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Bareiss algorithm - Wikipedia
During the execution of Bareiss algorithm, every integer that is computed is the determinant of a submatrix of the input matrix. This allows, using the Hadamard inequality, to bound the size of these integers. Otherwise, the Bareiss algorithm may be viewed as a variant of Gaussian elimination and needs roughly the same number of arithmetic operations.
nibble  ground-up  cs  tcs  algorithms  complexity  linear-algebra  numerics  sci-comp  fields
5 days ago by nhaliday
Teach debugging
A friend of mine and I couldn't understand why some people were having so much trouble; the material seemed like common sense. The Feynman Method was the only tool we needed.

1. Write down the problem
2. Think real hard
3. Write down the solution

The Feynman Method failed us on the last project: the design of a divider, a real-world-scale project an order of magnitude more complex than anything we'd been asked to tackle before. On the day he assigned the project, the professor exhorted us to begin early. Over the next few weeks, we heard rumors that some of our classmates worked day and night without making progress.

...

And then, just after midnight, a number of our newfound buddies from dinner reported successes. Half of those who started from scratch had working designs. Others were despondent, because their design was still broken in some subtle, non-obvious way. As I talked with one of those students, I began poring over his design. And after a few minutes, I realized that the Feynman method wasn't the only way forward: it should be possible to systematically apply a mechanical technique repeatedly to find the source of our problems. Beneath all the abstractions, our projects consisted purely of NAND gates (woe to those who dug around our toolbox enough to uncover dynamic logic), which outputs a 0 only when both inputs are 1. If the correct output is 0, both inputs should be 1. The input that isn't is in error, an error that is, itself, the output of a NAND gate where at least one input is 0 when it should be 1. We applied this method recursively, finding the source of all the problems in both our designs in under half an hour.
techtariat  dan-luu  engineering  programming  debugging  IEEE  reflection  stories  education  higher-ed  checklists  iteration-recursion  divide-and-conquer  thinking  ground-up  nitty-gritty  giants  feynman  error  input-output  structure  composition-decomposition  abstraction  systematic-ad-hoc  reduction  teaching  state
6 weeks ago by nhaliday
Hyperbolic angle - Wikipedia
A unit circle {\displaystyle x^{2}+y^{2}=1} x^2 + y^2 = 1 has a circular sector with an area half of the circular angle in radians. Analogously, a unit hyperbola {\displaystyle x^{2}-y^{2}=1} {\displaystyle x^{2}-y^{2}=1} has a hyperbolic sector with an area half of the hyperbolic angle.
nibble  math  trivia  wiki  reference  physics  relativity  concept  atoms  geometry  ground-up  characterization  measure  definition  plots  calculation  nitty-gritty  direction  metrics  manifolds
november 2017 by nhaliday
What is the connection between special and general relativity? - Physics Stack Exchange
Special relativity is the "special case" of general relativity where spacetime is flat. The speed of light is essential to both.
nibble  q-n-a  overflow  physics  relativity  explanation  synthesis  hi-order-bits  ground-up  gravity  summary  aphorism  differential  geometry
november 2017 by nhaliday
What is the difference between general and special relativity? - Quora
General Relativity is, quite simply, needed to explain gravity.

Special Relativity is the special case of GR, when the metric is flat — which means no gravity.

You need General Relativity when the metric gets all curvy, and when things start to experience gravitation.
nibble  q-n-a  qra  explanation  physics  relativity  synthesis  hi-order-bits  ground-up  gravity  summary  aphorism  differential  geometry
november 2017 by nhaliday
Genetics: CHROMOSOMAL MAPS AND MAPPING FUNCTIONS
Any particular gene has a specific location (its "locus") on a particular chromosome. For any two genes (or loci) alpha and beta, we can ask "What is the recombination frequency between them?" If the genes are on different chromosomes, the answer is 50% (independent assortment). If the two genes are on the same chromosome, the recombination frequency will be somewhere in the range from 0 to 50%. The "map unit" (1 cM) is the genetic map distance that corresponds to a recombination frequency of 1%. In large chromosomes, the cumulative map distance may be much greater than 50cM, but the maximum recombination frequency is 50%. Why? In large chromosomes, there is enough length to allow for multiple cross-overs, so we have to ask what result we expect for random multiple cross-overs.

1. How is it that random multiple cross-overs give the same result as independent assortment?

Figure 5.12 shows how the various double cross-over possibilities add up, resulting in gamete genotype percentages that are indistinguisable from independent assortment (50% parental type, 50% non-parental type). This is a very important figure. It provides the explanation for why genes that are far apart on a very large chromosome sort out in crosses just as if they were on separate chromosomes.

2. Is there a way to measure how close together two crossovers can occur involving the same two chromatids? That is, how could we measure whether there is spacial "interference"?

Figure 5.13 shows how a measurement of the gamete frequencies resulting from a "three point cross" can answer this question. If we would get a "lower than expected" occurrence of recombinant genotypes aCb and AcB, it would suggest that there is some hindrance to the two cross-overs occurring this close together. Crosses of this type in Drosophila have shown that, in this organism, double cross-overs do not occur at distances of less than about 10 cM between the two cross-over sites. ( Textbook, page 196. )

3. How does all of this lead to the "mapping function", the mathematical (graphical) relation between the observed recombination frequency (percent non-parental gametes) and the cumulative genetic distance in map units?

Figure 5.14 shows the result for the two extremes of "complete interference" and "no interference". The situation for real chromosomes in real organisms is somewhere between these extremes, such as the curve labelled "interference decreasing with distance".
org:junk  org:edu  explanation  faq  nibble  genetics  genomics  bio  ground-up  magnitude  data  flux-stasis  homo-hetero  measure  orders  metric-space  limits  measurement
october 2017 by nhaliday
Best Topology Olympiad ***EVER*** - Affine Mess - Quora
Most people take courses in topology, algebraic topology, knot theory, differential topology and what have you without once doing anything with a finite topological space. There may have been some quirky questions about such spaces early on in a point-set topology course, but most of us come out of these courses thinking that finite topological spaces are either discrete or only useful as an exotic counterexample to some standard separation property. The mere idea of calculating the fundamental group for a 4-point space seems ludicrous.

Only it’s not. This is a genuine question, not a joke, and I find it both hilarious and super educational. DO IT!!
nibble  qra  announcement  math  geometry  topology  puzzles  rec-math  oly  links  math.AT  ground-up  finiteness  math.GN
october 2017 by nhaliday
Resonance in a Pendulum - YouTube
The vibration of any given washer is able to transmit its energy only to another washer with exactly the same frequency. Since the length of a pendulum determines its frequency of vibration, each pendulum can only set another pendulum vibrating if it has the same length.
nibble  video  social  physics  mechanics  waves  oscillation  synchrony  flux-stasis  increase-decrease  concrete  ground-up  dirty-hands  phys-energy  frequency  spreading
september 2017 by nhaliday
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

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