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Linear and abstract algebra, operators from early analysis to a full-blown theory.
Surprisingly, there are differentiable manifolds that are homeomorphic (equivalent through continuous deformations) but not diffeomorphic (equivalent through smooth deformations) to the standard n-sphere. So basically they are spheres with nonstandard differential structure. Moreover, the diffeomorphism classes of oriented exotic spheres are the nontrivial elements in a finite abelian group (provided the dimension n does not equal 4, in which case little is known) with binary operation defined by the connected sum.
Jonathan Peters writing at Harvard Law & Policy Review's blog:
Using a high-security online drop box and a well-insulated website, WikiLeaks has published 76,000 classified U.S. documents about the war in Afghanistan, nearly 400,000 classified U.S. documents about the war in Iraq, and more than 2,000 U.S. diplomatic cables. In doing so, it has collaborated with some of the most powerful newspapers in the world, and it has rankled some of the most powerful people in the world. President Barack Obama said in July 2010, right after the release of the Afghanistan documents, that he was “concerned about the disclosure of sensitive information from the battlefield.” His concern spread quickly through the echelons of power, as WikiLeaks continued in the fall of 2010 to release caches of classified U.S. documents.
Secretary of State Hillary Clinton condemned the slow drip of diplomatic cables, saying it was “not just an attack on America’s foreign policy interests, it [was] an attack on the international community.” Director of National Intelligence James Clapper wrote in an e-mail to intelligence agencies that the “actions taken by WikiLeaks are not only deplorable, irresponsible, and reprehensible—they could have major impacts on our national security.” Members of Congress scrambled to respond to the website and its founder, Julian Assange, calling variously for a criminal prosecution, for an overhaul of the Espionage Act of 1917, and for a law that would make it illegal to publish the names of military and intelligence informants.
For his part, Attorney General Eric Holder announced in late November that the Justice Department and the Pentagon were investigating the circumstances surrounding the leaks to determine if criminal charges would be filed. Holder declined to say whether WikiLeaks or Assange were targets of the investigation. He said that anybody, regardless of citizenship or place of residence, could be a target, adding, “Let me be very clear . . . to the extent that we can find anybody who was involved in the breaking of American law . . . they will be held responsible.” Holder also said it would be a “misimpression” to think he was studying only the Espionage Act. Then, in early January 2011, the Justice Department subpoenaed records from Twitter about the account activity of several people connected to WikiLeaks. A federal grand jury reportedly has been meeting in Virginia to weigh the government’s evidence against WikiLeaks and Assange, in connection with the military’s case against Pfc. Bradley Manning, the Army intelligence analyst accused of leaking classified information to WikiLeaks.
This is largely unfamiliar territory for the Justice Department. As a result, the legal and constitutional issues are challenging and varied. This article focuses on one of them: When can the government, consonant with the First Amendment, punish the publication of classified information related to national security? To that end, Part I outlines the constitutional standards that could apply to such a prosecution of Assange or WikiLeaks. Part II discusses whether Assange and WikiLeaks are part of the press and whether that matters for constitutional purposes. Part III concludes by urging the Justice Department to proceed with caution.
Obscenity in the United States
Forty-two senators in the US have signed a letter to attorney general Holder asking for the DoJ to "combat the growing scourge of obscenity in America" by stepping up obscenity prosecutions (link).
This move can be seen in a larger context: the fighting against perceived social ills, with the accompanying belief that upholding particular freedoms and liberties is less important than addressing the harms they enable. Drugs, gambling, and sex are the big categories here. If I recall correctly, many southern States even banned sex toy shops. And there's major hypocrisy: for example, marijuana production, distribution, and possession are all illegal despite the general lack of harm while the legality of alcohol consumption allows (if not causes) huge damage to society and ruins people's lives.
I think many people conceive of legal authorities as having a duty to take up the "hero" role and intervene in people's lives, against their will, for their own good, or for the good of others. There's even a sentiment that they should act as sin regulators and make sure society's behavior is above a minimal threshold of purity, in essence if not in word. I view the latter political tendency as extremely dubious. I want to focus on "obscenity" in particular.
Some pornographers have made claims to the artistic merit of the works they publish. Although I think porn can be art and vice versa, I also think that a lot of porn is simply for libidinal purposes and has no artistically intended meaning - the merit claim might simply be ad hoc reinterpretation for legal defense, not real ideological position. Indeed, in porn sometimes it's the "obscenity" of given acts that make them eroticized for various demographics in the first place. A major psychological component of sex is the interplay between tension and relief, and one avenue to achieving this interplay is the exploration of natural feelings of shame and experiences associated with it. Hence, dirty talk and positive appraisal of so many related things "dirty" is (to an extent) mainstream and normal. Much less typical manifestations of this are seen in things like cuckolding, scat and watersports, bestiality, incest, and other extreme sexual activities plus many BDSM-themed ones, drawing upon humiliation for sexual excitement.
The general problem I see is this: why shouldn't the First Amendment protect obscene material just like any other? In practice obscenity has never been held as protected expression. There are a number of very important subissues, like with the vagueness doctrine, the merely hypothetical nature of the definition of obscene, and actionability of obscenity despite no interference with people's personal rights nor any direct injury caused by their production or distribution.
In Miller v. Californa, Chief Justice Burger wrote the court's opinion that define the modern Miller test:
The basic guidelines for the trier of fact must be: (a) whether 'the average person, applying contemporary community standards would find that the work, taken as a whole, appeals to the prurient interest, (b) whether the work depicts or describes, in a patently offensive way, sexual conduct specifically defined by the applicable state law; and (c) whether the work, taken as a whole, lacks serious literary, artistic, political, or scientific value.
Why must a material have one of those types of values in particular - and why must it have community-approved value at all to be allowed? Why single out offensive sexual conduct out of all potential offensive material, given that nonsexual offensive works harbor the very same potential for harm? Why are subjective, ever-changing community norms and mores the standard for liberty and freedom? Justice Douglas wrote a dissenting opinion:
The idea that the First Amendment permits government to ban publications that are 'offensive' to some people puts an ominous gloss on freedom of the press. That test would make it possible to ban any paper or any journal or magazine in some benighted place. The First Amendment was designed 'to invite dispute,' to induce 'a condition of unrest,' to 'create dissatisfaction with conditions as they are,' and even to stir 'people to anger.' The idea that the First Amendment permits punishment for ideas that are 'offensive' to the particular judge or jury sitting in judgment is astounding. No greater leveler of speech or literature has ever been designed. To give the power to the censor, as we do today, is to make a sharp and radical break with the traditions of a free society. The First Amendment was not fashioned as a vehicle for dispensing tranquilizers to the people. Its prime function was to keep debate open to 'offensive' as well as to 'staid' people. The tendency throughout history has been to subdue the individual and to exalt the power of government. The use of the standard 'offensive' gives authority to government that cuts the very vitals out of the First Amendment. As is intimated by the Court's opinion, the materials before us may be garbage. But so is much of what is said in political campaigns, in the daily press, on TV, or over the radio. By reason of the First Amendment—and solely because of it—speakers and publishers have not been threatened or subdued because their thoughts and ideas may be 'offensive' to some.
One person's trash is another's treasure, as the saying goes. I don't think it's the government's responsibility to take out the "trash" for vacuous puritanical reasons, especially when people are actively using said "trash" for their own private purposes. If the DoJ should expend its resources on expunging or combating harmful obscenity from society at all, it would be better spent on addressing the mud and slime in campaigns and general political discourse in the media rather than pornographic entertainment.
The recent letter cites pornographyharms.com as documentation of the harm caused by pornography. My synthesis is that the gist of the information on that website is as follows, in no rational order:
Sexual offenders consume hardcore pornography.
Children can easily come into contact with sexually explicit material via the internet.
Many people become addicted to pornography to the detriment of their relationships, potential or actual, or everyday activities, or personal self-esteem.
Consumption of some forms of pornography are followed by other forms as viewers (or readers) seek out "harder" material, potentially leading to child porn.
Sex trafficking is responsible for production of some pornography.
A large amount of pornography is generic, formulaic, one-dimensional and ultimately unrealistic.
Porn stars are at high risk for sexually transmitted diseases and often engage in destructive, hedonistic lifestyles.
Numerous messages are sent (or evoked) in porn that degrade or objectify women, sexualize children, rationalize exploitation or particular immoral acts and behaviors, or selectively emphasize certain points about sexual intimacy and normality. These messages are extremely prevalent.
These are not all meaningless issues. However, few are relevant to obscenity laws and actual real-world prosecutions, and of the points that are relevant, the question remains as to why consenting activity between adults should be legally actionable. Pornographic entertainment is not the only medium that can inspire harmful attitudes, perceptions, or behaviors - this is an inevitable outcome of any form of media ultimately - and literally the only quality that makes so-called "obscene" material a target for legal action, where other material with equal or greater potential for enabling harm is not, is that it is sexual in nature. Why should that matter, is this not an arbitrary double standard, does this not betray a shallow puritanical basis for obscenity laws in the first place? Movies, books, songs, newsletters, forums, and video games which are equally obscene - offensive to community standards and of no redeeming social value - are not outlawed: if anything their accessibility is simply restricted to adults. In the final analysis, if the First Amendment is powerful and broad enough to allow something as vitriolic as violent hate speech (which it does), it is absolutely inconsistent with the founding constitutional principles and spirited intentions underlying the amendment to outlaw abnormal or offensive pornography, a.k.a. "obscenity".
Math query answered on IRC
I posed a query on EFnet's #math channel and Breeden transferred the question over to Freenode's #math, where Nereid answered it in the affirmative. (link)
Theorem: if F is a field and φ: GL(n,F) → F* is a homomorphism then there exists a unique homomorphism β: F* → F* such that φ = β ∘ det. If φ is continuous, then so is β.
*laughter*
The barometer legend
"Describe how to determine the height of a skyscraper with a barometer."
One student replied:
"You tie a long piece of string to the neck of the barometer, then lower the barometer from the roof of the skyscraper to the ground. The length of the string plus the length of the barometer will equal the height of the building."
This highly original answer so incensed the examiner that the student was failed immediately. The student appealed on the grounds that his answer was indisputably correct, and the university appointed an independent arbiter to decide the case.
The arbiter judged that the answer was indeed correct, but did not display any noticeable knowledge of physics. To resolve the problem it was decided to call the student in and allow him six minutes in which to provide a verbal answer that showed at least a minimal familiarity with the basic principles of physics.
For five minutes the student sat in silence, forehead creased in thought. The arbiter reminded him that time was running out, to which the student replied that he had several extremely relevant answers, but couldn't make up his mind which to use. On being advised to hurry up the student replied as follows:
"Firstly, you could take the barometer up to the roof of the skyscraper, drop it over the edge, and measure the time it takes to reach the ground. The height of the building can then be worked out from the formula H = 0.5g x t squared. But bad luck on the barometer."
"Or if the sun is shining you could measure the height of the barometer, then set it on end and measure the length of its shadow. Then you measure the length of the skyscraper's shadow, and thereafter it is a simple matter of proportional arithmetic to work out the height of the skyscraper."
"But if you wanted to be highly scientific about it, you could tie a short piece of string to the barometer and swing it like a pendulum, first at ground level and then on the roof of the skyscraper. The height is worked out by the difference in the gravitational restoring force T =2 pi sqr root (l / g)."
"Or if the skyscraper has an outside emergency staircase, it would be easier to walk up it and mark off the height of the skyscraper in barometer lengths, then add them up."
"If you merely wanted to be boring and orthodox about it, of course, you could use the barometer to measure the air pressure on the roof of the skyscraper and on the ground, and convert the difference in millibars into feet to give the height of the building."
"But since we are constantly being exhorted to exercise independence of mind and apply scientific methods, undoubtedly the best way would be to knock on the janitor's door and say to him 'If you would like a nice new barometer, I will give you this one if you tell me the height of this skyscraper'."
The student was Niels Bohr, the only Dane to win the Nobel Prize for physics.
Cyclic changes in the shape of a quasi-rigid body on a curved manifold can lead to net translation and/or rotation of the body in the manifold. Presuming space-time is a curved manifold as portrayed by general relativity, translation in space can be accomplished by cyclic changes in the shape of a body, without any thrust or external forces.
Center of mass isn't well-defined in curved spacetime - this is what allows one to "swim" through space even without pushing off of anything at all.
A set of nontransitive dice is a set of dice for which the relation "is more likely to roll a higher number" is not transitive. This situation is similar to that in the game Rock, Paper, Scissors, in which each element has an advantage over one choice and a disadvantage to the other.
Venerating a superior being and blindly following its will is a natural human impulse, as it frees one of the heavy burden of decision-making and moral and intellectual judgment, and it also creates a feeling of safety and protection […] Even the most magnanimous leaders — perhaps especially them, given their belief in their own Goodness — are likely to veer into serious error, corruption and worse if they are liberated from a critical citizenry. […] subordinating your own critical faculties to a leader’s is, at all times, warped, self-destructive and dangerous.
- Glenn Greenwald, Salon.com (link)
Vector cross products
The usual binary cross product between vectors, u x v, is one which is thought to be specific to three-dimensional space, but it actually exists in seven dimensions too. But why only three or seven dimensions? Well I don't see any simple reason, but it's been proven apparently. Indeed, the cross product can be taken a step further as an r-fold operation (an action on r vectors instead of simply two), and to that end it is defined to satisfy three general properties: multilinearity, orthogonality, and what I call parallelo-volume, which means that the cross of u1, u2, ... ur has magnitude equal to the content (r-dimensional hypervolume) of the parallelopiped spanned by the vectors. One can even define the cross "product" of a single vector when r=1.
The r-fold cross product exists in d dimensions only for (r,d) = (1,2n); (2,3); (2,7); (3,8); and (n-1,n) for n=1,2,3,...
Source: Elduque
We propose that aesthetic pleasure is a function of the perceiver’s processing dynamics: The more fluently perceivers can process an object, the more positive their aesthetic response. We review variables known to influence aesthetic judgments, such as figural goodness, figure–ground contrast, stimulus repetition, symmetry, and prototypicality, and trace their effects to changes in processing fluency. Other variables that influence processing fluency, like visual or semantic priming, similarly increase judgments of aesthetic pleasure. Our proposal provides an integrative framework for the study of aesthetic pleasure and sheds light on the interplay between early preferences versus cultural influences on taste, preferences for both prototypical and abstracted forms, and the relation between beauty and truth. In contrast to theories that trace aesthetic pleasure to objective stimulus features per se, we propose that beauty is grounded in the processing experiences of the perceiver, which are in part a function of stimulus properties.
Lite version: A Perceptual Fluency Theory of Beauty
FOIA and academic freedom
In the wake of Scott Walker's bill stripping labor unions of collective bargaining rights, as well as other legislation ostensibly designed for budgetary considerations, University of Wisconsin-Madison professor and historian William Cronon did some research as to the source of the wave and stumbled upon a well-greased organization called the American Legislative Exchange Council (ALEC) and wrote a bright, balanced, well-documented post about it on his blog (link). The Wisconsin Republican Party's Deputy Executive Director Stephan Thompson swiftly filed an Open Records request for emails of Cronon's relating to the enigmatic organization, certain Republican figures and a number of political keywords, presumably so that the public employee could be caught using federally-funded university resources for partisan political purposes in contravention of university policy. Cronon was quick with a thorough, analytical rejoinder on his blog (link) inferring that the request effectively constituted an attempt to intimidate against criticism.
Looking at the relevant facts, it's clearcut either intimidation or retribution was the GOP's desire, whether or not they figured the maneuver had a component of legitimate investigative activism. Cronon makes the case that the Freedom Of Information Act (FOIA) - which does not require any statement of motive - enabling the GOP request should contain nuance for academic freedom: that its use in relation to university activity should be assessed based on if there is good reason to believe wrongdoing has occurred, and secondly that it should not be used to harass employees for unpalatable intellectual activity. The FOIA is a magnificently powerful democratic tool for transparency, yet its selective utilization means special ulterior motives can be satisfied to the potential detriment of totally legitimate pursuits.
Ah, where to strike a balance.
Harmonic Maps
A geodesic is roughly defined as a path in (not necessarily three-dimensional or Euclidean) space such that every sufficiently small segment of it minimizes the distance between its endpoints. As most know, in Euclidean space all geodesics are straight lines, but this truism does not generalize to curved spaces at all; straight lines don't even typically exist. Note that locality is featured but not globality: a geodesic between A and B is not necessarily the shortest between A and B, it's only the shortest path in all infinitesimally small neighborhoods of each point in the path.
The Euler-Lagrange equations from variational calculus and basic properties of metric tensors from differential geometry provide the groundwork necessary to derive the geodesic equation - a nonlinear system of partial differential equations which are satisfied by a path if and only if the path is geodesic. In practice, it is impossible (for all we know) to solve it exactly (meaning with an explicit symbolic formula), so we look for theoretical and numerical methods of approximating solutions.
Geodesics are one-dimensional objects imprinted on higher-dimensional objects called manifolds. The reverse is possible; one can "imprint" higher-dimensional spaces onto a one-dimensional continuum. These are called scalar functions.
Scalar functions take points in space and associate to them a single quantity, e.g. altitudes associated to coordinates in the xy plane which map out topography and landforms. Scalar functions have something called a Dirichlet energy associated to them, which is roughly a measure of how "variable" it is. The Euler-Lagrange equations can again be used to minimize Dirichlet energy; the energy E[f] equals zero when a scalar function f(x) satisfies Laplace's equation, which states that the flux divergence vanishes everywhere. Such functions are called harmonic functions - they share the description "harmonic" with musical notes because the latter exemplify harmonic functions on a 1-D space (associating to positions in space between source and receiver varying amounts of air pressure that make up soundwaves). Audio signals are linear combinations of sinusoids just like harmonic functions.
The figure above depicts a harmonic function defined on an annular region (two-dimensional doughnut) with special boundary conditions.
Axler, Bourdon, and Ramey have an excellent book on Harmonic Function Theory freely available in pdf format (link). It is actually one of my favorite digital textbooks so I highly recommend it the curious (and capable). Because of it I found out some gorgeous results on harmonic functions: automatic regularity, mean value property, minimum / maximum principle, Liouville's theorem, and how Poisson kernels resolve Dirichlet's problem on the ball. More generally, harmonic functions also lie in broader contexts of harmonic analysis (looking at signals as compositions of basic waves) and potential theory (physics-related issues, conformality).
So now we have geodesics which are 1-D lines mapped onto n-dimensional spaces, and harmonic functions which are n-dimensional spaces mapped onto 1-D continuums. It turns out these are both special cases of a more general object: harmonic maps. Again, roughly speaking - HMs map points on one (Riemannian) manifold to those on another, both of arbitrary dimension, in such a way that the HM optimizes the Dirichlet energy of the map compared to all maps which differ only infinitesimally from the HM. Wikipedia has a nifty way of visualizing this state of affairs. Call the harmonic map in question φ, mapping the manifold M to the manifold N.
We may imagine N made of marble and M made of rubber. A map from the latter to the former is considered a way to "apply" M to N. The Dirichlet energy E[φ] represents the total amount of elastic potential energy resulting from the tension in the rubber. The generalized analogue of Laplace's equation here is one which describes the tension field once again vanishing, τ(φ)=0. The harmonic map, then, is the application of M onto N which realizes an "energy-minimizing configuration".
A highlight of key issues on this subject is "Harmonic Maps" by Hélein and Wood (link). Way above my grade level and I sorely wish I could follow along beyond the first few pages. Someday.
PageRank and objectivity
Our search results are generated completely objectively and are independent of the beliefs and preferences of those who work at Google.
- The Google Team, 2007 (link)
That single point [Google considers its search results its opinions], which courts have agreed with, proves that there's no universally agreed-upon way to rank search results in response to a query. Therefore, web rankings (even if generated by an algorithm) are an expression of that search engine's particular philosophy.
- Google Associate Matt Curtis [2010?] (link)
By the way, I was pleasantly surprised awhile ago when I found out the core of Google's PageRank algorithm lies in (mostly) basic linear algebra with an extremely satisfying graph theoretic explanation. Hat tip: American Mathematical Society (link). Type and number of links on a webpage is nonrandom and isn't necessarily uniform with respect to "quality" or "relevance", so I'd be surprised if there wasn't some type of significant bias inherent in the internet itself as far as relational link structure is concerned.
Bureaucracy as Nash equilibrium
From Wikipedia:
In sociology, rationalization is the process where an increasing number of social actions become based on considerations of teleological efficiency or calculation rather than on motivations derived from morality, emotion, custom, or tradition. It is regarded as a central aspect of modernity, manifested especially in Western society; as a behaviour of the capitalist market; of rational administration in the state and bureaucracy; of the extension of modern science; and of the expansion of modern technology.
Many sociologists, critical theorists and contemporary philosophers have argued that rationalization, as falsely assumed progress, has a negative and dehumanizing effect on society, moving modernity away from the central tenets of enlightenment.
and
Iron cage, a sociological concept introduced by Max Weber, refers to the increased rationalization inherent in social life, particularly in Western capitalist societies. The "iron cage" thus traps individuals in systems based purely on teleological efficiency, rational calculation and control. Weber also described the bureaucratization of social order as "the polar night of icy darkness".
[...], there will be an evolution of an iron cage, which will be a technically ordered, rigid, dehumanized society. The iron cage is the one set of rules and laws that we are all subjected and must adhere to. Bureaucracy puts us in an iron cage, which limits individual human freedom and potential [...]
“Rational calculation... reduces every worker to a cog in this bureaucratic machine and, seeing himself in this light, he will merely ask how to transform himself… to a bigger cog… The passion for bureaucratization at this meeting drives us to despair.”
Loss of individuality and autonomy, depersonalization and hierarchical obsession derive from increasingly rationalized systems of human resources. Achieving large-scale efficiency may readily come with these hidden costs to people. As goal-oriented thinking dominates and socially oriented thinking wanes, increasing efficiency is prioritized over other values; bureaucracy is a naturally induced state of affairs given the right conditions and pressures.
Stability-threatening local deviations from assigned roles and approved modes of behavior are very likely to carry explicit or implicit penalties while relative acquiescence to the system will yield preferable payoffs. Bureaucracy manifests institutional Nash equilibrium: a set of working strategies adopted by a group of people such that any individual whose actions are knowingly under potential scrutiny and who is looking out for their own interests (or ass) has the optimal strategy of playing by the rules given that all others are doing so.
Personal detachment or even minor forms of learned helplessness - the crippling or cramping of ability to socially invest and participate purely volitionally if such unconstrained freedom were restored or realized elsewhere - are possible symptoms of sustained and consistent bureaucracy if one's will and desires are externally evaluable solely through the lens of efficiency. It is no surprise; humans are animals and similarly subject to behavioral conditioning.
Effective perturbation or institutional change in the system results from either altering the rules of the game or affecting the player's perceptions of the rules. Game-changing actions may instill systemic doubt about the optimality and preferableness of the usual considerations of bureaucratic restraint or they may reorient the goals and drives which the system is poised to operate towards. Ideally, efficiency and outcome are no longer weighted infinitely heavier than the 'spiritual' costs to members of the system but rather they are balanced fairly and ethically thus achieving both service to the outside and higher-ups as well as to the insiders at all levels.
The Typical Twelve
This is not my own work and I am uncertain of the author. It's about major methods of communication, specifically when someone comes to you with a problem, but it applies much more generally. They all have variable effects and implications for different persons and situations and predicting exactly what these are is extremely difficult.
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Ordering, directing, commanding: telling the other person to do something, giving them an order or a command.
Warning, admonishing, threatening: telling them what consequences will occur if they do something
Exhorting, moralizing, preaching: telling them what they should do
Advising, providing solutions or suggestions: telling them how to solve a problem, providing answers or solutions
Lecturing, teaching, logical argumentation: trying to influence them with facts, counterarguments, logic, information, or your own opinions
Judging, criticizing, disagreeing, blaming: making a negative judgment or evaluation, e.g. "You're not thinking clearly"
Praising, agreeing, positive evaluations: "I think you're pretty", "You have the ability to do well"
Name-calling, ridiculing, shaming: making the other person feel foolish, putting them into a category
Interpreting, analyzing, diagnosing: telling them what their motives are or analyzing why they are doing or saying something - communicating that you have them figured out or diagnosed
Reassuring, sympathizing, consoling, supporting: trying to make the other person feel better, talking them out of their feelings, trying to make the feelings go away or denying their strength, e.g. "All kids go through this sometime" or "You'll feel different tomorrow"
Probing, questioning, interrogating: trying to find reasons, motives, causes - searching for more information to help you solve the problem
Withdrawing, distracting, humoring, diverting: trying to get the other person away from the problem - withdrawing from the problem yourself - distracting or kidding them out of it, pushing the problem aside