483 research outputs found
Generalized explicit descent and its application to curves of genus 3
We introduce a common generalization of essentially all known methods for
explicit computation of Selmer groups, which are used to bound the ranks of
abelian varieties over global fields. We also simplify and extend the proofs
relating what is computed to the cohomologically-defined Selmer groups. Selmer
group computations have been practical for many Jacobians of curves over Q of
genus up to 2 since the 1990s, but our approach is the first to be practical
for general curves of genus 3. We show that our approach succeeds on some
genus-3 examples defined by polynomials with small coefficients.Comment: 58 pages; added a few references, and updated a few other
Prediction of Adolescent Sexual Reoffending: A Meta-Analysis of the J-SOAP-II, ERASOR, J-SORRAT-II, and Static-99
Several risk assessment tools, including the Juvenile Sex Offender Assessment Protocol-II (Prentky & Righthand, 2003), the Estimate of Risk of Adolescent Sexual Offense Recidivism (Worling & Curwen, 2001), the Juvenile Sexual Offense Recidivism Risk Assessment Tool-II (Epperson, ralston, Fowers, DeWitt, & Gore, 2006), and the Static-99 (Hanson & Thornton, 1999), have been used to assess reoffense risk among adolescents who have committed sexual offenses. Given that research on these tools has yielded somewhat mixed results, we empirically synthesized 33 published and unpublished studies involving 6,196 male adolescents who had committed a sexual offense. We conducted two separate meta-analyses, first with correlations and then with AUCs. Total scores on each of the tools significantly predicted sexual reoffending, with aggregated correlations ranging from .12 - .20 and aggregated area under the receiver operating characteristic curve (AUC) scores ranging from .64 - .67. In many cases, however, heterogeneity across studies was moderate to high. There were no significant differences between tools, and although the Static-99 was developed for adults, it achieved similar results as the adolescent tools. To help interpret these findings, results are compared to other meta-analyses of risk tools used in the area of violence risk assessment and in other fields
Incomplete Quadratic Exponential Sums in Several Variables
We consider incomplete exponential sums in several variables of the form
S(f,n,m) = \frac{1}{2^n} \sum_{x_1 \in \{-1,1\}} ... \sum_{x_n \in \{-1,1\}}
x_1 ... x_n e^{2\pi i f(x)/p}, where m>1 is odd and f is a polynomial of degree
d with coefficients in Z/mZ. We investigate the conjecture, originating in a
problem in computational complexity, that for each fixed d and m the maximum
norm of S(f,n,m) converges exponentially fast to 0 as n grows to infinity. The
conjecture is known to hold in the case when m=3 and d=2, but existing methods
for studying incomplete exponential sums appear to be insufficient to resolve
the question for an arbitrary odd modulus m, even when d=2. In the present
paper we develop three separate techniques for studying the problem in the case
of quadratic f, each of which establishes a different special case of the
conjecture. We show that a bound of the required sort holds for almost all
quadratic polynomials, a stronger form of the conjecture holds for all
quadratic polynomials with no more than 10 variables, and for arbitrarily many
variables the conjecture is true for a class of quadratic polynomials having a
special form.Comment: 31 pages (minor corrections from original draft, references to new
results in the subject, publication information
"The numerical accuracy of truncated Ewald sums for periodic systems with long-range Coulomb interactions"
Ewald summation is widely used to calculate electrostatic interactions in
computer simulations of condensed-matter systems. We present an analysis of the
errors arising from truncating the infinite real- and Fourier-space lattice
sums in the Ewald formulation. We derive an optimal choice for the
Fourier-space cutoff given a screening parameter . We find that the
number of vectors in Fourier space required to achieve a given accuracy scales
with . The proposed method can be used to determine computationally
efficient parameters for Ewald sums, to assess the quality of Ewald-sum
implementations, and to compare different implementations.Comment: 6 pages, 3 figures (Encapsulated PostScript), LaTe
Constructions of diagonal quartic and sextic surfaces with infinitely many rational points
In this note we construct several infinite families of diagonal quartic
surfaces \begin{equation*} ax^4+by^4+cz^4+dw^4=0, \end{equation*} where
with infinitely many rational points and
satisfying the condition . In particular, we present an
infinite family of diagonal quartic surfaces defined over \Q with Picard
number equal to one and possessing infinitely many rational points. Further, we
present some sextic surfaces of type , , , or
, with infinitely many rational points.Comment: revised version will appear in International Journal of Number Theor
On the Quantum Invariant for the Brieskorn Homology Spheres
We study an exact asymptotic behavior of the Witten-Reshetikhin-Turaev
invariant for the Brieskorn homology spheres by use of
properties of the modular form following a method proposed by Lawrence and
Zagier. Key observation is that the invariant coincides with a limiting value
of the Eichler integral of the modular form with weight 3/2. We show that the
Casson invariant is related to the number of the Eichler integrals which do not
vanish in a limit . Correspondingly there is a
one-to-one correspondence between the non-vanishing Eichler integrals and the
irreducible representation of the fundamental group, and the Chern-Simons
invariant is given from the Eichler integral in this limit. It is also shown
that the Ohtsuki invariant follows from a nearly modular property of the
Eichler integral, and we give an explicit form in terms of the L-function.Comment: 26 pages, 2 figure
Constructing Elliptic Curves over with Moderate Rank
We give several new constructions for moderate rank elliptic curves over
. In particular we construct infinitely many rational elliptic
surfaces (not in Weierstrass form) of rank 6 over using
polynomials of degree two in . While our method generates linearly
independent points, we are able to show the rank is exactly 6 \emph{without}
having to verify the points are independent. The method generalizes; however,
the higher rank surfaces are not rational, and we need to check that the
constructed points are linearly independent.Comment: 11 page
- …