5,730 research outputs found
Difference Ramsey Numbers and Issai Numbers
We present a recursive algorithm for finding good lower bounds for the
classical Ramsey numbers. Using notions from this algorithm we then give some
results for generalized Schur numbers, which we call Issai numbers.Comment: 10 page
On Range Searching with Semialgebraic Sets II
Let be a set of points in . We present a linear-size data
structure for answering range queries on with constant-complexity
semialgebraic sets as ranges, in time close to . It essentially
matches the performance of similar structures for simplex range searching, and,
for , significantly improves earlier solutions by the first two authors
obtained in~1994. This almost settles a long-standing open problem in range
searching.
The data structure is based on the polynomial-partitioning technique of Guth
and Katz [arXiv:1011.4105], which shows that for a parameter , , there exists a -variate polynomial of degree such that
each connected component of contains at most points
of , where is the zero set of . We present an efficient randomized
algorithm for computing such a polynomial partition, which is of independent
interest and is likely to have additional applications
Evaluating the Impact of Treating the Optimal Subgroup
Suppose we have a binary treatment used to influence an outcome. Given data
from an observational or controlled study, we wish to determine whether or not
there exists some subset of observed covariates in which the treatment is more
effective than the standard practice of no treatment. Furthermore, we wish to
quantify the improvement in population mean outcome that will be seen if this
subgroup receives treatment and the rest of the population remains untreated.
We show that this problem is surprisingly challenging given how often it is an
(at least implicit) study objective. Blindly applying standard techniques fails
to yield any apparent asymptotic results, while using existing techniques to
confront the non-regularity does not necessarily help at distributions where
there is no treatment effect. Here we describe an approach to estimate the
impact of treating the subgroup which benefits from treatment that is valid in
a nonparametric model and is able to deal with the case where there is no
treatment effect. The approach is a slight modification of an approach that
recently appeared in the individualized medicine literature
Fast integer multiplication using generalized Fermat primes
For almost 35 years, Sch{\"o}nhage-Strassen's algorithm has been the fastest
algorithm known for multiplying integers, with a time complexity O(n
log n log log n) for multiplying n-bit inputs. In 2007, F{\"u}rer
proved that there exists K > 1 and an algorithm performing this operation in
O(n log n K log n). Recent work by Harvey, van der Hoeven,
and Lecerf showed that this complexity estimate can be improved in order to get
K = 8, and conjecturally K = 4. Using an alternative algorithm, which relies on
arithmetic modulo generalized Fermat primes, we obtain conjecturally the same
result K = 4 via a careful complexity analysis in the deterministic multitape
Turing model
Survey on counting special types of polynomials
Most integers are composite and most univariate polynomials over a finite
field are reducible. The Prime Number Theorem and a classical result of
Gau{\ss} count the remaining ones, approximately and exactly.
For polynomials in two or more variables, the situation changes dramatically.
Most multivariate polynomials are irreducible. This survey presents counting
results for some special classes of multivariate polynomials over a finite
field, namely the the reducible ones, the s-powerful ones (divisible by the
s-th power of a nonconstant polynomial), the relatively irreducible ones
(irreducible but reducible over an extension field), the decomposable ones, and
also for reducible space curves. These come as exact formulas and as
approximations with relative errors that essentially decrease exponentially in
the input size.
Furthermore, a univariate polynomial f is decomposable if f = g o h for some
nonlinear polynomials g and h. It is intuitively clear that the decomposable
polynomials form a small minority among all polynomials. The tame case, where
the characteristic p of Fq does not divide n = deg f, is fairly
well-understood, and we obtain closely matching upper and lower bounds on the
number of decomposable polynomials. In the wild case, where p does divide n,
the bounds are less satisfactory, in particular when p is the smallest prime
divisor of n and divides n exactly twice. The crux of the matter is to count
the number of collisions, where essentially different (g, h) yield the same f.
We present a classification of all collisions at degree n = p^2 which yields an
exact count of those decomposable polynomials.Comment: to appear in Jaime Gutierrez, Josef Schicho & Martin Weimann
(editors), Computer Algebra and Polynomials, Lecture Notes in Computer
Scienc
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