156,272 research outputs found
Reliability and reproducibility of Atlas information
We discuss the reliability and reproducibility of much of the information
contained in the Atlas of Finite Groups
Sato-Tate distributions of twists of y^2=x^5-x and y^2=x^6+1
We determine the limiting distribution of the normalized Euler factors of an
abelian surface A defined over a number field k when A is isogenous to the
square of an elliptic curve defined over k with complex multiplication. As an
application, we prove the Sato-Tate Conjecture for Jacobians of Q-twists of the
curves y^2=x^5-x and y^2=x^6+1, which give rise to 18 of the 34 possibilities
for the Sato-Tate group of an abelian surface defined over Q. With twists of
these two curves one encounters, in fact, all of the 18 possibilities for the
Sato-Tate group of an abelian surface that is isogenous to the square of an
elliptic curve with complex multiplication. Key to these results is the
twisting Sato-Tate group of a curve, which we introduce in order to study the
effect of twisting on the Sato-Tate group of its Jacobian.Comment: minor edits, 42 page
The Brauer characters of the sporadic simple Harada-Norton group and its automorphism group in characteristics 2 and 3
We determine the 2-modular and 3-modular character tables of the sporadic
simple Harada-Norton group and its automorphism group.Comment: 29 page
Fast and Powerful Hashing using Tabulation
Randomized algorithms are often enjoyed for their simplicity, but the hash
functions employed to yield the desired probabilistic guarantees are often too
complicated to be practical. Here we survey recent results on how simple
hashing schemes based on tabulation provide unexpectedly strong guarantees.
Simple tabulation hashing dates back to Zobrist [1970]. Keys are viewed as
consisting of characters and we have precomputed character tables
mapping characters to random hash values. A key
is hashed to . This schemes is
very fast with character tables in cache. While simple tabulation is not even
4-independent, it does provide many of the guarantees that are normally
obtained via higher independence, e.g., linear probing and Cuckoo hashing.
Next we consider twisted tabulation where one input character is "twisted" in
a simple way. The resulting hash function has powerful distributional
properties: Chernoff-Hoeffding type tail bounds and a very small bias for
min-wise hashing. This also yields an extremely fast pseudo-random number
generator that is provably good for many classic randomized algorithms and
data-structures.
Finally, we consider double tabulation where we compose two simple tabulation
functions, applying one to the output of the other, and show that this yields
very high independence in the classic framework of Carter and Wegman [1977]. In
fact, w.h.p., for a given set of size proportional to that of the space
consumed, double tabulation gives fully-random hashing. We also mention some
more elaborate tabulation schemes getting near-optimal independence for given
time and space.
While these tabulation schemes are all easy to implement and use, their
analysis is not
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