intersection-connectedness   22

State (polity) - Wikipedia
In the medieval period (500-1400) in Europe, there were a variety of authority forms throughout the region. These included feudal lords, empires, religious authorities, free cities, and other authorities.[42] Often dated to the 1648 Peace of Westphalia, there began to be the development in Europe of modern states with large-scale capacity for taxation, coercive control of their populations, and advanced bureaucracies.[43] The state became prominent in Europe over the next few centuries before the particular form of the state spread to the rest of the world via the colonial and international pressures of the 19th century and 20th century.[44] Other modern states developed in Africa and Asia prior to colonialism, but were largely displaced by colonial rule.[45]


Two related theories are based on military development and warfare, and the role that these forces played in state formation. Charles Tilly developed an argument that the state developed largely as a result of "state-makers" who sought to increase the taxes they could gain from the people under their control so they could continue fighting wars.[42] According to Tilly, the state makes war and war makes states.[49] In the constant warfare of the centuries in Europe, coupled with expanded costs of war with mass armies and gunpowder, warlords had to find ways to finance war and control territory more effectively. The modern state presented the opportunity for them to develop taxation structures, the coercive structure to implement that taxation, and finally the guarantee of protection from other states that could get much of the population to agree.[50] Taxes and revenue raising have been repeatedly pointed out as a key aspect of state formation and the development of state capacity. Economist Nicholas Kaldor emphasized on the importance of revenue raising and warned about the dangers of the dependence on foreign aid.[51] Tilly argues, state making is similar to organized crime because it is a "quintessential protection racket with the advantage of legitimacy."[52]

State of nature:
Thomas Hobbes
The pure state of nature or "the natural condition of mankind" was deduced by the 17th century English philosopher Thomas Hobbes, in Leviathan and in his earlier work On the Citizen.[4] Hobbes argued that all humans are by nature equal in faculties of body and mind (i.e., no natural inequalities are so great as to give anyone a "claim" to an exclusive "benefit"). From this equality and other causes [example needed]in human nature, everyone is naturally willing to fight one another: so that "during the time men live without a common power to keep them all in awe, they are in that condition which is called warre; and such a warre as is of every man against every man". In this state every person has a natural right or liberty to do anything one thinks necessary for preserving one's own life; and life is "solitary, poor, nasty, brutish, and short" (Leviathan, Chapters XIII–XIV). Hobbes described this natural condition with the Latin phrase bellum omnium contra omnes (meaning war of all against all), in his work De Cive.

Within the state of nature there is neither personal property nor injustice since there is no law, except for certain natural precepts discovered by reason ("laws of nature"): the first of which is "that every man ought to endeavour peace, as far as he has hope of obtaining it" (Leviathan, Ch. XIV); and the second is "that a man be willing, when others are so too, as far forth as for peace and defence of himself he shall think it necessary, to lay down this right to all things; and be contented with so much liberty against other men as he would allow other men against himself" (loc. cit.). From here Hobbes develops the way out of the state of nature into political society and government, by mutual contracts.

According to Hobbes the state of nature exists at all times among independent countries, over whom there is no law except for those same precepts or laws of nature (Leviathan, Chapters XIII, XXX end). His view of the state of nature helped to serve as a basis for theories of international law and relations.[5]

John Locke
John Locke considers the state of nature in his Second Treatise on Civil Government written around the time of the Exclusion Crisis in England during the 1680s. For Locke, in the state of nature all men are free "to order their actions, and dispose of their possessions and persons, as they think fit, within the bounds of the law of nature." (2nd Tr., §4). "The state of Nature has a law of Nature to govern it", and that law is reason. Locke believes that reason teaches that "no one ought to harm another in his life, liberty, and or property" (2nd Tr., §6) ; and that transgressions of this may be punished. Locke describes the state of nature and civil society to be opposites of each other, and the need for civil society comes in part from the perpetual existence of the state of nature.[6] This view of the state of nature is partly deduced from Christian belief (unlike Hobbes, whose philosophy is not dependent upon any prior theology).

Although it may be natural to assume that Locke was responding to Hobbes, Locke never refers to Hobbes by name, and may instead have been responding to other writers of the day, like Robert Filmer.[7] In fact, Locke's First Treatise is entirely a response to Filmer's Patriarcha, and takes a step by step method to refuting Filmer's theory set out in Patriarcha. The conservative party at the time had rallied behind Filmer's Patriarcha, whereas the Whigs, scared of another prosecution of Anglicans and Protestants, rallied behind the theory set out by Locke in his Two Treatises of Government as it gave a clear theory as to why the people would be justified in overthrowing a monarchy which abuses the trust they had placed in it.[citation needed]


Jean-Jacques Rousseau
Hobbes' view was challenged in the eighteenth century by Jean-Jacques Rousseau, who claimed that Hobbes was taking socialized people and simply imagining them living outside of the society in which they were raised. He affirmed instead that people were neither good nor bad, but were born as a blank slate, and later society and the environment influence which way we lean. In Rousseau's state of nature, people did not know each other enough to come into serious conflict and they did have normal values. The modern society, and the ownership it entails, is blamed for the disruption of the state of nature which Rousseau sees as true freedom.[9]
Ulpian's statements were known in medieval Europe, but sovereignty was an important concept in medieval times.[1] Medieval monarchs were not sovereign, at least not strongly so, because they were constrained by, and shared power with, their feudal aristocracy.[1] Furthermore, both were strongly constrained by custom.[1]

Sovereignty existed during the Medieval period as the de jure rights of nobility and royalty, and in the de facto capability of individuals to make their own choices in life.[citation needed]



Sovereignty reemerged as a concept in the late 16th century, a time when civil wars had created a craving for stronger central authority, when monarchs had begun to gather power onto their own hands at the expense of the nobility, and the modern nation state was emerging. Jean Bodin, partly in reaction to the chaos of the French wars of religion, presented theories of sovereignty calling for strong central authority in the form of absolute monarchy. In his 1576 treatise Les Six Livres de la République ("Six Books of the Republic") Bodin argued that it is inherent in the nature of the state that sovereignty must be:[1]

- Absolute: On this point he said that the sovereign must be hedged in with obligations and conditions, must be able to legislate without his (or its) subjects' consent, must not be bound by the laws of his predecessors, and could not, because it is illogical, be bound by his own laws.
- Perpetual: Not temporarily delegated as to a strong leader in an emergency or to a state employee such as a magistrate. He held that sovereignty must be perpetual because anyone with the power to enforce a time limit on the governing power must be above the governing power, which would be impossible if the governing power is absolute.

Bodin rejected the notion of transference of sovereignty from people to the ruler (also known as the sovereign); natural law and divine law confer upon the sovereign the right to rule. And the sovereign is not above divine law or natural law. He is above (ie. not bound by) only positive law, that is, laws made by humans. He emphasized that a sovereign is bound to observe certain basic rules derived from the divine law, the law of nature or reason, and the law that is common to all nations (jus gentium), as well as the fundamental laws of the state that determine who is the sovereign, who succeeds to sovereignty, and what limits the sovereign power. Thus, Bodin’s sovereign was restricted by the constitutional law of the state and by the higher law that was considered as binding upon every human being.[1] The fact that the sovereign must obey divine and natural law imposes ethical constraints on him. Bodin also held that the lois royales, the fundamental laws of the French monarchy which regulated matters such as succession, are natural laws and are binding on the French sovereign.


Age of Enlightenment
During the Age of Enlightenment, the idea of sovereignty gained both legal and moral force as the main Western description of the meaning and power of a State. In particular, the "Social contract" as a mechanism for establishing sovereignty was suggested and, by 1800, widely accepted, especially in the new United States and France, though also in Great Britain to a lesser extent.

Thomas Hobbes, in Leviathan (1651) arrived a conception of sovereignty similar to Bodin's, which had just achieved legal… [more]
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august 2018 by nhaliday
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
Deadweight loss - Wikipedia
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
inequalities - Is the Jaccard distance a distance? - MathOverflow
Steinhaus Transform
the referenced survey:

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
Wizard War | West Hunter
Some of his successes were classically thin, as when he correctly analyzed the German two-beam navigation system (Knickebein). He realize that the area of overlap of two beams could be narrow, far narrower than suggested by the Rayleigh criterion.

During the early struggle with the Germans, the “Battle of the Beams”, he personally read all the relevant Enigma messages. They piled up on his desk, but he could almost always pull out the relevant message, since he remembered the date, which typewriter it had been typed on, and the kind of typewriter ribbon or carbon. When asked, he could usually pick out the message in question in seconds. This system was deliberate: Jones believed that the larger the field any one man could cover, the greater the chance of one brain connecting two facts – the classic approach to a ‘thick’ problem, not that anyone seems to know that anymore.

All that information churning in his head produced results, enough so that his bureaucratic rivals concluded that he had some special unshared source of information. They made at least three attempts to infiltrate his Section to locate this great undisclosed source. An officer from Bletchley Park was offered on a part-time basis with that secret objective. After a month or so he was called back, and assured his superiors that there was no trace of anything other than what they already knew. When someone asked ‘Then how does Jones do it? ‘ he replied ‘Well, I suppose, Sir, he thinks!’
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november 2016 by nhaliday
Son of low-hanging fruit | West Hunter
You see, you can think of the thunderstorm, after a ground discharge, as a vertical dipole. Its electrical field drops as the cube of altitude. The threshold voltage for atmospheric breakdown is proportional to pressure, while pressure drops exponentially with altitude: and as everyone knows, a negative exponential drops faster than any power.

The curves must cross. Electrical breakdown occurs. Weird lightning, way above the clouds.

As I said, people reported sprites at least a hundred years ago, and they have probably been observed occasionally since the dawn of time. However, they’re far easier to see if you’re above the clouds – pilots often do.

Pilots also learned not to talk about it, because nobody listened. Military and commercial pilots have to pass periodic medical exams known as ‘flight physicals’, and there was a suspicion that reporting glowing red cephalopods in the sky might interfere with that. Generally, you had to see the things that were officially real (whether they were really real or not), and only those things.

Sprites became real when someone recorded one by accident on a fast camera in 1989. Since then it’s turned into a real subject, full of strangeness: turns out that thunderstorms sometimes generate gamma-rays and even antimatter.
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november 2016 by nhaliday
Coefficient of relationship - Wikipedia, the free encyclopedia
relatedness by consanguinity

Average percent DNA shared between relatives – 23andMe Customer Care:
summary of relatedness by consanguinity
shouldn't it be 2^-4 ~ 6% for first cousins?
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july 2016 by nhaliday

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