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Topics - Agrul

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Agrulian Archives / Abstract Algebra & Galois Theory (grad student level)
« on: December 12, 2013, 02:16:16 PM »
Brief Subject Overview: abstract algebra identifies the properties that make elementary algebra on the real numbers work as it does, and uplifts these properties to the status of axioms; it then explores the consequences of various combinations of these properties. Central objects of study are groups, rings, fields, etc.; straight-forward questions of algebraic interest include, for example, whether and when a polynomial equation has a solution in some underlying set (the reals being the most familiar example) on which we're capable of doing algebra. Abstract algebra intertwines with many other aras of higher mathematics, as, for example, in the definition of Lie groups, which combines ideas from differential topology and abstract algebra. So far I've taken a one-semester course in this subject, and hope to take a second semester of it in the spring; however, IIRC we won't be covering much of chapter 13 (field theory) or any of chapter 14 (galois theory), and I've always wanted to at least understand the proof that quintic polynomials don't have a closed-form solution (which I believe has something to do with galois theory), so I'd like to teach this subarea to myself. We will be covering chapter 9 (polynomials over fields) in the spring, so the plan of study will be a bit redundant with that, but there's nothing like repetition to breed mastery.

Text(s): Dummit & Foote's "Abstract Algebra". Focus on chapters 9, 13, 14.

Assigned problems:

13,14,15,16,17 p. 298-299
4,11 p. 301-303
1,2,4,5 p. 306-307
4,7,11,17 p. 311-313
1,5,6,7 p. 315
6, 8, 13, 22, 29, 33, 34, 35, 43 p. 330-335

1,5,7,8 p. 519
3,7,12,20 p. 529-531
1,2,4 p. 535-536
1,2,5,6 p. 545
1,6,10,11 p. 551-552
1,2,3,9,10 p. 555-557

1,3,5,7,9 p. 566-567
2,5,7,20,31 p. 581-585
8,11,12,15,17 p. 589-591
3,5,7,8 p. 595-596
6,10,11,12,13 p. 603-606
13, 15, 18, 33, 37, 38, 44, 51 p. 617-624
2,3,4,5 p. 635-639
2,5,6 p. 644-645
2,8,12,14,15 p. 652-654

Replies will contain worked solutions, discussion, etc.

General Discussion / Extreme Diets can Quickly alter Gut Bacteria
« on: December 12, 2013, 01:00:48 AM »

With all the talk lately about how the bacteria in the gut affect health and disease, it's beginning to seem like they might be in charge of our bodies. But we can have our say, by what we eat. For the first time in humans, researchers have shown that a radical change in diet can quickly shift the microbial makeup in the gut and also alter what those bacteria are doing. The study takes a first step toward pinpointing how these microbes, collectively called the gut microbiome, might be used to keep us healthy.

"It's a landmark study," says Rob Knight, a microbial ecologist at the University of Colorado, Boulder, who was not involved with the work. "It changes our view of how rapidly the microbiome can change."

Almost monthly, a new study suggests a link between the bacteria living in the gut and diseases ranging from obesity to autism, at least in mice. Researchers have had trouble, however, pinning down connections between health and these microbes in humans, in part because it’s difficult to make people change their diets for the weeks and months researchers thought it would take to alter the gut microbes and see an effect on health.

But in 2009, Peter Turnbaugh, a microbiologist at Harvard University, demonstrated in mice that a change in diet affected the microbiome in just a day. So he and Lawrence David, now a computational biologist at Duke University in Durham, North Carolina, decided to see if diet could have an immediate effect in humans as well. They recruited 10 volunteers to eat only what the researchers provided for 5 days. Half ate only animal products—bacon and eggs for breakfast; spareribs and brisket for lunch; salami and a selection of cheeses for dinner, with pork rinds and string cheese as snacks. The other half consumed a high-fiber, plants-only diet with grains, beans, fruits, and vegetables. For the several days prior to and after the experiment, the volunteers recorded what they ate so the researchers could assess how food intake differed.

The scientists isolated DNA and other molecules, as well as bacteria, from stool samples from before, during, and after the experiment. In this way, they could determine which bacterial species were present in the gut and what they were producing. The researchers also looked at gene activity in the microbes.

Within each diet group, differences between the microbiomes of the volunteers began to disappear. The types of bacteria in the guts didn't change very much, but the abundance of those different types did, particularly in the meat-eaters, David, Turnbaugh, and their colleagues report online today in Nature. In 4 days, bacteria known to tolerate high levels of bile acids increased significantly in the meat-eaters. (The body secretes more bile to digest meat.) Gene activity, which reflects how the bacteria were metabolizing the food, also changed quite a bit. In those eating meat, genes involved in breaking down proteins increased their activity, while in those eating plants, other genes that help digest carbohydrates surfaced. "What was really surprising is that the gene [activity] profiles conformed almost exactly to what [is seen] in herbivores and carnivores," David says. This rapid shift even occurred in the long-term vegetarian who switched to meat for the study, he says. "I was really surprised how quickly it happened.”

From an evolutionary perspective, the fact that gut bacteria can help buffer the effects of a rapid change in diet, quickly revving up different metabolic capacities depending on the meal consumed, may have been quite helpful for early humans, David says. But this flexibility also has possible implications for health today.

"This is a very important aspect of a very hot area of science," writes Colin Hill, a microbiologist at University College Cork in Ireland, who was not involved with the work. "Perhaps by adjusting diet, one can shape the microbiome in a way that can promote health," adds Sarkis Mazmanian, a microbiologist at the California Institute of Technology in Pasadena, also unaffiliated with the study.

But how it should be shaped is still up in the air. "We're not yet at a point where we can make sensible dietary recommendations aimed at 'improving' the microbiota (and the host)," Hill writes. He and others are cautious, for example, about the implications of the increase seen in one bacteria, Bilophila wadsworthia, in the meat-eaters that in mice is associated with inflammatory bowel disease and high-fat diets. Says Knight, "There's still a long way to go before causality is established."

So Hill's best advice for now: "People should ideally consume a diverse diet, with adequate nutrients and micronutrients—whether it's derived from animal or plant or a mixed diet."

Agrulian Archives / Electricity & Magnetism (undergrad student level)
« on: December 11, 2013, 09:37:52 PM »
Brief Subject Overview: the study of electricity, magnetism and --- on realizing their intimate connection --- electromagnetism, is a very old area of classical physics, and one of the most successful, as judged by the limited need for changes to it in light of new evidence. In fact, Maxwell's equations, which (together with Newton's laws) determine the behavior of electromagnetic systems, were a central part of Einstein's inspiration in generating relativistic mechanics, and in my dim understanding essentially needed no revision because of their key role in this development. Anyway, electromagnetism is the study of precisely what it sounds like, primarily on length and speed scales that are within the realm of classical mechanics. An interesting feature, to me, of electromagnetcs is its tireless, creative use of the vector calculus; although gradients always seemed to me to have an obvious interpretation, one often doesn't hear very lucid explanations for the intuition behind vector operations like the curl and divergence. Here we see exactly that, as those operators are fundamental to the models handled.

Text(s): Purcell & Morin's "Electricity and Magnetism".

Assigned problems:

7,29,10,17,25 from Ch 1
26, 14, 12, 28, 7, 1, 17, 16 from Ch 2

Replies will contain worked solutions, discussion, etc.

Agrulian Archives / Statistical Mechanics (undergrad student level)
« on: December 11, 2013, 09:31:46 PM »
Brief Subject Overview: statistical mechanics is the branch of physics that deals with many-particle systems, and with the properties of matter and physical experience that arise from the statistical, average properties of many individual particles interacting. In a sense statistical mechanics is not a "true" foundational area of physics, in that it is in principle derivable entirely from the laws of classical, quantum, electromagnetic, and relativistic mechanics. Of course, in practice nobody can solve the systems of equations that would result from attempting such a reduction, and so in practice statistical mechanics stands on its own as the study of such properties as pressure, heat, phase transition in matter, etc., that arise fundamentally from the interactions of many particles, typically on the order of Avogrado's constant, about 623.

Text(s): Mandl's "Statistical Physics", Kittel & Kroemer's "Thermal Physics", Penrose's "Foundations of Stat Mech". Not really sure which of these books I like best so far; Mandl seems too terse, and assumes a lot of prior physics background. Currently focusing on K&K; jury's still out on how that'll work out. Penrose from what I recall is clear and satisfyingly deductive but assumes familiarity with the Hamiltonian formulation of classical mechanics, which I don't have yet; may return to his book once I work through that chapter in Class Mech.

Assigned problems:

2,5 from Ch 1 (Levin)
TBA (Penrose)

Replies will contain worked solutions, discussion, etc.

Agrulian Archives / Quantum Mechanics (undergrad student level)
« on: December 11, 2013, 09:24:55 PM »
Brief Subject Overview: quantum mechanics is the physics of the very small, on the order of Planck's constant, or about 6.62 x 10-34 (squared meter)-kilograms per second. When this number doesn't seem negligibly small in comparison to the lengths of the things with which you're working, then quantum effects will probably start to become important. Quantum physics contains a number of major deviations from classical physics; most texts seem to begin by explaining particle-wave duality, particularly as applied to light, and motivated by a sequence of experiments with a pedigree of several hundred years, maybe the most iconic of which is the two-slit experiment.

Text(s): Levin's "An Introduction to Quantum Theory" & Griffith's "An Intro to Quantum Mech". Was originally working out of Levin, but later read Griffiths and find him, at least thus far, both clearer and more concise, so using Griffiths as the primary text now.

Assigned problems:

7,8,13,15,19 from quantum Ch1 (Levin)
TBA (Griffiths)

Replies will contain worked solutions, discussion, etc.

Agrulian Archives / Classical Mechanics (undergrad student level)
« on: December 11, 2013, 09:17:42 PM »
Brief Subject Overview: classical mechanics is, in my understanding, the foundation upon which most of a typical physics education is built. It is Newton's physics of motion, or Lagrange's, or Hamilton's --- all equivalent formulations of the same set of physical laws. They describe motion (position, velocity, acceleration, etc) with great fidelity for nearly all sorts of physical things over a very broad range of masses, sizes, and speeds. For the very small and the very quickly moving classical mechanics breaks down and gives way to quantum mechanics and (special/general) relativistic mechanics, but for most everyday ranges it works as an exceedingly successful approximation. Moreover, these other subject areas tend to explain themselves by contrast with and through examples familiar from classical mechanics; so, classical mechanics is important for understanding physics generally. This applies equally well to other areas of physics: for example, Maxwell's laws and the theory of electromagnetism describe the behavior of electromagnetic forces, but nevertheless obey Newton's three laws (in the realm of classical approximation), and one could hardly make practical use of Coloumb's Law, which gives the force exerted by one electric non-moving charge on another, without knowing, for example, that F=ma, from classical mechanics.

Text(s): Taylor's "Classical Mechanics".

Assigned problems: (NOTE -- odds only bc Taylor only has odd solutions)

10,24,43,44 from ClassMech Ch1
1,9,19,25,39,49,50,53 from Ch2
5,7,21 from Ch3
27,31,45 from Ch4
11,41,45 from Ch5

Replies will contain worked solutions, discussion, etc.

Agrulian Archives / Vibrations and Waves (high school level)
« on: December 11, 2013, 09:05:43 PM »
Brief Subject Overview: vibrations and waves is about the physics, and to some extent the mathematics, of waves that move in various media---light waves, sound waves, etc.

Text(s): King's "Vibrations and Waves".

Assigned problems:

8,9,10 from Ch 1

Replies will contain worked solutions, discussion, etc.

Agrulian Archives / i dont know how to sticky threads (fffffuuuuuu level)
« on: December 11, 2013, 08:54:53 PM »

Agrulian Archives / Copulas (grad student level)
« on: December 11, 2013, 04:55:44 PM »
Brief Subject Overview: in probability theory, multivariate probability distributions or density functions describe the probability that vectors of, say, N variables, will occur together; i.e., the probabiltiy that variable 1 will have value x at the same time that variable 2 will have value y at the same time that variable 3 will have value z, and so on. In general knowledge of the value of one variable may tell us something about the likely values of another variable; that is, the probability distribution may contain a dependence between its variables. Copulas are a standard way of expressing and analyzing this dependence structure. They have become widely used in, for example, quantitative finance, as a result of which the Gaussian copula (which is just a single, widely used kind of copula) and its limitations became the focus of at least one scathing article in the wake of the financial crisis.

Text(s): Nelsen's "An Introduction to Copulas".

Assigned problems:

Text contains no problems! Will assign theorems/propositions/examples to be worked out in detail instead.

Replies will contain worked solutions, discussion, etc.

Brief Subject Overview: differential topology and the theory of (differentiable) manifolds is mostly concerned with studying topological or differentiable 'manifolds.' Topological manifolds are structures that look locally like Euclidean N-space in the sense that, for each point of the manifold and some neighborhood around that point, there is a continuous bijection with continuous inverse (a homeomorphism) between that neighborhood and some open ball in Euclidean N-space. Differentiable manifolds are topological manifolds with an extra property: if a homeomorphism f and the inverse of another homeomorphism g, as defined above, happen to have overlapping domains (as they often will in practice), then taking f(g^{-1}()) gives us a map from R^N -> R^N and we can impose the condition that this map be differentiable (one, two, etc times, as desired). This kind of differentiability condition on a manifold gives it a great deal of extra structure, and essentially lets us "do calculus" in any space that "looks smooth like R^N" if you get close enough to it.


Introduction to Topological Manifolds by Lee (Lee1).
Introduction to Topological Manifolds by Schutz.
Introduction to Smooth Manifolds by Lee (Lee2).

Assigned problems:

4,14,19 from Lee1, Ch1
Ex. 2.1 from Schutz

Appendices (topol-alg-calc) proofs of statements/theorems/exercises, p. 540-596, Lee2
Ex. 1.1-1.6 from Lee2, (middle of) Ch 1
1,3,4,5,9 from Lee2, (end of) Ch1

Replies will contain worked solutions, discussion, etc.

Agrulian Archives / Measure Theory (grad student level)
« on: December 11, 2013, 04:38:54 PM »
Brief Subject Overview: meaure theory studies, as the name suggests, the construction and properties of "measures." Measures are a generalization of probability distributions, and measure theory helps to place probability theory on a clear basis theoretically (it was rather informal prior to the theory of measures, I believe). Measure theory also helps us to define more general notions of integration than the basic Riemann integration; by partitioning the range rather than the domain of a function before performing the finite summations that form the basis for our infinite summation (which is all integration is), we become capable of integrating over even very badly behaved functions like the Dirichlet function, which has the value 1 on all rational numbers and 0 on all irrational numbers, and so is ugly as shit because it oscillates very wildly between 1 and 0---doing so infinitely often within any arbtirarily small intervl, and in fact uncountably often there---and is  effectively impossible to visualize or graph properly. Measure theory is also a pretty rich source of examples with interesting and theoretically useful behavior, such as the Cantor function (which itself forms a key illustration in the theory of fractals) or the "non-measurable" sets that we know must exist, such as Vitali's sets. As an aside, my adviser---who is a bayesian statistician by training---has emphasized to me that he thinks measure theory is mostly used by probabilists to make their work more intimidating and impenetrable, and not because it actually helps them get anything extra done. I don't really know enough to weigh in one way or the other on that, but I think it's certainly true, for the time being and on net, of some areas that I've studied.

Text(s): Adams & Guillemin's "Measure Theory & Probability".

Assigned problems:

Sec. 1.1 # 16, 8, 11, 12, 19
Sec. 1.2 # 5, 8, 9, 13
Sec. 1.3 # 1, 5, 10, 20
Sec. 1.4 # 4, 6, 8, 14, 16
Sec 2.1 # 1, 6, 9, 10
Sec 2.2 # 2, 3, 6, 7, 12
Sec 2.3 # 1, 2, 12, 13, 14
Sec 2.4 # 1, 2, 3, 6
Sec 2.5 # 3, 7, 9, 11, 12
Sec 2.6 # 7, 8, 9, 10
Sec 2.7 # 2, 6, 7
Sec 2.8 # 1, 2, 3
Sec 3.1 # 1, 5, 7, 8
Sec 3.2 # 3, 6, 8
Sec 3.3 # 1, 4, 5, 8, 11
Sec 3.4 # 3, 5, 6, 7, 9
Sec 3.5 # 3, 5, 9, 12, 14
Sec 3.6 # 2, 6, 8, 9
Sec 3.7 # 3, 5, 6, 8, 9
Sec 3.8 # (write out proof of CLT in detail)

Replies will contain worked solutions, discussion, etc.

Brief Subject Overview: computational complexity theory is the studied of computational problems, particularly those expressible in the binary language of digital computers. Complexity theory is concerned primarily with classifying tasks according to their difficulty (and, of course, solving those that can be solved); to achieve this, it assigns problems to complexity classes, such as the famous classes P and NP. There are many more classes as well: PLS, co-NP, PPAD, FIXP, etc. Defining new classes is often the first step in a complexity theorist's attempt to understand how hard a problem is to solve. Generally, problems in P are efficiently solvable; problems in NP are not. There also exist easier problem classes than P, and harder ones than NP; the most extreme example of the latter is the class of "intractable" problems, which cannot be solved by any computer, even in principle, with an arbitrarily large amount of time and space in which to compute, etc. Recently complexity theory has also been applied to problems outside of standard computer science, such as the complexity of finding solution concepts in game theory and economics, with the basic idea being that problems without efficient solutions are not good solution concepts, since nobody could be expected to find them in practice. A curious feature of computational complexity theory is that it is the source of many unsolved conjectures which are nevertheless strongly believed to be true; i.e., nobody has shown but most theorists strongly believe that P does not equal NP, and likewise that NP does not equal co-NP.

Text(s): Rich's "Automata, Computability, and Complexity".

Assigned problems:


Replies will contain worked solutions, discussion, etc.

Agrulian Archives / Chaos Theory (undergrad student level)
« on: December 11, 2013, 04:15:39 PM »
Brief Subject Overview: 'chaos' is a phenomenon found in discrete-time and continuous-time, deterministic and stochastic dynamical systems, i.e., systems that update their state from time period to time period according to some predetermined rule (with the rule defining a probability of moving from each state to another in stochastic systems), either in time periods 1,2,3,4... or in time periods indexed by all time points in an interval, such as [0,∞). Intuitively, chaos is a situation in which the deterministic behavior of the system leads to seemingly random, unpredictable behavior in the large; formally, chaos has a number of different (and not all equivalent) definitions. Most of these definitions have as their centerpiece some kind of "topological mixing;" i.e., chaos requires that all solutions starting in some open set eventually end up in any other open set, given enough time. Another common, better known condition is that solutions starting arbitrarily close together should separate from one another exponentially fast in time (up to some limit, at least, if the state space itself is bounded in size). Chaos theory is about defining and proving the existence of chaos in formal systems, understanding its determinants and behavior, undertanding what limitations chaos does or doesn't imply for predictability, and identifying chaos in practice in real-world systems. It also concerns the formulation of various definitions of chaos, and studying whether and when they are or aren't equivalent. The text I use is on the low end of technical difficulty in this subject area, primarily because it deals with discrete-time systems; this makes it accessible and easy to get into, which is nice; also, the author provides exercises to work, which most authors on continuous-time chaos seem not to do for some reason.

Text(s): Elaydi's "Discrete Chaos".

Assigned problems:

3,6,11 from Chaos 1.1-1.3
1,2,4 from 1.4-1.5
4,12,14 from 1.6
4,8,13 from 1.7
5,6,16 from 1.8
1,2 from 1.9
2,3,15 from 2.1-2.2
3,6,15 from 2.3-2.4
5,6,7 from 2.5
1,2,3 from 2.6
1,15,18 from 3.1-3.2
3,5 from 3.3
10,14 from 3.4
1,3,11 from 3.5
6,7,10 from 3.6
1,14,15 from 3.7
1,2,6 from 4.1-4.2
1,3,4 from 4.3-4.4
8,10,12 from 4.5-4.7
8,13,15 from 4.8
2,13,14 from 4.9-4.10
2,6,7 from 4.11
1,4,6 from 5.1
2,9,13 from 5.2
5,8,9 from 5.3
1,5,9 from 5.4-5.5
10,15,16 from 6.1-6.2
3,7,15 from 6.3
1,5,10 from 6.4
8,11 from 7.1-7.2
4,7,9 from7.3-7.4
8,9 from 7.5
4,9 from 7.6

Replies will contain worked solutions, discussion, etc.

Agrulian Archives / Topology (grad student level)
« on: December 11, 2013, 03:40:18 PM »
Brief Subject Overview: topology is the study of open sets, independent of any concrete notion of distance; its starting point is in identifying some properties of open sets in familiar settings like real/Euclidean N-space and defining open sets in more general spaces as any collection of sets that have those properties. Topology allows us to talk about, for example, continuity of functions or the number of holes in very abstract settings and spaces, where a familiar notion of distance may not be available, and artificially imposing one may be awkward. In short topology helps us to identify properties that depend only on the structure imposed on a space by its open sets, so that we do not need to worry about unimportant, extraneous detail in describing these properties.

Text(s): Munkres "Topology".

Assigned problems:

1,3,7 p. 83
4,5,7,9 p. 91-92
5,7,11,12,18 p. 100-102
1,3,8,11,12 p. 111-112
2,4,8,10 p. 118
6,7,9,10,11 p. 126-129
2,4,6,9,12 p. 133-136
1,3,5,6 p. 144-145
1,3,4,7 p. 145-146 (supp: topol groups)

2,4,5,11 p. 152
5,8,10,12 p. 157-159
1,6,7,8,9 p. 162-163
7,9,10,11,13 p. 170-172
1,3,4,6 p. 177-178
3,4,5,7 p. 181-182
4,7,8,11 p. 186
1,4,5,10 p. 187-188 (supp: nets)

1,3,11,13,15 p. 194-195
1,5,7 p. 199-200
4,5,6,10 p. 205-207
1,2,3,7,10 p. 212-214
1,3,4,6 p. 218
1,3,5,7,8 p. 223-224
1,3,4 p. 227
1,3,4,7 p. 228-229 (supp: basics review)

1,2,4 p. 235-237
3,4,5,9 p. 241-242

2,5,6 p. 248
1,4,8 p. 260-261
2 p. 262

2,3,7 p. 270-271
1,2,3 p. 274-275
3,4,5,7,8 p. 280-281
2,4,6,9 p. 288-290
2,3 p. 292-293

5,9,10,11,12 p. 298-300
1,2 p. 304
1,3,7 p. 315-316
3,7,8 p. 316-318

Ch9~ (Algebraic Topology)
1,2,3 p. 330
1,2,6 p. 334-335
3,4,5 p. 341
1,4,7,8 p. 347-348
1,2,4 p. 353
1,2, p. 356
1,2,4 p. 359
1,4,5,9 p. 366-367
2,3,4 p. 370
3,4,5 p. 375

1,2 p. 380-381
2,3,4,5,6 p. 384-385
1,2,3 p. 393-394
1 p. 398
1,2 p. 406

1,2,5,6 p. 411-412
2,3,4 p. 421
1,3,4 p. 425
1,2,3 p. 433
1,2,4,5 p. 438
1,2,3 p. 441
1,3,4 p. 445

2,3,4,5 p. 453-454
1,2,4 p. 457
1,2 p. 462
1,2,4 p. 470-471
2,4,5 p. 476

3,5,6,7 p. 483-484
1 p. 487
1,2,6 p. 492-494
2 p. 499
1,3 p. 499-500 (supp: topol props and pi_1)

1,2 p. 505-506
2,3 p. 513
1,2,3 p. 515

Replies will contain worked solutions, discussion, etc.

Agrulian Archives / 'Light' Textbook Walkthroughs of Tough Subjects
« on: December 11, 2013, 03:21:11 PM »
It is often the case that, for a tough, formal subject, textbooks come in at least two flavors: the dense, encyclopedic, painful, rewarding-if-engaged kind, and a lighter kind, with a style dealing in more prose and somewhat less math, and particularly less of a rigid theorem-lemma-proof format. I often find it is helpful to have both kinds of books, since the dense ones are tough to simply read and it can be easy to get lost in their swamp of details if you are not tremendously on top of your game; while the dense tomes are the only path to full understanding of a subject, the lighter froo-froo texts can be awesome for getting a bird's-eye view of the landscape and a quick, often exceedingly useful intuition about a discipline and its tools. Of course these categories are somewhat fuzzy, but I think it is usually clear enough where a text falls.

In short the emphasis in the 'light/Type 2' books is on developing a key nugget of intuition and some appreciation for the most important or main results of a subject, while in the former 'dense/Type 1' books it is on rigorous, exhaustive understanding. Please note that this thread is not for 'popular' books on a subject, which might be called books of Type 3; Type 2 books are still textbooks, and still engage with the material in a rigorous, formal way, albeit less so than Type 1 books. With those descriptions & caveats in mind, this thread is devoted to the latter kind of textbook, in whatever subject, because they can be somewhat hard to find.

Here's a list of 'light/Type 2' texts, all much more accessible than the standard in their literature:

Abstract Algebra :
A Book of Abstract Algebra by Pinter

Differentiable Topology/Manifolds :
Geometrical Methods of Mathematical Physics by Schutz
Differential Topology with a View to Applications* by Chillingworth

Stochastic Calculus :
An Introduction to the Mathematics of Financial Derivatives

Vector Calculus :
Div, Grad, Curl, and All That (I think this fits; haven't read it, AD may correct me)

Quantum Physics :
Understanding Quantum Physics by Morrison and Its Sequel

Number Theory :
Excursions into Number Theory by Ogilvy

Logic :
Godel's Proof by Newman & Nagel

Computational Complexity:
Computers & Intractability* by Garey & Johnson

Derivatives Theory:
Derivatives by Wilmott (note: broader coverage than typical of Type 2, but same style)

Linear Algebra:
Introduction to Linear Algebra by Strang

Might follow-up later with summaries/comments, not sure. Lemme know if you have anything you'd like to add to this list.

* denotes a book that is also widely cited in the literature. Always find it weird when a 'classic' just so happens to be eminently accessible & readable too.

Agrulian Archives / Set Valued Analysis (grad student level)
« on: December 11, 2013, 03:08:30 PM »
Brief Subject Overview: analysis is the "theory of calculus," and set valued analysis concerns the generalization of this theory from the standard calculus, which focuses on functions mapping elements of a set S to individual real numbers, to a broader setting in which functions map a single element of a set S to multiple elements in some other set T. More general definitions are given in set valued analysis for their corresponding notions in the more usual "real-valued" analysis/calculus, such as for limits (two different notions of limit that are equivalent in real-valued analysis turn out to be non-equivalent in set-valued analysis) of sequences, functions, etc. Set valued analysis is the proper area in which proofs of Kakutani's Fixed Point Theorem emerges, which is used in many settings, and in particular in game theory, to show that a problem has a solution.

Text(s): Aubin & Frankowska's "Set Valued Analysis".

Assigned problems:

Text doesn't contain any problems! Will assign theorems to work through in detail.

Replies will contain worked solutions, discussion, etc.

Agrulian Archives / Functional Analysis (grad student level)
« on: December 11, 2013, 03:04:25 PM »
Brief Subject Overview: functional analysis is concerned with limiting operations---the first example of which anybody who's taken Calculus 1 has met---and their behavior in very general spaces. Functional analysis takes as its starting point the consideration of spaces in which every limit of a sequence of elements in the space converges to another element in the space, so-called "complete spaces." The real numbers and integers are both complete in this sense, for example, but the rationals are not. Problems in functional analysis often deal with *function spaces* and *sequence spaces*, in which the elements of the space are not individual numbers, but functions; this step-up in abstraction can be challenging to come to grips with, but starts to feel familiar as you work through more problems and theorems.

Text(s): Kreyszig's "Introductory Functional Analysis with Applications".

Assigned problems:

7,8 from FuncAnal, Ch 1.1
3,8,11 from 1.2
3,12,14 from 1.3
1,2 from 1.4
1,11,12,15 from 1.5
5,11,13,15 from 1.6
8,12,14,15 from 2.1
6,7,9,13,15 from 2.2
2,4,6,15 from 2.3
1,5,7,8 from 2.4
1,3,5,9 from 2.5
4,8,12,14 from 2.6
1,6,8 from 2.7
2,3,10,14,15 from 2.8
4,5,6,7,8 from 2.9
8,9,10,13,15 from 2.10

Replies will contain worked solutions, discussion, etc.

Agrulian Archives / Fourier Analysis (grad student level)
« on: December 11, 2013, 03:02:56 PM »
Brief Subject Overview: fourier analysis is concerned with finding conditions under which a function f(.) can be decomposed and expressed as an infinite sum of the simplest trigonmetric functions, sin(.) and cos(.).

Text(s): Stein & Shakarchi's "Fourier Analysis".

Assigned problems:

1,9 from Ch1

Replies will contain worked solutions, discussion, etc.

Agrulian Archives / Vector Calculus (undergrad student level)
« on: December 11, 2013, 03:01:29 PM »
Brief Subject Overview: vector calculus is the generalization of the differential (calculus 1) and integral calculus (calc 2) beyond just multiple variables (calculus 3) and on into vector spaces. Some of the most interesting and useful theorems in vector calculus concern the generalization of integrals from "infinite sums over intervals" to "infinite sums over curves;" that is, we can define path integrals, surface integrals, etc., which use integration to sum over a function defined on a curvy path or surface of some sort. The primary theorems of vector calculus generalize those of basic calculus to this broader setting, and often enable us to calculate complex curve/path/surface integrals in terms of simpler, standard integrals more familiar from basic multivariate calculus.

Text(s): Stewart's Calculus. Emphasis on ch. 17.

Assigned problems: TBA

Replies will contain worked solutions, discussion, etc.

Agrulian Archives / Forum Purpose:
« on: December 10, 2013, 11:27:00 PM »
The point of this forum is, interpreted narrowly, to work on problems, meaning the analysis of formal models, whether math problems, physics problems, chemistry problems, finance problems, etc etc. Yours, mine, whoever's. This is the primary point of the forum in my mind, and I'll be using it to organize my self-study (in the obvious thread), sharing/posting problems as I go, and occasionally posting commentaries on/summaries of the books/problems. (Although I might carve the posted problems up into other threads by topic so they're easy to find.) More broadly than problems, the forum's also about studying in general. Whether what you have to do for school or self-initiated, and whether it's formal models you're solving or something else -- learning a language, taking careful notes on a dense text, learning a new programming language, outlining a careful empirical analysis, ranting about your research bullshit, glancing over a research article, & so on. All well within the scope of what I have in mind.

That said I'm not mod'ing shit so post whatever you want or nothin' at all.

Brief video behind the link. What he says verbatim is 'I think you have to connect with women on an emotional level."

As Republican candidates figure out how to best win over women voters, Iowa GOP Senate candidate Mark Jacobs thinks he has the answer: appeal to their emotions.

In an interview Sunday with WHO-TV in Des Moines, host Dave Price asked Jacobs what the "biggest difference between men and women" is, in terms of reaching out to them as voters.

"I think you have to connect with women on an emotional level," said Jacobs. "And with a wife of 25 years and an 18-year-old daughter, I've had a lot of coaching on that."

Last week, Politico reported that the National Republican Congressional Committee and House Speaker John Boehner's (R-Ohio) office are "meeting with top aides of sitting Republicans to teach them what to say -- or not to say -- on the trail, especially when their boss is running against a woman."

"[We're] trying to get them to be a little more sensitive," Boehner said of fellow Republicans at a press conference on Thursday. "You know, you look around the Congress, there are a lot more females in the Democratic caucus than there are in the Republican conference. And some of our members just aren't as sensitive as they ought to be."

Jennifer Lawless, the director of the Women and Politics Institute at American University, told The Huffington Post that like men, women voters want to know their candidates are "competent, can lead and have a sense of empathy and integrity."

"Certainly, there can be gender gaps on issue salience -- women, for example, might be more concerned than men about issues affecting women, families, and children," she said. "But it’s the attention candidates spend on those issues and their ability to demonstrate that they understand challenges women face that matter.

"It’s not about talking to the female electorate as though you are their husband or father," she added. "In fact, doing so plays into damaging stereotypes and reinforces the notion that women need to be treated in a way that is somehow less serious and cerebral."

Neither Jacobs' office nor the Republicans' Senate campaign arm -- the National Republican Senatorial Committee -- returned a request for comment.

Jacobs, a former energy CEO, is one of seven Republicans running for the seat being vacated by Sen. Tom Harkin (D-Iowa). Rep. Bruce Braley (D-Iowa) is running on the Democratic side.

In the 2012 elections, Republican Senate candidates often attracted attention for their inability to reach women's voters, with former Rep. Todd Akin (R-Mo.), then running against Sen. Claire McCaskill (D-Mo.), famously saying he believed women were able to stop themselves from getting pregnant after a "legitimate rape."

Obama won 55 percent of the female vote in 2012.

"GOP Senate Candidate Mark Jacobs, Rep. Boehner and the whole Republican Party know from experience that it is not a good election strategy to demean woman voters -- and yet they seem committed to continuing to do that," said Nita Chaudhary, co-founder of UltraViolet, which advocates for women's rights through online activism. "The way to talk to women is to treat them with respect and understand that we are adults. For some reason, the GOP has a hard time doing that."

"Ultimately, when it comes to winning votes, actions speak louder than words," she added. "So we are eager to hear Mark Jacobs' plan for ensuring women earn equal pay for equal work and guaranteeing women have the ability to control their own health care decisions.”

General Discussion / We Say We Like Creativity, But Really We Don't
« on: December 09, 2013, 09:40:45 PM »
In the United States we are raised to appreciate the accomplishments of inventors and thinkers—creative people whose ideas have transformed our world. We celebrate the famously imaginative, the greatest artists and innovators from Van Gogh to Steve Jobs. Viewing the world creatively is supposed to be an asset, even a virtue. Online job boards burst with ads recruiting “idea people” and “out of the box” thinkers. We are taught that our own creativity will be celebrated as well, and that if we have good ideas, we will succeed.

It’s all a lie. This is the thing about creativity that is rarely acknowledged: Most people don’t actually like it. Studies confirm what many creative people have suspected all along: People are biased against creative thinking, despite all of their insistence otherwise.

“We think of creative people in a heroic manner, and we celebrate them, but the thing we celebrate is the after-effect,” says Barry Staw, a researcher at the University of California–Berkeley business school who specializes in creativity.

Staw says most people are risk-averse.
He refers to them as satisfiers. “As much as we celebrate independence in Western cultures, there is an awful lot of pressure to conform,” he says. Satisfiers avoid stirring things up, even if it means forsaking the truth or rejecting a good idea. 

Even people who say they are looking for creativity react negatively to creative ideas, as demonstrated in a 2011 study from the University of Pennsylvania. Uncertainty is an inherent part of new ideas, and it’s also something that most people would do almost anything to avoid. People’s partiality toward certainty biases them against creative ideas and can interfere with their ability to even recognize creative ideas.

A close friend of mine works for a tech startup. She is an intensely creative and intelligent person who falls on the risk-taker side of the spectrum. Though her company initially hired her for her problem-solving skills, she is regularly unable to fix actual problems because nobody will listen to her ideas. “I even say, ‘I’ll do the work. Just give me the go ahead and I’ll do it myself,’ ” she says. “But they won’t, and so the system stays less efficient.”

In the documentary The September Issue, Anna Wintour systematically rejects the ideas of her creative director Grace Coddington, seemingly with no reason aside from asserting her power.

Social rejection is not actually bad for the creative process—and can even facilitate it.
This is a common and often infuriating experience for a creative person. Even in supposedly creative environments, in the creative departments of advertising agencies and editorial meetings at magazines, I've watched people with the most interesting—the most “out of the box”—ideas be ignored or ridiculed in favor of those who repeat an established solution.

“Everybody hates it when something’s really great,” says essayist and art critic Dave Hickey. He is famous for his scathing critiques against the art world, particularly against art education, which he believes institutionalizes mediocrity through its systematic rejection of good ideas. Art is going through what Hickey calls a “stupid phase.”

In fact, everyone I spoke with agreed on one thing—unexceptional ideas are far more likely to be accepted than wonderful ones.

Staw was asked to contribute to a 1995 book about creativity in the corporate world. Fed up with the hypocrisy he saw, he called his chapter “Why No One Really Wants Creativity.” The piece was an indictment of the way our culture deals with new ideas and creative people”


In terms of decision style, most people fall short of the creative ideal … unless they are held accountable for their decision-making strategies, they tend to find the easy way out—either by not engaging in very careful thinking or by modeling the choices on the preferences of those who will be evaluating them.
Unfortunately, the place where our first creative ideas go to die is the place that should be most open to them—school. Studies show that teachers overwhelmingly discriminate against creative students, favoring their satisfier classmates who more readily follow directions and do what they’re told.

Even if children are lucky enough to have a teacher receptive to their ideas, standardized testing and other programs like No Child Left Behind and Race to the Top (a program whose very designation is opposed to nonlinear creative thinking) make sure children’s minds are not on the “wrong” path, even though adults’ accomplishments are linked far more strongly to their creativity than their IQ. It’s ironic that even as children are taught the accomplishments of the world’s most innovative minds, their own creativity is being squelched.

All of this negativity isn’t easy to digest, and social rejection can be painful in some of the same ways physical pain hurts. But there is a glimmer of hope in all of this rejection. A Cornell study makes the case that social rejection is not actually bad for the creative process—and can even facilitate it. The study shows that if you have the sneaking suspicion you might not belong, the act of being rejected confirms your interpretation. The effect can liberate creative people from the need to fit in and allow them to pursue their interests.

Perhaps for some people, the pain of rejection is like the pain of training for a marathon—training the mind for endurance. Research shows you’ll need it. Truly creative ideas take a very long time to be accepted. The better the idea, the longer it might take. Even the work of Nobel Prize winners was commonly rejected by their peers for an extended period of time.

Most people agree that what distinguishes those who become famously creative is their resilience. While creativity at times is very rewarding, it is not about happiness. Staw says a successful creative person is someone “who can survive conformity pressures and be impervious to social pressure.”

To live creatively is a choice. You must make a commitment to your own mind and the possibility that you will not be accepted. You have to let go of satisfying people, often even yourself.

Spamalot / DESTROYED my abstract algebra final today
« on: December 09, 2013, 09:04:47 PM »
unique factorization domains can suck my dick


Click link for video. TL;DR summary is kind of obvious.

Imaginations everywhere have been stoked since Amazon CEO Jeff Bezos announced his company plans to start offering 30-minute deliveries via drone-like "octocopters."

What's not fascinating about a near future in which fleets of whirring sky robots can drop our every impulse buy on our doorstep faster than we can get Chinese delivered? (You know, aside from accidental strayings into restricted air space or the rise of the machines.)

But when Bezos took to "60 Minutes" on Sunday to introduce the world to Amazon Prime Air, his idea prompted more questions than it provided answers.

So how close are we, really, to door-to-door drones becoming a reality? And how would they work?

We reached out to Amazon, where official details are still scarce, and chatted with drone expert Missy Cummings, an associate professor at MIT and one of the Navy's first female fighter pilots. Here's some of what we've been able to piece together on a project that Amazon says is, at the very least, a couple of years away from takeoff.

Could drones really be delivering packages by 2015?

That's what Bezos said is the best possible scenario. But Cummings, a longtime advocate for the commercial use of drones, thinks that's optimistic.

The Federal Aviation Administration needs to sign off on Amazon's flight plans, and Cummings says the agency hasn't been quick to move on the domestic use of drones.

"I think they (Amazon) are stepping out in a typically naive way, (but) maybe they have some secret insight to the FAA that I don't have," she said.

Cummings predicts the company will get approval to start Prime Air in other countries before the United States, but she says that having a retail and technology giant like Amazon pushing for it could speed things up for everyone.

"I don't want anybody to think this is right around the corner," Bezos warned during the "60 Minutes" interview.

How will I know if I'm eligible for a drone visit?

Bezos said the octocopters will have a 10-mile radius. So, it's likely that folks in big cities near Amazon distribution sites would be a lot more likely to qualify than those in more remote areas.

He says they'll initially carry items up to five pounds, which is roughly 86% of all deliveries Amazon makes.

The best Twitter jokes about Amazon's drones

But for even that 10-mile range to work, Amazon better be onto something about battery life that the rest of us don't know. Cummings said drones the size of the octocopters have a battery life of about 30 minutes, and the weight of their cargo could make that even shorter.

What will keep people from shooting them down?

OK, it's perhaps a little off-topic. But every single conversation we've had about the Amazon drones has, at some point, ended up focused on the innate human desire to knock stuff out of the sky, preferably with a loud bang.

Cummings joked about producing a reality show in which marksmen from different states compete to see how many octocopter targets they can bag. At least, we're pretty sure it was a joke.

Perhaps not surprisingly, Amazon doesn't directly address its drones becoming high-tech clay pigeons in a statement about safety.

"The FAA is actively working on rules and an approach for unmanned aerial vehicles that will prioritize public safety. Safety will be our top priority, and our vehicles will be built with multiple redundancies and designed to commercial aviation standards," the statement reads.

But Cummings says it's a real issue.

"It's not just people who hate drones," she said. "It's people who want those packages."

She speculated the drones will need to fly at an altitude of at least 300 feet for as long as possible to avoid attracting pot shots from target shooters or thieves. She also envisions safe "drop spots," at least at first, instead of delivery to any address within range.

"There are lots of details that need to be worked out, but nothing that is technologically overwhelming," she said.

Will the drones work when the weather is bad?

Amazon's official statement doesn't address this obvious question. But Cummings says that to make the drones reliable in most weather conditions, Amazon would need to improve on currently available technology.

"They can fly in some precipitation, but certainly not heavy precipitation," she said. "Sleet or snow ... would obscure some of the sensors. It's hard to make it a really solid business if the weather holds you back. They're going to have to work on that."

What could come next?

Amazon isn't the only company at least toying with the idea of using unmanned aerial vehicles for commercial purposes. Domino's posted video of the "DomiCopter" delivering two pizzas in the United Kingdom earlier this year. In June, the Burrito Bomber, the creation of a couple of engineers from Yelp, demoed its ability to fly that tasty treat to your doorstep as well.

And in Australia, Zookal, a textbook company, is already using drones for deliveries.
Cummings hopes that's all just the beginning. Using drones for beneficial civic or commercial purposes, instead of military actions, is a growing trend.

"Medical supplies, wildlife monitoring, cargo, firefighting -- it's a pretty long list of things that drones can do," she said. "It's reinvigorating a dying aerospace industry."

Trading, and to some extent investing, is all about knowing when markets are moving with the wisdom of the crowds and when they're moving with the madness of the crowds. In recent years, there has seemed to be much more madness than wisdom (a statement which can probably be generalized beyond the financial markets themselves, come to think of it). Where do we stand now?

I think a recent letter by John Hussman of Hussman Strategic Advisors, entitled "An Open Letter to the FOMC: Recognizing the Valuation Bubble In Equities," is worth reading. Hussman is far from the only person, nor even the most influential investor, questioning the valuation of equities at the moment.
Our own valuation models have had the projected 10-year compounded real return of equities below 3% for several years, and below 2% since late April. For a time, that may have been sustainable because of the overall low level of real rates, but since the summertime rates selloff the expected equity premium has been below 1.5% per annum, compounded - and is now below 1% (see chart, source Enduring Investments).

Hussman shows a number of other ways of looking at the data, all of which suggest that equity prices are unsustainable in the long run. But what really caught my eye was the section "Textbook speculative features," where he cites none other than Didier Sornette. Sornette wrote a terrific book called Why Stock Markets Crash: Critical Events in Complex Financial Systems, in which he argues that markets at increased risk of failure demonstrate certain regular characteristics. There is now a considerable literature on non-linear dynamics in complex systems, including Ubiquity: Why Catastrophes Happen by Mark Buchanan and Paul Ormerod's Why Most Things Fail: Evolution, Extinction and Economics. But Sornette's book is one of the better balances between accessibility to the non-mathematician and utility to the financial practitioner. But Hussman is the first investor I've seen to publicly apply Sornette's method to imply a point of singularity to markets in real time. While the time of "breakage" of the markets cannot be assessed with any more, and probably less, confidence than one can predict a precise time that a certain material will break under load - and Hussman, it should be noted, "emphatically" does not lay out an explicit time path for prices - his assessment puts Sornette dates between mid-December and January.

Hussman, like me, is clearly of the belief that we are well beyond the wisdom of crowds, into the madness thereof.

One might reasonably ask "what could cause such a crash to happen?" My pat response is that I don't know what will trigger such a crash, but the cause would be the extremely high valuations. The trigger and the cause are separate discussions. I can imagine a number of possibilities, including something as innocuous as a bad "catch-up" CPI print or two that produces a resurgence of taper talk or an ill-considered remark from Janet Yellen. But speculating on a specific trigger event is madness in itself. Again, the cause is valuations that imply poor equity returns over the long term; of the many paths that lead to poor long-term returns, some include really bad short-term returns and then moderate or even good returns thereafter.

I find this thought process of Hussman's interesting because it seems consonant with another notion: that the effectiveness of QE might be approaching zero asymptotically as well. That is, if each increment of QE is producing smaller and smaller improvements in the variables of interest (depending who you are, that might mean equity prices, long-term interest rates, bank lending, unemployment, etc), then at some point the ability of QE to sustain highly speculative valuations goes away and we're left with the coyote-running-over-the-cliff scenario. Some Fed officials have been expressing opinions about the declining efficacy of QE, and Janet Yellen comes to office on February 2nd. I suspect the market is likely to test her very early.

None of this means that stocks cannot go straight up from here for much longer. There's absolutely nothing to keep stock prices from doubling or tripling from here, except the rationality of investors. And as Mackay said, "Men, it has been well said, think in herds; it will be seen that they go mad in herds, while they only recover their senses slowly, and one by one." Guessing at the date on which the crowd will toggle back from "madness" to "wisdom" is inherently difficult. What is interesting about the Sornette work, via Hussman, is that it circles a high-risk period on the calendar.

For two days in a row now, I've discussed other people's views. On Wednesday or Thursday, I'll share my own thoughts - about the possible effects of Obamacare on measured medical care inflation.

Another article on the same topic:

Is it possible to detect a financial bubble and predict when it will burst? Dr. Didier Sornette, a former physicist who is the director of the Financial Crisis Observatory in Switzerland, developed a statistical model designed to do just that. Sornette and his colleague Anders Johansen determined in 2004 that in two thirds of the cases where financial assets suffered extremely large drawdowns, market prices followed a "super-exponential" behavior prior to their occurances. According to mutual fund manager and former finance professor Dr. John Hussman, the Sornette model is now predicting a stock market crash as early as next year.

TL;DR: I've posted about Didier Sornette's work here before; he's developed a particular, non-linear/stochastic dynamical model of markets that makes use of common features of macronomic theory (eg rational expectations) but that predicts that bubbles have meaningful leading indicators and can be forecasted ahead of time. He made the news a year or two? ago for making some high-profile predictions about bubbles in particular markets and sealing them away ahead of time on arXiv somewhere. Recently an equities trader has apparently tried to use Sornette's models, amongst other things, to argue that there's a bubble in equities. I like to keep up w/ Sornette and other complex systems economics work so fuck you now it's on TZT.

MSNBC host Martin Bashir has resigned from the network following controversial comments about former Alaska Gov. Sarah Palin (R), Mediate reported Wednesday:

"After making an on-air apology, I asked for permission to take some additional time out around the Thanksgiving holiday.

Upon further reflection, and after meeting with the President of MSNBC, I have tendered my resignation. It is my sincere hope that all of my colleagues, at this special network, will be allowed to focus on the issues that matter without the distraction of myself or my ill-judged comments.

I deeply regret what was said, will endeavor to work hard at making constructive contributions in the future and will always have a deep appreciation for our viewers – who are the smartest, most compassionate and discerning of all television audiences. I would also wish to express deepest gratitude to my immediate colleagues, and our contributors, all of whom have given so much of themselves to our broadcast.’"

During a segment discussing Palin's comparison of the national debt to slavery last month, Bashir suggested the governor be subjected to certain disciplinary tactics used by a slave owner.

"In 1756, he records that a slave named Darby 'catched eating kanes had him well flogged and pickled, then made Hector, another slave, s-h-i-t in his mouth,'" Bashir said. "When Mrs. Palin invokes slavery, she doesn’t just prove her rank ignorance. She confirms if anyone truly qualified for a dose of discipline from Thomas Thistlewood, she would be the outstanding candidate."

Bashir's outing follows the cancellation of fellow host Alec Baldwin's show "Up Late" after the actor received criticism for using a gay slur.

General Discussion / Could aggregate fiscal decisions ever be delegated?
« on: December 04, 2013, 02:53:53 AM »
The political battle over delegating decisions over monetary policy to central banks has been fought and won. There may be serious concerns about accountability in some countries, and mandates in others, but there seems to be a political consensus in most places that delegation in this respect is a good thing. (I know some readers disagree with this consensus, but this post is a question about what could happen, rather than what ought to happen.)

There is no major country which delegates decisions over aggregate fiscal policy. I stress aggregate here: I’m not suggesting decisions about particular tax rates or types of spending could be delegated. Instead an independent fiscal institution could set a target level for the budget deficit, and leave it up to the government how that target was achieved. Furthermore the choice between meeting the deficit target using tax changes or spending changes would remain with politicians, so key questions about the size of the state would stay under democratic control.

I’m reminded of this question not by the impending UK autumn statement, but because I have just received my copy of a new collection of essays edited by George Kopits. Its title is “Restoring Public Debt Sustainability: The Role of Independent Fiscal Institutions”. The story behind the book is interesting in itself. Its basis is a conference in Budapest organised by the former Hungarian Fiscal Council. Although a few fiscal councils [1] existed a decade ago, in the last ten years many more have been established, and that included one in Hungary that George chaired. All such councils are advisory - none can tell governments what to do. The meeting in Budapest was I believe the first international gathering of these councils, as well as a few academics that had a particular interest in these institutions. (It is what led me to create this website.)

The conference was a prelude to both success and failure. The failure was that soon after the conference the Hungarian Fiscal Council was effectively abolished by a new government. For that government this act was a good indication of things to come, as others have documented. The brief story of Hungary’s Fiscal Council is told in one of the chapters of this book. However, the success is that, with George’s help, the OECD took on the task of holding regular gatherings of fiscal councils, and it has issued a statement of principles which are an appendix to the book’s introduction.

A few of the essays in the book touch on the question I posed at the beginning of this post, including my own, which compares the delegation of monetary and fiscal policy. In a sense the demise of Hungary’s fiscal council explains why most of the discussion at the conference was happy to see such councils as advisory only. Giving governments advice they may well not want to hear is difficult and dangerous enough, and so fiscal councils need to be well established (and therefore less vulnerable) before we can think of going any further. One step at a time.

Yet once these councils have been established, it becomes easier to imagine the possibility that delegation could go beyond advice to actual control. Take the UK case for example. The government sets its fiscal mandate (cyclically adjusted current balance in 5 years time), just as it does the inflation target. The OBR then tells the government what it needs to do to meet that mandate. So, having set the mandate, the amount of aggregate discretion left to the government in each budget is limited. It would seem quite a small step to let the OBR decide how quickly the mandate should be achieved. Another small step would be for the government and OBR to negotiate over the mandate itself (just as the central bank and government negotiate over the inflation target in New Zealand).

Small steps, but much too large in political terms right now, as I once discovered when giving evidence to the Treasury Select Committee. (See the second footnote to this post.) Yet in ten or so year’s time, when more of these councils are well established, I can see things might be quite different for two reasons. First, when the recession is finally over there will be a clear consensus that a slow (and state contingent) reduction in net debt levels is required, yet some governments may start to waver from this task for short term political gain. Second, it will have become even clearer that governments, by undertaking austerity at just the wrong time, inflicted substantial damage on their economies, and that maybe everyone would be better off if they were not given that opportunity again.

[1] I use the term fiscal council to cover much the same set that George calls Independent Fiscal Institutions. His term is probably more accurate, but I still prefer fiscal council!

Spamalot / Yo AD [ANIME]
« on: December 01, 2013, 12:22:00 AM »
Have you kept up with Kill La Kill?

I have. Still enjoying it but not nearly as impressed as I was with Gurrenn.

I do think it has the potential to match Gurrenn, though with a totally different kind of emotion; it can do the same sort of furious, ambitious, unhinged drive, but it can't make that the entire point.

But, yeah -- I think there is something there. I just don't think they've found it yet. The technical quality behind the episodes is clear, but they seem to dick around as often as not doing nothing at all, with uninteresting side stories, without developing the world and without giving us any kind of gripping drive to their vision. It needs them to give a shit about it more carefully -- not more deeply, but more carefully -- than they seem to right now.

edit: Will say that Shingeki was unquestionably the blow-out & best anime of the season. Nothing else came close. There were a few others essentially comparable to KlK, but Shingeki was fuck-out dominant.

edit2: I really wish they would cut it the fuck out with the cutesy perv jokes. If you want to make hentai, just fucking make hentai. It is not improving your non-pornographic show.

Agrulian Archives / dat HMWK + AUTO-DIDACTY THRAED
« on: November 29, 2013, 01:03:56 AM »
prescript:  thread must be hidden from the potentially idle threat of synth-nazism. i shall store this copy here

idle threat went active. code maroon, code maroon. this is now the only thread.

A mathematician wrote on the inside cover of one of the books I own but can't locate right now something to the effect of: "Math is something you do, not something you know. If you do not do the math, you will at best achieve a second-rate understanding." That's the point of this thread: To improve understanding by doing math.

Also covered will be anything and everything that seems interesting and lends itself in its established form to study via the analysis of formal models and abstractions, i.e. physics, logic, theoretical computer science, econ, finance, PChem, OChem, basic chem, game theory, etc etc etc etc etc. Anyone is welcome to join or not; this is my self-study structure for myself, but it's open to commentary/participation ofc.

General Discussion / The Different Sizes of Infinity
« on: November 27, 2013, 02:10:40 AM »
Infinity is a powerful concept. Philosophers, artists, theologians, scientists, and people from all walks of life have struggled with ideas of the infinite and the eternal throughout history.

Infinity is also an extremely important concept in mathematics. Infinity shows up almost immediately in dealing with infinitely large sets — collections of numbers that go on forever, like the natural, or counting numbers: 1, 2, 3, 4, 5, and so on.

Infinite sets are not all created equal, however. There are actually many different sizes or levels of infinity; some infinite sets are vastly larger than other infinite sets.

The theory of infinite sets was developed in the late nineteenth century by the brilliant mathematician Georg Cantor. Many of Cantor's ideas and theorems sit at the foundation of modern mathematics. One of Cantor's coolest innovations was a way to compare the sizes of infinite sets, and to use this idea to show that there are many infinities.

To see how Cantor's theory works, we start out by saying that two sets are the same size if we can make a one to one correspondence, or pairing up, of the elements of the two sets. We can start small — the sets {a, b, c} and {1, 2, 3} are the same size, since I can pair up their elements:

This is a little overcomplicated for comparing two small finite sets like these — it is obvious that they both have three elements, and so are the same size. However, when we are looking at infinite sets, we cannot just look at the sets and count up the numbers of elements, since the sets go on forever. So, this more formal definition will be very helpful.

Countably Infinite Sets
Our baseline level of infinity will come from our most basic infinite set: the previously mentioned natural numbers. A set that is the same size as the natural numbers — that can be put into a one to one correspondence with the natural numbers — is called a countably infinite set.

A surprising number of infinite sets are actually countable. At first glance, the set of integers, made up of the natural numbers, their negative number counterparts, and zero, looks like it should be bigger than the naturals. After all, for each of our natural numbers, like 2 or 10, we just added a negative number, -2 or -10. But the integers are countable — we can find a way to assign exactly one integer to each natural number by bouncing back and forth between positive and negative numbers:

If we continue the pattern suggested above, we end up assigning exactly one integer to each natural number, with each integer assigned to a natural number, giving us the kind of one to one pairing that means the two sets are the same size.

This is a little freaky, since the natural numbers are a subset of the integers — each natural number is also an integer. But even though the natural numbers are fully contained in the integers, the two sets actually do have the same size.

The rational numbers are those numbers that can be written as a fraction, or ratio, of two integers: 1/2, -5/4, 3 (which can be written as 3/1), and the like. This is another infinite set that looks like it should be bigger than the natural numbers — between any two natural numbers, we have infinitely many fractions.

But as with the integers, we can still make a one to one pairing, assigning exactly one natural number to each rational number. Start by making a grid of the rationals: each row has a particular natural number in the bottom part of the fraction — the denominators of the first row are all 1's, and the 2nd row all 2's. Each column has a particular number in the top part of the fraction — the numerators of the first column are all 1's, and the second column all 2's. This grid covers all of the positive rational numbers, since any ratio of two positive integers will show up somewhere in the grid:

We get our correspondence between the rationals and the naturals by moving in a zig-zag pattern through the grid and counting. Fractions like 2/2 and 4/6 that are just alternate representations of numbers we have already seen (2/2 is the same as 1/1, and 4/6 is the same as 2/3) are skipped over:

So, the first rational number is 1/1, the second is 2/1, the third is 1/2, the fourth is 1/3, we skip 2/2 since this just reduces to 1/1, the fifth is 3/1, and so on.

Continuing like this, every rational number will be assigned a unique natural number, showing that, like the integers, the rationals are also a countably infinite set.

Even though we have added all these fractions and negative numbers to our original basic natural number set, we are still at our first, baseline, level of infinity.

Uncountably Infinite Sets
Now we consider the real numbers. The real numbers are the collection of numbers that can be written out with some kind of decimal expansion. The real numbers include the rational numbers — any fraction of two integers can be divided out and turned into a decimal. 1/2 = 0.5 and 1/3 = 0.3333..., with the latter continuing on with 3's forever. The real numbers also include irrational numbers, or decimals that go on forever without settling into a repeated pattern or ending. π is irrational — its decimal expansion starts out with the familiar 3.14159... but keeps going on forever, its digits veering around wildly.

We were able to come up with clever correspondences with the natural numbers for the integers and the rationals, showing that they are all countably infinite and the same size. Given that, we might think that we can do something similar with the real numbers.

This is, however, impossible. The real numbers are an uncountably infinite set — there actually are far more real numbers than there are natural numbers, and there is no way to line up the reals and the naturals so that we are assigning exactly one real number to each natural number.

To see this, we use an extremely powerful technique in mathematics: proof by contradiction. We will start out by hypothesizing that the opposite of our claim is true — that the real numbers are countably infinite, and so there is a way to line up all the reals with the naturals in a one to one correspondence. We will see that it doesn't matter exactly what this correspondence looks like, so let's say that the first few pairs in the correspondence are the following:

Our big assumption here is that each and every real number appears somewhere on this list. We are now going to show that this is in fact wrong by making a new number that does not show up in the list.

For each natural number n, we look at the corresponding real number on the list, and take the digit n places to the right of the real number's decimal point. So, take the first digit of the first number, the second digit of the second number, the third digit of the third number, and so on:

From our first real number we get a 5, our second number a 3, and our third number a 1. We make a new number by taking each of these digits, and adding 1 to them (flipping around to a 0 if my original digit is 9), giving us the number 0.64207..., continuing on for all the other numbers on our list.

This new "diagonal" number is definitely a real number — it has a decimal expansion. But it is different from all the numbers on the list: its first digit is different from the first digit of our first number, its second digit is different from the second digit of our second number, and so on.

We have made a new real number that does not show up on our list. This contradicts our main assumption that every real number appears somewhere in the correspondence.

We mentioned before that the details of the correspondence did not matter. This is because, no matter what alignment we try between the real numbers and the natural numbers, we can do the same diagonal trick above, making a number that does not show up in the correspondence.

This shows that the reals are not countably infinite. No matter what we try, there is no way to make a one to one pairing up of the natural numbers and the real numbers. These two sets are not the same size. This leads to the profound and somewhat uncomfortable realization that there must be multiple levels of infinity — the natural numbers and the real numbers are both infinite sets, but the reals form a set that is vastly larger than the naturals — they represent some "higher level" of infinity.

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