nhaliday : mental-math   32

exponential function - Feynman's Trick for Approximating \$e^x\$ - Mathematics Stack Exchange
1. e^2.3 ~ 10
2. e^.7 ~ 2
3. e^x ~ 1+x
e = 2.71828...

errors (absolute, relative):
1. +0.0258, 0.26%
2. -0.0138, -0.68%
3. 1 + x approximates e^x on [-.3, .3] with absolute error < .05, and relative error < 5.6% (3.7% for [0, .3]).
nibble  q-n-a  overflow  math  feynman  giants  mental-math  calculation  multiplicative  AMT  identity  objektbuch  explanation  howto  estimate  street-fighting  stories  approximation  data  trivia  nitty-gritty
october 2019 by nhaliday
Introduction to Scaling Laws
http://galileo.phys.virginia.edu/classes/304/scaling.pdf

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  elegance
august 2017 by nhaliday
Kelly criterion - Wikipedia
In probability theory and intertemporal portfolio choice, the Kelly criterion, Kelly strategy, Kelly formula, or Kelly bet, is a formula used to determine the optimal size of a series of bets. In most gambling scenarios, and some investing scenarios under some simplifying assumptions, the Kelly strategy will do better than any essentially different strategy in the long run (that is, over a span of time in which the observed fraction of bets that are successful equals the probability that any given bet will be successful). It was described by J. L. Kelly, Jr, a researcher at Bell Labs, in 1956.[1] The practical use of the formula has been demonstrated.[2][3][4]

The Kelly Criterion is to bet a predetermined fraction of assets and can be counterintuitive. In one study,[5][6] each participant was given \$25 and asked to bet on a coin that would land heads 60% of the time. Participants had 30 minutes to play, so could place about 300 bets, and the prizes were capped at \$250. Behavior was far from optimal. "Remarkably, 28% of the participants went bust, and the average payout was just \$91. Only 21% of the participants reached the maximum. 18 of the 61 participants bet everything on one toss, while two-thirds gambled on tails at some stage in the experiment." Using the Kelly criterion and based on the odds in the experiment, the right approach would be to bet 20% of the pot on each throw (see first example in Statement below). If losing, the size of the bet gets cut; if winning, the stake increases.
nibble  betting  investing  ORFE  acm  checklists  levers  probability  algorithms  wiki  reference  atoms  extrema  parsimony  tidbits  decision-theory  decision-making  street-fighting  mental-math  calculation
august 2017 by nhaliday
The Flynn effect for verbal and visuospatial short-term and working memory: A cross-temporal meta-analysis
Specifically, the Flynn effect was found for forward digit span (r = 0.12, p < 0.01) and forward Corsi block span (r = 0.10, p < 0.01). Moreover, an anti-Flynn effect was found for backward digit span (r = − 0.06, p < 0.01) and for backward Corsi block span (r = − 0.17, p < 0.01). Overall, the results support co-occurrence theories that predict simultaneous secular gains in specialized abilities and declines in g. The causes of the differential trajectories are further discussed.

http://www.unz.com/jthompson/working-memory-bombshell/
https://www.newscientist.com/article/2146752-we-seem-to-be-getting-stupider-and-population-ageing-may-be-why/
study  psychology  cog-psych  psychometrics  iq  trends  dysgenics  flynn  psych-architecture  meta-analysis  multi  albion  scitariat  summary  commentary  blowhards  mental-math  science-anxiety  news  org:sci
august 2017 by nhaliday
A sense of where you are | West Hunter
Nobody at the Times noticed it at first. I don’t know that they ever did notice it by themselves- likely some reader brought it to their attention. But this happens all the time, because very few people have a picture of the world in their head that includes any numbers. Mostly they don’t even have a rough idea of relative size.

In much the same way, back in the 1980s,lots of writers were talking about 90,000 women a year dying of anorexia nervosa, another impossible number. Then there was the great scare about 1,000,000 kids being kidnapped in the US each year – also impossible and wrong. Practically all the talking classes bought into it.

You might think that the people at the top are different – but with a few exceptions, they’re just as clueless.
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may 2017 by nhaliday
Pearson correlation coefficient - Wikipedia
https://en.wikipedia.org/wiki/Coefficient_of_determination
deleted but it was about the Pearson correlation distance: 1-r
I guess it's a metric

https://en.wikipedia.org/wiki/Explained_variation

http://infoproc.blogspot.com/2014/02/correlation-and-variance.html
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  dependence-independence
may 2017 by nhaliday
Learning mathematics in a visuospatial format: A randomized, controlled trial of mental abacus instruction
We asked whether MA improves students’ mathematical abilities, and whether expertise – which requires sustained practice of mental imagery – is driven by changes to basic cognitive capacities like working memory. MA students improved on arithmetic tasks relative to controls, but training was not associated with changes to basic cognitive abilities. Instead, differences in spatial working memory at the beginning of the study mediated MA learning. We conclude that MA expertise can be achieved by many children in standard classrooms and results from efficient use of pre-existing abilities.

Cohen’s d = .60 (95% CI: .30 - .89) for arithmetic, .24 (-.05 - .52) for WJ-III, and .28 (.00 - .57) for place value

This finding suggests that the development of MA expertise is mediated by children’s pre-existing cognitive abilities, and thus that MA may not be suitable for all K-12 classroom environments, especially in groups of children who have low spatial working memory or attentional capacities (which may have been the case in our study). Critically, this does not mean that MA expertise depends on unusually strong cognitive abilities. Perhaps because we studied children from relatively disadvantaged backgrounds, few Mental Abacus 21 children in our sample had SWM capacities comparable to those seen among typical children in the United States.
study  psychology  cog-psych  visuo  spatial  iq  mental-math  field-study  education  learning  intelligence  nitty-gritty  india  asia  psych-architecture  c:**  intervention  effect-size  flexibility  quantitative-qualitative  input-output
march 2017 by nhaliday
Sir Ronald Aylmer Fisher | West Hunter
In 1930 he published The Genetical Theory of Natural Selection, which completed the fusion of Darwinian natural selection with Mendelian inheritance. James Crow said that it was ‘arguably the deepest and most influential book on evolution since Darwin’. In it, Fisher analyzed sexual selection, mimicry, and sex ratios, where he made some of the first arguments using game theory. The book touches on many other topics. As was the case with his other works, The Genetical Theory is a dense book, not easy for most people to understand. Fisher’s tendency to leave out mathematical steps that he deemed obvious (a leftover from his early training in mental mathematics) frustrates many readers.

The Genetical Theory is of particular interest to us because Fisher there lays out his ideas on how population size can speed up evolution. As we explain elsewhere, more individuals mean there will be more mutations, including favorable mutations, and so Fisher expected more rapid evolution in larger populations. This idea was originally suggested, in a nonmathematical way, in Darwin’s Origin of Species.

Although Fisher was fiercely loyal to friends and could be very charming, he had a quick temper and was a fine hater. The same uncompromising spirit that fostered his originality led to constant conflict with authority. He had a long conflict with Karl Pearson, who had also played an important part in the development of mathematical statistics. In this case, Pearson was more at fault, resisting the advent of a more talented competitor, as well as being an eminently hateable person in general. Over time Fisher also became increasing angry at Sewall Wright (another one of the founders of population genetics) due to scientific disagreements – and this was just wrong, because Wright was a sweetheart.

Fisher’s personality decreased his potential influence. He was not a school-builder, and was impatient with administrators. He expected to find some form of war-work in the Second World War, but his characteristics had alienated too many people, and thus his team dispersed to other jobs during the war. He returned to Rothamsted for the duration. This was a difficult time for him: his marriage disintegrated and his oldest son, an RAF pilot, was killed in the war.

...

Fisher’s ideas in genetics have taken an odd path. The Genetical Theory was not widely read, sold few copies, and has never been translated. Only gradually did its ideas find an audience. Of course, that audience included people like Bill Hamilton, the greatest mathematical biologist of the last half of the 20th century, who was strongly influenced by Fisher’s work. Hamilton said “By the time of my ultimate graduation,will I have understood all that is true in this book and will I get a First? I doubt it. In some ways some of us have overtaken Fisher; in many, however, this brilliant, daring man is still far in front.“

In fact, over the past generation, much of Fisher’s work has been neglected – in the sense that interest in population genetics has decreased (particularly interest in selection) and fewer students are exposed to his work in genetics in any way. Ernst Mayr didn’t even mention Fisher in his 1991 book One Long Argument: Charles Darwin and the Genesis of Modern Evolutionary Thought, while Stephen Jay Gould, in The Structure of Evolutionary Theory, gave Fisher 6 pages out of 1433. Of course Mayr and Gould were both complete chuckleheads.

Fisher’s work affords continuing insight, including important implications concerning human evolution that have emerged more than 50 years after his death. We strongly discourage other professionals from learning anything about his ideas.
west-hunter  history  bio  evolution  genetics  population-genetics  profile  giants  people  mostly-modern  the-trenches  innovation  novelty  britain  fisher  mental-math  narrative  scitariat  old-anglo  world-war  pre-ww2  scale  population  pop-structure  books  classic  speed  correlation  mutation  personality
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
Shut Up And Guess - Less Wrong
At what confidence level do you guess? At what confidence level do you answer "don't know"?

I took several of these tests last month, and the first thing I did was some quick mental calculations. If I have zero knowledge of a question, my expected gain from answering is 50% probability of earning one point and 50% probability of losing one half point. Therefore, my expected gain from answering a question is .5(1)-.5(.5)= +.25 points. Compare this to an expected gain of zero from not answering the question at all. Therefore, I ought to guess on every question, even if I have zero knowledge. If I have some inkling, well, that's even better.

You look disappointed. This isn't a very exciting application of arcane Less Wrong knowledge. Anyone with basic math skills should be able to calculate that out, right?

I attend a pretty good university, and I'm in a postgraduate class where most of us have at least a bachelor's degree in a hard science, and a few have master's degrees. And yet, talking to my classmates in the cafeteria after the first test was finished, I started to realize I was the only person in the class who hadn't answered "don't know" to any questions.

even more interesting stories in the comments
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september 2016 by nhaliday
Doomsday rule - Wikipedia, the free encyclopedia
It takes advantage of each year having a certain day of the week, called the doomsday, upon which certain easy-to-remember dates fall; for example, 4/4, 6/6, 8/8, 10/10, 12/12, and the last day of February all occur on the same day of the week in any year. Applying the Doomsday algorithm involves three steps:
1. Determination of the anchor day for the century.
2. Calculation of the doomsday for the year from the anchor day.
3. Selection of the closest date out of those that always fall on the doomsday, e.g., 4/4 and 6/6, and count of the number of days (modulo 7) between that date and the date in question to arrive at the day of the week.

This technique applies to both the Gregorian calendar A.D. and the Julian calendar, although their doomsdays are usually different days of the week.

Easter date: https://en.wikipedia.org/wiki/Computus
https://www.tondering.dk/claus/cal/easter.php
Easter Sunday is the first Sunday after the first full moon on or after the vernal equinox.

The calculation of Easter is complicated because it is linked to (an inaccurate version of) the Hebrew calendar.

...

It was therefore decided to make Easter Sunday the first Sunday after the first full moon after vernal equinox. Or more precisely: Easter Sunday is the first Sunday after the “official” full moon on or after the “official” vernal equinox.

The official vernal equinox is always 21 March.

The official full moon may differ from the real full moon by one or two days.

...

The full moon that precedes Easter is called the Paschal full moon. Two concepts play an important role when calculating the Paschal full moon: The Golden Number and the Epact. They are described in the following sections.

...

*What is the Golden Number?*
Each year is associated with a Golden Number.

Considering that the relationship between the moon’s phases and the days of the year repeats itself every 19 years (as described in the section about astronomy), it is natural to associate a number between 1 and 19 with each year. This number is the so-called Golden Number. It is calculated thus:

GoldenNumber=(year mod 19) + 1

However, 19 tropical years is 234.997 synodic months, which is very close to an integer. So every 19 years the phases of the moon fall on the same dates (if it were not for the skewness introduced by leap years). 19 years is called a Metonic cycle (after Meton, an astronomer from Athens in the 5th century BC).

So, to summarise: There are three important numbers to note:

A tropical year is 365.24219 days.
A synodic month is 29.53059 days.
19 tropical years is close to an integral number of synodic months.]

In years which have the same Golden Number, the new moon will fall on (approximately) the same date. The Golden Number is sufficient to calculate the Paschal full moon in the Julian calendar.

...

Under the Gregorian calendar, things became much more complicated. One of the changes made in the Gregorian calendar reform was a modification of the way Easter was calculated. There were two reasons for this. First, the 19 year cycle of the phases of moon (the Metonic cycle) was known not to be perfect. Secondly, the Metonic cycle fitted the Gregorian calendar year worse than it fitted the Julian calendar year.

It was therefore decided to base Easter calculations on the so-called Epact.

*What is the Epact?*
Each year is associated with an Epact.

The Epact is a measure of the age of the moon (i.e. the number of days that have passed since an “official” new moon) on a particular date.

...

In the Julian calendar, the Epact is the age of the moon on 22 March.

In the Gregorian calendar, the Epact is the age of the moon at the start of the year.

The Epact is linked to the Golden Number in the following manner:

Under the Julian calendar, 19 years were assumed to be exactly an integral number of synodic months, and the following relationship exists between the Golden Number and the Epact:

Epact=(11 × (GoldenNumber – 1)) mod 30

...

In the Gregorian calendar reform, some modifications were made to the simple relationship between the Golden Number and the Epact.

In the Gregorian calendar the Epact should be calculated thus: [long algorithm]

...

Suppose you know the Easter date of the current year, can you easily find the Easter date in the next year? No, but you can make a qualified guess.

If Easter Sunday in the current year falls on day X and the next year is not a leap year, Easter Sunday of next year will fall on one of the following days: X–15, X–8, X+13 (rare), or X+20.

...

If you combine this knowledge with the fact that Easter Sunday never falls before 22 March and never falls after 25 April, you can narrow the possibilities down to two or three dates.
tricks  street-fighting  concept  wiki  reference  cheatsheet  trivia  nitty-gritty  objektbuch  time  calculation  mental-math  multi  religion  christianity  events  howto  cycles
august 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
Guess the Correlation
some basic rules?
- more trouble w/ high than low end (maybe because I'm just guessing slope/omitting outliers?)
- should try out w/ correlated Gaussians to get some intuition
games  learning  stats  intuition  thinking  hmm  street-fighting  correlation  instinct  mental-math  nitty-gritty  simulation  operational  todo  spock  quantitative-qualitative  dependence-independence
july 2016 by nhaliday

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