2,543 research outputs found
GKZ-Generalized Hypergeometric Systems in Mirror Symmetry of Calabi-Yau Hypersurfaces
We present a detailed study of the generalized hypergeometric system
introduced by Gel'fand, Kapranov and Zelevinski (GKZ-hypergeometric system) in
the context of toric geometry. GKZ systems arise naturally in the moduli theory
of Calabi-Yau toric varieties, and play an important role in applications of
the mirror symmetry. We find that the Gr\"obner basis for the so-called toric
ideal determines a finite set of differential operators for the local solutions
of the GKZ system. At the special point called the large radius limit, we find
a close relationship between the principal parts of the operators in the GKZ
system and the intersection ring of a toric variety. As applications, we
analyze general three dimensional hypersurfaces of Fermat and non-Fermat types
with Hodge numbers up to . We also find and analyze several non
Landau-Ginzburg models which are related to singular models.Comment: 55 pages, 3 Postscript figures, harvma
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
Strongly internal sets and generalized smooth functions
Based on a refinement of the notion of internal sets in Colombeau's theory,
so-called strongly internal sets, we introduce the space of generalized smooth
functions, a maximal extension of Colombeau generalized functions. Generalized
smooth functions as morphisms between sets of generalized points form a
sub-category of the category of topological spaces. In particular, they can be
composed unrestrictedly.Comment: 17 pages, some minor correction
On Smarandache's form of the individual Fermat-Euler theorem
In the paper it is shown how a form of the classical FERMAT-EULER Theorem discovered by F. SMARANDACHE fits into the generalizations found by S.SCHWARZ, M.LASSAK and the author. Then we show how SMARANDACHE'S
algorithm can be used to effective computations of the so called group membership
On -isogenies over of elliptic curves with rational -invariants
The main goal of this paper is to determine for which prime numbers
can an elliptic curve~ defined over have an -isogeny over
. We study this question under various assumptions on the
2-torsion of . Apart from being a natural question itself, the mod~
representations attached to such arise in the Darmon program for the
generalized Fermat equation of signature , playing a key role in the
proof of modularity of certain Frey varieties in the recent work of Billerey,
Chen, Dieulefait and Freitas.Comment: 8 pages. This was previously an appendix to arXiv:2205.1586
Topological and algebraic structures on the ring of Fermat reals
The ring of Fermat reals is an extension of the real field containing
nilpotent infinitesimals, and represents an alternative to Synthetic
Differential Geometry in classical logic. In the present paper, our first aim
is to study this ring from using standard topological and algebraic structures.
We present the Fermat topology, generated by a complete pseudo-metric, and the
omega topology, generated by a complete metric. The first one is closely
related to the differentiation of (non standard) smooth functions defined on
open sets of Fermat reals. The second one is connected to the differentiation
of smooth functions defined on infinitesimal sets. Subsequently, we prove that
every (proper) ideal is a set of infinitesimals whose order is less than or
equal to some real number. Finally, we define and study roots of
infinitesimals. A computer implementation as well as an application to
infinitesimal Taylor formulas with fractional derivatives are presented.Comment: 43 page
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