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Introduction to Scaling Laws

Galileo’s Discovery of Scaling Laws: https://www.mtholyoke.edu/~mpeterso/classes/galileo/scaling8.pdf
Days 1 and 2 of Two New Sciences

An example of such an insight is “the surface of a small solid is comparatively greater than that of a large one” because the surface goes like the square of a linear dimension, but the volume goes like the cube.5 Thus as one scales down macroscopic objects, forces on their surfaces like viscous drag become relatively more important, and bulk forces like weight become relatively less important. Galileo uses this idea on the First Day in the context of resistance in free fall, as an explanation for why similar objects of different size do not fall exactly together, but the smaller one lags behind.
nibble  org:junk  exposition  lecture-notes  physics  mechanics  street-fighting  problem-solving  scale  magnitude  estimate  fermi  mental-math  calculation  nitty-gritty  multi  scitariat  org:bleg  lens  tutorial  guide  ground-up  tricki  skeleton  list  cheatsheet  identity  levers  hi-order-bits  yoga  metabuch  pdf  article  essay  history  early-modern  europe  the-great-west-whale  science  the-trenches  discovery  fluid  architecture  oceans  giants  tidbits 
august 2017 by nhaliday
st.statistics - Lower bound for sum of binomial coefficients? - MathOverflow
- basically approximate w/ geometric sum (which scales as final term) and you can get it up to O(1) factor
- not good enough for many applications (want 1+o(1) approx.)
- Stirling can also give bound to constant factor precision w/ more calculation I believe
- tighter bound at Section 7.3 here: http://webbuild.knu.ac.kr/~trj/Combin/matousek-vondrak-prob-ln.pdf
q-n-a  overflow  nibble  math  math.CO  estimate  tidbits  magnitude  concentration-of-measure  stirling  binomial  metabuch  tricki  multi  tightness  pdf  lecture-notes  exposition  probability  probabilistic-method  yoga 
february 2017 by nhaliday
probability - How to prove Bonferroni inequalities? - Mathematics Stack Exchange
- integrated version of inequalities for alternating sums of (N choose j), where r.v. N = # of events occuring
- inequalities for alternating binomial coefficients follow from general property of unimodal (increasing then decreasing) sequences, which can be gotten w/ two cases for increasing and decreasing resp.
- the final alternating zero sum property follows for binomial coefficients from expanding (1 - 1)^N = 0
- The idea of proving inequality by integrating simpler inequality of r.v.s is nice. Proof from CS 150 was more brute force from what I remember.
q-n-a  overflow  math  probability  tcs  probabilistic-method  estimate  proofs  levers  yoga  multi  tidbits  metabuch  monotonicity  calculation  nibble  bonferroni  tricki  binomial  s:null 
january 2017 by nhaliday
cv.complex variables - Absolute value inequality for complex numbers - MathOverflow
In general, once you've proven an inequality like this in R it holds automatically in any Euclidean space (including C) by averaging over projections. ("Inequality like this" = inequality where every term is the length of some linear combination of variable vectors in the space; here the vectors are a, b, c).

I learned this trick at MOP 30+ years ago, and don't know or remember who discovered it.
q-n-a  overflow  math  math.CV  estimate  tidbits  yoga  oly  mathtariat  math.FA  metabuch  inner-product  calculation  norms  nibble  tricki 
january 2017 by nhaliday
Information Processing: Oppenheimer on Bohr (1964 UCLA)
I find it strange that psychometricians usually define "verbal ability" over a vocabulary set that excludes words from mathematics and other scientific areas. A person's verbal score is enhanced by knowing many (increasingly obscure) words for the same concept, as opposed to knowing words which describe new concepts beyond those which appear in ordinary language.
hsu  scitariat  thinking  metabuch  language  neurons  psychometrics  dimensionality  concept  list  critique  conceptual-vocab  quotes  giants  discussion  wordlessness  novelty  tricki 
january 2017 by nhaliday
soft question - Thinking and Explaining - MathOverflow
- good question from Bill Thurston
- great answers by Terry Tao, fedja, Minhyong Kim, gowers, etc.

Terry Tao:
- symmetry as blurring/vibrating/wobbling, scale invariance
- anthropomorphization, adversarial perspective for estimates/inequalities/quantifiers, spending/economy

fedja walks through his though-process from another answer

Minhyong Kim: anthropology of mathematical philosophizing

Per Vognsen: normality as isotropy
comment: conjugate subgroup gHg^-1 ~ "H but somewhere else in G"

gowers: hidden things in basic mathematics/arithmetic
comment by Ryan Budney: x sin(x) via x -> (x, sin(x)), (x, y) -> xy
I kinda get what he's talking about but needed to use Mathematica to get the initial visualization down.
To remind myself later:
- xy can be easily visualized by juxtaposing the two parabolae x^2 and -x^2 diagonally
- x sin(x) can be visualized along that surface by moving your finger along the line (x, 0) but adding some oscillations in y direction according to sin(x)
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january 2017 by nhaliday
gt.geometric topology - Intuitive crutches for higher dimensional thinking - MathOverflow
Terry Tao:
I can't help you much with high-dimensional topology - it's not my field, and I've not picked up the various tricks topologists use to get a grip on the subject - but when dealing with the geometry of high-dimensional (or infinite-dimensional) vector spaces such as R^n, there are plenty of ways to conceptualise these spaces that do not require visualising more than three dimensions directly.

For instance, one can view a high-dimensional vector space as a state space for a system with many degrees of freedom. A megapixel image, for instance, is a point in a million-dimensional vector space; by varying the image, one can explore the space, and various subsets of this space correspond to various classes of images.

One can similarly interpret sound waves, a box of gases, an ecosystem, a voting population, a stream of digital data, trials of random variables, the results of a statistical survey, a probabilistic strategy in a two-player game, and many other concrete objects as states in a high-dimensional vector space, and various basic concepts such as convexity, distance, linearity, change of variables, orthogonality, or inner product can have very natural meanings in some of these models (though not in all).

It can take a bit of both theory and practice to merge one's intuition for these things with one's spatial intuition for vectors and vector spaces, but it can be done eventually (much as after one has enough exposure to measure theory, one can start merging one's intuition regarding cardinality, mass, length, volume, probability, cost, charge, and any number of other "real-life" measures).

For instance, the fact that most of the mass of a unit ball in high dimensions lurks near the boundary of the ball can be interpreted as a manifestation of the law of large numbers, using the interpretation of a high-dimensional vector space as the state space for a large number of trials of a random variable.

More generally, many facts about low-dimensional projections or slices of high-dimensional objects can be viewed from a probabilistic, statistical, or signal processing perspective.

Scott Aaronson:
Here are some of the crutches I've relied on. (Admittedly, my crutches are probably much more useful for theoretical computer science, combinatorics, and probability than they are for geometry, topology, or physics. On a related note, I personally have a much easier time thinking about R^n than about, say, R^4 or R^5!)

1. If you're trying to visualize some 4D phenomenon P, first think of a related 3D phenomenon P', and then imagine yourself as a 2D being who's trying to visualize P'. The advantage is that, unlike with the 4D vs. 3D case, you yourself can easily switch between the 3D and 2D perspectives, and can therefore get a sense of exactly what information is being lost when you drop a dimension. (You could call this the "Flatland trick," after the most famous literary work to rely on it.)
2. As someone else mentioned, discretize! Instead of thinking about R^n, think about the Boolean hypercube {0,1}^n, which is finite and usually easier to get intuition about. (When working on problems, I often find myself drawing {0,1}^4 on a sheet of paper by drawing two copies of {0,1}^3 and then connecting the corresponding vertices.)
3. Instead of thinking about a subset S⊆R^n, think about its characteristic function f:R^n→{0,1}. I don't know why that trivial perspective switch makes such a big difference, but it does ... maybe because it shifts your attention to the process of computing f, and makes you forget about the hopeless task of visualizing S!
4. One of the central facts about R^n is that, while it has "room" for only n orthogonal vectors, it has room for exp⁡(n) almost-orthogonal vectors. Internalize that one fact, and so many other properties of R^n (for example, that the n-sphere resembles a "ball with spikes sticking out," as someone mentioned before) will suddenly seem non-mysterious. In turn, one way to internalize the fact that R^n has so many almost-orthogonal vectors is to internalize Shannon's theorem that there exist good error-correcting codes.
5. To get a feel for some high-dimensional object, ask questions about the behavior of a process that takes place on that object. For example: if I drop a ball here, which local minimum will it settle into? How long does this random walk on {0,1}^n take to mix?

Gil Kalai:
This is a slightly different point, but Vitali Milman, who works in high-dimensional convexity, likes to draw high-dimensional convex bodies in a non-convex way. This is to convey the point that if you take the convex hull of a few points on the unit sphere of R^n, then for large n very little of the measure of the convex body is anywhere near the corners, so in a certain sense the body is a bit like a small sphere with long thin "spikes".
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december 2016 by nhaliday
Information Processing: Bounded cognition
Many people lack standard cognitive tools useful for understanding the world around them. Perhaps the most egregious case: probability and statistics, which are central to understanding health, economics, risk, crime, society, evolution, global warming, etc. Very few people have any facility for calculating risk, visualizing a distribution, understanding the difference between the average, the median, variance, etc.

Risk, Uncertainty, and Heuristics: http://infoproc.blogspot.com/2018/03/risk-uncertainty-and-heuristics.html
Risk = space of outcomes and probabilities are known. Uncertainty = probabilities not known, and even space of possibilities may not be known. Heuristic rules are contrasted with algorithms like maximization of expected utility.

How do smart people make smart decisions? | Gerd Gigerenzer

Helping Doctors and Patients Make Sense of Health Statistics: http://www.ema.europa.eu/docs/en_GB/document_library/Presentation/2014/12/WC500178514.pdf
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july 2016 by nhaliday
For potential Ph.D. students
Ravi Vakil's advice for PhD students

General advice:
Think actively about the creative process. A subtle leap is required from undergraduate thinking to active research (even if you have done undergraduate research). Think explicitly about the process, and talk about it (with me, and with others). For example, in an undergraduate class any Ph.D. student at Stanford will have tried to learn absolutely all the material flawlessly. But in order to know everything needed to tackle an important problem on the frontier of human knowledge, one would have to spend years reading many books and articles. So you'll have to learn differently. But how?

Don't be narrow and concentrate only on your particular problem. Learn things from all over the field, and beyond. The facts, methods, and insights from elsewhere will be much more useful than you might realize, possibly in your thesis, and most definitely afterwards. Being broad is a good way of learning to develop interesting questions.

When you learn the theory, you should try to calculate some toy cases, and think of some explicit basic examples.

Talk to other graduate students. A lot. Organize reading groups. Also talk to post-docs, faculty, visitors, and people you run into on the street. I learn the most from talking with other people. Maybe that's true for you too.

Specific topics:
- seminars
- giving talks
- writing
- links to other advice
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may 2016 by nhaliday
Answer to What is it like to understand advanced mathematics? - Quora
thinking like a mathematician

some of the points:
- small # of tricks (echoes Rota)
- web of concepts and modularization (zooming out) allow quick reasoning
- comfort w/ ambiguity and lack of understanding, study high-dimensional objects via projections
- above is essential for research (and often what distinguishes research mathematicians from people who were good at math, or majored in math)
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may 2016 by nhaliday

bundles : abstractmathproblem-solvingthinkingworrydream

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