nhaliday + intersection   18

gn.general topology - Pair of curves joining opposite corners of a square must intersect---proof? - MathOverflow
In his 'Ordinary Differential Equations' (sec. 1.2) V.I. Arnold says "... every pair of curves in the square joining different pairs of opposite corners must intersect".

This is obvious geometrically but I was wondering how one could go about proving this rigorously. I have thought of a proof using Brouwer's Fixed Point Theorem which I describe below. I would greatly appreciate the group's comments on whether this proof is right and if a simpler proof is possible.

...

Since the full Jordan curve theorem is quite subtle, it might be worth pointing out that theorem in question reduces to the Jordan curve theorem for polygons, which is easier.

Suppose on the contrary that the curves A,BA,B joining opposite corners do not meet. Since A,BA,B are closed sets, their minimum distance apart is some ε>0ε>0. By compactness, each of A,BA,B can be partitioned into finitely many arcs, each of which lies in a disk of diameter <ε/3<ε/3. Then, by a homotopy inside each disk we can replace A,BA,B by polygonal paths A′,B′A′,B′ that join the opposite corners of the square and are still disjoint.

Also, we can replace A′,B′A′,B′ by simple polygonal paths A″,B″A″,B″ by omitting loops. Now we can close A″A″ to a polygon, and B″B″ goes from its "inside" to "outside" without meeting it, contrary to the Jordan curve theorem for polygons.

- John Stillwell
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october 2017 by nhaliday
'Capital in the Twenty-First Century' by Thomas Piketty, reviewed | New Republic
by Robert Solow (positive)

The data then exhibit a clear pattern. In France and Great Britain, national capital stood fairly steadily at about seven times national income from 1700 to 1910, then fell sharply from 1910 to 1950, presumably as a result of wars and depression, reaching a low of 2.5 in Britain and a bit less than 3 in France. The capital-income ratio then began to climb in both countries, and reached slightly more than 5 in Britain and slightly less than 6 in France by 2010. The trajectory in the United States was slightly different: it started at just above 3 in 1770, climbed to 5 in 1910, fell slightly in 1920, recovered to a high between 5 and 5.5 in 1930, fell to below 4 in 1950, and was back to 4.5 in 2010.

The wealth-income ratio in the United States has always been lower than in Europe. The main reason in the early years was that land values bulked less in the wide open spaces of North America. There was of course much more land, but it was very cheap. Into the twentieth century and onward, however, the lower capital-income ratio in the United States probably reflects the higher level of productivity: a given amount of capital could support a larger production of output than in Europe. It is no surprise that the two world wars caused much less destruction and dissipation of capital in the United States than in Britain and France. The important observation for Piketty’s argument is that, in all three countries, and elsewhere as well, the wealth-income ratio has been increasing since 1950, and is almost back to nineteenth-century levels. He projects this increase to continue into the current century, with weighty consequences that will be discussed as we go on.

...

Now if you multiply the rate of return on capital by the capital-income ratio, you get the share of capital in the national income. For example, if the rate of return is 5 percent a year and the stock of capital is six years worth of national income, income from capital will be 30 percent of national income, and so income from work will be the remaining 70 percent. At last, after all this preparation, we are beginning to talk about inequality, and in two distinct senses. First, we have arrived at the functional distribution of income—the split between income from work and income from wealth. Second, it is always the case that wealth is more highly concentrated among the rich than income from labor (although recent American history looks rather odd in this respect); and this being so, the larger the share of income from wealth, the more unequal the distribution of income among persons is likely to be. It is this inequality across persons that matters most for good or ill in a society.

...

The data are complicated and not easily comparable across time and space, but here is the flavor of Piketty’s summary picture. Capital is indeed very unequally distributed. Currently in the United States, the top 10 percent own about 70 percent of all the capital, half of that belonging to the top 1 percent; the next 40 percent—who compose the “middle class”—own about a quarter of the total (much of that in the form of housing), and the remaining half of the population owns next to nothing, about 5 percent of total wealth. Even that amount of middle-class property ownership is a new phenomenon in history. The typical European country is a little more egalitarian: the top 1 percent own 25 percent of the total capital, and the middle class 35 percent. (A century ago the European middle class owned essentially no wealth at all.) If the ownership of wealth in fact becomes even more concentrated during the rest of the twenty-first century, the outlook is pretty bleak unless you have a taste for oligarchy.

Income from wealth is probably even more concentrated than wealth itself because, as Piketty notes, large blocks of wealth tend to earn a higher return than small ones. Some of this advantage comes from economies of scale, but more may come from the fact that very big investors have access to a wider range of investment opportunities than smaller investors. Income from work is naturally less concentrated than income from wealth. In Piketty’s stylized picture of the United States today, the top 1 percent earns about 12 percent of all labor income, the next 9 percent earn 23 percent, the middle class gets about 40 percent, and the bottom half about a quarter of income from work. Europe is not very different: the top 10 percent collect somewhat less and the other two groups a little more.

You get the picture: modern capitalism is an unequal society, and the rich-get-richer dynamic strongly suggest that it will get more so. But there is one more loose end to tie up, already hinted at, and it has to do with the advent of very high wage incomes. First, here are some facts about the composition of top incomes. About 60 percent of the income of the top 1 percent in the United States today is labor income. Only when you get to the top tenth of 1 percent does income from capital start to predominate. The income of the top hundredth of 1 percent is 70 percent from capital. The story for France is not very different, though the proportion of labor income is a bit higher at every level. Evidently there are some very high wage incomes, as if you didn’t know.

This is a fairly recent development. In the 1960s, the top 1 percent of wage earners collected a little more than 5 percent of all wage incomes. This fraction has risen pretty steadily until nowadays, when the top 1 percent of wage earners receive 10–12 percent of all wages. This time the story is rather different in France. There the share of total wages going to the top percentile was steady at 6 percent until very recently, when it climbed to 7 percent. The recent surge of extreme inequality at the top of the wage distribution may be primarily an American development. Piketty, who with Emmanuel Saez has made a careful study of high-income tax returns in the United States, attributes this to the rise of what he calls “supermanagers.” The very highest income class consists to a substantial extent of top executives of large corporations, with very rich compensation packages. (A disproportionate number of these, but by no means all of them, come from the financial services industry.) With or without stock options, these large pay packages get converted to wealth and future income from wealth. But the fact remains that much of the increased income (and wealth) inequality in the United States is driven by the rise of these supermanagers.

and Deirdre McCloskey (p critical): https://ejpe.org/journal/article/view/170
nice discussion of empirical economics, economic history, market failures and statism, etc., with several bon mots

Piketty’s great splash will undoubtedly bring many young economically interested scholars to devote their lives to the study of the past. That is good, because economic history is one of the few scientifically quantitative branches of economics. In economic history, as in experimental economics and a few other fields, the economists confront the evidence (as they do not for example in most macroeconomics or industrial organization or international trade theory nowadays).

...

Piketty gives a fine example of how to do it. He does not get entangled as so many economists do in the sole empirical tool they are taught, namely, regression analysis on someone else’s “data” (one of the problems is the word data, meaning “things given”: scientists should deal in capta, “things seized”). Therefore he does not commit one of the two sins of modern economics, the use of meaningless “tests” of statistical significance (he occasionally refers to “statistically insignificant” relations between, say, tax rates and growth rates, but I am hoping he does not suppose that a large coefficient is “insignificant” because R. A. Fisher in 1925 said it was). Piketty constructs or uses statistics of aggregate capital and of inequality and then plots them out for inspection, which is what physicists, for example, also do in dealing with their experiments and observations. Nor does he commit the other sin, which is to waste scientific time on existence theorems. Physicists, again, don’t. If we economists are going to persist in physics envy let us at least learn what physicists actually do. Piketty stays close to the facts, and does not, for example, wander into the pointless worlds of non-cooperative game theory, long demolished by experimental economics. He also does not have recourse to non-computable general equilibrium, which never was of use for quantitative economic science, being a branch of philosophy, and a futile one at that. On both points, bravissimo.

...

Since those founding geniuses of classical economics, a market-tested betterment (a locution to be preferred to “capitalism”, with its erroneous implication that capital accumulation, not innovation, is what made us better off) has enormously enriched large parts of a humanity now seven times larger in population than in 1800, and bids fair in the next fifty years or so to enrich everyone on the planet. [Not SSA or MENA...]

...

Then economists, many on the left but some on the right, in quick succession from 1880 to the present—at the same time that market-tested betterment was driving real wages up and up and up—commenced worrying about, to name a few of the pessimisms concerning “capitalism” they discerned: greed, alienation, racial impurity, workers’ lack of bargaining strength, workers’ bad taste in consumption, immigration of lesser breeds, monopoly, unemployment, business cycles, increasing returns, externalities, under-consumption, monopolistic competition, separation of ownership from control, lack of planning, post-War stagnation, investment spillovers, unbalanced growth, dual labor markets, capital insufficiency (William Easterly calls it “capital fundamentalism”), peasant irrationality, capital-market imperfections, public … [more]
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april 2017 by nhaliday
example:
Deadweight loss created by a binding price ceiling. Producer surplus is necessarily decreased, while consumer surplus may or may not increase; however the decrease in producer surplus must be greater than the increase (if any) in consumer surplus.
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february 2017 by nhaliday
The genetics of politics: discovery, challenges, and progress
Figure 1. Summary of relative genetic and environmental influences on political traits.

- heritability increases discontinuously on leaving home
- pretty big range of heritability for different particular traits (party identification is lowest w/ largest shared environment by far)
- overall ideology quite highly heritable
- social trust is surprisingly highly compared other measurements I've seen...
- ethnocentrism quite low (sample-dependent?)
- authoritarianism and traditionalism quite high
- voter turnout quite high

Genes, psychological traits and civic engagement: http://rstb.royalsocietypublishing.org/content/370/1683/20150015
We show an underlying genetic contribution to an index of civic engagement (0.41), as well as for the individual acts of engagement of volunteering for community or public service activities (0.33), regularly contributing to charitable causes (0.28) and voting in elections (0.27). There are closer genetic relationships between donating and the other two activities; volunteering and voting are not genetically correlated. Further, we show that most of the correlation between civic engagement and both positive emotionality and verbal IQ can be attributed to genes that affect both traits.

Are Political Orientations Genetically Transmitted?: http://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1006&context=poliscifacpub
TABLE 1. Genetic and Environmental Influences on Political Attitudes: The 28 Individual Wilson–Patterson Items

The origins of party identification and its relationship to political orientations: http://sci-hub.tw/http://www.sciencedirect.com/science/article/pii/S0191886915002470

All models showed a good overall fit (see Table 3). The data indicate that party identification is substantially heritable, with about 50% of the variation in PID attributable to additive genetic effects. Moreover, the results indicate that the non-genetic influences on party identification stem primarily from unique environmental factors rather than shared ones such as growing up in the same family. This too is not consistent with the Michigan model.

Table 3 also indicates that genetic influences explained about 50% of the variance in liberalism–conservatism. This estimate is similar to previous behavior genetic findings on political attitudes (e.g., Alford et al., 2005; Bouchard, 2004; Hatemi et al., 2014; Kandler, Bleidorn, & Riemann, 2012). The remaining variance was again due primarily to nonshared environmental influences. The latter finding indicates that the Michigan hypothesis that partisan social influences affect political orientations may have some merit, although the substantial level of heritability for this variable suggests that genetic effects also play an important role.

...

As Table 4 reveals, the best fitting model indicates that 100% of the genetic variance in PID is held in common with liberalism–conservatism ([aC2]/[aC2 + aPID2] = 1.00). Similarly, 73% of the environmental variation in PID is shared with liberalism–conservatism ([eC2]/[eC2 + ePID2] = .73). All told, only 13% of the total variance in PID cannot be explained by variation in liberalism–conservatism (1 [aC2 + eC2] = .13), as illustrated in Fig. 3. Since only a small proportion of the variance in PID cannot be explained by liberalism– conservatism, the findings are consistent with the hypothesis that genetic and environmental factors influence liberalism–conservatism, which in turn affects party identification. However, as discussed below, other causal scenarios cannot be ruled out.

Table 4 and Fig. 3 also show that 55% of the total variance in liberalism–conservatism cannot be accounted for by variance in PID

Fig. 3. Venn diagram mapping the common and specific variance in party
identification and liberalism–conservatism.

intuition for how you can figure out overlap of variance: look at how corr(PID, liberal-conservative) differs between MZ and DZ twin pairs, etc., fit structural equational model

p_k,i,j = r_A a_k,i,j,p + r_C c_k,i,p + r_E e_k,i,j,p (k=MZ or DZ, i=1..n_k, j=1,2, p=PID or LC value)

c_k,i,j,p = r_{C,p} c'_k,i,p + r_{C,common} c'_k,i,common (ditto)
e_k,i,j,p = r_{E,p} e'_k,i,j,p + r_{E,common} e'_k,i,j,common (ditto)

MZ twins:
a_MZ,i,j,p = r_{A,p} a'_MZ,i,p + r_{A,common} a'_MZ,i,common (i=1..n_k, j=1,2 p=PID or LC value)

DZ twins:
a_DZ,i,j,p = r_{A,p} (1/2 a'_DZ,i,p + 1/2 a'_DZ,i,j,p) + r_{A,common} (1/2 a'_DZ,i,common + 1/2 a'_DZ,i,j,common) (i=1..n_k, j=1,2 p=PID or LC value)

Gaussian distribution for the underlying a', c' and e' variables, maximum likelihood, etc.

see page 9 here: https://pinboard.in/u:nhaliday/b:70f8b5b559a9

basically:
1. calculate population means μ from data (so just numbers)
2. calculate covariance matrix Σ in terms of latent parameters r_A, r_C, etc. (so variable correlations)
3. assume observed values are Gaussian with those parameters μ, Σ
4. maximum likelihood to figure out the parameters r_A, r_C, etc.

A Genetic Basis of Economic Egalitarianism: http://sci-hub.tw/10.1007/s11211-017-0297-y
Our results show that the large portion of the variance in a four-item economic egalitarianism scale can be attributed to genetic factor. At the same time, shared environment, as a socializing factor, has no significant effect. The effect of environment seems to be fully reserved for unique personal experience. Our findings further problematize a long-standing view that social justice attitudes are dominantly determined by socialization.

published in the journal "Social Justice Research" by some Hungarians, lol

various political science findings, w/ a few behavioral genetic, focus on Trump, right-wing populism/authoritarianism, and polarization: http://www.nationalaffairs.com/blog/detail/findings-a-daily-roundup/a-bridge-too-far
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february 2017 by nhaliday
inequalities - Is the Jaccard distance a distance? - MathOverflow
Steinhaus Transform
the referenced survey: http://kenclarkson.org/nn_survey/p.pdf

It's known that this transformation produces a metric from a metric. Now if you take as the base metric D the symmetric difference between two sets, what you end up with is the Jaccard distance (which actually is known by many other names as well).
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february 2017 by nhaliday
MinHash - Wikipedia
- goal: compute Jaccard coefficient J(A, B) = |A∩B| / |A∪B| in sublinear space
- idea: pick random injective hash function h, define h_min(S) = argmin_{x in S} h(x), and note that Pr[h_min(A) = h_min(B)] = J(A, B)
- reduce variance w/ Chernoff bound
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february 2017 by nhaliday
mg.metric geometry - Pushing convex bodies together - MathOverflow
- volume of intersection of colliding, constant-velocity convex bodies is unimodal
- pf by Brunn-Minkowski inequality
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january 2017 by nhaliday
Projections onto convex sets - Wikipedia, the free encyclopedia
straightforward method to find point in intersection of convex sets, w/ some convergence guarantees
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june 2016 by nhaliday

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