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# Solving Cubic Polynomials

2010-11-17 <NAME>

Although a closed form solution exists for the roots of polynomials of degree ≤ 4, the general formulae for cubics (and quartics) is ugly. Various simplifications can be made; commonly, the cubic $a_3 x^3 + a_2 x^2 + a_1 x + a_0$ is transformed by substituting $x = t - a_2 / 3 a_3$, giving

$t^3 + p t + q = 0,$

where

\begin{aligned} p &= \frac{1}{a_3} \left( a_1 - \frac{a_2^2}{3 a_3} \right) \\ q &= \frac{1}{a_3} \left( a_0 - \frac{a_2 a_1}{3 a_3} - \frac{2 a_2^3}{27 a_3^2} \right). \end{aligned}

Several strategies exist for solving this depressed cubic, but most involve dealing with complex numbers, even when we only want to find real roots. Fortunately, there are a couple ways to find solutions with only real arithmetic. First, we need to know the value of the discriminant $\Delta = 4 p^3 + 27 q^2$.

If $\Delta < 0$, then there are three real roots; this is the tough case. Luckily, we can use the uncommon trigonometric identity $4 \cos^3(\theta) - 3 \cos(\theta) - \cos(3 \theta) = 0$ to calculate the roots with trig functions. Making another substitution, $t = 2 \sqrt{-p/3} \cos(\theta)$, gives us

$4 \cos^3(\theta) - 3 \cos(\theta) - \frac{3 q}{2 p} \sqrt{\frac{-3}{p}} = 0,$

so

$\cos(3 \theta) = \frac{3 q}{2 p} \sqrt{\frac{-3}{p}}.$

Thus the three roots are:

$t_n = 2 \sqrt{\frac{-p}{3}} \cos \left( \frac{1}{3} \cos^{-1} \left( \frac{3 q}{2 p} \sqrt{\frac{-3}{p}} \right) - \frac{2 \pi n}{3} \right),$

for $n \in \{0, 1, 2\}$. We can save a cosine calculation by noting that $t_2 = -t_0 |_{q = -q}$ and $t_1 = -(t_0 + t_2)$. When generated in this way, the roots are ordered from smallest to largest ($t_0 < t_1 < t_2$).

For the $\Delta > 0$ case, there is a single real root. We can apply the general cubic formula and avoid dealing with complex numbers, provided we choose the signs right. In this case,

$t = -\mathrm{sgn}(q) \left( C - \frac{p}{3 C} \right), \text{ where } C = \sqrt[3]{\sqrt{\frac{\Delta}{108}} + \frac{|q|}{2}}.$

When $\Delta = 0$, there is a root of at least multiplicity 2, and we can avoid calculating any radicals. If $p = 0$, the only root is $t_0 = 0$; otherwise, the duplicate root is $-3 q /2 p$ and the simple root is $3 q / p$.

My C implementation of this solution method can be seen in the dmnsn_solve_cubic() function here.


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# A bus stop is barricaded from the remaining part of the road, by using 50 hollow cones made of recycled cardboard. Each cone has a base diameter of 40 cm and height 1 m. - Mathematics

A bus stop is barricaded from the remaining part of the road, by using 50 hollow cones made of recycled cardboard. Each cone has a base diameter of 40 cm and height 1 m. If the outer side of each of the cones is to be painted and the cost of painting is Rs 12 per m2, what will be the cost of painting all these cones?

("Use "π = 3.14" and take "sqrt1.04= 1.02)

#### Solution

Radius (r) of cone = 40/2 = 20cm = 0.2m

Height (h) of cone = 1 m

Slant height (l) of cone= sqrt(h^2+r^2)

=[sqrt((1)^2+(0.2)^2)]m=sqrt1.04m=1.02m

CSA of each cone = πrl

= (3.14 × 0.2 × 1.02) m2 = 0.64056 m2

CSA of 50 such cones = (50 × 0.64056) m2

= 32.028 m2

Cost of painting 1 m2 area = Rs 12

Cost of painting 32.028 m2 area = Rs (32.028 × 12)

= Rs 384.336

= Rs 384.34 (approximately)

Therefore, it will cost Rs 384.34 in painting 50 such hollow cones.

Concept: Surface Area of a Right Circular Cone
Is there an error in this question or solution?

#### APPEARS IN

NCERT Class 9 Maths
Chapter 13 Surface Area and Volumes
Exercise 13.3 | Q 8 | Page 221

format: plain


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Corpus ID: 201640476

# Fourier multipliers on compact Lie groups

@inproceedings{Ruzhansky2015FourierMO,
title={Fourier multipliers on compact Lie groups},
author={<NAME> and <NAME>},
year={2015}
}
• Published 2015
In this paper we prove L p Fourier multiplier theorems for invariant and also noninvariant operators on compactLie groups in the spirit of thewell-knownHörmander–Mikhlin theorem on Rn and its variants on tori Tn . We also give applications to a-priori estimates for non-hypoelliptic operators. Already in the case of tori we get an interesting refinement of the classical multiplier theorem.
18 Citations


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# Groups becoming algebraic groups

Let $G$ be an algebraic variety over an algebraically closed field $k$ (any characteristic). Suppose that: (1) the set of $k$-points has the structure of a group. (2) for any $g\in G$ the right-multiplication by $g$ is a morphism of algebraic varieties $G\to G$. (3) the inverse map is a morphism $G\to G$.

Does it imply that $G$ is an algebraic group? (i.e. is the multiplication $G\times G\to G$ a morphism?)

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you might specify if you have in mind characteristic zero or arbitrary characteristic. –  YCor Nov 17 '12 at 18:01
What set are you using to be the elements of the group? –  <NAME> Nov 17 '12 at 18:03
The multplicative group of quaternions (presented as $\mathbb{C}\times \mathbb{C} \setminus \{0\}$ yields a non-algebraic example of your (1)+(2); yet it fails (3). –  <NAME> Nov 17 '12 at 19:02
@Mikhail: In char. $p$ such near-misses (losing (3)) abound. Let $H$ be a linear algebraic group and $H'$ an infinitesimal non-normal closed subgroup scheme of $H$. The quotient scheme $G = H'\backslash H$ is a smooth affine variety and the natural map $H(k) \rightarrow G(k)$ is bijective. If $G(k)$ is thereby equipped with the group structure of $H(k)$ then this is non-algebraic (as otherwise the quotient map $H \rightarrow G$ would be a homomorphism of algebraic groups and so its schematic kernel $H'$ would be normal), and (1) and (2) hold but (3) fails. –  user28172 Nov 17 '12 at 19:25
Since there was some discussion about the extent to which the group structure is needed, the following might be relevant. If $k$ is a countably infinite field, then there is a function $f:k\times k\to k$ that is not a polynomial even though, for each fixed value of $x$ in $k$, $f(x,y)$ is a polynomial function of $y$ and, for each fixed $y\in k$, $f(x,y)$ is a polynomial function of $x$. (There is no such $f$ when $k$ is uncountable.) –  <NAME> Nov 18 '12 at 2:21

## 1 Answer

I predict that in whatever is the situation of motivating interest, you know more: for any algebraically closed field $K/k$ you likewise have a group structure on $G(K)$ functorially in $K/k$ making the translations and inversions by $G(K)$ also be morphisms on $G_K$. Under this additional condition the answer is always affirmative, by using a trick that I believe goes back to Hasse, namely base change to the function field of $G$ (or rather, in our situation, an algebraic closure thereof). This is a Weil-style way of getting at Yoneda's Lemma.

The inversion hypothesis implies that left-translations are also morphisms, and we will use this condition instead of the inversion hypothesis (and deduce at the end that inversion is a morphism too). In fact, in char. 0 we won't even need this condition with the left-translations (or inversion) at all, but we are using a stronger assumption on the existence of group laws on $K$-points for lots of $K/k$. This stronger initial assumption across many $K/k$ bypasses Bondarko's examples in char. 0. The near-miss examples avoiding the inversion condition in char. $p$ in the comments are not ruled out by our stronger hypothesis across all $K/k$, but they are ruled out by the left-translations being morphisms (!), so matters will be more delicate in positive characteristic. We need the left-translation condition in positive characteristic to circumvent some inseparability issues with function fields of $k$-varieties.

Before we begin the actual arguments, note that since $k$ is algebraically closed, so $G(k)$ is Zariski-dense in $G$, for any field extension $F/k$ the subset $G(k)$ inside $G(F) = G_F(F)$ is Zariski-dense $G_F$. Hence, $F$-morphisms among $G_F$, $G_F \times G_F$, etc. are determined by their effect on $k$-points promoted to $F$-points. I mention this at the outset for peace of mind in some later discussions.

The "translation is a morphism" hypothesis ensures that $G$ is smooth, so its connected components are irreducible. I will assume that the variety $G$ is connected, a harmless assumption since (i) for $g \in G^0(k)$ the right-translation morphism by $g$ preserves $G^0$ because it carries $e$ to $g$, (ii) inversion preserves $G^0$ since it fixes $e$, (iii) the morphism property on $G^0$ clearly implies the general case by using the "morphism" property for a few translations by points in the various connected components of $G$. Thus, now $G$ is irreducible and so has a "function field" $k(G)$.

By applying the functoriality in $K/k$ to $k$-automorphisms of $K$, we see via Galois descent that the initial hypothesis for algebraically closed extensions is actually valid for all perfect extensions of $k$ as well. Let $K$ be the perfect closure of $k(G)$, and let $\eta \in G(k(G)) \subset G(K)$ correspond to the generic point. This induces a right-translation morphism $\rho_K:G_K \simeq G_K$ (computing right-translation by $\eta_K$). Since $G$ is finite type over $k$, by expressing $K$ as a directed union of finite purely inseparable extensions of $k(G)$ we obtain a finite purely inseparable extension $F$ of $k(G)$ and an $F$-morphism $\rho_F:G_F \simeq G_F$ that descends $\rho_K$ and clearly computes right-translation by $\eta_F$. The normalization $X$ of $G$ in the finite extension $F/k(G)$ is a $k$-variety finite radiciel over $G$, and $X_{\eta} = {\rm{Spec}}(F)$ over $k(G)$. Thus, by expressing $k(G)$ as the directed union of the coordinate rings of affine opens around $\eta$ in $G$, we find such an open $U$ so that over its preimage $U'$ in $X$ (which is finite radiciel over $U$) there is a $U'$-morphism $\rho_{U'}:G_{U'} \simeq G_{U'}$ extending $\rho_F$.

The map $U'(k) \rightarrow U(k)$ is bijective, so for each $g \in U(k)$ let $g' \in U'(k)$ be the unique point over $g$. Consider the specialization $\rho_{g'}:G \simeq G$ of $\rho_{U'}$ over $g'$. I claim that this is the right-translation by $g$. To see this, pick $h \in G(k)$ and consider the morphism of left-translation $\ell_h:G \simeq G$ by $h$. This carries $\eta$ to another $k(G)$-point of $G$ (over $k$) that "spreads out" to a $U$-point of $G$ whose specialization at $g \in U(k)$ is $\ell_h(g) = hg$ (so if we work instead with the $U'$-point over this via the canonical $U' \rightarrow U$ then its specialization at $g' \in U'(k)$ is also $\ell_h(g) = hg$). Now inside the group $G(K)$ we have $\ell_h(\eta_K) = \rho_K(h_K)$, so $\rho_{U'}(h_{U'})$ is the $U'$-point obtained from applying $\ell_h$ to the canonical $U'$-point of $G$ (spreading out $\eta_F$) because this comparison of $U'$-points may be checked by working at the generic point of $U'$ (and this generic point is dominated by the canonical $K$-point, over which we have the equality $\ell_h(\eta_K) = \rho_K(h_K)$). Specializing this equality of $U'$-points at the point $g' \in U'(k)$ over $g \in U(k) \subset G(k)$ gives the equality of $k$-points $\rho_{g'}(h) = \ell_h(g) = hg$, as desired.

To summarize, we have constructed a $U'$-morphism $\rho:G_{U'} \simeq G_{U'}$ whose specialization at any $g' \in U'(k)$ is right-translation on $G$ by the corresponding point $g \in U(k)$. Suppose ${\rm{char}}(k) = 0$, so $U' = U$ and the composite morphism $${\rm{pr}}_1 \circ \rho:G \times U \rightarrow G$$ on $k$-points is the group law restricted to $U(k)$ in the 2nd variable. For any $g \in G(k)$ the right translation morphism $\rho_g$ carries $U$ to another open subscheme $\rho_g(U)$, and I claim that these opens cover $G$. It suffices to check the covering property on $k$-points, so for $h \in G(k)$ we seek $g \in G(k)$ such that $h \in \rho_g(U)$, or equivalently $hg^{-1} \in U(k)$. Pick any $u \in U(k)$ and let $g = u^{-1}h$. This proves the covering property, and the multiplication map $G(k) \times U(k)g \rightarrow G(k)$ arises from a morphism, namely the composition of ${\rm{pr}}_1 \circ \rho$ and the morphism $\rho_g$. We conclude (in char. 0) that the group law on $G(k)$ is induced by a morphism $G \times G \rightarrow G$.

Now we can deduce in char. 0 that inversion is a morphism (so we have a group variety), even though we never used the left-translation or inversion conditions. Indeed, the "universal right-translation" $G \times G \rightarrow G \times G$ defined by $(x,y) \mapsto (xy,y)$ between fppf $G$-schemes (via ${\rm{pr}}_2$) is a scheme isomorphism between fibers over all points in $G(k)$ and therefore is a scheme isomorphism (by fibral isomorphism criteria, adapted to the peculiarities of $k$-points when $k$ is alg. closed). This yields the morphism property for inversion for free! This was a char-free argument, but it required the composition law to be a morphism. Anyway, char. 0 is now settled.

Assume ${\rm{char}}(k) = p > 0$, so for sufficiently large $n \ge 0$ the finite flat $n$-fold relative Frobenius morphism $G^{(1/p^n)} \rightarrow G$ of $G^{(1/p^n)}$ dominates $U'$ over $U$ (namely, pick $n \ge 0$ so that the finite purely inseparable extension $F/k(G)$ is contained inside $k(G)^{1/p^n}$). Since the initial choice of $F$ could be replaced with a finite purely inseparable extension at the outset if we wish, we may therefore assume that $U'$ is open inside $G^{(1/p^n)}$. Using a covering and translation argument similar to characteristic 0, we arrive at a morphism $$m_r:G \times G^{(1/p^n)} \rightarrow G$$ that recovers the given group law on $k$-points via the natural identification of $G^{(1/p^n)}(k)$ with $G(k)$ (for some $n \ge 0$). Our problem is precisely to show that $m_r$ factors (in the sense of morphisms of varieties) through the $n$-fold relative Frobenius morphism in the 2nd variable.

Now we shall use that left-translations are morphisms too. At the cost of increasing our $n$ if necessary, we can run through the same arguments with "left" instead of "right" to arrive at another morphism $$m_{\ell}:G^{(1/p^n)} \times G \rightarrow G$$ which recovers the given group law on $k$-points (with the same $n$).

Returning to our task of checking that $m_r$ factors through the appropriate iterated Frobenius in the 2nd variable, since that iterated Frobenius is fppf (as $G$ is smooth!) we conclude via fppf descent that it is harmless to check the existence of such a factorization after precomposing $m_r$ with an fppf morphism in the first variable. So let's compose with the same iterated Frobenius in the first variable, arriving at a morphism $$G^{(1/p^n)} \times G^{(1/p^n)} \rightarrow G$$ that recovers the composition law of $G(k)$ on $k$-points. It suffices to show that this latter map factors through the $n$-fold Frobenius in its 2nd variable, but this factorization is clear: it is the composition of $m_{\ell}$ with that iterated Frobenius (as we may check by computing on $k$-points, since we're working with reduced $k$-schemes of finite type). QED

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This is a very nice answer ! –  <NAME> Nov 18 '12 at 2:41


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## Sunday, August 14, 2011 ... /////

### Ex-alarmist Pawlenty ends presidential campaign

In the morning, I suffered from one of my bad habits - visiting the blog of Capitalist Imperialist Pig, something that I have done many times since the first moments when the anonymous far left alarmist anti-string New Mexico professor began to attack your humble correspondent on a regular basis.

Then I clicked at a random link on that blog and it led me here:

<NAME>: Pawlenty in Trouble
The only purpose of this left-wing blogger was to argue - without arguments - that Pawlenty is not in trouble and that his distant third place (after Bachmann and Paul) among the U.S. presidential candidates wasn't bad.

He wrote, among other things:
Wait a second. Everyone expected Bachmann to win, and likewise, everyone knows that <NAME> has a demonstrated (and meaningless) ability to round up his tiny band of fanatical troops for events like this. So third place isn't too bad for Pawlenty, is it? Or was the "Pawlenty in trouble" media narrative already so set in stone that it didn't matter how well he did?
I was thinking: holy cow. What kind of people may read garbage blogs such as this one ("Mother Jones"). There is absolutely no merit or logic in his text. The only purpose is to spin, distort, misinterpret, or downright deny the reality.

Of course that his bad showing in Iowa was an extremely bad sign that indicated that he was not viable - and only crazy people could deny that it was a problem for Pawlenty. He got a lukewarm support from 14% of the voters which is one-half of a tiny band - because <NAME> received 28% and this figure is equal to "one tiny band", if we use Drum's silly units. Paul's gain is not only twice as big as Pawlenty's gain; but his supporters are more enthusiastic, as Drum admits as well.

Moreover, Pawlenty was the kind of a candidate that was expected to have a strong beginning - he was a potential McCain running mate in 2008 - and a declining trend. So if he has comparably bad poll numbers already now, it's a reason to give up.

Of course, I only needed to wait for several hours and Pawlenty ended his campaign, kind of showing that Drum has really no clue about the U.S. (or any other) politics. It's just shocking that babbling of people like Drum has ever been taken as a serious political punditry.

Drum tries to mask his preferences but he highlights his propaganda and logically indefensible spin, distortions, and misinterpretations. My principles are the opposite ones: I always try to acknowledge all my preferences that are relevant, and avoid misinterpretations of any facts, whether they're pleasant or unpleasant to me.

So let me also admit that I view his surrender as good news. This guy is clearly no conservative. But that wouldn't be the ultimate problem. But he has really been crazy in the environmentalist policymaking. The green people often disagree whether they should introduce a "cap" or a "tax". Pawlenty introduced both of them - "cap and tax"! :-)

The video above is quite revealing. With those naive beliefs, he could join Greenpeace if he wanted. It's not just the bad policymaking he has been associated with; what's also annoying is his worshiping of the scientific hired guns. I mean all those amazing hypernatural superscientists in the Academy who believe in the judgement day. Those who really drove this movement are not scientists, at least not good ones, OK?

His flip-flops have been spectacular, too. Has he learned from the mistakes? I am not sure. It's more likely that he is an opportunist who recently realized that global warming is no longer the winning card. But I think that the U.S. may still find many politicians who have never jumped on the fanatical anti-carbon bandwagon. The people discredited by the global warming hoax should be gradually phased out of the U.S. politics and the politics of other countries in the world, too. Their mistake has been way too grave for them to stay.

In this particular case, Pawlenty's political capital is likely and hopefully going to be recycled by <NAME> III - <NAME> just joined the G.O.P. presidential campaign. ;-)

Global warming saves dozens of lives in D.C.

Despite its name, global warming hasn't managed to "warm the globe" at least for 10 years. But it has achieved many other things. No U.S. vice-president who considers himself an expert in climatology has any doubts that the recent heat wave in much of the U.S. was caused by global warming. But what does the heat wave do to human lives?

It has actually saved dozens of lives.
D.C.'s murder rate plummets as temperatures soar
The murders in Washington D.C. by July dropped from 32 to 18 - by a whopping 44 percent. In a similar way, the number of robberies, sex abuses, and assaults with guns dropped by dozens of percent, too. Even in mid August, the drop of the murder rate dropped 43 percent.

In July, it was appreciated that the improvement of the security situation in the U.S. capital was caused by global warming:
As temperatures climbed relentlessly this summer, homicides in the District fell just as dramatically, something a leading expert says may not be a coincidence.
However, the mayor has figured out a method how to harvest the fruits that were produced by global warming. In the newer article, "Mayor <NAME> is crediting the city's summer crime-fighting initiative." ;-)

Well, people know how to deal with global warming. When it causes something good, they say that it's their own achievement. When they do something wrong, they say that it's the fault of global warming. Clever! You may see that global warming plays the same role as Jesus Christ, adopting all sins of the mankind and bringing everyone the benefits people want to come true! :-)

#### snail feedback (2) :

Very nice results for <NAME>. But I am affraid the others won't let <NAME> to be president. He want to cancel the FED, stop foreign wars, cut taxes, ballance the budget. That makes too many enemies for him. For me he would be the n.1 choise.

reader الرجل ذبح بعضهم البعض ولكن الخيول باهظة الثمن said...

non sequitur

ergo

Kamil has left a new comment on the post "Ex-alarmist Pawlenty ends presidential campaign":

Very nice results for <NAME>. But I am affraid the others won't let <NAME> to be president. He want to cancel the FED, stop foreign wars, cut taxes, ballance the budget. That makes too many enemies for him. For me he would be the n.1 choise.

....well ideologic bastards

commies ex-commmies neo-nazis

soc's soc's dem and so weiter

all the same

the gregarius bunch


text:
College Physics (4th Edition)

The work done on the stone is $151~J$
We can find the work done on the stone: $Work = \tau ~\Delta \theta$ $Work = r~F ~\Delta \theta$ $Work = (0.10~m)(20.0~N)(12~rev)(2\pi~rad/rev)$ $Work = 151~J$ The work done on the stone is $151~J$.


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# Characteristic not implies quasiautomorphism-invariant

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., characteristic subgroup) need not satisfy the second subgroup property (i.e., quasiautomorphism-invariant subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about characteristic subgroup|Get more facts about quasiautomorphism-invariant subgroup
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property characteristic subgroup but not quasiautomorphism-invariant subgroup|View examples of subgroups satisfying property characteristic subgroup and quasiautomorphism-invariant subgroup

## Statement

It is possible to have a group $G$ and a characteristic subgroup $H$ of $G$ such that $H$ is not a quasiautomorphism-invariant subgroup, i.e., there are quasiautomorphisms of $G$ that do not send $H$ to itself.

## Proof

Further information: inner automorphism group of wreath product of groups of order p

Suppose $p$ is a prime number greater than $3$. Let $G$ be the group isomorphic to the inner automorphism group of the wreath product of two groups of order $p$. $G$ is a group of order $p^p$ with an elementary abelian normal subgroup $N$ of order $p^{p-1}$, an element of order $p$ acting on it from outside, and every non-identity element of $G$ has order $p$.

Consider the subgroup $H = [G,G]$. $H$ is a group of order $p^{p-2}$, contained inside the elementary abelian normal subgroup. It is a characteristic subgroup.

However, we can construct a quasiautomorphism $\sigma$ of $G$ that does not preserve $H$ as follows: the restriction of $\sigma$ to $N$ is an automorphism of $N$ that fixes the center $Z(G)$ (which is cyclic of order $p$) but does not send $H$ to itself, and $\sigma$ fixes every element of $G$ outside $H$. Note that we need $p > 3$ to ensure that $H$ is strictly bigger than $Z(G)$, which is necessary to be able to construct a $\sigma$ with the desired specifications.


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# What is a reasonable amount of population growth for 900 years?

Starting with about 500 humans and allowing for 900 years to pass, about how many people would there be at the end? (Just asking for a rough estimate, maybe a range.) Assuming:

1. People are generally in good health, drink clean water and eat well.

2. People naturally live to around 60.

3. Contraceptives and abortions don't exist.

4. Incest isn't a problem.

5. No wars, minimal violent crime.

And what other factors might affect the end number? I know I've got to be missing some important ones.

• Is there any technological progress during these 900 years or is it stagnant ? It might also help to have the starting level of technology. From your description Considering you question I guess you have a level of tech form the 1950-1960. or it's alternate history ? Dec 9 '14 at 5:53
• – user
Dec 9 '14 at 9:05
• Could you explain why incest isn't a problem? With a population of 500 you're right up against the commonly estimated value for the lower bound of the Minimum Viable Population which assumes maximum genetic disparity in offspring. You'd quite literally need a breeding program to ensure that your population's gene pool remains diverse enough to guarantee long-term existence. While there's a lot of debate on the 500-number, Traill, Bradshaw and Brook find that "the MVP for most species will exceed a few thousand individuals". Dec 9 '14 at 16:06
• What sort of medical technology do your people have? Maternal death in childbirth was once a significant cause of death, and lots of children used to die young, but those aren't major factors anymore. If every woman can produce 10 healthy children, your population can increase by a factor of 5 every generation. Dec 9 '14 at 16:39
• @MikeNichols Correct, apparently I've had it explained to me incorrectly. The linked answer mentions a number as low as 80 as long as genetic diversity is optimised. As long as the population has the standard taboo on incest, 500 people is likely enough to establish a solid gene pool without too many deleterious genes working their way in. Dec 22 '14 at 21:02

Others have looked at this from various theoretical points of view. I want to look at it from the point of view of our own Earth's history, and what would be the impact if your population followed the same growth rate seen on Earth. Do note that for the purposes of this answer, I am ignoring the minimum viable population; in the real world, that's something you'd have to consider, and your 500 humans might not be large enough starting out particularly if they aren't selected based on genetic criteria for a maximally genetically diverse population and that diversity is carefully managed and monitored.

The Real Population Problem on the blog Do The Math comes in handy, as it gives population graphs and growth rates for the human population during different eras in a convenient form. The numbers are basically (all numbers are approximate, obviously):

• From 10000 BC to 3000 BC: about 0.03% population growth per year

Here is an alternative graph (Kremer and Vermillion) which shows the period 2500 BC to 2000 AD. Notice the negative population growth around 1300 AD, and that it dips to right around zero on several occasions. The dip around 1300 AD might be explainable by the beginnings of the Little Ice Age and the Bubonic plague, although strictly speaking that is speculation on my part.

The thesis posed in the Do The Math post to explain the jumps in growth rate is:

We can perhaps attribute the 1700 jump to the Renaissance and scientific progress. We learned to wash our hands after wrestling with our pigs, and that diseases were not caused by bad vapors conjured by impure thoughts. The jump around 1870 corresponds to the Industrial Revolution, in which coal transformed the production of steel (providing agricultural tools), rail transport of commodities, and began to mechanize agriculture in a limited way. 1950 marks the Green Revolution: full-scale petrolification of agriculture, accompanied by massive fertilizer campaigns using natural gas as the chemical feedstock.

This leads to a rather simple thesis: the surplus energy presented to us by fossil fuels allowed us to feed people more easily the world over. The bounty of fossil-fuel-turned-food encouraged an explosion in birth rates, as happens for virtually all organisms given similar circumstances. It’s so blindingly obvious that I am embarrassed to have belabored the point as long as I have.

We can also use these numbers to answer your question with a reasonable degree of accuracy. For example, for a Middle Ages level society, the human population will grow at about 0.12% every year on average. (If you want this to be realistic, don't forget to throw in the occasional boom year as well as the occasional plague or die-off due to a few years of crop failure!) Starting with 500 humans and letting 900 years pass, we end up with $$500 \times 1.0012^{900} \approx 1471$$ or let's round that off to 1,500 humans. Not a whole lot. Under those conditions, your colony is extremely vulnerable to die-off; a single serious disease can easily take out a large fraction of your entire population.

If instead we use the current era, during which we have gone to the Moon, (even human) spaceflight is so routine that it often isn't reported in the news, energy has been plentiful, intercontinental travel and trade is something the world could barely survive without, and so on, then the same calculation becomes $$500 \times 1.017^{900} \approx 1.94 \times 10^9$$ or just under two billion people.

The above calculations assume no significant leaps in technology or society during the interim period as described by the blog post. For a reasonably developed society and over such a long period of time, this appears an unrealistic assumption. If we instead take the 900 year period from 1100 AD to 2000 AD and use the above figures, then the calculation becomes slightly more involved, but significantly more realistic.

• 600 years from 1100 AD to 1700 AD at 0.12%: $500 \times 1.0012^{600} \approx 1027$
• 170 years from 1700 AD to 1870 AD at 0.41%: $1027 \times 1.0041^{170} \approx 2059$
• 80 years from 1870 AD to 1950 AD at 0.82%: $2059 \times 1.0082^{80} \approx 3957$
• 50 years from 1950 AD to 2000 AD at 1.7%: $3957 \times 1.017^{50} \approx 9192$

for a final population of about 9,200 people. Frankly this sounds low, but that's part of the problem with the exponential function: it works slowly at first and with small inputs, then takes to the skies when the input grows. Note that the first doubling took about 600 years, whereas the last doubling happened in less than 50 years.

You can plug these equations into a spreadsheet and play with the numbers to see if you can get the effect you are after. For example, if instead of starting with 500 people at year 1100 AD we start with 10,000 people and calculate over the same period, the result becomes quite different:

• 600 years from 1100 AD to 1700 AD at 0.12%: $10000 \times 1.0012^{600} \approx 20535$
• 170 years from 1700 AD to 1870 AD at 0.41%: $20535 \times 1.0041^{170} \approx 41170$
• 80 years from 1870 AD to 1950 AD at 0.82%: $41170 \times 1.0082^{80} \approx 79125$
• 50 years from 1950 AD to 2000 AD at 1.7%: $79125 \times 1.017^{50} \approx 183807$

The population grows by the same factor in both cases (about 18x) but since the starting population size is larger, the resulting population also obviously is larger.

Looking at your median age at death of 60 years, we can also look at life expectancy variation over time and conclude that this is similar to a Medieval Britain (approximately 500 - 1500 AD) level of society (at age 21, life expectancy was to a total age of 64 years). Our handy-dandy table above doesn't include specific figures covering that period, but around 0.1% population growth per year appears to be a reasonable extrapolation based on the data we do have. That also appears to match reasonably well with the Kremer and Vermillion graph posted on the History Stack Exchange. In the present day, we see such life expectancies primarily in Africa south of the equator, and slightly longer life expectancies in Russia, including Asian Russia. The effect of diseases on life expectancy is particularly pronounced in Africa, however; according to World Health Organization data as quoted on Wikipedia, the life expectancy in Botswana and Zimbabwe would more than double were it not for HIV/AIDS. Current day countries which have a life expectancy of exactly 60 years at birth are Kenya, Rwanda and Afghanistan (again WHO data, dated 2012). Perhaps interestingly, neither of these show a large difference between genders; they are all listed as 59 years for men and 61 years for women.

It is also important to keep in mind that the population growth rate is going to be heavily impacted by culture as well. If the culture is one that encourages people to have lots of children, the overall population growth rate obviously goes up; if the culture actively or passively encourages people to have fewer children (as for example is the case with China's family planning policy), the number will be lower or could even lead to a population decrease over time. A population that reduces in size over the long haul is obviously not sustainable, but in extreme situations it could become necessary to take such measures in order to avoid even worse consequences of resource overshoot. The culture can also, of course, change over time; 900 years is a fairly long time for any society when looking at it in terms of a human lifespan.

You can decide on population growth rates for different periods and perform the same types of calculations yourself to get population sizes even down to per year if you are so inclined. If you do, I encourage you to make sure that there are die-offs as well; there are going to be years when quite a number of people die, especially in a society that doesn't have access to the advanced health care technology of today, and those are going to make a major dent in the population curve. For further added realism, consider demographics as well (maybe it is a disease that kills off the young more than the adults; also considering that you don't have many people who are actually old). The effects of such a die-off could last for decades before the population pyramid is somewhat back in shape.

• In your current era calculation, shouldn't the result be 2 billion people? Dec 9 '14 at 13:24
• Life expectancy at the high end has little direct influence on population growth. Assuming peak childbearing at 20 - 35, after about 55 adult death doesn't much affect things, since the youngest children will be grown by then. Jun 3 '15 at 4:23
• @WhatRoughBeast I'm not quite sure what you are referring to. This answer is based on historic overall population growth rates, which should already account for such factors as well as things like infant mortality. If you feel this answer is in need of improvement, then please state which part you are referring to so I can fix that.
– user
Jun 3 '15 at 7:59
• No, your analysis is fine. I'm just pointing out that specifying a "normal" lifespan" of 60 doesn't have a lot of effect. 50 would do just as well. As long as the parents live long enough for the kids to grow up, anything else is frosting on the cake. Jun 3 '15 at 12:13

From looking into my Crystal Ball - no, that's not Wikipedia on my laptop, now pay attention here - the answer to your question would be 'virtually any amount'. If you look at historical populations. (ok, fine, I admit it), particularly about half way down the page, you can see a list of estimated human populations through history. In 2000 BC, the estimated world population was 27 million. 1000 years later, 50 million. That's less than a 2 fold increase. And from year 1AD to 1000, it went 200 to 265 million - it didn't even double. But from 1000AD to 2000... it went from 265 million to 6 billion. A huge increase.

From that, you can see the range over 900 years might be from somewhere between virtually no increase in population, to 25 times or more. Obviously when talking human history you have wars and plagues to deal with. So you might raise the projections if your world doesn't have them. And technology - particularly medical technology - is an important factor. If people die after catching the sniffles, its going to have a serious impact on population growth.

My recommendation is to find a year range in that chart that best represents your world's level of development, and work from that as your base guess, adjusting as desired. You gave a life expectancy of 60, so you might base your estimates around real world population increases from when the real-world's life expectancy was 60.

Other factors might include available land/resources - is it a tough life or not? Also wealth, culture, luxury (modern 'western' nations have lower birth rates than other areas of the world).

Your question is quite general so I'll try to give you a range.

The absolute maximum would probably involve an industrial baby farm type system, where the limit is basically a woman's fertility (since even with only a few men there would be plenty of sperm). If you always have 50% women then you start with 250 of them. Let's say every woman can have 1 baby a year, every year, for 50 years of her life. This leads to an absurdly high population of at least 10^32, which would be impossible once you take into account environmental effects.

A more reasonable system where everyone has a large family: Suppose everyone pairs up when they're 20 and has 6 children, who also grow up til they're 20, pair up and have 6 children then die, turning 2 people into 6 people. Every 20 years your population multiplies by 3. Simply enough, after 900 = 45*20 years you have a population of 500*3^45 = 1.48*10^24, or 1.48 million billion billion. Still ridiculous.

A more likely scenario: Couples have on average 2.4 children. That's a population increase by a factor of 1.2 every generation. Again, assuming the generation is about 20 years, that's 500*1.2^45 = 1,828,630 or 1.83 million, a much more reasonable figure. That's probably your best estimate.

• There's something seriously wrong with your program if it ran into the billions after only a single decade. At that point, none of the new babies have reached reproductive age, so you'd only have 250 + 13 * 250 = 3500 people. To a rough approximation your 1st model can be represented as 16year generations (56 over 900 years) and each generation being 33x larger than the one before. This gives an estimated final population of 33^56 ~ 10^85. Dec 9 '14 at 14:09
• By the time @DanNeely mentions 10^85, I feel compelled to point out that the observable universe is estimated at 10^80 baryons. A baryon is composed of three quarks, with a quark of course being an elementary particle. In other words, we'd be incredibly likely to hit hard physical limits long before we even approach 10^85 people, even if we could figure out how to feed everyone.
– user
Nov 9 '15 at 19:34
• Just some data points - geneticists looking at human prehistory generally assume 29 years per generation. The age of the parents at the age of their average child not their first child, is what matters. Peak fertility per woman is about 7-8 per lifetime for a large population if not genetically twin-prone. Lifetime fertility is lower for hunter-gatherers due to longer nursing related infertility viz farmers. But, there's huge variation is mortality. About 2.1 kids per woman per lifetime is replacement rate today, but you need more historically with high rates of infant and childhood mortality. Jan 6 '17 at 1:12

Assuming no wars or serious plagues, anywhere from 0 to 18 quadrillion – or more.

If each couple has an average of four children, at about the age of twenty, then after 900 years the population will be 17,592,000,000,000,000. If each couple has an average of one child, at the age of twenty, humanity will end in about 250 years.

The problem with the 18 quadrillion estimate is the lack of land and resources. live science theorizes that the Earth can support a population of up to 10 billion, long-term. If people will completely ignore sustainability, I'd guess that the limit is about 100 billion.

However, if technology improves significantly, or many other planets are colonized, than 18 quadrillion is plausible. In fact, you could go even higher – a nontillion (1,000,000,000,000,000,000,000,000,000,000) people is doable with just six children per family.

Typical population growth assuming no limiting factors is roughly 1.7%. This is based on modern population growth I copied from Michael Kjörlings answer. The real maximum could be higher, maximum measured in the real world was 2.1%.

But here is the really important part: In practice the population growth has always been limited by some other factor. If you want to find out a plausible rate of population growth, you have to actually find out what is limiting the growth and derive the rate from that.

For most of history, until industrialization or so, population was limited by available food. When access was gained to new resources, more food, population could increase rapidly, but otherwise population would be relatively stable or even decline due to decreased acricultural productivity. Soil quality in acricultural areas would decline significantly due to erosion and, on irrigated areas, salinity. During antiquity and medieval times until the black death people actually considered world to be full, which had significant effects on society.

In urbanized societies where a significant parts of population lived in areas of high population density and trade was a significant factor disease would become significant factor. More in the form of occasional drops from epidemics than from adding a limit beyond the limits of available food. The drops would have an effect in the long term population growth, so for a period of 900 years you need to consider the level of medical technology and sanitation. Also a small starting population might cause an increase in the level of genetic disease, which could affect the growth for the first few centuries.

In modern societies social effects have become a major limiting factor on population growth. Even in societies where food is plentiful people postpone having children and choose not to have large families. This is essentially a life style choice. In other countries people are encouraged to have smaller families to prevent poverty and improve resource management.

From the parameters you give the colonists would seem to have a decent level of technology. So the limiting factor on the population growth would probably be the rate they can increase the available resources and expand their infrastructure. This could be close to that 2% mark, even above if the society is really focused on growing. Lack of contraceptives implies they would be, but that might not be sustainable over 900 years. Fast population growth without contraceptives or abortions would be pretty harsh on women. As such the limiting factor is probably going to be social. Note that the time women spend having children is time they can't contribute to increasing the available resources, so it would probably be counter-productive to reduce women into baby-making machines. Not to mention it would add an flavor you might not want into the society.

The rule given in GURPS Space: "On a wholly earth-like planet, with medical technology of at least TL5, a human population will increase by a factor of 10 every 100 years, up to the maximum population for the planet." This fits what I wrote above (giving independent confirmation) and gives a very simple formula, that is roughly equivalent of 2.3% growth a year. Which seems bit high, but given that I expected the number to be something slightly over 2% not implausible. The formula gives final population of 500,000,000,000. Essentially this means that provided that the colony becomes stable within the first century, it can grow to fill the maximum supported population of your planet.

• I wouldn't say at all that we currently have "no factors limiting population growth" (not an exact quote from your answer; paraphrasing).
– user
Dec 11 '14 at 10:58
• @MichaelKjörling I wouldn't either. Merely that current growth is a good guess for a value to use. (Which I forgot to give a reason for...) My answer specifically lists social factors that limit current growth. The paragraph start with "In modern societies..." Dec 11 '14 at 11:03

"Reasonable" depends of cultural expectations.

If you go for the max:

With in-vitro fertilization, you can select gender.

Female is fertile until 40 (increased chance of genetic defects). So let's assume 20 daughters and single son per generation, and 20 years per generation = 450 generations, and all females are fertile and willing to breed.

Each woman from your starting population can have 20**450 descendants. FYI that is approximately 2.9 followed by 585 zeroes. 2.9e+585 That's quite a lot mouths to feed. You will need huge robot army to work millions of solar systems all over galaxy. And they will be crowded.

Without IVF, if we assume 10 daughters per generation, you have 10**450 per original female.

Assuming a Doubling of the Population every 27.5 years (the medium Human Generation Time) with 32.72 Doublings from the original 500 will grow to 3,537,296,484,444 without me taking the time to calculate the exact population by only subtracting the deaths. But let's use the 1 Year Growth Rate the numbers are:

1. 1%: 3,874,417
2. 2%: 2,748,6825,799
3. 3%: 178,843,120,520,152

And then it just gets stupid.....

Doubling the Population calculated by Generation Times result in:

1. 10 years: 618,970,019,642,690,137,449,562,112,000
2. 25 years: 34,359,738,368,000
3. 27.5 years (as above): 3,537,296,484,444
4. 30 yeas: 524,288,000
5. 50 years: 131,072,000

Again, I did not calculate death-rates.


text:
#### 11th Standard Accounts Final Accounts of Sole Proprietors - II English Medium Free Online Test One Mark Questions 2020 - 2021

11th Standard

Reg.No. :
•
•
•
•
•
•

Accountancy

Time : 00:10:00 Hrs
Total Marks : 10

10 x 1 = 10
1. Net profit is

(a)

Debited to capital account

(b)

Credited to capital account

(c)

Debited to drawings account

(d)

Credited to drawings account

2. Accrued interest on investment will be shown

(a)

On the credit side of profit and loss account

(b)

On the assets side of balance sheet

(c)

Both (a) and (b)

(d)

None of these

3. The income which has been earned during the current accounting year but .not received till the end of that accounting year is called as _____

(a)

Outstanding expenses

(b)

Prepaid expenses

(c)

accrued income

(d)

4. _______ is a loss for the business and should be adjusted against profit.

(a)

Interest on Capital

(b)

Depreciation

(c)

Debtors

(d)

5. The Trial Balance shows bank loan Rs.5,00,000 at 12% on 01-01-2015. Interest paid Rs.40,000. Interest outstanding is ______ as on 31-12-2015.

(a)

Rs.60,000

(b)

Rs.40,000

(c)

Rs.27,600

(d)

Rs.20,000

6. All the items given in the adjustment will appear at ______ in the Filial accounts.

(a)

One place

(b)

Two places

(c)

Three places

(d)

None of these

7. Net profit is transferred from profit and loss account ______ account.

(a)

Capital

(b)

Drawings

(c)

Bank overdraft

(d)

Sundry creditors

8. Unexpired expenses are also called as_________.

(a)

Outstanding expenses

(b)

Prepaid expenses

(c)

Accrued income

(d)

9. The decrease in book value of fixed assets due to usage or passage of time is called_________.

(a)

(b)

Depreciation

(c)

Closing stock

(d)

Capital

10. Commission on net profit after charging such commission:

(a)

$\frac{Rate\ of\ Commission}{(100+Rate\ of\ commission)}$

(b)

$\frac{Rate\ of\ Commission}{100}$

(c)

$\frac{Rate\ of\ Commission}{(100-Rate\ of\ commission)}$

(d)

None of these


text:
# Efficient pen-and-paper calculation of the Galois Field multiplication?

I have been reading on multiplication over GF for the Inverse Matrix Step in AES, and I feel like converting the hex numbers to binary, multiplying them and then doing the modifications required for GF arithmetic is too time-consuming when done by hand.

It would be very helpful if someone could help me with a faster way to do it.

• "converting the hex numbers to binary", you wouldn't need that if you're implementing AES in C language, as integer types are represented in memory as bit strings (only converted to hexadecimal or decimal when outputting). What language are you using to implement AES? – DannyNiu Feb 27 at 4:18
• And it seems the problem you're facing is completely about programming - in which case it belongs to SO. – DannyNiu Feb 27 at 4:21
• I am sorry if I made some mistake while writing, but I meant doing the calculations manually. – user3656142 Feb 27 at 5:03
• This might be helpful for you. It suggest to use $x2$ if you know how to compute it. – kelalaka Feb 27 at 8:18
• Would using a logarithm table (and an antilogarithm table) be acceptable? That way, you can reduce the Galois field multiplication to three table lookups and a modular addition. – <NAME> Feb 27 at 10:21

Probably the easiest way to do finite field multiplication by hand is using discrete logarithm and antilogarithm tables. For example, here's a pair of such tables for the AES field (using 3 as the base), generated using this Python code:

log| _0 _1 _2 _3 _4 _5 _6 _7 _8 _9 _A _B _C _D _E _F
---+------------------------------------------------
0_ | -- 00 19 01 32 02 1A C6 4B C7 1B 68 33 EE DF 03
1_ | 64 04 E0 0E 34 8D 81 EF 4C 71 08 C8 F8 69 1C C1
2_ | 7D C2 1D B5 F9 B9 27 6A 4D E4 A6 72 9A C9 09 78
3_ | 65 2F 8A 05 21 0F E1 24 12 F0 82 45 35 93 DA 8E
4_ | 96 8F DB BD 36 D0 CE 94 13 5C D2 F1 40 46 83 38
5_ | 66 DD FD 30 BF 06 8B 62 B3 25 E2 98 22 88 91 10
6_ | 7E 6E 48 C3 A3 B6 1E 42 3A 6B 28 54 FA 85 3D BA
7_ | 2B 79 0A 15 9B 9F 5E CA 4E D4 AC E5 F3 73 A7 57
8_ | AF 58 A8 50 F4 EA D6 74 4F AE E9 D5 E7 E6 AD E8
9_ | 2C D7 75 7A EB 16 0B F5 59 CB 5F B0 9C A9 51 A0
A_ | 7F 0C F6 6F 17 C4 49 EC D8 43 1F 2D A4 76 7B B7
B_ | CC BB 3E 5A FB 60 B1 86 3B 52 A1 6C AA 55 29 9D
C_ | 97 B2 87 90 61 BE DC FC BC 95 CF CD 37 3F 5B D1
D_ | 53 39 84 3C 41 A2 6D 47 14 2A 9E 5D 56 F2 D3 AB
E_ | 44 11 92 D9 23 20 2E 89 B4 7C B8 26 77 99 E3 A5
F_ | 67 4A ED DE C5 31 FE 18 0D 63 8C 80 C0 F7 70 07
exp| _0 _1 _2 _3 _4 _5 _6 _7 _8 _9 _A _B _C _D _E _F
---+------------------------------------------------
0_ | 01 03 05 0F 11 33 55 FF 1A 2E 72 96 A1 F8 13 35
1_ | 5F E1 38 48 D8 73 95 A4 F7 02 06 0A 1E 22 66 AA
2_ | E5 34 5C E4 37 59 EB 26 6A BE D9 70 90 AB E6 31
3_ | 53 F5 04 0C 14 3C 44 CC 4F D1 68 B8 D3 6E B2 CD
4_ | 4C D4 67 A9 E0 3B 4D D7 62 A6 F1 08 18 28 78 88
5_ | 83 9E B9 D0 6B BD DC 7F 81 98 B3 CE 49 DB 76 9A
6_ | B5 C4 57 F9 10 30 50 F0 0B 1D 27 69 BB D6 61 A3
7_ | FE 19 2B 7D 87 92 AD EC 2F 71 93 AE E9 20 60 A0
8_ | FB 16 3A 4E D2 6D B7 C2 5D E7 32 56 FA 15 3F 41
9_ | C3 5E E2 3D 47 C9 40 C0 5B ED 2C 74 9C BF DA 75
A_ | 9F BA D5 64 AC EF 2A 7E 82 9D BC DF 7A 8E 89 80
B_ | 9B B6 C1 58 E8 23 65 AF EA 25 6F B1 C8 43 C5 54
C_ | FC 1F 21 63 A5 F4 07 09 1B 2D 77 99 B0 CB 46 CA
D_ | 45 CF 4A DE 79 8B 86 91 A8 E3 3E 42 C6 51 F3 0E
E_ | 12 36 5A EE 29 7B 8D 8C 8F 8A 85 94 A7 F2 0D 17
F_ | 39 4B DD 7C 84 97 A2 FD 1C 24 6C B4 C7 52 F6(01)

To use these tables to multiply two hexadecimal numbers in the AES field, follow these steps:

1. If either of the numbers is 0, the result will also be 0, because that's how multiplication by zero works in general. In that case, you can stop here. Otherwise, follow the steps below:

2. Look up both of the numbers in the first table (the one with log in the top left corner). Specifically, for each number, find the entry at the intersection of the row matching the first hex digit and the column matching the second hex digit of the number being looked up. This is the (base 3) discrete logarithm of the number in the AES field.

For example, let's say we want to multiply 84 and 5A. Looking at row 8_ and column _4 of the log table, we find that the base 3 discrete logarithm of 84 is F4. Doing the same for 5A, we find that its discrete logarithm (at the intersection of row 5_ and column _A) is E2.

3. Add the results of the previous step together modulo 255. That is to say, first add them together normally, and if this results in a three hex digit sum, subtract 255 (hex FF) from it. Or, equivalently, just drop the first digit (which will always be 1) and add one to what's left.

(Technically, if the sum happens to equal FF, we should also replace it with zero. But we can ignore that special case and instead just add an extra entry into the bottom right corner of the antilogarithm table, as I have done above.)

Continuing the example above, F4 + E2 equals 1D6. (If you can't easily do hexadecimal addition in your head, you may want to also print out a table for it.) Since this is a three-digit number, we drop the first digit and add 1, giving D6 + 1 = D7.

4. Finally, look up the result of the previous step in the antilogarithm table (i.e. the second table, labeled exp, above). This is the result of the multiplication.

For example, looking at row D_ and column _7 in the second table above, we find that the antilogarithm of D7 (and thus the product of 84 and 5A) in the AES field is 91.

BTW, if you're worried about making mistakes in the table lookups (e.g. looking in the wrong row or column), a useful property of the two tables above is that they are (by definition) inverses of each other. Thus, you can verify a log table lookup by looking the result up in the exp table and checking that you get the original number back, and vice versa.

(The one minor exception to this symmetry is the extra entry for exp(FF) = exp(00) that I included in the exp table for convenience. If you look up FF in the exp table, you get 01, for which the log table gives the canonical logarithm 00.)

You also can use these tables for Galois field division (i.e. multiplication by inverse) by replacing the addition in step 3 above by subtraction (and adding 255 = FF to the result if it's negative). And, of course, if you want to calculate the AES Galois field inverse of a number, you can do that by dividing 01 with it. Or, equivalently, just look up its logarithm, subtract it from FF, and look up the antilogarithm of the result.

Let's multiply 0x84 and 0x5a using keyboard and monitor (paper and pen works the same):

Converting into binary you can do digit by digit independently and should get 10000100 and 01011010.

The minimal polynomial taken for the field with 256 elements is 0x11b for AES. You should read that as a bit in the position of $$2^8$$ is the same as 0x1b = 00011011 binary.

The first step is to create a multiplication table for one of the factors, let's say 0x84. The first row is the number in binary. From row to row you just append a zero, and drop the first digit (shifting all other digits one to the left). But if the first digit is 1 (corresponding to $$2^8$$ or $$X^8$$, if you remember that we do polynomial multiplication modulo the AES-polynomial), then you have to add 0x1b = 00011011. Adding is without carries, i.e., xor. You stop as soon you have 8 rows, as we work with 8-bit values.

10000100
00001000 + 00011011 = 00010011
00100110
01001100
10011000
00110000 + 00011011 = 00101011
01010110
10101100

If you've written down maybe two or three multiplication tables then the time it takes to write down such a table is limited by your typing/writing speed. Not much brain activity is needed for the arithmetic.

Now look at the binary representation of the other factor 0x5a = 01011010. Each of its digit corresponds to a row of the table: The last digit to the first row, the 2nd last to the second, and so on. Cross out every row that corresponds to a digit 0, and add up each row that corresponds to a digit 1 (sorry that 5a is symmetric, for 5b = 01011011 the first row would have a 1 in front):

0: 10000100
1: 00001000 + 00011011 = 00010011
0: 00100110
1: 01001100
1: 10011000
0: 00110000 + 00011011 = 00101011
1: 01010110
0: 10101100

As I don't know how to cross out here, I just copied the rows with a 1 at the beginning and added up each column writing the result below the line. Adding is again without carry: Write 0 if you see an even number of 1s above, and 1 if it's an odd number of 1s in the column.

00010011
01001100
10011000
01010110
--------
10010001

Convert to hex 10010001 = 0x91 to get the same result as Ilmari with his logarithm tables.

If you multiply several times with the same number, of course you can recycle the multiplication tables.


text:
maximal spacing around order statistics [#2]

June 7, 2018
By

(This article was originally published at R – Xi'an's Og, and syndicated at StatsBlogs.)

The proposed solution of the riddle from the Riddler discussed here a few weeks ago is rather approximative, in that the distribution of

$\Delta_n=\max_i\,\min_j\,|X_{i}-X_{j}|$

when the n-sample is made of iid Normal variates is (a) replaced with the distribution of one arbitrary minimum and (b) the distribution of the minimum is based on an assumption of independence between the absolute differences. Which does not hold, as shown by the above correlation matrix (plotted via corrplot) for N=11 and 10⁴ simulations. One could think that this correlation decreases with N, but it remains essentially 0.2 for larger values of N. (On the other hand, the minima are essentially independent.)

Please comment on the article here: R – Xi'an's Og

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text:
## Flexibility 2

1 = √(1)
= √(-1)(-1)
= (√-1)(√-1)
= i.i
= i$^{2}$
= -1
Is this a correct equation??
anythings wrong with this?
i think theoretically it is correct but it seems like
√(1) = √(-1)(-1)
√(1) = √(1)(1) also!!
so how to explain this??

 Quote by <NAME> 1 = √(1) = √(-1)(-1) = (√-1)(√-1) = i.i = i$^{2}$ = -1 Is this a correct equation?? anythings wrong with this? i think theoretically it is correct but it seems like √(1) = √(-1)(-1) √(1) = √(1)(1) also!! so how to explain this??

Squaring is a double branched complex variable function, and thus "usual" properties in real numbers can fail miserably.

In this case the problem appears in the third equality: it isn't true without imposing certain restrictive conditions.

DonAntonio


text:
## HarmonicRegression (C++/Python)

This C++ library with python bindings implements spherical harmonics and an associated least-squares regression to recover a smooth surface description from point-cloud data. These methods are described in our paper

"Grasp Moduli Spaces and Spherical Harmonics", <NAME>, <NAME>, <NAME>, IEEE ICRA, 2014

This software is freely available for academic/research purposes, just drop me an email to get the code: fpokorny (at) kth.se

## Introduction

Consider the space $L^2(\mathbb{S}^2)$ of integrable functions on the sphere $\mathbb{S}^2\subset \mathbb{R}^3$. We choose a coordinate chart $p(\theta, \phi) = (\sin \theta \cos \phi, \sin \theta \sin \phi, \cos\theta)$ on the sphere, with $(\theta, \phi) \in [0, \pi]\times [0, 2\pi)$. For integers $l, m$, the real valued spherical harmonic function of degree $l$ and order $m$, $|m|\le l$, is defined by $Y_{l,m}(\theta, \phi) = \begin{cases} c_{l,m}P_l^{|m|}(\cos \theta)\sin (|m|\phi) & -l\le m \le -1\\ \frac{c_{l,m}}{\sqrt{2}}P_l^0(\cos \theta) & m = 0\\ c_{l,m}P_l^{m}(\cos \theta)\cos(m\phi) & 1\le m \le l, \end{cases}$ where $c_{l,m} = \sqrt{\frac{2l+1}{2\pi}\frac{(l-|m|)!}{(l+|m|)!}}$ and $P_l^m$ denotes the associated Legendre polynomial of order $m$ and degree $l$. The functions $Y_{l,m}$ arise as eigen-functions of the Laplacian $\Delta=\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}(\sin\theta \frac{\partial}{\partial\theta})+\frac{1}{\sin^2\theta}\frac{\partial^2}{\partial^2\phi}$ on $\mathbb{S}^2$ For fixed $l$, there are $2l+1$ eigenfunctions $\{Y_{l,m}: m\in\{-l, \ldots, l\}\}$ satisfying $\Delta Y_{l,m} = \lambda_l Y_{l,m},$ where $\lambda_l = l(l+1)$ denotes the corresponding eigen-value. These eigen-functions, for all $l, m$, form an infinite orthonormal basis for $L^2(\mathbb{S}^2)$ with respect to the standard inner product $\langle f, g\rangle = \int_{\mathbb{S}^2} fg \,dVol$, for $f, g\in L^2(\mathbb{S}^2)$ and where $dVol = \sin \theta d \theta d \phi$.

A visualization of the functions $Y_{l, m}$, where the degree changes from 0 to 4 along the horizontal axis
and the order $m$ changes from $-l$ to $l$ along the vertical axis.

This library can be used to compute the functions $Y_{l, m}$ and their derivatives. Additionally, a least squares regression of point-cloud data which can be mapped onto a sphere can be performed easily with the provided code. The figure below illustrates an example reconstruction of a smooth surface from point-cloud data where all $Y_{l, m}$ up to degree $L$ are used.

Point cloud P with 25000 points and reconstructed surfaces, where all eigen-spaces up to degree $L$ are used.


text:
# $$\frac{1000}{1}$$! Oops

Algebra Level 4

Let $$s_n$$ be the sum of the digits of a natural number $$n$$. If $$\dfrac ab$$ satisfies the condition of the smallest value of $$\dfrac n{s_n}$$, where $$n$$ is a four-digit number, find $$a+b$$.

×


text: Suppose $r$ is an estimate of the exact value of $\sqrt{240}$. Find an expression, in terms of $r$, for a better estimate. How quickly converging an expression can you find?


text:
## Directions

We are located at the 6th floor of the G.O. Jones Building on the Mile End Campus, midway between Stepney Green and Mile End Tube stations, approximately 15-20 minutes from central London on the Central or District lines. If exiting Stepney Green tube station, turn left and walk along the Mile End Road for approximately 300 metres. The G.O. Jones (Physics) building is to the right of the main college building, which is fronted by a clocktower and lawn. If exiting Mile End tube station, turn left and walk approximately 300 metres until you are opposite the main college building. A more detailed description can be found here.

## Seminars at Queen Mary University of London

Found at least 20 result(s)

### 29.07.2021 (Thursday)

#### Integrability and chaos in SYM theories from anomalous-dimension spectra

Regular Seminar Anne Spiering (Trinity College Dublin)

 at: 14:00 QMWroom zoom abstract: The discovery of integrability in planar N=4 SYM theory led to considerable advances in the computation of its planar anomalous dimension spectrum. Less is known at the non-planar level where the theory is assumed to be non-integrable. I will show how statistical properties of numerical anomalous dimension spectra can give insight into the symmetries of the underlying model and that the N=4 SYM non-planar spectrum and its beta-deformed version are well described by random matrix theory, indicating their quantum-chaotic nature. Doing so I will also discuss on-going work on using on-shell methods to obtain the dilatation operator for deformed versions of N=4 SYM theory. [for zoom link please contact jung-wook(dot)kim(at)qmul(dot)ac(dot)uk]

### 22.07.2021 (Thursday)

#### Tidal effects from quantum field theory

 at: 14:00 QMWroom zoom abstract: Recently there has been great success in describing the inspiral phase of compact binary coalescence in General Relativity (GR) using scattering amplitudes. These efforts aim to improve the precision of gravitational wave (GW) templates used by detectors to identify GW events and test GR. The majority of this work has focused on describing a pair of objects that do not tidally deform. However real objects do tidally deform, and future GW detectors will be increasingly sensitive to such deformations. It has been shown that scattering amplitudes are applicable to the description of tidal effects as well. In this talk we discuss the application of the Hilbert series to the characterization of an entire class of tidal effects: those involving two powers of the Weyl tensor and objects with spin 0 and 1/2. The Hilbert series guides us in the construction of actions describing these effects, which we then use to calculate the leading-PM tidal scattering amplitude. With the amplitude in hand, we compute several classical quantities and compare to the literature where possible. [for zooom link please contact jung-wook(dot)kim(at)qmul(dot)ac(dot)uk]

### 13.05.2021 (Thursday)

#### BPS states and RG flows

Regular Seminar <NAME> (Uppsala U.)

 at: 14:00 QMWroom zoom abstract: The study of possible constraints on RG flows is an important question in our understanding of quantum field theory. In this talk I will discuss a constraint which arises by considering the spectrum of BPS excitations (an H-theorem). Examples are found in the context of four-dimensional supersymmetric quantum field theories with enough supersymmetry that the BPS spectrum is explicitly computable. Among others, we will discuss applications in the context of Maruyoshi-Song flows. [for zoom link please contact jung-wook(dot)kim(at)qmul(dot)ac(dot)uk]

### 06.05.2021 (Thursday)

#### Towards a general map from Navier-Stokes to Maxwell via Einstein

Regular Seminar <NAME> (ASU)

 at: 15:00 QMWroom zoom abstract: [for zoom link please email s.nagyATqmul.ac.uk] After a brief review of the cutoff-surface formulation of fluid-gravity duality, we explore the square root'' of the fluid-dual metrics via the classical-double copy, highlighting the constant vorticity flows and potential flows which have algebraically special Weyl double copy fields. We then present progress towards building the same map, from fluid solutions to gravitational solutions to Maxwell solutions, for generic fluids in 2+1 dimensions. This talk is based on JHEP 08 (2020) 147 published with ASU students <NAME> and <NAME>, and forthcoming work.

### 29.04.2021 (Thursday)

#### The Topology of Data: from String Theory to Cosmology to Phases of Matter

Regular Seminar <NAME> (Wisconsin U., Madison)

 at: 15:00 QMWroom zoom abstract: Abstract: We are faced with an explosion of data in many areas of physics, but very so often, it is not the size but the complexity of the data that makes extracting physics from big datasets challenging. As I will discuss in this talk, data has shape and the shape of data encodes the underlying physics. Persistent homology is a tool in computational topology developed for quantifying the shape of data. I will discuss three applications of topological data analysis: 1) identifying structure of the string landscape, 2) constraining cosmological parameters from CMB measurements and large scale structures data, and 3) detecting and classifying phases of matter. Persistent homology condenses these datasets into their most relevant (and interpretable) features, so that simple statistical pipelines are sufficient in these contexts. This suggests that TDA can be used in conjunction with machine learning algorithms and improves their architecture. [for zoom link please contact jung-wook(dot)kim(at)qmul(dot)ac(dot)uk]

### 22.04.2021 (Thursday)

#### Boundary-bulk relations in conformal/topological field theories

Regular Seminar <NAME> (Shenzhen Institute of Quantum Science and Engineering, SUSTech)

 at: 14:00 QMWroom zoom abstract: [There will be a pre-seminar for students starting at 13:30. For zoom link please email s.nagyATqmul.ac.uk] The relation between the physics of the bulk and that of a boundary plays an important role in quantum field theories, quantum gravity and condensed matter physics. In this talk, I will review a manifestation of this relation that can be summarized as a short statement: "the bulk is the center of a boundary". I will explain the meaning of "center" and provide a formal proof of this statement (arXiv:1702.00673). By including higher codimensional domain walls between boundaries, the boundary-bulk relation can be formulated mathematically as a higher functor. In lower dimensional cases, it becomes precise mathematical theorems. In the end, I will discuss the significances of this relation in the study of topological orders and topological phase transitions.

### 15.04.2021 (Thursday)

#### A CFT Distance Conjecture

Regular Seminar <NAME> (IPhT)

 at: 14:00 QMWroom zoom abstract: [for zoom link, please email s.nagyATqmul.ac.uk] We formulate conjectures relating the geometry of conformal manifolds to the spectrum of local operators in conformal field theories in d>2 spacetime dimensions. We focus on conformal manifolds with limiting points at infinite distance with respect to the Zamolodchikov metric. Our central conjecture is that all theories at infinite distance possess an emergent higher-spin symmetry, generated by an infinite tower of currents whose anomalous dimensions vanish exponentially in the distance. Our conjectures are related to the Distance Conjecture in the swampland program. We discuss the supporting evidence, their holographic interpretation, and implications for superconformal field theories.

### 08.04.2021 (Thursday)

#### Moduli space of vortices and 3d supersymmetric gauge theories

Regular Seminar <NAME> (Rutgers)

 at: 14:00 QMWroom zoom abstract: [For zoom link please email s.nagyATqmul.ac.uk There will be a pre-seminar for students at 13:30] I will discuss the geometric interpretation of the twisted index of 3d supersymmetric gauge theories on a closed Riemann surface. I will show that the twisted index reproduces the virtual Euler characteristic of the moduli space of solutions to vortex equations on the Riemann surface. I will also discuss 3d N = 4 mirror symmetry in this context, which implies non-trivial relations between enumerative invariants associated to the moduli space of vortices. Finally, I will comment on level structures and a wall-crossing formula of the twisted indices derived from the gauge theory point of view.

### 01.04.2021 (Thursday)

#### Scrambling in Yang-Mills

Regular Seminar <NAME> (Witwatersrand U.)

 at: 14:00 QMWroom zoom abstract: We study operators with a bare dimension that grows as N^2 in the large N limit. These operators are labeled by a Young diagram with p long rows, as well as a graph, with p nodes. The dilatation operator describing the mixing of these operators defines a Hamiltonian for excitations hopping on this graph. The scrambling and equilibration of the resulting dynamics is studied. [For zoom link please contact jung-wook(dot)kim(at)qmul(dot)ac(dot)uk]

### 25.03.2021 (Thursday)

#### Another dimension of Kerr

Regular Seminar <NAME> (Edinburgh U.)

 at: 14:00 QMWroom zoom abstract: [there will be a 30min pre-seminar for students. For zoom link, please contact s.nagyATqmul.ac.uk] Abstract: Recent work has broadened the scope of scattering amplitudes to include large, classical objects such as the Kerr black hole. This new perspective illuminates surprising aspects of classical gravity, such as the Newman-Janis shift relating Kerr to Schwarzschild. I will describe the connection between amplitudes and classical physics, focussing on the case of Kerr.

### 11.03.2021 (Thursday)

#### QCD instantons and Hadron Colliders

Regular Seminar <NAME> (Durham U.)

 at: 14:00 QMWroom zoom abstract: QCD instantons are arguably the best motivated yet unobserved nonperturbative effects predicted by the Standard Model. A discovery and detailed study of instanton-generated processes at colliders would provide a new window into the phenomenological exploration of QCD and a vastly improved fundamental understanding of its non-perturbative dynamics. We present for the first time a full calculation of QCD instanton-induced processes in proton-proton collisions accounting for quantum corrections due to both initial and final state gluon interactions. Although QCD instanton processes are predicted to be produced with a large scattering cross-section at small centre-of-mass partonic energies, discovering them at hadron colliders is a challenging task that requires dedicated search strategies. [Based on https://arxiv.org/abs/2010.02287 and https://arxiv.org/abs/1911.09726.] [for zoom link contact jung-wook(dot)kim(at)qmul(dot)ac(dot)uk]

### 25.02.2021 (Thursday)

#### Energy correlations at conformal collider

Regular Seminar <NAME> (IPhT Saclay)

 at: 14:00 QMWroom zoom abstract: [For zoom link please contact s.nagyATqmul.ac.uk. There will be a pre-seminar for students at 13:30] The energy-energy correlation (EEC) measures the angular distribution of the energy that flows through two calorimeters separated by some relative angle in the final state created by a source. We present a new approach to computing this observable that exploits the relation between the energy correlations and four-point correlation functions of conserved currents. In the limit of small and large angles, when EEC describes the correlation between particles belonging, respectively, to the same jet and to two almost back-to-back jets, we obtain a concise representation of the EEC in terms of the conformal data of twist-two operators and verify it by comparing with the results of explicit calculation at next-to-next-to-leading order in maximally supersymmetric Yang-Mills theory. As a byproduct of our analysis, we predict the maximal weight part of the analogous QCD expression.

### 18.02.2021 (Thursday)

#### OPE for form factors

Regular Seminar <NAME> (Tel Aviv U.)

 at: 16:00 QMWroom zoom abstract: We propose an operator product expansion for planar form factors of local operators in N = 4 SYM theory. This expansion is based on the dual conformal symmetry of these objects or, equivalently, the conformal symmetry of their dual description in terms of periodic Wilson loops. A form factor is decomposed into a sequence of known pentagon transitions and a new universal object that we call the â€œform factor transitionâ€. This transition is subject to a set of non-trivial bootstrap constraints, which allows us to bootstrap it at any value of the coupling. We evaluate the form factor transition for MHV form factors of the chiral half of the stress tensor supermultiplet at leading order in perturbation theory and use it to produce OPE predictions at any loop order. We match the one-loop and two-loop predictions with data available in the literature. [for zoom link contact jung-wook(dot)kim(at)qmul(dot)ac(dot)uk]

### 11.02.2021 (Thursday)

#### Knot Theory and Machine Learning

Regular Seminar <NAME> (CERN)

 at: 14:00 QMWroom Zoom abstract: [For zoom details please email s.nagyATqmul.ac.uk There will be a pre-seminar for students at 13:30] Knot theory plays an important role in physics, mathematics and biology. Characterizing knots is, however, a difficult task. There are different ways to represent a knot (e.g. via braids, Gauss codes, Dowker-Thistlethwaite notation), and many knot invariants exist (e.g. the Alexander polynomial, Jones polynomial, determinant, slice genus). However, it is not known whether these can be used to identify a trivial knot, the so-called unknot. We use different machine learning techniques to tackle this question. First, we use a very recent neural network architecture developed for natural language processing, called the reformer, to decide whether a given knot is the unknot. We also apply Reinforcement Learning to solve the harder problem of finding a set of Reidemeister/Markov moves that explicitly simplify a given knot as much as possible. If the algorithm finds a sequence of moves that removes all crossings of a knot in a given representation, then this knot is provably the unknot.

### 03.02.2021 (Wednesday)

#### Quantum Information Theory of the Gravitational Anomaly

Triangular Seminar <NAME> (Tokyo U., IPMU)

 at: 15:00 QMWroom Zoom abstract: I will explain that the notion of quantum entanglement is not defined for gravitationally anomalous two-dimensional theories, because they do not admit a local tensor factorization of the Hilbert space into local Hilbert spaces. Qualitatively, the modular flow cannot act consistently and unitarily in a finite region, if there are different numbers of states with a given energy traveling in the two opposite directions. I will make this precise by decomposing it into two observations: First, a two-dimensional conformal field theory admits a consistent quantization on a space with boundary only if it is not anomalous. Second, a local tensor factorization always leads to a definition of consistent, unitary, energy-preserving boundary condition. As a corollary I will establish a generalization of the Nielsen-Ninomiya theorem to all two-dimensional unitary local quantum field theories: No continuum quantum field theory in two dimensions can admit a lattice regulator unless its gravitational anomaly vanishes. I will advocate that these points be used to reinterpret the gravitational anomaly quantum-information-theoretically, as a fundamental obstruction to the localization of quantum information. [Zoom link: please ask your local triangle organizer]

### 03.02.2021 (Wednesday)

#### Islands in black holes and cosmology

Triangular Seminar <NAME> (Cornell)

 at: 16:30 QMWroom Zoom abstract: I will summarize recent developments on the information paradox involving "islands" and replica wormholes, and discuss the extent to which these ideas can be applied to cosmology. [Zoom link: please ask your local triangle organizer]

### 28.01.2021 (Thursday)

#### Quantum Natural Language Processing on NISQ hardware

Regular Seminar <NAME> (Oxford)

 at: 14:00 QMWroom Zoom abstract: [Please inqure jung-wook(dot)kim(at)qmul(dot)ac(dot)uk for the zoom link] Taking advantage of mathematical similarities between grammar models and quantum theory, we present a canonical way of instantiating grammatical sentences as quantum circuits. In the context of quantum machine learning, we define toy natural language processing tasks and implement them on readily accessible quantum processors, such as those provided by IBMQ.

### 21.01.2021 (Thursday)

#### Unimodular vs Nilpotent Superfield Approach to Pure dS Supergravity

Regular Seminar <NAME> (Chulalongkorn University)

 at: 14:00 QMWroom Zoom abstract: [there will be a 15 min pre-seminar for students. For zoom link, please email s.nagyATqmul.ac.uk] Abstract: Recent progress in understanding de Sitter spacetime in supergravity and string theory has led to the development of a 4D supergravity with spontaneously broken supersymmetry allowing for de Sitter vacua, also called de Sitter supergravity. One approach makes use of constrained (nilpotent) superfields, while an alternative one couples supergravity to a locally supersymmetric generalization of the Volkov-Akulov goldstino action. These two approaches have been shown to give rise to the same 4D action. A novel approach to de Sitter vacua in supergravity involves the generalisation of unimodular gravity to supergravity using a super-StÃ¼ckelberg mechanism. We make a connection between this new approach and the previous two. We show that upon appropriate field redefinitions, the 4D actions match up to the cubic order in the fields. This points at the possible existence of a more general framework to obtain de Sitter spacetimes from high-energy theories.

### 03.12.2020 (Thursday)

#### Topologically Ordered Matter and Why You Should be Interested

Regular Seminar <NAME> (Oxford)

 at: 14:00 QMWroom zoom abstract: Abstract: In two dimensional topologically ordered matter, processes depend on gross topology rather than detailed geometry. Thinking in 2+1 dimensions, particle world lines can be interpreted as knots or links, and the amplitude for certain processes becomes a topological invariant of that link. While sounding rather exotic, we believe that such phases of matter not only exist, but have actually been observed in quantum Hall experiments, and could provide a route to building a quantum computer. Possibilities have also been proposed for creating similar physics in systems ranging from superfluid helium to topological superconductors to semiconductor-superconductor junctions to quantum wires to spin systems to graphene to cold atoms. Please contact jung-wook(dot)kim(at)qmul(dot)ac(dot)uk for the zoom link.