Is the bounty system effective? - Meta Stack Exchange

november 2019 by nhaliday

https://math.meta.stackexchange.com/questions/20155/how-effective-are-bounties

could do some kinda econometric analysis using the data explorer to determine this once and for all: https://pinboard.in/u:nhaliday/b:c0cd449b9e69

maybe some kinda RDD in time, or difference-in-differences?

I don't think answer quality/quantity by time meets the common trend assumption for DD, tho... Questions that eventually receive bounty are prob higher quality in the first place, and higher quality answers accumulate more and better answers regardless. Hmm.

q-n-a
stackex
forum
community
info-foraging
efficiency
cost-benefit
data
analysis
incentives
attention
quality
ubiquity
supply-demand
multi
math
causation
endogenous-exogenous
intervention
branches
control
tactics
sleuthin
hmm
idk
todo
data-science
overflow
dbs
regression
shift
methodology
econometrics
could do some kinda econometric analysis using the data explorer to determine this once and for all: https://pinboard.in/u:nhaliday/b:c0cd449b9e69

maybe some kinda RDD in time, or difference-in-differences?

I don't think answer quality/quantity by time meets the common trend assumption for DD, tho... Questions that eventually receive bounty are prob higher quality in the first place, and higher quality answers accumulate more and better answers regardless. Hmm.

november 2019 by nhaliday

exponential function - Feynman's Trick for Approximating $e^x$ - Mathematics Stack Exchange

october 2019 by nhaliday

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
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]).

october 2019 by nhaliday

Stack Exchange Data Explorer

october 2019 by nhaliday

- nice way to practice SQL and learn useful info at the same time

- flavor is T-SQL

- can do graphs

https://data.stackexchange.com/help

https://meta.stackexchange.com/questions/256115/information-about-database-indexes-in-the-data-explorer

https://blog.exsilio.com/all/understanding-query-execution-plans-in-sql-server-management-studio/

https://developer.rackspace.com/blog/understanding-a-sql-server-execution-plan/

https://www.mssqltips.com/sqlservertip/1873/how-to-read-sql-server-graphical-query-execution-plans/

https://www.sqlshack.com/sql-server-query-execution-plans-understanding-reading-plans/

https://www.sqlshack.com/how-to-analyze-sql-execution-plan-graphical-components/

database
info-foraging
info-dynamics
stackex
overflow
forum
community
q-n-a
dataset
tools
data
let-me-see
dbs
data-science
calculator
dynamic
interview-prep
multi
documentation
reference
howto
visualization
DSL
sleuthin
objektbuch
explanation
jargon
notation
traces
performance
measurement
microsoft
dotnet
- flavor is T-SQL

- can do graphs

https://data.stackexchange.com/help

https://meta.stackexchange.com/questions/256115/information-about-database-indexes-in-the-data-explorer

https://blog.exsilio.com/all/understanding-query-execution-plans-in-sql-server-management-studio/

https://developer.rackspace.com/blog/understanding-a-sql-server-execution-plan/

https://www.mssqltips.com/sqlservertip/1873/how-to-read-sql-server-graphical-query-execution-plans/

https://www.sqlshack.com/sql-server-query-execution-plans-understanding-reading-plans/

https://www.sqlshack.com/how-to-analyze-sql-execution-plan-graphical-components/

october 2019 by nhaliday

online resources - How to write special set notation by hand? - Mathematics Stack Exchange

october 2019 by nhaliday

Your ℕN is “incorrect” in that a capital N in any serif font has the diagonal thickened, not the verticals. In fact, the rule (in Latin alphabet) is that negative slopes are thick, positive ones are thin. Verticals are sometimes thin, sometimes thick. Unique exception: Z. Just look in a newspaper at A, V, X, M, and N.

nibble
q-n-a
overflow
math
writing
notetaking
howto
pic
notation
trivia
october 2019 by nhaliday

How to fool the "try some test cases" heuristic: Algorithms that appear correct, but are actually incorrect - Computer Science Stack Exchange

nibble q-n-a overflow cs programming correctness error measurement random checking examples counterexample rigor rand-approx greedy debugging math.NT multiplicative algorithms intricacy heuristic methodology

august 2019 by nhaliday

nibble q-n-a overflow cs programming correctness error measurement random checking examples counterexample rigor rand-approx greedy debugging math.NT multiplicative algorithms intricacy heuristic methodology

august 2019 by nhaliday

big list - Are there proofs that you feel you did not "understand" for a long time? - MathOverflow

nibble q-n-a overflow soft-question big-list math proofs expert-experience heavyweights gowers mathtariat reflection learning intricacy grokkability intuition algebra math.GR motivation math.GN topology synthesis math.CT computation tcs logic iteration-recursion math.CA extrema smoothness span-cover grokkability-clarity

august 2019 by nhaliday

nibble q-n-a overflow soft-question big-list math proofs expert-experience heavyweights gowers mathtariat reflection learning intricacy grokkability intuition algebra math.GR motivation math.GN topology synthesis math.CT computation tcs logic iteration-recursion math.CA extrema smoothness span-cover grokkability-clarity

august 2019 by nhaliday

galois theory - Existence of irreducible polynomial of arbitrary degree over finite field without use of primitive element theorem? - Mathematics Stack Exchange

nibble q-n-a overflow math math.CA algebra multiplicative tidbits proofs existence pigeonhole-markov estimate fields identity measure

july 2019 by nhaliday

nibble q-n-a overflow math math.CA algebra multiplicative tidbits proofs existence pigeonhole-markov estimate fields identity measure

july 2019 by nhaliday

The Existential Risk of Math Errors - Gwern.net

july 2019 by nhaliday

How big is this upper bound? Mathematicians have often made errors in proofs. But it’s rarer for ideas to be accepted for a long time and then rejected. But we can divide errors into 2 basic cases corresponding to type I and type II errors:

1. Mistakes where the theorem is still true, but the proof was incorrect (type I)

2. Mistakes where the theorem was false, and the proof was also necessarily incorrect (type II)

Before someone comes up with a final answer, a mathematician may have many levels of intuition in formulating & working on the problem, but we’ll consider the final end-product where the mathematician feels satisfied that he has solved it. Case 1 is perhaps the most common case, with innumerable examples; this is sometimes due to mistakes in the proof that anyone would accept is a mistake, but many of these cases are due to changing standards of proof. For example, when David Hilbert discovered errors in Euclid’s proofs which no one noticed before, the theorems were still true, and the gaps more due to Hilbert being a modern mathematician thinking in terms of formal systems (which of course Euclid did not think in). (David Hilbert himself turns out to be a useful example of the other kind of error: his famous list of 23 problems was accompanied by definite opinions on the outcome of each problem and sometimes timings, several of which were wrong or questionable5.) Similarly, early calculus used ‘infinitesimals’ which were sometimes treated as being 0 and sometimes treated as an indefinitely small non-zero number; this was incoherent and strictly speaking, practically all of the calculus results were wrong because they relied on an incoherent concept - but of course the results were some of the greatest mathematical work ever conducted6 and when later mathematicians put calculus on a more rigorous footing, they immediately re-derived those results (sometimes with important qualifications), and doubtless as modern math evolves other fields have sometimes needed to go back and clean up the foundations and will in the future.7

...

Isaac Newton, incidentally, gave two proofs of the same solution to a problem in probability, one via enumeration and the other more abstract; the enumeration was correct, but the other proof totally wrong and this was not noticed for a long time, leading Stigler to remark:

...

TYPE I > TYPE II?

“Lefschetz was a purely intuitive mathematician. It was said of him that he had never given a completely correct proof, but had never made a wrong guess either.”

- Gian-Carlo Rota13

Case 2 is disturbing, since it is a case in which we wind up with false beliefs and also false beliefs about our beliefs (we no longer know that we don’t know). Case 2 could lead to extinction.

...

Except, errors do not seem to be evenly & randomly distributed between case 1 and case 2. There seem to be far more case 1s than case 2s, as already mentioned in the early calculus example: far more than 50% of the early calculus results were correct when checked more rigorously. Richard Hamming attributes to Ralph Boas a comment that while editing Mathematical Reviews that “of the new results in the papers reviewed most are true but the corresponding proofs are perhaps half the time plain wrong”.

...

Gian-Carlo Rota gives us an example with Hilbert:

...

Olga labored for three years; it turned out that all mistakes could be corrected without any major changes in the statement of the theorems. There was one exception, a paper Hilbert wrote in his old age, which could not be fixed; it was a purported proof of the continuum hypothesis, you will find it in a volume of the Mathematische Annalen of the early thirties.

...

Leslie Lamport advocates for machine-checked proofs and a more rigorous style of proofs similar to natural deduction, noting a mathematician acquaintance guesses at a broad error rate of 1/329 and that he routinely found mistakes in his own proofs and, worse, believed false conjectures30.

[more on these "structured proofs":

https://academia.stackexchange.com/questions/52435/does-anyone-actually-publish-structured-proofs

https://mathoverflow.net/questions/35727/community-experiences-writing-lamports-structured-proofs

]

We can probably add software to that list: early software engineering work found that, dismayingly, bug rates seem to be simply a function of lines of code, and one would expect diseconomies of scale. So one would expect that in going from the ~4,000 lines of code of the Microsoft DOS operating system kernel to the ~50,000,000 lines of code in Windows Server 2003 (with full systems of applications and libraries being even larger: the comprehensive Debian repository in 2007 contained ~323,551,126 lines of code) that the number of active bugs at any time would be… fairly large. Mathematical software is hopefully better, but practitioners still run into issues (eg Durán et al 2014, Fonseca et al 2017) and I don’t know of any research pinning down how buggy key mathematical systems like Mathematica are or how much published mathematics may be erroneous due to bugs. This general problem led to predictions of doom and spurred much research into automated proof-checking, static analysis, and functional languages31.

[related:

https://mathoverflow.net/questions/11517/computer-algebra-errors

I don't know any interesting bugs in symbolic algebra packages but I know a true, enlightening and entertaining story about something that looked like a bug but wasn't.

Define sinc𝑥=(sin𝑥)/𝑥.

Someone found the following result in an algebra package: ∫∞0𝑑𝑥sinc𝑥=𝜋/2

They then found the following results:

...

So of course when they got:

∫∞0𝑑𝑥sinc𝑥sinc(𝑥/3)sinc(𝑥/5)⋯sinc(𝑥/15)=(467807924713440738696537864469/935615849440640907310521750000)𝜋

hmm:

Which means that nobody knows Fourier analysis nowdays. Very sad and discouraging story... – fedja Jan 29 '10 at 18:47

--

Because the most popular systems are all commercial, they tend to guard their bug database rather closely -- making them public would seriously cut their sales. For example, for the open source project Sage (which is quite young), you can get a list of all the known bugs from this page. 1582 known issues on Feb.16th 2010 (which includes feature requests, problems with documentation, etc).

That is an order of magnitude less than the commercial systems. And it's not because it is better, it is because it is younger and smaller. It might be better, but until SAGE does a lot of analysis (about 40% of CAS bugs are there) and a fancy user interface (another 40%), it is too hard to compare.

I once ran a graduate course whose core topic was studying the fundamental disconnect between the algebraic nature of CAS and the analytic nature of the what it is mostly used for. There are issues of logic -- CASes work more or less in an intensional logic, while most of analysis is stated in a purely extensional fashion. There is no well-defined 'denotational semantics' for expressions-as-functions, which strongly contributes to the deeper bugs in CASes.]

...

Should such widely-believed conjectures as P≠NP or the Riemann hypothesis turn out be false, then because they are assumed by so many existing proofs, a far larger math holocaust would ensue38 - and our previous estimates of error rates will turn out to have been substantial underestimates. But it may be a cloud with a silver lining, if it doesn’t come at a time of danger.

https://mathoverflow.net/questions/338607/why-doesnt-mathematics-collapse-down-even-though-humans-quite-often-make-mista

more on formal methods in programming:

https://www.quantamagazine.org/formal-verification-creates-hacker-proof-code-20160920/

https://intelligence.org/2014/03/02/bob-constable/

https://softwareengineering.stackexchange.com/questions/375342/what-are-the-barriers-that-prevent-widespread-adoption-of-formal-methods

Update: measured effort

In the October 2018 issue of Communications of the ACM there is an interesting article about Formally verified software in the real world with some estimates of the effort.

Interestingly (based on OS development for military equipment), it seems that producing formally proved software requires 3.3 times more effort than with traditional engineering techniques. So it's really costly.

On the other hand, it requires 2.3 times less effort to get high security software this way than with traditionally engineered software if you add the effort to make such software certified at a high security level (EAL 7). So if you have high reliability or security requirements there is definitively a business case for going formal.

WHY DON'T PEOPLE USE FORMAL METHODS?: https://www.hillelwayne.com/post/why-dont-people-use-formal-methods/

You can see examples of how all of these look at Let’s Prove Leftpad. HOL4 and Isabelle are good examples of “independent theorem” specs, SPARK and Dafny have “embedded assertion” specs, and Coq and Agda have “dependent type” specs.6

If you squint a bit it looks like these three forms of code spec map to the three main domains of automated correctness checking: tests, contracts, and types. This is not a coincidence. Correctness is a spectrum, and formal verification is one extreme of that spectrum. As we reduce the rigour (and effort) of our verification we get simpler and narrower checks, whether that means limiting the explored state space, using weaker types, or pushing verification to the runtime. Any means of total specification then becomes a means of partial specification, and vice versa: many consider Cleanroom a formal verification technique, which primarily works by pushing code review far beyond what’s humanly possible.

...

The question, then: “is 90/95/99% correct significantly cheaper than 100% correct?” The answer is very yes. We all are comfortable saying that a codebase we’ve well-tested and well-typed is mostly correct modulo a few fixes in prod, and we’re even writing more than four lines of code a day. In fact, the vast… [more]

ratty
gwern
analysis
essay
realness
truth
correctness
reason
philosophy
math
proofs
formal-methods
cs
programming
engineering
worse-is-better/the-right-thing
intuition
giants
old-anglo
error
street-fighting
heuristic
zooming
risk
threat-modeling
software
lens
logic
inference
physics
differential
geometry
estimate
distribution
robust
speculation
nonlinearity
cost-benefit
convexity-curvature
measure
scale
trivia
cocktail
history
early-modern
europe
math.CA
rigor
news
org:mag
org:sci
miri-cfar
pdf
thesis
comparison
examples
org:junk
q-n-a
stackex
pragmatic
tradeoffs
cracker-prog
techtariat
invariance
DSL
chart
ecosystem
grokkability
heavyweights
CAS
static-dynamic
lower-bounds
complexity
tcs
open-problems
big-surf
ideas
certificates-recognition
proof-systems
PCP
mediterranean
SDP
meta:prediction
epistemic
questions
guessing
distributed
overflow
nibble
soft-question
track-record
big-list
hmm
frontier
state-of-art
move-fast-(and-break-things)
grokkability-clarity
technical-writing
trust
1. Mistakes where the theorem is still true, but the proof was incorrect (type I)

2. Mistakes where the theorem was false, and the proof was also necessarily incorrect (type II)

Before someone comes up with a final answer, a mathematician may have many levels of intuition in formulating & working on the problem, but we’ll consider the final end-product where the mathematician feels satisfied that he has solved it. Case 1 is perhaps the most common case, with innumerable examples; this is sometimes due to mistakes in the proof that anyone would accept is a mistake, but many of these cases are due to changing standards of proof. For example, when David Hilbert discovered errors in Euclid’s proofs which no one noticed before, the theorems were still true, and the gaps more due to Hilbert being a modern mathematician thinking in terms of formal systems (which of course Euclid did not think in). (David Hilbert himself turns out to be a useful example of the other kind of error: his famous list of 23 problems was accompanied by definite opinions on the outcome of each problem and sometimes timings, several of which were wrong or questionable5.) Similarly, early calculus used ‘infinitesimals’ which were sometimes treated as being 0 and sometimes treated as an indefinitely small non-zero number; this was incoherent and strictly speaking, practically all of the calculus results were wrong because they relied on an incoherent concept - but of course the results were some of the greatest mathematical work ever conducted6 and when later mathematicians put calculus on a more rigorous footing, they immediately re-derived those results (sometimes with important qualifications), and doubtless as modern math evolves other fields have sometimes needed to go back and clean up the foundations and will in the future.7

...

Isaac Newton, incidentally, gave two proofs of the same solution to a problem in probability, one via enumeration and the other more abstract; the enumeration was correct, but the other proof totally wrong and this was not noticed for a long time, leading Stigler to remark:

...

TYPE I > TYPE II?

“Lefschetz was a purely intuitive mathematician. It was said of him that he had never given a completely correct proof, but had never made a wrong guess either.”

- Gian-Carlo Rota13

Case 2 is disturbing, since it is a case in which we wind up with false beliefs and also false beliefs about our beliefs (we no longer know that we don’t know). Case 2 could lead to extinction.

...

Except, errors do not seem to be evenly & randomly distributed between case 1 and case 2. There seem to be far more case 1s than case 2s, as already mentioned in the early calculus example: far more than 50% of the early calculus results were correct when checked more rigorously. Richard Hamming attributes to Ralph Boas a comment that while editing Mathematical Reviews that “of the new results in the papers reviewed most are true but the corresponding proofs are perhaps half the time plain wrong”.

...

Gian-Carlo Rota gives us an example with Hilbert:

...

Olga labored for three years; it turned out that all mistakes could be corrected without any major changes in the statement of the theorems. There was one exception, a paper Hilbert wrote in his old age, which could not be fixed; it was a purported proof of the continuum hypothesis, you will find it in a volume of the Mathematische Annalen of the early thirties.

...

Leslie Lamport advocates for machine-checked proofs and a more rigorous style of proofs similar to natural deduction, noting a mathematician acquaintance guesses at a broad error rate of 1/329 and that he routinely found mistakes in his own proofs and, worse, believed false conjectures30.

[more on these "structured proofs":

https://academia.stackexchange.com/questions/52435/does-anyone-actually-publish-structured-proofs

https://mathoverflow.net/questions/35727/community-experiences-writing-lamports-structured-proofs

]

We can probably add software to that list: early software engineering work found that, dismayingly, bug rates seem to be simply a function of lines of code, and one would expect diseconomies of scale. So one would expect that in going from the ~4,000 lines of code of the Microsoft DOS operating system kernel to the ~50,000,000 lines of code in Windows Server 2003 (with full systems of applications and libraries being even larger: the comprehensive Debian repository in 2007 contained ~323,551,126 lines of code) that the number of active bugs at any time would be… fairly large. Mathematical software is hopefully better, but practitioners still run into issues (eg Durán et al 2014, Fonseca et al 2017) and I don’t know of any research pinning down how buggy key mathematical systems like Mathematica are or how much published mathematics may be erroneous due to bugs. This general problem led to predictions of doom and spurred much research into automated proof-checking, static analysis, and functional languages31.

[related:

https://mathoverflow.net/questions/11517/computer-algebra-errors

I don't know any interesting bugs in symbolic algebra packages but I know a true, enlightening and entertaining story about something that looked like a bug but wasn't.

Define sinc𝑥=(sin𝑥)/𝑥.

Someone found the following result in an algebra package: ∫∞0𝑑𝑥sinc𝑥=𝜋/2

They then found the following results:

...

So of course when they got:

∫∞0𝑑𝑥sinc𝑥sinc(𝑥/3)sinc(𝑥/5)⋯sinc(𝑥/15)=(467807924713440738696537864469/935615849440640907310521750000)𝜋

hmm:

Which means that nobody knows Fourier analysis nowdays. Very sad and discouraging story... – fedja Jan 29 '10 at 18:47

--

Because the most popular systems are all commercial, they tend to guard their bug database rather closely -- making them public would seriously cut their sales. For example, for the open source project Sage (which is quite young), you can get a list of all the known bugs from this page. 1582 known issues on Feb.16th 2010 (which includes feature requests, problems with documentation, etc).

That is an order of magnitude less than the commercial systems. And it's not because it is better, it is because it is younger and smaller. It might be better, but until SAGE does a lot of analysis (about 40% of CAS bugs are there) and a fancy user interface (another 40%), it is too hard to compare.

I once ran a graduate course whose core topic was studying the fundamental disconnect between the algebraic nature of CAS and the analytic nature of the what it is mostly used for. There are issues of logic -- CASes work more or less in an intensional logic, while most of analysis is stated in a purely extensional fashion. There is no well-defined 'denotational semantics' for expressions-as-functions, which strongly contributes to the deeper bugs in CASes.]

...

Should such widely-believed conjectures as P≠NP or the Riemann hypothesis turn out be false, then because they are assumed by so many existing proofs, a far larger math holocaust would ensue38 - and our previous estimates of error rates will turn out to have been substantial underestimates. But it may be a cloud with a silver lining, if it doesn’t come at a time of danger.

https://mathoverflow.net/questions/338607/why-doesnt-mathematics-collapse-down-even-though-humans-quite-often-make-mista

more on formal methods in programming:

https://www.quantamagazine.org/formal-verification-creates-hacker-proof-code-20160920/

https://intelligence.org/2014/03/02/bob-constable/

https://softwareengineering.stackexchange.com/questions/375342/what-are-the-barriers-that-prevent-widespread-adoption-of-formal-methods

Update: measured effort

In the October 2018 issue of Communications of the ACM there is an interesting article about Formally verified software in the real world with some estimates of the effort.

Interestingly (based on OS development for military equipment), it seems that producing formally proved software requires 3.3 times more effort than with traditional engineering techniques. So it's really costly.

On the other hand, it requires 2.3 times less effort to get high security software this way than with traditionally engineered software if you add the effort to make such software certified at a high security level (EAL 7). So if you have high reliability or security requirements there is definitively a business case for going formal.

WHY DON'T PEOPLE USE FORMAL METHODS?: https://www.hillelwayne.com/post/why-dont-people-use-formal-methods/

You can see examples of how all of these look at Let’s Prove Leftpad. HOL4 and Isabelle are good examples of “independent theorem” specs, SPARK and Dafny have “embedded assertion” specs, and Coq and Agda have “dependent type” specs.6

If you squint a bit it looks like these three forms of code spec map to the three main domains of automated correctness checking: tests, contracts, and types. This is not a coincidence. Correctness is a spectrum, and formal verification is one extreme of that spectrum. As we reduce the rigour (and effort) of our verification we get simpler and narrower checks, whether that means limiting the explored state space, using weaker types, or pushing verification to the runtime. Any means of total specification then becomes a means of partial specification, and vice versa: many consider Cleanroom a formal verification technique, which primarily works by pushing code review far beyond what’s humanly possible.

...

The question, then: “is 90/95/99% correct significantly cheaper than 100% correct?” The answer is very yes. We all are comfortable saying that a codebase we’ve well-tested and well-typed is mostly correct modulo a few fixes in prod, and we’re even writing more than four lines of code a day. In fact, the vast… [more]

july 2019 by nhaliday

data structures - Why are Red-Black trees so popular? - Computer Science Stack Exchange

june 2019 by nhaliday

- AVL trees have smaller average depth than red-black trees, and thus searching for a value in AVL tree is consistently faster.

- Red-black trees make less structural changes to balance themselves than AVL trees, which could make them potentially faster for insert/delete. I'm saying potentially, because this would depend on the cost of the structural change to the tree, as this will depend a lot on the runtime and implemntation (might also be completely different in a functional language when the tree is immutable?)

There are many benchmarks online that compare AVL and Red-black trees, but what struck me is that my professor basically said, that usually you'd do one of two things:

- Either you don't really care that much about performance, in which case the 10-20% difference of AVL vs Red-black in most cases won't matter at all.

- Or you really care about performance, in which you case you'd ditch both AVL and Red-black trees, and go with B-trees, which can be tweaked to work much better (or (a,b)-trees, I'm gonna put all of those in one basket.)

--

> For some kinds of binary search trees, including red-black trees but not AVL trees, the "fixes" to the tree can fairly easily be predicted on the way down and performed during a single top-down pass, making the second pass unnecessary. Such insertion algorithms are typically implemented with a loop rather than recursion, and often run slightly faster in practice than their two-pass counterparts.

So a RedBlack tree insert can be implemented without recursion, on some CPUs recursion is very expensive if you overrun the function call cache (e.g SPARC due to is use of Register window)

--

There are some cases where you can't use B-trees at all.

One prominent case is std::map from C++ STL. The standard requires that insert does not invalidate existing iterators

...

I also believe that "single pass tail recursive" implementation is not the reason for red black tree popularity as a mutable data structure.

First of all, stack depth is irrelevant here, because (given log𝑛 height) you would run out of the main memory before you run out of stack space. Jemalloc is happy with preallocating worst case depth on the stack.

nibble
q-n-a
overflow
cs
algorithms
tcs
data-structures
functional
orders
trees
cost-benefit
tradeoffs
roots
explanans
impetus
performance
applicability-prereqs
programming
pls
c(pp)
ubiquity
- Red-black trees make less structural changes to balance themselves than AVL trees, which could make them potentially faster for insert/delete. I'm saying potentially, because this would depend on the cost of the structural change to the tree, as this will depend a lot on the runtime and implemntation (might also be completely different in a functional language when the tree is immutable?)

There are many benchmarks online that compare AVL and Red-black trees, but what struck me is that my professor basically said, that usually you'd do one of two things:

- Either you don't really care that much about performance, in which case the 10-20% difference of AVL vs Red-black in most cases won't matter at all.

- Or you really care about performance, in which you case you'd ditch both AVL and Red-black trees, and go with B-trees, which can be tweaked to work much better (or (a,b)-trees, I'm gonna put all of those in one basket.)

--

> For some kinds of binary search trees, including red-black trees but not AVL trees, the "fixes" to the tree can fairly easily be predicted on the way down and performed during a single top-down pass, making the second pass unnecessary. Such insertion algorithms are typically implemented with a loop rather than recursion, and often run slightly faster in practice than their two-pass counterparts.

So a RedBlack tree insert can be implemented without recursion, on some CPUs recursion is very expensive if you overrun the function call cache (e.g SPARC due to is use of Register window)

--

There are some cases where you can't use B-trees at all.

One prominent case is std::map from C++ STL. The standard requires that insert does not invalidate existing iterators

...

I also believe that "single pass tail recursive" implementation is not the reason for red black tree popularity as a mutable data structure.

First of all, stack depth is irrelevant here, because (given log𝑛 height) you would run out of the main memory before you run out of stack space. Jemalloc is happy with preallocating worst case depth on the stack.

june 2019 by nhaliday

What's the expected level of paper for top conferences in Computer Science - Academia Stack Exchange

june 2019 by nhaliday

Top. The top level.

My experience on program committees for STOC, FOCS, ITCS, SODA, SOCG, etc., is that there are FAR more submissions of publishable quality than can be accepted into the conference. By "publishable quality" I mean a well-written presentation of a novel, interesting, and non-trivial result within the scope of the conference.

...

There are several questions that come up over and over in the FOCS/STOC review cycle:

- How surprising / novel / elegant / interesting is the result?

- How surprising / novel / elegant / interesting / general are the techniques?

- How technically difficult is the result? Ironically, FOCS and STOC committees have a reputation for ignoring the distinction between trivial (easy to derive from scratch) and nondeterministically trivial (easy to understand after the fact).

- What is the expected impact of this result? Is this paper going to change the way people do theoretical computer science over the next five years?

- Is the result of general interest to the theoretical computer science community? Or is it only of interest to a narrow subcommunity? In particular, if the topic is outside the STOC/FOCS mainstream—say, for example, computational topology—does the paper do a good job of explaining and motivating the results to a typical STOC/FOCS audience?

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analogy
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My experience on program committees for STOC, FOCS, ITCS, SODA, SOCG, etc., is that there are FAR more submissions of publishable quality than can be accepted into the conference. By "publishable quality" I mean a well-written presentation of a novel, interesting, and non-trivial result within the scope of the conference.

...

There are several questions that come up over and over in the FOCS/STOC review cycle:

- How surprising / novel / elegant / interesting is the result?

- How surprising / novel / elegant / interesting / general are the techniques?

- How technically difficult is the result? Ironically, FOCS and STOC committees have a reputation for ignoring the distinction between trivial (easy to derive from scratch) and nondeterministically trivial (easy to understand after the fact).

- What is the expected impact of this result? Is this paper going to change the way people do theoretical computer science over the next five years?

- Is the result of general interest to the theoretical computer science community? Or is it only of interest to a narrow subcommunity? In particular, if the topic is outside the STOC/FOCS mainstream—say, for example, computational topology—does the paper do a good job of explaining and motivating the results to a typical STOC/FOCS audience?

june 2019 by nhaliday

classification - ImageNet: what is top-1 and top-5 error rate? - Cross Validated

june 2019 by nhaliday

Now, in the case of top-1 score, you check if the top class (the one having the highest probability) is the same as the target label.

In the case of top-5 score, you check if the target label is one of your top 5 predictions (the 5 ones with the highest probabilities).

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In the case of top-5 score, you check if the target label is one of your top 5 predictions (the 5 ones with the highest probabilities).

june 2019 by nhaliday

linear algebra - Recommendations for a usable, fast C++ matrix library? - Computational Science Stack Exchange

may 2019 by nhaliday

Basically the same question popped up on SO:

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may 2019 by nhaliday

soft question - What are good non-English languages for mathematicians to know? - MathOverflow

february 2019 by nhaliday

I'm with Deane here: I think learning foreign languages is not a very mathematically productive thing to do; of course, there are lots of good reasons to learn foreign languages, but doing mathematics is not one of them. Not only are there few modern mathematics papers written in languages other than English, but the primary other language they are written (French) in is pretty easy to read without actually knowing it.

Even though I've been to France several times, my spoken French mostly consists of "merci," "si vous plait," "d'accord" and some food words; I've still skimmed 100 page long papers in French without a lot of trouble.

If nothing else, think of reading a paper in French as a good opportunity to teach Google Translate some mathematical French.

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Even though I've been to France several times, my spoken French mostly consists of "merci," "si vous plait," "d'accord" and some food words; I've still skimmed 100 page long papers in French without a lot of trouble.

If nothing else, think of reading a paper in French as a good opportunity to teach Google Translate some mathematical French.

february 2019 by nhaliday

orbit - Best approximation for Sun's trajectory around galactic center? - Astronomy Stack Exchange

december 2017 by nhaliday

The Sun orbits in the Galactic potential. The motion is complex; it takes about 230 million years to make a circuit with an orbital speed of around 220 km/s, but at the same time it oscillates up and down with respect to the Galactic plane every ∼70∼70 million years and also wobbles in and out every ∼150∼150 million years (this is called epicyclic motion). The spatial amplitudes of these oscillations are around 100 pc vertically and 300 pc in the radial direction inwards and outwards around an average orbital radius (I am unable to locate a precise figure for the latter).

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december 2017 by nhaliday

light - Why doesn't the moon twinkle? - Astronomy Stack Exchange

december 2017 by nhaliday

As you mention, when light enters our atmosphere, it goes through several parcels of gas with varying density, temperature, pressure, and humidity. These differences make the refractive index of the parcels different, and since they move around (the scientific term for air moving around is "wind"), the light rays take slightly different paths through the atmosphere.

Stars are point sources

…the Moon is not

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Stars are point sources

…the Moon is not

december 2017 by nhaliday

galaxy - How do astronomers estimate the total mass of dust in clouds and galaxies? - Astronomy Stack Exchange

december 2017 by nhaliday

Dust absorbs stellar light (primarily in the ultraviolet), and is heated up. Subsequently it cools by emitting infrared, "thermal" radiation. Assuming a dust composition and grain size distribution, the amount of emitted IR light per unit dust mass can be calculated as a function of temperature. Observing the object at several different IR wavelengths, a Planck curve can be fitted to the data points, yielding the dust temperature. The more UV light incident on the dust, the higher the temperature.

The result is somewhat sensitive to the assumptions, and thus the uncertainties are sometimes quite large. The more IR data points obtained, the better. If only one IR point is available, the temperature cannot be calculated. Then there's a degeneracy between incident UV light and the amount of dust, and the mass can only be estimated to within some orders of magnitude (I think).

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The result is somewhat sensitive to the assumptions, and thus the uncertainties are sometimes quite large. The more IR data points obtained, the better. If only one IR point is available, the temperature cannot be calculated. Then there's a degeneracy between incident UV light and the amount of dust, and the mass can only be estimated to within some orders of magnitude (I think).

december 2017 by nhaliday

general relativity - What if the universe is rotating as a whole? - Physics Stack Exchange

november 2017 by nhaliday

To find out whether the universe is rotating, in principle the most straightforward test is to watch the motion of a gyroscope relative to the distant galaxies. If it rotates at an angular velocity -ω relative to them, then the universe is rotating at angular velocity ω. In practice, we do not have mechanical gyroscopes with small enough random and systematic errors to put a very low limit on ω. However, we can use the entire solar system as a kind of gyroscope. Solar-system observations put a model-independent upper limit of 10^-7 radians/year on the rotation,[Clemence 1957] which is an order of magnitude too lax to rule out the Gödel metric.

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november 2017 by nhaliday

What is the connection between special and general relativity? - Physics Stack Exchange

november 2017 by nhaliday

Special relativity is the "special case" of general relativity where spacetime is flat. The speed of light is essential to both.

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november 2017 by nhaliday

references - Mathematician wants the equivalent knowledge to a quality stats degree - Cross Validated

nibble q-n-a overflow lens acm stats hypothesis-testing limits confluence books recommendations list top-n accretion data-science roadmap p:whenever p:someday reading quixotic advanced markov monte-carlo convexity-curvature optimization topics linear-models linear-algebra machine-learning classification random rand-approx martingale regression time-series no-go

november 2017 by nhaliday

nibble q-n-a overflow lens acm stats hypothesis-testing limits confluence books recommendations list top-n accretion data-science roadmap p:whenever p:someday reading quixotic advanced markov monte-carlo convexity-curvature optimization topics linear-models linear-algebra machine-learning classification random rand-approx martingale regression time-series no-go

november 2017 by nhaliday

Expected Value of Random Walk - Mathematics Stack Exchange

october 2017 by nhaliday

cf Section 3.10 in Grimmett-Stirzaker or Section III.3 in Feller, Vol 1

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october 2017 by nhaliday

gn.general topology - Pair of curves joining opposite corners of a square must intersect---proof? - MathOverflow

october 2017 by nhaliday

In his 'Ordinary Differential Equations' (sec. 1.2) V.I. Arnold says "... every pair of curves in the square joining different pairs of opposite corners must intersect".

This is obvious geometrically but I was wondering how one could go about proving this rigorously. I have thought of a proof using Brouwer's Fixed Point Theorem which I describe below. I would greatly appreciate the group's comments on whether this proof is right and if a simpler proof is possible.

...

Since the full Jordan curve theorem is quite subtle, it might be worth pointing out that theorem in question reduces to the Jordan curve theorem for polygons, which is easier.

Suppose on the contrary that the curves A,BA,B joining opposite corners do not meet. Since A,BA,B are closed sets, their minimum distance apart is some ε>0ε>0. By compactness, each of A,BA,B can be partitioned into finitely many arcs, each of which lies in a disk of diameter <ε/3<ε/3. Then, by a homotopy inside each disk we can replace A,BA,B by polygonal paths A′,B′A′,B′ that join the opposite corners of the square and are still disjoint.

Also, we can replace A′,B′A′,B′ by simple polygonal paths A″,B″A″,B″ by omitting loops. Now we can close A″A″ to a polygon, and B″B″ goes from its "inside" to "outside" without meeting it, contrary to the Jordan curve theorem for polygons.

- John Stillwell

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This is obvious geometrically but I was wondering how one could go about proving this rigorously. I have thought of a proof using Brouwer's Fixed Point Theorem which I describe below. I would greatly appreciate the group's comments on whether this proof is right and if a simpler proof is possible.

...

Since the full Jordan curve theorem is quite subtle, it might be worth pointing out that theorem in question reduces to the Jordan curve theorem for polygons, which is easier.

Suppose on the contrary that the curves A,BA,B joining opposite corners do not meet. Since A,BA,B are closed sets, their minimum distance apart is some ε>0ε>0. By compactness, each of A,BA,B can be partitioned into finitely many arcs, each of which lies in a disk of diameter <ε/3<ε/3. Then, by a homotopy inside each disk we can replace A,BA,B by polygonal paths A′,B′A′,B′ that join the opposite corners of the square and are still disjoint.

Also, we can replace A′,B′A′,B′ by simple polygonal paths A″,B″A″,B″ by omitting loops. Now we can close A″A″ to a polygon, and B″B″ goes from its "inside" to "outside" without meeting it, contrary to the Jordan curve theorem for polygons.

- John Stillwell

october 2017 by nhaliday

multivariate analysis - Is it possible to have a pair of Gaussian random variables for which the joint distribution is not Gaussian? - Cross Validated

october 2017 by nhaliday

The bivariate normal distribution is the exception, not the rule!

It is important to recognize that "almost all" joint distributions with normal marginals are not the bivariate normal distribution. That is, the common viewpoint that joint distributions with normal marginals that are not the bivariate normal are somehow "pathological", is a bit misguided.

Certainly, the multivariate normal is extremely important due to its stability under linear transformations, and so receives the bulk of attention in applications.

note: there is a multivariate central limit theorem, so those such applications have no problem

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It is important to recognize that "almost all" joint distributions with normal marginals are not the bivariate normal distribution. That is, the common viewpoint that joint distributions with normal marginals that are not the bivariate normal are somehow "pathological", is a bit misguided.

Certainly, the multivariate normal is extremely important due to its stability under linear transformations, and so receives the bulk of attention in applications.

note: there is a multivariate central limit theorem, so those such applications have no problem

october 2017 by nhaliday

Karl Pearson and the Chi-squared Test

october 2017 by nhaliday

Pearson's paper of 1900 introduced what subsequently became known as the chi-squared test of goodness of fit. The terminology and allusions of 80 years ago create a barrier for the modern reader, who finds that the interpretation of Pearson's test procedure and the assessment of what he achieved are less than straightforward, notwithstanding the technical advances made since then. An attempt is made here to surmount these difficulties by exploring Pearson's relevant activities during the first decade of his statistical career, and by describing the work by his contemporaries and predecessors which seem to have influenced his approach to the problem. Not all the questions are answered, and others remain for further study.

original paper: http://www.economics.soton.ac.uk/staff/aldrich/1900.pdf

How did Karl Pearson come up with the chi-squared statistic?: https://stats.stackexchange.com/questions/97604/how-did-karl-pearson-come-up-with-the-chi-squared-statistic

He proceeds by working with the multivariate normal, and the chi-square arises as a sum of squared standardized normal variates.

You can see from the discussion on p160-161 he's clearly discussing applying the test to multinomial distributed data (I don't think he uses that term anywhere). He apparently understands the approximate multivariate normality of the multinomial (certainly he knows the margins are approximately normal - that's a very old result - and knows the means, variances and covariances, since they're stated in the paper); my guess is that most of that stuff is already old hat by 1900. (Note that the chi-squared distribution itself dates back to work by Helmert in the mid-1870s.)

Then by the bottom of p163 he derives a chi-square statistic as "a measure of goodness of fit" (the statistic itself appears in the exponent of the multivariate normal approximation).

He then goes on to discuss how to evaluate the p-value*, and then he correctly gives the upper tail area of a χ212χ122 beyond 43.87 as 0.000016. [You should keep in mind, however, that he didn't correctly understand how to adjust degrees of freedom for parameter estimation at that stage, so some of the examples in his papers use too high a d.f.]

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methodology
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degrees-of-freedom
limits
original paper: http://www.economics.soton.ac.uk/staff/aldrich/1900.pdf

How did Karl Pearson come up with the chi-squared statistic?: https://stats.stackexchange.com/questions/97604/how-did-karl-pearson-come-up-with-the-chi-squared-statistic

He proceeds by working with the multivariate normal, and the chi-square arises as a sum of squared standardized normal variates.

You can see from the discussion on p160-161 he's clearly discussing applying the test to multinomial distributed data (I don't think he uses that term anywhere). He apparently understands the approximate multivariate normality of the multinomial (certainly he knows the margins are approximately normal - that's a very old result - and knows the means, variances and covariances, since they're stated in the paper); my guess is that most of that stuff is already old hat by 1900. (Note that the chi-squared distribution itself dates back to work by Helmert in the mid-1870s.)

Then by the bottom of p163 he derives a chi-square statistic as "a measure of goodness of fit" (the statistic itself appears in the exponent of the multivariate normal approximation).

He then goes on to discuss how to evaluate the p-value*, and then he correctly gives the upper tail area of a χ212χ122 beyond 43.87 as 0.000016. [You should keep in mind, however, that he didn't correctly understand how to adjust degrees of freedom for parameter estimation at that stage, so some of the examples in his papers use too high a d.f.]

october 2017 by nhaliday

self study - Looking for a good and complete probability and statistics book - Cross Validated

october 2017 by nhaliday

I never had the opportunity to visit a stats course from a math faculty. I am looking for a probability theory and statistics book that is complete and self-sufficient. By complete I mean that it contains all the proofs and not just states results.

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october 2017 by nhaliday

correlation - Variance of product of dependent variables - Cross Validated

october 2017 by nhaliday

cov[X^2,Y^2] + (var[X]+(E[X])^2)(var[Y]+(E[Y])^2) − (cov[X,Y]+E[X]E[Y])^2

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october 2017 by nhaliday

Variance of product of multiple random variables - Cross Validated

october 2017 by nhaliday

prod_i (var[X_i] + (E[X_i])^2) - prod_i (E[X_i])^2

two variable case: var[X] var[Y] + var[X] (E[Y])^2 + (E[X])^2 var[Y]

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probability
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two variable case: var[X] var[Y] + var[X] (E[Y])^2 + (E[X])^2 var[Y]

october 2017 by nhaliday

forces - The Time That 2 Masses Will Collide Due To Newtonian Gravity - Physics Stack Exchange

october 2017 by nhaliday

If two particles of dust are placed in an empty universe 1 light year apart from each other, how long will it take for them to collide due to the effects of gravity?: https://www.reddit.com/r/theydidthemath/comments/3rum1p/request_if_two_particles_of_dust_are_placed_in_an/

How long for 2 particles to collide due to gravity?: https://www.physicsforums.com/threads/how-long-for-2-particles-to-collide-due-to-gravity.698767/

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How long for 2 particles to collide due to gravity?: https://www.physicsforums.com/threads/how-long-for-2-particles-to-collide-due-to-gravity.698767/

october 2017 by nhaliday

newtonian gravity - Newton's original proof of gravitation for non-point-mass objects - Physics Stack Exchange

september 2017 by nhaliday

This theorem is Proposition LXXI, Theorem XXXI in the Principia. To warm up, consider the more straightforward proof of the preceding theorem, that there's no inverse-square force inside of a spherical shell:

picture

The crux of the argument is that the triangles HPI and LPK are similar. The mass enclosed in the small-but-near patch of sphere HI goes like the square of the distance HP, while the mass enclosed in the large-but-far patch of sphere KL, with the same solid angle, goes like the square of the distance KP. This mass ratio cancels out the distance-squared ratio governing the strength of the force, and so the net force from those two patches vanishes.

For a point mass outside a shell, Newton's approach is essentially the same as the modern approach:

picture

One integral is removed because we're considering a thin spherical shell rather than a solid sphere. The second integral, "as the semi-circle AKB revolves about the diameter AB," trivially turns Newton's infinitesimal arcs HI and KL into annuli.

The third integral is over all the annuli in the sphere, over 0≤ϕ≤τ/20≤ϕ≤τ/2 or over R−r≤s≤R+rR−r≤s≤R+r. This one is a little bit hairy, even with the advantage of modern notation.

Newton's clever trick is to consider the relationship between the force due to the smaller, nearer annulus HI and the larger, farther annulus KL defined by the same viewing angle (in modern notation, dθdθ). If I understand correctly he argues again, based on lots of similar triangles with infinitesimal angles, that the smaller-but-nearer annulus and the larger-but-farther annulus exert the same force at P. Furthermore, he shows that the force doesn't depend on the distance PF, and thus doesn't depend on the radius of the sphere; the only parameter left is the distance PS (squared) between the particle and the sphere's center. Since the argument doesn't depend on the angle HPS, it's true for all the annuli, and the theorem is proved.

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The crux of the argument is that the triangles HPI and LPK are similar. The mass enclosed in the small-but-near patch of sphere HI goes like the square of the distance HP, while the mass enclosed in the large-but-far patch of sphere KL, with the same solid angle, goes like the square of the distance KP. This mass ratio cancels out the distance-squared ratio governing the strength of the force, and so the net force from those two patches vanishes.

For a point mass outside a shell, Newton's approach is essentially the same as the modern approach:

picture

One integral is removed because we're considering a thin spherical shell rather than a solid sphere. The second integral, "as the semi-circle AKB revolves about the diameter AB," trivially turns Newton's infinitesimal arcs HI and KL into annuli.

The third integral is over all the annuli in the sphere, over 0≤ϕ≤τ/20≤ϕ≤τ/2 or over R−r≤s≤R+rR−r≤s≤R+r. This one is a little bit hairy, even with the advantage of modern notation.

Newton's clever trick is to consider the relationship between the force due to the smaller, nearer annulus HI and the larger, farther annulus KL defined by the same viewing angle (in modern notation, dθdθ). If I understand correctly he argues again, based on lots of similar triangles with infinitesimal angles, that the smaller-but-nearer annulus and the larger-but-farther annulus exert the same force at P. Furthermore, he shows that the force doesn't depend on the distance PF, and thus doesn't depend on the radius of the sphere; the only parameter left is the distance PS (squared) between the particle and the sphere's center. Since the argument doesn't depend on the angle HPS, it's true for all the annuli, and the theorem is proved.

september 2017 by nhaliday

electricity - Why is AC more "dangerous" than DC? - Physics Stack Exchange

september 2017 by nhaliday

One of the reasons that AC might be considered more dangerous is that it arguably has more ways of getting into your body. Since the voltage alternates, it can cause current to enter and exit your body even without a closed loop, since your body (and what ground it's attached to) has capacitance. DC cannot do that. Also, AC is quite easily stepped up to higher voltages using transformers, while with DC that requires some relatively elaborate electronics. Finally, while your skin has a fairly high resistance to protect you, and the air is also a terrific insulator as long as you're not touching any wires, sometimes the inductance of AC transformers can cause high-voltage sparks that break down the air and I imagine can get through your skin a bit as well.

Also, like you mentioned, the heart is controlled by electric pulses and repeated pulses of electricity can throw this off quite a bit and cause a heart attack. However, I don't think that this is unique to alternating current. I read once about an unfortunate young man that was learning about electricity and wanted to measure the resistance of his own body. He took a multimeter and set a lead to each thumb. By accident or by stupidity, he punctured both thumbs with the leads, and the small (I imagine it to be 9 V) battery in the multimeter caused a current in his bloodstream, and he died on the spot. So maybe ignorance is more dangerous than either AC or DC.

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Also, like you mentioned, the heart is controlled by electric pulses and repeated pulses of electricity can throw this off quite a bit and cause a heart attack. However, I don't think that this is unique to alternating current. I read once about an unfortunate young man that was learning about electricity and wanted to measure the resistance of his own body. He took a multimeter and set a lead to each thumb. By accident or by stupidity, he punctured both thumbs with the leads, and the small (I imagine it to be 9 V) battery in the multimeter caused a current in his bloodstream, and he died on the spot. So maybe ignorance is more dangerous than either AC or DC.

september 2017 by nhaliday

Why is Earth's gravity stronger at the poles? - Physics Stack Exchange

september 2017 by nhaliday

The point is that if we approximate Earth with an oblate ellipsoid, then the surface of Earth is an equipotential surface,11 see e.g. this Phys.SE post.

Now, because the polar radius is smaller than the equatorial radius, the density of equipotential surfaces at the poles must be bigger than at the equator.

Or equivalently, the field strength22 gg at the poles must be bigger than at the equator.

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Now, because the polar radius is smaller than the equatorial radius, the density of equipotential surfaces at the poles must be bigger than at the equator.

Or equivalently, the field strength22 gg at the poles must be bigger than at the equator.

september 2017 by nhaliday

What is recommended number of latent factors for the implicit collaborative filtering using ALS? - Quora

q-n-a qra expert machine-learning acm model-class best-practices matrix-factorization ranking matching generalization nibble multi overflow data-science data optimization expert-experience human-ml

august 2017 by nhaliday

q-n-a qra expert machine-learning acm model-class best-practices matrix-factorization ranking matching generalization nibble multi overflow data-science data optimization expert-experience human-ml

august 2017 by nhaliday

Is it possible to recover Classical Mechanics from Schrödinger's equation? - Physics Stack Exchange

august 2017 by nhaliday

Classical limit of quantum mechanics: https://physics.stackexchange.com/questions/32112/classical-limit-of-quantum-mechanics

https://physics.stackexchange.com/questions/108222/from-quantum-mechanics-to-classical-mechanics

Classical Limit of Quantum Mechanics: https://mathoverflow.net/questions/102313/classical-limit-of-quantum-mechanics

How/when does quantum mechanics become classical mechanics?: https://www.quora.com/How-when-does-quantum-mechanics-become-classical-mechanics

Remarks concerning the status & some ramifications of EHRENFEST’S THEOREM: http://www.reed.edu/physics/faculty/wheeler/documents/Quantum%20Mechanics/Miscellaneous%20Essays/Ehrenfest's%20Theorem.pdf

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https://physics.stackexchange.com/questions/108222/from-quantum-mechanics-to-classical-mechanics

Classical Limit of Quantum Mechanics: https://mathoverflow.net/questions/102313/classical-limit-of-quantum-mechanics

How/when does quantum mechanics become classical mechanics?: https://www.quora.com/How-when-does-quantum-mechanics-become-classical-mechanics

Remarks concerning the status & some ramifications of EHRENFEST’S THEOREM: http://www.reed.edu/physics/faculty/wheeler/documents/Quantum%20Mechanics/Miscellaneous%20Essays/Ehrenfest's%20Theorem.pdf

august 2017 by nhaliday

diffusion - Surviving under water in air bubble - Physics Stack Exchange

august 2017 by nhaliday

I get d≈400md≈400m.

It's interesting to note that this is independent of pressure: I've neglected pressure dependence of DD and human resilience to carbon dioxide, and the maximum safe concentration of carbon dioxide is independent of pressure, just derived from measurements at STP.

Finally, a bubble this large will probably rapidly break up due to buoyancy and Plateau-Rayleigh instabilities.

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It's interesting to note that this is independent of pressure: I've neglected pressure dependence of DD and human resilience to carbon dioxide, and the maximum safe concentration of carbon dioxide is independent of pressure, just derived from measurements at STP.

Finally, a bubble this large will probably rapidly break up due to buoyancy and Plateau-Rayleigh instabilities.

august 2017 by nhaliday

How to create recommender system that integrates both collaborative filtering and content features? - Cross Validated

nibble q-n-a overflow data-science machine-learning acm model-class matrix-factorization matching ranking todo project pinboard exocortex exploratory features papers recommendations research human-ml latent-variables

august 2017 by nhaliday

nibble q-n-a overflow data-science machine-learning acm model-class matrix-factorization matching ranking todo project pinboard exocortex exploratory features papers recommendations research human-ml latent-variables

august 2017 by nhaliday

rotational dynamics - Why do non-rigid bodies try to increase their moment of inertia? - Physics Stack Exchange

august 2017 by nhaliday

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.

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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.

august 2017 by nhaliday

gravity - Gravitational collapse and free fall time (spherical, pressure-free) - Physics Stack Exchange

august 2017 by nhaliday

the parenthetical regarding Gauss's law just involves noting a shell of radius r + symmetry (so single parameter determines field along shell)

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august 2017 by nhaliday

co.combinatorics - Classification of Platonic solids - MathOverflow

july 2017 by nhaliday

My question is very basic: where can I find a complete (and hopefully self-contained) proof of the classification of Platonic solids? In all the references that I found, they use Euler's formula v−e+f=2v−e+f=2 to show that there are exactly five possible triples (v,e,f)(v,e,f). But of course this is not a complete proof because it does not rule out the possibility of different configurations or deformations. Has anyone ever written up a complete proof of this statement?!

...

This is a classical question. Here is my reading of it: Why is there a unique polytope with given combinatorics of faces, which are all regular polygons? Of course, for simple polytopes (tetrahedron, cube, dodecahedron) this is clear, but for the octahedron and icosahedron this is less clear.

The answer lies in the Cauchy's theorem. It was Legendre, while writing his Elements of Geometry and Trigonometry, noticed that Euclid's proof is incomplete in the Elements. Curiously, Euclid finds both radii of inscribed and circumscribed spheres (correctly) without ever explaining why they exist. Cauchy worked out a proof while still a student in 1813, more or less specifically for this purpose. The proof also had a technical gap which was found and patched up by Steinitz in 1920s.

The complete (corrected) proof can be found in the celebrated Proofs from the Book, or in Marcel Berger's Geometry. My book gives a bit more of historical context and some soft arguments (ch. 19). It's worth comparing this proof with (an erroneous) pre-Steinitz exposition, say in Hadamard's Leçons de Géométrie Elémentaire II, or with an early post-Steinitz correct but tedious proof given in (otherwise, excellent) Alexandrov's monograph (see also ch.26 in my book which compares all the approaches).

P.S. Note that Coxeter in Regular Polytopes can completely avoid this issue but taking a different (modern) definition of the regular polytopes (which are symmetric under group actions). For a modern exposition and the state of art of this approach, see McMullen and Schulte's Abstract Regular Polytopes.

https://en.wikipedia.org/wiki/Platonic_solid#Classification

https://mathoverflow.net/questions/46502/on-the-number-of-archimedean-solids

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This is a classical question. Here is my reading of it: Why is there a unique polytope with given combinatorics of faces, which are all regular polygons? Of course, for simple polytopes (tetrahedron, cube, dodecahedron) this is clear, but for the octahedron and icosahedron this is less clear.

The answer lies in the Cauchy's theorem. It was Legendre, while writing his Elements of Geometry and Trigonometry, noticed that Euclid's proof is incomplete in the Elements. Curiously, Euclid finds both radii of inscribed and circumscribed spheres (correctly) without ever explaining why they exist. Cauchy worked out a proof while still a student in 1813, more or less specifically for this purpose. The proof also had a technical gap which was found and patched up by Steinitz in 1920s.

The complete (corrected) proof can be found in the celebrated Proofs from the Book, or in Marcel Berger's Geometry. My book gives a bit more of historical context and some soft arguments (ch. 19). It's worth comparing this proof with (an erroneous) pre-Steinitz exposition, say in Hadamard's Leçons de Géométrie Elémentaire II, or with an early post-Steinitz correct but tedious proof given in (otherwise, excellent) Alexandrov's monograph (see also ch.26 in my book which compares all the approaches).

P.S. Note that Coxeter in Regular Polytopes can completely avoid this issue but taking a different (modern) definition of the regular polytopes (which are symmetric under group actions). For a modern exposition and the state of art of this approach, see McMullen and Schulte's Abstract Regular Polytopes.

https://en.wikipedia.org/wiki/Platonic_solid#Classification

https://mathoverflow.net/questions/46502/on-the-number-of-archimedean-solids

july 2017 by nhaliday

distributions - In linear regression, when is it appropriate to use the log of an independent variable instead of the actual values? - Cross Validated

q-n-a overflow nibble data-science stats methodology multiplicative regression linear-models best-practices checklists econometrics positivity outliers street-fighting nitty-gritty reference signum

april 2017 by nhaliday

q-n-a overflow nibble data-science stats methodology multiplicative regression linear-models best-practices checklists econometrics positivity outliers street-fighting nitty-gritty reference signum

april 2017 by nhaliday

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