an amount:
$$T = Dv/c^2$$

What does it mean?

If 2 clocks are moving at the same spd, shouldn't they run in the same time as well and should be synchronized?

Any help will be appreciated.

2. Jan 15, 2007

Hootenanny

Staff Emeritus
Your solution to the first question is correct. With respect to the second question. Imagine the clocks are synchronised in their rest frame, but are moving with respect to you. You are stood directly in between the two clocks (at D/2) such such that one is moving away from you at v m/s and one is approaching you at the same speed. Now, imagine that these clocks are digital and emit a flash every second. Which flash will you see first, the flash from the clock travelling away from you, or towards you?

3. Jan 15, 2007

Delzac

ohh......i get it, it has to do with the simultaneity of the events isn't it?

So i will actually see the events of the front clock first before the rear clock.

But one question, did the event actually happen at the same time? Is there a different simultaneity because light enter my eyes at different times although both events happen at the same time?

4. Jan 15, 2007

Hootenanny

Staff Emeritus
Yes, its closely related.
It's actually the other way round, but that's not the point.
Both events are simultaneous in the clocks reference frame, but they are not simultaneous in your reference frame.

5. Jan 15, 2007

Delzac

But the isn't reason i don't see them as simultaneous because of the mystical property of light? Since light only helps us pass information only.

Anyway,

Clock2 ---------------me------------clock1 --------> direction of motion

I believe that events that occur in clock1 will reach me first, is that correct?

6. Jan 15, 2007

Delzac

Last edited by a moderator: May 2, 2017
7. Jan 15, 2007

Hootenanny

Staff Emeritus
There is nothing 'mystical' about it, it is simply a Lorentz transformation.
No, that is not correct. Lets set up two reference frames. Let the S' frame be the rest frame of the clocks; let the S frame be your rest frame such that the S' frame is moving with a velocity v from left to right in your diagram. So, by the Lorentz transformation we have;

$$\Delta t = \gamma\left(\Delta t' +\frac{v\Delta x'}{c^2}\right)$$

Let us define $\Delta t:=t_2-t_1$, where t1 t2 are the times when clocks one and two emit their flashes respectively in the S frame. Similarity, $\Delta t':=t_2'-t_1'$, where t1' t2' are the times when clocks one and two emit their flashes respectively in the S' frame. Now, since the two clocks are synchronised in their rest frame (S'), the interval $\Delta t' = 0$. Thus we obtain;

$$\Delta t = \gamma\left(0 +\frac{v\Delta x'}{c^2}\right) = \Delta t = -\gamma\frac{v\Delta x'}{c^2}$$

Which is not only non-zero, but since $\Delta t < 0 \Rightarrow t_2-t_1 < 0 \Rightarrow t_2 < t_1$. Therefore, you will observe the flash from clock number 2 first.

Do you follow?

8. Jan 15, 2007

Delzac

Ok i get what you are saying, but that seem to contradict the picture from the lecture slide. what was in the picture was :

Train moving in -------> direction

lighting bolt struck-----------me--------------lighting bolt struck(front)

And it wrote there that me(Mavis) saw the lighting bolt struck the front of the train first, then saw the LB struck the end of the train later.

Is this true?

9. Jan 15, 2007

Hootenanny

Staff Emeritus
Yes, the picture is correct, but is different from the clocks situation. In the train situation, mavis is moving towards event B and away from event A. However, in the clocks situation, clock 2 is moving towards you and clock 1 is moving away from you. Note, that in both cases you observe the event you are moving towards first.

Does that make sense?

10. Jan 15, 2007

Delzac

Ahh.......yes it makes better sense now.

But can i argue by saying that, well after all to see the event happening for the clock case, light must be transmitted to my eyes.

Which is the same(maybe?) as the lighting bolt case since :
direction of motion ------->
Light from clock2---------> me <----------light from clock1

11. Jan 15, 2007

Hootenanny

Staff Emeritus
No, it is not the same, it is the opposite as I have just explained. Consider what is moving towards you and what is moving away from you.

12. Jan 15, 2007

Delzac

Clock 1 is moving away from me
Clock 2 is moving towards me.

But can't it be that the light waves from clock 1 is moving also towards me?

13. Jan 15, 2007

Hootenanny

Staff Emeritus
Yes, but you are moving away from the source! In your lecture slide, what about the light from event A, isn't that moving towards mavis?

14. Jan 15, 2007

Delzac

oh ok, i get what you are driving at. got it now.
haha, thx for the help. :P

15. Jan 15, 2007

Hootenanny

Staff Emeritus
Twas a pleasure


text:
# Keeping it simple and getting things done: The move to a plaintext GTD system

I've recently experimented with a Getting Things Done (GTD) that uses no apps but just two plain text files. It's different, radically simple, and awesome.

Update: After experimenting with todo.txt, I eventually moved back to using ToDoist. This post explains why; but still, the temporary switch to plain text was interesting and useful for my overall productivity, so it's worth reading about IMO.

As readers of this blog know, I'm a big proponent of Getting Things Done, the time/task/project/life management method detailed in David Allen's book Getting Things Done: The Art of Stress-Free Productivity. I've practiced GTD for almost 10 years and written a lot about it here on the website. One of the points I have made about GTD, especially for beginners, is the importance of not confusing GTD with the tools used to implement GTD. Many GTD newbies get caught up with GTD apps rather than GTD itself and fail to adopt the habits of mind needed to really succeed with GTD. The key is to develop the behaviors needed for good GTD practice first and then choose the simplest, most trusted system you can to implement it.

When I started using GTD, I used the venerable Kinkless GTD system that is just a collection of Applescripts that work with OmniOutliner files. Kinkless morphed into what's now known as OmniFocus, and I was an alpha tester on the original, buggy-as-heck-but-super-promising OmniFocus releases. I used OmniFocus for 4-5 years before switching to Nozbe (I had traded in my iPhone for an Android phone and needed something non-iOS). For various reasons I got tired of Nozbe after a couple of years and switched to ToDoist, which has been my main platform for GTD...

...Until recently. I've switched again based on some new concerns I have had, not about ToDoist (which is IMO the best paid GTD app out there) but about the relationship that has existed between my GTD data --- my "stuff" --- and the apps that work with it. TL;DR -- I have been experimenting with todo.txt, a minimalist GTD implementation that uses only plain text files, and it is pretty awesome.

## The problem with tools

What got me thinking about my tools was the recent purchase of Wunderlist by Microsoft. Basically, Microsoft bought Wunderlist and promptly killed it, and reanimated its zombie corpse into something called Microsoft ToDo. This "new" tool sort of looks like Wunderlist and has a handful of the basic features of Wunderlist but none of the features that made it really useful for GTD.

While I wasn't using Wunderlist when this happened, I felt the pain of all those Wunderlist users who were now scrambling to find a new tool to use and hoping that they can import their data into it successfully. This made me think: What if I had stuck with Wunderlist for my GTD system? How much time and energy would I be spending trying to maintain my system rather than actually getting things done? And who's to say that what happened with Wunderlist won't also happen with ToDoist or any other app that I use --- being bought and killed off by a larger company, or being incompatible with the next macOS or iOS upgrade, or simply going out of business?

In some ways you can never get away from those concerns. I always have to trust the hardware and software that I choose. But I began to wonder: What tools for implementing GTD would give me the best balance of control, simplicity, and future-proof-ness --- and decouple my data from apps? After doing some research and some thinking about this, I realized that the optimum solution was right under my nose: The humble plain text file.

## The case for plain text

There are a lot of reasons to like plain text --- including formats like Markdown and LaTeX that are based on plain text --- for, really, anything:

• Text files are future-proof. They have been around since the beginning of computing and will be around long after all the apps we currently use are dead/gone/bought out by Microsoft.
• Text files are the ultimate in simplicity, containing precisely zero formatting and zero fluff.
• Text files are lightweight. You can have thousands of them on a storage device before they even make a dent, and sharing their contents is trivial.
• Text files are absolutely platform-independent, with high-quality text editors available on every OS and device[1]. They can also be turned into any other kind of format you want using simple tools like pandoc.

There is nothing inherent in GTD practice that says we can't use something as simple as plain text files to handle our stuff. In fact in Leo Babauta's book Zen To Done (which was the basis for my GTD for Academics series) he makes the point strongly that we have to keep the GTD system as simple as possible, and all that's really needed are basic lists. That's easily done with text files -- or even just a notebook and a pen. The trick is in having a system for working with lists in text files to make it easy to search and filter stuff within the lists. It turns out there's a simple, platform-agnostic system out there in use that works very well, and it's called todo.txt.

## todo.txt

Todo.txt is a system of formatting plain text files for task and project management. It does not require any particular app and uses no proprietary file format. It's merely a layer of syntax used in a plain text file for task management. Tasks are housed in a plain text file called todo.txt, one task per line and formatted using three different kinds of data:

• Priority, indicated using capital letters in parentheses, (A) through (Z).
• Context. These are indicated with an @ symbol, for example @phone. As in canonical GTD, context can refer to any sort of tag you wish to put on a task and is used for filtering tasks by physical location or tools available. I explained my use of contexts in this older post.
• Projects, indicated with a + sign, for example +BookReview.

Tasks can be labeled with any combination of priority, context, or project, including none of those. The system is flexible and you can use as many or as few of these data points as you want. In fact the only rules for formatting tasks are:

• Each task should appear on its own line.
• If a task has a priority, the priority is to be listed first.
• Contexts and projects may appear anywhere in the line but are to appear after the priority and start date.
• To mark a task as completed, put an x with a space at the beginning and put the date of completion directly following the x.

For example, here are correctly formatted tasks in todo.txt:

 (A) Post blog article about todo.txt @computer +Blogging
(C) Call mom due:2017-07-15
x 2017-07-07 Draft article about todo.txt @computer +Blogging

The first task has a priority, context and project. The second has a priority and a due date but nothing else. The third has a start date and context but no priority or project. The last one is a completed task.

The other part of the todo.txt overall system is a second text file called done.txt that is used to house all your completed tasks. This is an optional piece of the overall system but most apps (see below) use it.

So, the whole system consists two plain text files, one holding the current uncompleted tasks and the other holding the completed ones, with simple syntax for managing the contents. That's it. There are no apps, no proprietary formatting or rules, and no worries about portability or compatibility.

## Really? No apps?

Well, there are actually plenty of apps for working with todo.txt. These are listed at the todo.txt main page and include a wide variety of approaches and platforms including natives apps for Mac, Windows, and Linux; Chrome extensions; mobile apps for iOS and Android; and more. All of these apps are merely graphical interfaces to the todo.txt file and none of them are necessary --- you could just work with the plain text file in an editor of your choice if you wanted.

For me, the two most compelling "apps" for working with todo.txt for me are the todo.sh command line script and the SublimeTodoTxt plugin for the Sublime Text editor. The todo.sh script simply allows manipulation of the todo.txt file --- adding and editing tasks, listing tasks filtered by search queries, marking tasks as done and archiving them to done.txt, and more --- from command line entries. The SublimeTodoTxt plugin provides syntax highlighting, and I can bring the full power of Sublime Text's features to bear on my tasks.

The header image on this post shows the top 1/3 of my current todo.txt file in Sublime Text so you can see what it looks like. It definitely satsifies that part of my brain --- a large part --- that craves simplicity.

## My plaintext workflow (so far)

I started working with todo.txt a couple of months ago when my semester ended, because my workload this summer is low and so it's a good time to experiment. I am still figuring some parts of my workflow out, but here's what I have so far that's worked pretty well.

I have the two essential files todo.txt and done.txt in a special Dropbox folder for syncing purposes[2]. I also introduced two other text files to help with my workflow: A file called inbox.txt for capturing incoming thoughts (I treat this as just another inbox, along with my email accounts, Google Keep, and my physical inboxes); and a file called masterlist.txt which contains a master list of all my projects and all the tasks associated with those projects.

My "stuff" goes through a progression of these four files --- inbox, masterlist, todo, and done --- as follows:

1. Incoming "stuff" shows up gets captured into an inbox (not necessarily inbox.txt).
2. On a regular basis --- usually at my Weekly Review --- I process all the stuff that's accumulated in my inboxes. As discussed in this post on processing, items that are actionable and take more than 2 minutes and aren't delegated to a calendar or another person end up in a task list --- masterlist.txt in this case. Tasks going into masterlist.txt are given project/context/priorities and due dates if applicable. While todo.txt allows for up to 26 levels of priority ((A) through (Z)), I only use three: (A) designates most important things (MITs) for the day, (B) designates other stuff that are not MITs but which I would like to do today, and (C) designates stuff to do this week. Tasks that need my attention but don't fall into any of those categories show up in todo.txt as tasks without priorities.
3. At the weekly review, I process through masterlist.txt and if there are tasks that need my attention, I cut and paste them into todo.txt. So all my tasks are split between masterlist.txt and todo.txt: The ones on my radar screen are in todo and the tasks that will eventually be on the radar screen, but not yet, remain in masterlist.
4. On a daily basis, I have todo.txt open and make choices about what to do based on what's in that file. Just as with any other GTD implementation, I make those choices based on my physical context, energy level, and time available. I have contexts set up to help me search and filter: They are the same contexts as I used in ToDoist, which I spelled out in this post.

The use of masterlist.txt in conjunction with todo.txt is my invention. In previous GTD implementations I would often find myself completely paralyzed trying to decide on a task to do, because I used my system to hold all the tasks I could possibly ever do on a project, and it would overflow my buffers with tasks that didn't really need to be in my attention. I felt I needed to split the complete list of tasks into the ones needing attention and the ones that could wait. I did this in ToDoist with labels, but that wasn't enough: I needed those tasks that didn't belong on the radar screen to be out of sight and out of mind. For me, masterlist.txt is the staging area while todo.txt is where my attention is directed.

As far as apps go, I've tried out several and have gravitated toward these:

• On my phone (a Samsung Galaxy S8) I use the original Todo.txt Android app.
• On my iPad, it's SwifTodo.
• On my Mac, I go back and forth between TodoTxtMac and using just the text file itself in Sublime Text and the todo.sh command line interface.

But note, none of these apps is necessary. I could easily just work with the text file in Sublime Text and many times I do just that.

## Pros/cons

So far here's what I like about todo.txt and what I don't like as much.

I really love the minimalism of todo.txt. Working with plain text files is the ultimate in simplicity and control and portability. It gives me options for how I want to work with my data, and those data are not married to anybody's app. Going with plain text does mean forgoing some features you'd get in a GTD app (see below) but what you get in return is a completely future-proof, fool-proof system.

I also love the fact that going plaintext makes me think a lot more about the nature of my projects and tasks and what role they play in the big picture. I spend a lot more time now thinking about why tasks appear in my system in the first place and whether they should have my attention or not. Before with ToDoist, there was so little friction in using the system that I could just forward any old email into my system without stopping to think about whether that email should go in the system. Now I think there's an appropriate amount of friction there, and the things that end up in todo.txt truly belong there.

I've also been surprised at how little I notice the changes I've made. Aside from a few missing features that I've had to work around, I find that I am able to do everything I need to do without all the frills.

There are some things about todo.txt I'm not a fan of:

• There's definitely not the vibrant community of developers you'll find with ToDoist or OmniFocus. In fact there's not much going on at all with todo.txt in terms of app development. Some of the apps mentioned on todotxt.com no longer exist, and some of Gina Trapani's code for todo.txt on GitHub hasn't seen a commit in 4-5 years. Using some of the todo.txt apps is like stepping back in time 6-7 years and I don't mean that in a good way. SwifTodo, which is maintained by Michael Descy, is a big exception: He returns tweets minutes after sending and updates the app about once every six weeks, and the app -- while quite basic -- looks and behaves like a modern iOS app.
• Most of the todo.txt apps are based on syncing the todo.txt file via Dropbox, and this has led to some scary sync conflicts and data loss a few times for me.
• Editing text files on a smartphone (at least on Android) has not yet evolved into a pleasant experience, and that's your only option on mobile if you don't want to deal with an app.

## What's next

I really like todo.txt and plan on continuing with it when I emerge from this summer and into my sabbatical, when I'll have a real workload again and the system can really be put through its paces. In the meanwhile I am going to try to tweak and optimize my system. For example, while I have a reasonable rationale for using the masterlist.txt file, I'm not certain it's necessary; and I am becoming fairly certain that I don't need inbox.txt at all.

I also plan on doing some development of todo.txt tools, especially workflows for Editorial, my go-to text editor on iOS. Right now there is just one todo.txt workflow available (that marks tasks as done). I use my iPad more and more as my main computer for all but coding purposes and it would be nice to use Editorial as my todo.txt "app" (even though SwifTodo is a very good app by itself).

Finally, I'll just note that my move to todo.txt is part of a larger push to move everything I do to plain text files, patterned after this Plaintext Productivity system outlined by SwifTodo's Michael Descy. I'll have more on that big picture in a later post.

1. Except Android apparently. If I'm wrong on that, give me your recommendations. ↩

2. You could argue that using Dropbox defeats the whole "decouple your data from apps" ethos here and there's some truth to this. I suppose I could self-host the data, but (1) most of the apps for todo.txt use Dropbox for syncing, so if you use the apps you need to use Dropbox and (2) call me naive, but I trust Dropbox from having used it for almost a decade. ↩


text:
# In Exercises $1-18$, find the area and perimeter of each figure.

###### Question:

In Exercises $1-18$, find the area and perimeter of each figure.

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# How to solve an integral with the Inverse error function

1. May 16, 2014

### matteo86bo

Hi,
this is not a homework and my problem is much bigger for me to give full details here. I came across this integral

$\mathcal{I}(\xi)=\int^{\xi_c}_{\xi}{\rm d}\xi^\prime\exp\left[\sqrt{2}\sigma\,{\rm Erf}^{-1}\left(1-\frac{8\pi}{3}{\xi^\prime}^3\right)\right]$

where Erf$^{-1}$ is the inverse error function and

$\xi_c=\left[\frac{3}{8\pi}\left(1-{\rm Erf}\left(\frac{\sigma^2-\sqrt{2}\sigma\,{\rm Erf^{-1}(2\beta-1)}}{\sqrt{2}\sigma}\right)\right)\right]^{1/3}$

with $0\le\beta\le1$.

I would like get an analytical approximation but I can't figure out a way to do that, even with software like Mathematica. I tried solving the integral numerically and I find a reliable solution, however, I'm mostly interested in points where $\xi\to\xi_c^-$, and here the inverse error function diverges.

Do you have any ideas on how to approximate this integral?

2. May 17, 2014

### Simon Bridge

Use a taylor series?

3. May 17, 2014

### matteo86bo

Thank you for your answer. But I don't understand how I can Taylor expand where the integrand diverges.
Also, do you see a way to normalize the integral and/or make it more simple to solve it numerically.
What I find challenging is that for every value of beta I have to numerically solve the integral. It would be better to normalize somehow the result with beta and then do the integral only once.

4. May 17, 2014

### Simon Bridge

The inverse errorfunction can be defined as a Maclaurin Series.
That seems like a way to get rid of some awkwardness there, but I don't know for sure that it is the best way to go.

But I don't immediately see how to make it much simpler besides that.

5. May 17, 2014

### matteo86bo

Thanks, the problem seems very complicated and I think I have to resort to a numerical integration.
However, that would be easier if one could write

$\mathcal{I}(\xi,\beta)=f(\beta)\mathcal{I}^{\prime}(\xi)$

so this way I would have to integrate only once. Do you see a way to possibly achieve that?

6. Feb 2, 2017

### robkay

I understand this is quite an old post but: any luck with computing this integral? I have come across something similar recently.