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Power of a point - Wikipedia
The power of point P (see in Figure 1) can be defined equivalently as the product of distances from the point P to the two intersection points of any ray emanating from P.
nibble  math  geometry  spatial  ground-up  concept  metrics  invariance  identity  atoms  wiki  reference  measure  yoga  calculation 
september 2017 by nhaliday
rotational dynamics - Why do non-rigid bodies try to increase their moment of inertia? - Physics Stack Exchange
This happens to isolated rotating system that is not a rigid body.

Inside such a body (for example, steel chain in free fall) the parts move relatively to each other and there is internal friction that dissipates kinetic energy of the system, while angular momentum is conserved. The dissipation goes on until the parts stop moving with respect to each other, so body rotates as a rigid body, even if it is not rigid by constitution.

The rotating state of the body that has the lowest kinetic energy for given angular momentum is that in which the body has the greatest moment of inertia (with respect to center of mass). For example, a long chain thrown into free fall will twist and turn until it is all straight and rotating as rigid body.


If LL is constant (net torque of external forces acting on the system is zero) and the constitution and initial conditions allow it, the system's dissipation will work to diminish energy until it has the minimum value, which happens for maximum IaIa possible.
nibble  q-n-a  overflow  physics  mechanics  tidbits  spatial  rigidity  flexibility  invariance  direction  stylized-facts  dynamical  volo-avolo  street-fighting  yoga 
august 2017 by nhaliday
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
Inscribed angle - Wikipedia
- for triangle w/ one side = a diameter, draw isosceles triangle and use supplementary angle identities
- otherwise draw second triangle w/ side = a diameter, and use above result twice
nibble  math  geometry  spatial  ground-up  wiki  reference  proofs  identity  levers  yoga 
august 2017 by nhaliday
Analysis of variance - Wikipedia
Analysis of variance (ANOVA) is a collection of statistical models used to analyze the differences among group means and their associated procedures (such as "variation" among and between groups), developed by statistician and evolutionary biologist Ronald Fisher. In the ANOVA setting, the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether or not the means of several groups are equal, and therefore generalizes the t-test to more than two groups. ANOVAs are useful for comparing (testing) three or more means (groups or variables) for statistical significance. It is conceptually similar to multiple two-sample t-tests, but is more conservative (results in less type I error) and is therefore suited to a wide range of practical problems.

good pic: https://en.wikipedia.org/wiki/Analysis_of_variance#Motivating_example

tutorial by Gelman: http://www.stat.columbia.edu/~gelman/research/published/econanova3.pdf

so one way to think of partitioning the variance:
y_ij = alpha_i + beta_j + eps_ij
Var(y_ij) = Var(alpha_i) + Var(beta_j) + Cov(alpha_i, beta_j) + Var(eps_ij)
and alpha_i, beta_j are independent, so Cov(alpha_i, beta_j) = 0

can you make this work w/ interaction effects?
data-science  stats  methodology  hypothesis-testing  variance-components  concept  conceptual-vocab  thinking  wiki  reference  nibble  multi  visualization  visual-understanding  pic  pdf  exposition  lecture-notes  gelman  scitariat  tutorial  acm  ground-up  yoga 
july 2017 by nhaliday
Pearson correlation coefficient - Wikipedia
what does this mean?: https://twitter.com/GarettJones/status/863546692724858880
deleted but it was about the Pearson correlation distance: 1-r
I guess it's a metric


A less misleading way to think about the correlation R is as follows: given X,Y from a standardized bivariate distribution with correlation R, an increase in X leads to an expected increase in Y: dY = R dX. In other words, students with +1 SD SAT score have, on average, roughly +0.4 SD college GPAs. Similarly, students with +1 SD college GPAs have on average +0.4 SAT.

this reminds me of the breeder's equation (but it uses r instead of h^2, so it can't actually be the same)

stats  science  hypothesis-testing  correlation  metrics  plots  regression  wiki  reference  nibble  methodology  multi  twitter  social  discussion  best-practices  econotariat  garett-jones  concept  conceptual-vocab  accuracy  causation  acm  matrix-factorization  todo  explanation  yoga  hsu  street-fighting  levers  🌞  2014  scitariat  variance-components  meta:prediction  biodet  s:**  mental-math  reddit  commentary  ssc  poast  gwern  data-science  metric-space  similarity  measure 
may 2017 by nhaliday
Strings, periods, and borders
A border of x is any proper prefix of x that equals a suffix of x.

...overlapping borders of a string imply that the string is periodic...

In the border array ß[1..n] of x, entry ß[i] is the length
of the longest border of x[1..i].
pdf  nibble  slides  lectures  algorithms  strings  exposition  yoga  atoms  levers  tidbits  sequential 
may 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 - Variance of maximum of Gaussian random variables - Cross Validated
In full generality it is rather hard to find the right order of magnitude of the variance of a Gaussien supremum since the tools from concentration theory are always suboptimal for the maximum function.

order ~ 1/log n
q-n-a  overflow  stats  probability  acm  orders  tails  bias-variance  moments  concentration-of-measure  magnitude  tidbits  distribution  yoga  structure  extrema  nibble 
february 2017 by nhaliday
6.896: Essential Coding Theory
- probabilistic method and Chernoff bound for Shannon coding
- probabilistic method for asymptotically good Hamming codes (Gilbert coding)
- sparsity used for LDPC codes
mit  course  yoga  tcs  complexity  coding-theory  math.AG  fields  polynomials  pigeonhole-markov  linear-algebra  probabilistic-method  lecture-notes  bits  sparsity  concentration-of-measure  linear-programming  linearity  expanders  hamming  pseudorandomness  crypto  rigorous-crypto  communication-complexity  no-go  madhu-sudan  shannon  unit  p:** 
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
Computational Complexity: Favorite Theorems: The Yao Principle
The Yao Principle applies when we don't consider the algorithmic complexity of the players. For example in communication complexity we have two players who each have a separate half of an input string and they want to compute some function of the input with the minimum amount of communication between them. The Yao principle states that the best probabilistic strategies for the players will achieve exactly the communication bounds as the best deterministic strategy over a worst-case distribution of inputs.

The Yao Principle plays a smaller role where we measure the running time of an algorithm since applying the Principle would require solving an extremely large linear program. But since so many of our bounds are in information-based models like communication and decision-tree complexity, the Yao Principle, though not particularly complicated, plays an important role in lower bounds in a large number of results in our field.
tcstariat  tcs  complexity  adversarial  rand-approx  algorithms  game-theory  yoga  levers  communication-complexity  random  lower-bounds  average-case  nibble  org:bleg 
january 2017 by nhaliday
CS 731 Advanced Artificial Intelligence - Spring 2011
- statistical machine learning
- sparsity in regression
- graphical models
- exponential families
- variational methods
- dimensionality reduction, eg, PCA
- Bayesian nonparametrics
- compressive sensing, matrix completion, and Johnson-Lindenstrauss
course  lecture-notes  yoga  acm  stats  machine-learning  graphical-models  graphs  model-class  bayesian  learning-theory  sparsity  embeddings  markov  monte-carlo  norms  unit  nonparametric  compressed-sensing  matrix-factorization  features 
january 2017 by nhaliday
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