scholarly journals Keplerian trigonometry

2021 ◽  
Author(s):  
Alessandro Gambini ◽  
Giorgio Nicoletti ◽  
Daniele Ritelli
Keyword(s):  
Elliptic Curve ◽  
Real Interval ◽  
Planar Curve ◽  
Signed Area

AbstractTaking the hint from usual parametrization of circle and hyperbola, and inspired by the pathwork initiated by Cayley and Dixon for the parametrization of the “Fermat” elliptic curve $$x^3+y^3=1$$ x 3 + y 3 = 1 , we develop an axiomatic study of what we call “Keplerian maps”, that is, functions $${{\,\mathrm{{\mathbf {m}}}\,}}(\kappa )$$ m ( κ ) mapping a real interval to a planar curve, whose variable $$\kappa $$ κ measures twice the signed area swept out by the O-ray when moving from 0 to $$\kappa $$ κ . Then, given a characterization of k-curves, the images of such maps, we show how to recover the k-map of a given parametric or algebraic k-curve, by means of suitable differential problems.

scholarly journals A complete diophantine characterization of the rational torsion of an elliptic curve

2011 ◽  
Vol 28 (1) ◽  
pp. 83-96 ◽  
Author(s):  
Irene García-Selfa ◽  
José M. Tornero
Keyword(s):  
Elliptic Curve

scholarly journals CHARACTERIZING O-MINIMAL GROUPS IN TAME EXPANSIONS OF O-MINIMAL STRUCTURES

2019 ◽  
pp. 1-26
Author(s):  
Pantelis E. Eleftheriou
Keyword(s):  
Elliptic Curve ◽  
Independent Set ◽  
Dimension Theory ◽  
Maximal Dimension ◽  
Minimal Group ◽  
Full Characterization ◽  
Minimal Structure ◽  
Definable Groups ◽  
Topological Closure

We establish the first global results for groups definable in tame expansions of o-minimal structures. Let ${\mathcal{N}}$ be an expansion of an o-minimal structure ${\mathcal{M}}$ that admits a good dimension theory. The setting includes dense pairs of o-minimal structures, expansions of ${\mathcal{M}}$ by a Mann group, or by a subgroup of an elliptic curve, or a dense independent set. We prove: (1) a Weil’s group chunk theorem that guarantees a definable group with an o-minimal group chunk is o-minimal, (2) a full characterization of those definable groups that are o-minimal as those groups that have maximal dimension; namely, their dimension equals the dimension of their topological closure, (3) as an application, if ${\mathcal{N}}$ expands ${\mathcal{M}}$ by a dense independent set, then every definable group is o-minimal.

scholarly journals A characterization of chaos

1986 ◽  
Vol 34 (2) ◽  
pp. 283-292 ◽  
Author(s):  
K. Janková ◽  
J. Smítal
Keyword(s):  
Topological Entropy ◽  
Real Interval ◽  
Complex Behaviour ◽  
Positive Topological Entropy ◽  
Continuous Map ◽  
Continuous Mappings

Consider the continuous mappings f from a compact real interval to itself. We show that when f has a positive topological entropy (or equivalently, when f has a cycle of order ≠ 2n, n = 0, 1, 2, …) then f has a more complex behaviour than chaoticity in the sense of Li and Yorke: something like strong or uniform chaoticity, distinguishable on a certain level ɛ > 0. Recent results of the second author then imply that any continuous map has exactly one of the following properties: It is either strongly chaotic or every trajectory is approximable by cycles. Also some other conditions characterizing chaos are given.

scholarly journals Tropical mirror symmetry for elliptic curves

2017 ◽  
Vol 2017 (732) ◽  
pp. 211-246 ◽  
Author(s):  
Janko Böhm ◽  
Kathrin Bringmann ◽  
Arne Buchholz ◽  
Hannah Markwig
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Mirror Symmetry ◽  
Alternative Proof ◽  
Hurwitz Numbers ◽  
Feynman Integrals ◽  
Feynman Graphs ◽  
Correspondence Theorem ◽  
Theoretical Results

Abstract Mirror symmetry relates Gromov–Witten invariants of an elliptic curve with certain integrals over Feynman graphs [10]. We prove a tropical generalization of mirror symmetry for elliptic curves, i.e., a statement relating certain labeled Gromov–Witten invariants of a tropical elliptic curve to more refined Feynman integrals. This result easily implies the tropical analogue of the mirror symmetry statement mentioned above and, using the necessary Correspondence Theorem, also the mirror symmetry statement itself. In this way, our tropical generalization leads to an alternative proof of mirror symmetry for elliptic curves. We believe that our approach via tropical mirror symmetry naturally carries the potential of being generalized to more adventurous situations of mirror symmetry. Moreover, our tropical approach has the advantage that all involved invariants are easy to compute. Furthermore, we can use the techniques for computing Feynman integrals to prove that they are quasimodular forms. Also, as a side product, we can give a combinatorial characterization of Feynman graphs for which the corresponding integrals are zero. More generally, the tropical mirror symmetry theorem gives a natural interpretation of the A-model side (i.e., the generating function of Gromov–Witten invariants) in terms of a sum over Feynman graphs. Hence our quasimodularity result becomes meaningful on the A-model side as well. Our theoretical results are complemented by a Singular package including several procedures that can be used to compute Hurwitz numbers of the elliptic curve as integrals over Feynman graphs.

THE PLANARITY PROBLEM FOR SIGNED GAUSS WORDS

1993 ◽  
Vol 02 (04) ◽  
pp. 359-367 ◽  
Author(s):  
GRANT CAIRNS ◽  
DANIEL M. ELTON
Keyword(s):  
Necessary Condition ◽  
Sufficient Condition ◽  
Original Condition ◽  
Necessary And Sufficient Condition ◽  
Planar Curve ◽  
Necessary And Sufficient ◽  
Number Of Authors

C.F. Gauss gave a necessary condition for a word to be the intersection word of a closed normal planar curve and he gave an example which showed that his condition was not sufficient. M. Dehn provided a solution to the planarity problem [3] and subsequently, different solutions have been given by a number of authors (see [9]). However, all of these solutions are algorithmic in nature. As B. Grünbaum remarked in [7], “they are of the same aesthetically unpleasing character as MacLane’s [1937] criterion for planarity of graphs. A characterization of Gauss codes in the spirit of the Kuratowski criterion for planarity of graphs is still missing”. In this paper we use the work of J. Scott Carter [2] to give a necessary and sufficient condition for planarity of signed Gauss words which is analogous to Gauss’s original condition.

scholarly journals On a characterization of Shimura's elliptic curve over ℚ(√37)

1996 ◽  
Vol 77 (2) ◽  
pp. 157-171 ◽  
Author(s):  
Masanari Kida
Keyword(s):  
Elliptic Curve

scholarly journals Explicit characterization of the torsion growth of rational elliptic curves with complex multiplication over quadratic fields

2021 ◽  
Vol 56 (1) ◽  
pp. 47-61
Author(s):  
Enrique González-Jiménez ◽  
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Complex Multiplication ◽  
Number Fields ◽  
Quadratic Fields ◽  
Fixed Degree ◽  
Explicit Characterization

In a series of papers we classify the possible torsion structures of rational elliptic curves base-extended to number fields of a fixed degree. In this paper we turn our attention to the question of how the torsion of an elliptic curve with complex multiplication defined over the rationals grows over quadratic fields. We go further and we give an explicit characterization of the quadratic fields where the torsion grows in terms of some invariants attached to the curve.

Sign in / Sign up

Export Citation Format

Share Document