polynomials   133
[1705.00098] Xorshift random number generators from primitive polynomials
A class of xorshift random number generators (RNGs) are introduced by Marsaglia. We have proposed an algorithm which constructs a full period xorshift RNG from a given primitive polynomial. It is shown there is a weakness present in those RNGs and is suggested its improvement. A separate algorithm is also proposed which returns a full period xorshift generator with desired number of xorshift operations.%We also introduce the notion of tweaked primitive multiple-recursive matrix method with improved linear complexity.
cryptography  algorithms  number-theory  polynomials  rather-interesting  performance-measure  nudge-targets  consider:stress-testing
may 2017 by Vaguery
[1109.2396] New Solutions of \$d=2x^3+y^3+z^3\$
We discuss finding large integer solutions of d=2x3+y3+z3 by using Elsenhans and Jahnel's adaptation of Elkies' LLL-reduction method. We find 28 first solutions for |d|<10000.
number-theory  algebra  polynomials  constraint-satisfaction  rather-interesting  to-write-about  nudge-targets  consider:looking-to-see  stamp-collecting  algorithms
april 2017 by Vaguery
Peter Norvig, the meaning of polynomials, debugging as psychotherapy | Quomodocumque
He briefly showed a demo where, given values of a polynomial, a machine can put together a few lines of code that successfully computes the polynomial. But the code looks weird to a human eye. To compute some quadratic, it nests for-loops and adds things up in a funny way that ends up giving the right output. So has it really ”learned” the polynomial? I think in computer science, you typically feel you’ve learned a function if you can accurately predict its value on a given input. For an algebraist like me, a function determines but isn’t determined by the values it takes; to me, there’s something about that quadratic polynomial the machine has failed to grasp. I don’t think there’s a right or wrong answer here, just a cultural difference to be aware of. Relevant: Norvig’s description of “the two cultures” at the end of this long post on natural language processing (which is interesting all the way through!)
mathtariat  org:bleg  nibble  tech  ai  talks  summary  philosophy  lens  comparison  math  cs  tcs  polynomials  nlp  debugging  psychology  cog-psych  complex-systems  deep-learning  analogy  legibility  interpretability
march 2017 by nhaliday
6.896: Essential Coding Theory
- probabilistic method and Chernoff bound for Shannon coding
- probabilistic method for asymptotically good Hamming codes (Gilbert coding)
- sparsity used for LDPC codes
mit  course  yoga  tcs  complexity  coding-theory  math.AG  fields  polynomials  pigeonhole-markov  linear-algebra  probabilistic-method  lecture-notes  bits  sparsity  concentration-of-measure  linear-programming  linearity  expanders  hamming  pseudorandomness  crypto  rigorous-crypto  communication-complexity  no-go  madhu-sudan  shannon  unit  p:**
february 2017 by nhaliday
What is the relationship between information theory and Coding theory? - Quora
basically:
- finite vs. asymptotic
- combinatorial vs. probabilistic (lotsa overlap their)
- worst-case (Hamming) vs. distributional (Shannon)

Information and coding theory most often appear together in the subject of error correction over noisy channels. Historically, they were born at almost exactly the same time - both Richard Hamming and Claude Shannon were working at Bell Labs when this happened. Information theory tends to heavily use tools from probability theory (together with an "asymptotic" way of thinking about the world), while traditional "algebraic" coding theory tends to employ mathematics that are much more finite sequence length/combinatorial in nature, including linear algebra over Galois Fields. The emergence in the late 90s and first decade of 2000 of codes over graphs blurred this distinction though, as code classes such as low density parity check codes employ both asymptotic analysis and random code selection techniques which have counterparts in information theory.

They do not subsume each other. Information theory touches on many other aspects that coding theory does not, and vice-versa. Information theory also touches on compression (lossy & lossless), statistics (e.g. large deviations), modeling (e.g. Minimum Description Length). Coding theory pays a lot of attention to sphere packing and coverings for finite length sequences - information theory addresses these problems (channel & lossy source coding) only in an asymptotic/approximate sense.
q-n-a  qra  math  acm  tcs  information-theory  coding-theory  big-picture  comparison  confusion  explanation  linear-algebra  polynomials  limits  finiteness  math.CO  hi-order-bits  synthesis  probability  bits  hamming  shannon  intricacy  nibble  s:null  signal-noise
february 2017 by nhaliday
Ehrhart polynomial - Wikipedia
In mathematics, an integral polytope has an associated Ehrhart polynomial that encodes the relationship between the volume of a polytope and the number of integer points the polytope contains. The theory of Ehrhart polynomials can be seen as a higher-dimensional generalization of Pick's theorem in the Euclidean plane.
math  math.MG  trivia  polynomials  discrete  wiki  reference  atoms  geometry  spatial  nibble  curvature  convexity-curvature
january 2017 by nhaliday

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