scholarly journals On elliptic curves with a closed component passing through a hexagon

2019 ◽  
Vol 27 (2) ◽  
pp. 67-82
Author(s):  
Miroslav Kureš
Keyword(s):  
Elliptic Curves ◽  
Numerical Experiments ◽  
Algebraic Curves ◽  
Closed Curve ◽  
Special Shape ◽  
The Third ◽  
Convex Pentagon

AbstractIn general, there exists an ellipse passing through the vertices of a convex pentagon, but any ellipse passing through the vertices of a convex hexagon does not have to exist. Thus, attention is turned to algebraic curves of the third degree, namely to the closed component of certain elliptic curves. This closed curve will be called the spekboom curve. Results of numerical experiments and some hypotheses regarding hexagons of special shape connected with the existence of this curve passing through the vertices are presented and suggested. Some properties of the spekboom curve are described, too.

scholarly journals Hyperelliptic uniformization of algebraic curves of the third order

2015 ◽  
Vol 284 ◽  
pp. 38-49 ◽  
Author(s):  
A.I. Aptekarev ◽  
D.N. Toulyakov ◽  
W. Van Assche
Keyword(s):  
Algebraic Curves ◽  
Third Order ◽  
The Third

scholarly journals INFINITUDE OF ELLIPTIC CARMICHAEL NUMBERS

2012 ◽  
Vol 92 (1) ◽  
pp. 45-60 ◽  
Author(s):  
AARON EKSTROM ◽  
CARL POMERANCE ◽  
DINESH S. THAKUR
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Residue Class ◽  
Complex Multiplication ◽  
Weak Form ◽  
Arithmetic Progressions ◽  
Carmichael Numbers ◽  
The Third ◽  
Fermat’S Little Theorem

AbstractIn 1987, Gordon gave an integer primality condition similar to the familiar test based on Fermat’s little theorem, but based instead on the arithmetic of elliptic curves with complex multiplication. We prove the existence of infinitely many composite numbers simultaneously passing all elliptic curve primality tests assuming a weak form of a standard conjecture on the bound on the least prime in (special) arithmetic progressions. Our results are somewhat more general than both the 1999 dissertation of the first author (written under the direction of the third author) and a 2010 paper on Carmichael numbers in a residue class written by Banks and the second author.

scholarly journals Numerical investigation on the effect of specimen gripping arrangement on dynamic shear characterization using Torsion Split Hopkinson Bar

2021 ◽  
Vol 250 ◽  
pp. 02032
Author(s):  
Bhaskar Ramagiri ◽  
Chandra Sekher Yerramalli
Keyword(s):  
Numerical Experiments ◽  
Pure Shear ◽  
Hopkinson Bar ◽  
Shear Loading ◽  
Dynamic Shear ◽  
Split Hopkinson Bar ◽  
The Third ◽  
Split Hopkinson ◽  
Tubular Specimens

Torsion Split Hopkinson Bar (TSHB) is widely used in the dynamic shear characterization of material under pure shear loading. In TSHB, tubular specimens with either circular or hexagonal flanges are used. The specimens with circular flanges are generally bonded using adhesive to the incident and transmission bars. The specimens with hexagonal flanges are gripped into the hexagonal holders that are fixed onto incident and transmission bars. In the current study, numerical simulations are carried out to see the effect of gripping arrangements on the dynamic shear characterization quality. Numerical experiments with three gripping configurations are studied—the first gripping configuration with a direct bond (numerically-tie) between specimen and bars. The second configuration with the specimen gripped by hexagonal holders fixed to bars. The third configuration with specimen directly gripped into the incident and transmission bars having hexagonal slots.

scholarly journals SUPERSINGULAR EDWARDS CURVES AND EDWARDS CURVE POINTS COUNTING METHOD OVER FINITE FIELD

2020 ◽  
pp. 68-88
Author(s):  
Ruslan Skuratovskii
Keyword(s):  
Finite Field ◽  
Elliptic Curves ◽  
Algebraic Curves ◽  
Discrete Logarithm ◽  
Edwards Curves ◽  
Projective Curves ◽  
Birational Equivalence ◽  
Supersingular Curves ◽  
Edwards Curve ◽  
Embedding Degree

We consider problem of order counting of algebraic affine and projective curves of Edwards [2, 8] over the finite field $F_{p^n}$. The complexity of the discrete logarithm problem in the group of points of an elliptic curve depends on the order of this curve (ECDLP) [4, 20] depends on the order of this curve [10]. We research Edwards algebraic curves over a finite field, which are one of the most promising supports of sets of points which are used for fast group operations [1]. We construct a new method for counting the order of an Edwards curve over a finite field. It should be noted that this method can be applied to the order of elliptic curves due to the birational equivalence between elliptic curves and Edwards curves. We not only find a specific set of coefficients with corresponding field characteristics for which these curves are supersingular, but we additionally find a general formula by which one can determine whether a curve $E_d [F_p]$ is supersingular over this field or not. The embedding degree of the supersingular curve of Edwards over $F_{p^n}$ in a finite field is investigated and the field characteristic, where this degree is minimal, is found. A birational isomorphism between the Montgomery curve and the Edwards curve is also constructed. A one-to-one correspondence between the Edwards supersingular curves and Montgomery supersingular curves is established. The criterion of supersingularity for Edwards curves is found over $F_{p^n}$.

A NEW REPRESENTATION FOR QUARTIC CURVES AND COMPLETE SETS OF GEOMETRIC INVARIANTS

1999 ◽  
Vol 13 (08) ◽  
pp. 1137-1149 ◽  
Author(s):  
MUSTAFA UNEL ◽  
WILLIAM A. WOLOVICH
Keyword(s):  
Algebraic Curve ◽  
Closed Curve ◽  
Free Form ◽  
Zero Sets ◽  
Geometric Invariants ◽  
The Third ◽  
Open Curve ◽  
Complete Sets ◽  
Quartic Curves

Many free-form object boundaries can be modeled by quartics with bounded zero sets. The fact that any nondegenerate closed-bounded algebraic curve of even degree n=2p can be expressed as the product of p conics, which are real ellipses, plus a remaining polynomial of degree n-2,12 can be utilized to express a nondegenerate quartic as the product of two leading ellipses plus a third conic which might be either a closed curve (an ellipse) or an open curve (a hyperbola). However, it can be shown that the leading ellipses can be modified with appropriate constants by constraining the third conic to be a circle, thus implying a 2-ellipse and 1-circle; i.e. an elliptical-circular(E2C)representation of the quartic. The use of such representations is to simplify the analysis of quartics by exploiting the well-known properties of conics and to develop a set of functionally independent geometric invariants for recognition purposes. Also, it is shown that the underlying Euclidean transformation between two configurations of the same quartic can be determined using the centers of the three conics.

scholarly journals Spectral discretizations of the Darcy's equations with non standard boundary conditions

2018 ◽  
Vol 10 (1) ◽  
Author(s):  
Bernard Jean-Marie
Keyword(s):  
Error Estimate ◽  
Numerical Experiments ◽  
Flow In Porous Media ◽  
Optimal Error Estimate ◽  
Discrete Solution ◽  
Optimal Error ◽  
The Third ◽  
Darcy Problem ◽  
Darcy’S Equations ◽  
Variational Formulations

This paper is devoted to the approximation of anonstandard Darcy problem, which modelizes the flow in porous media, byspectral methods: the pressure is assigned on a part of the boundary.We propose two variational formulations, as well as three spectraldiscretizations. The second discretization improves the approximation of thedivergence-free condition, but the error estimate on the pressure is notoptimal, while the third one leads to optimal error estimate with adivergence-free discrete solution, which is important for someapplications. Next, their numerical analysis is performed in detailand we present some numerical experiments which confirm the interestof the third discretization.

scholarly journals Overcoming a stack of rough cylinders by a six-legged robot

2021 ◽  
pp. 1-28
Author(s):  
Yury Filippovich Golubev ◽  
Victor Vladimirovich Korianov
Keyword(s):  
Computer Modeling ◽  
Dry Friction ◽  
Horizontal Plane ◽  
Numerical Experiments ◽  
Rolling Friction ◽  
Legged Robot ◽  
Equilibrium Conditions ◽  
Modeling Tools ◽  
The Third

A method has been developed for overcoming obstacles in the form of a stack of rough cylinders of different radii by a six-legged robot. Two of the cylinders are lying on a horizontal plane and the third one is lying on the two mentioned above. The cylinders are under the influence of gravity, dry friction, and rolling friction. The equilibrium conditions of the system were found when the system is under the influence of a force due to the support of the robot on the system of cylinders. Using computer modeling tools, an algorithm for overcoming the stack that does not destroy the structure of the obstacle was developed. The results of numerical experiments and corresponding video materials are presented.

scholarly journals Criterions of Supersinguliarity and Groups of Montgomery and Edwards Curves in Cryptography

2021 ◽  
Vol 19 ◽  
pp. 709-722
Author(s):  
Ruslan Skuratovskii ◽  
Volodymyr Osadchyy
Keyword(s):  
Normal Form ◽  
Finite Field ◽  
Elliptic Curves ◽  
Algebraic Curves ◽  
Minimum Degree ◽  
Basic Algorithm ◽  
Edwards Curves ◽  
Edwards Curve ◽  
Birational Isomorphism ◽  
Twisted Edwards Curve

We consider the algebraic affine and projective curves of Edwards over the finite field Fpn. It is well known that many modern cryptosystems can be naturally transformed into elliptic curves. The criterions of the supersingularity of Montgomery and Edwards curves are found. In this paper, we extend our previous research into those Edwards algebraic curves over a finite field and we construct birational isomorphism of them with cubic in Weierstrass normal form. One class of twisted Edwards is researched too. We propose a novel effective method of point counting for both Edwards and elliptic curves. In addition to finding a specific set of coefficients with corresponding field characteristics for which these curves are supersingular, we also find a general formula by which one can determine whether or not a curve Ed[Fp] is supersingular over this field. The method proposed has complexity O( p log2 2 p ) . This is an improvement over both Schoof’s basic algorithm and the variant which makes use of fast arithmetic (suitable for only the Elkis or Atkin primes numbers) with complexities O(log8 2 pn) and O(log4 2 pn) respectively. The embedding degree of the supersingular curve of Edwards over Fpn in a finite field is additionally investigated. Singular points of twisted Edwards curve are completely described. Due existing the birational isomorphism between twisted Edwards curve and elliptic curve in Weierstrass normal form the result about order of this curve over finite field is extended on cubic in Weierstrass normal form. Also it is considered minimum degree of an isogeny (distance) between curves of this two classes when such isogeny exists. We extend the existing isogenous of elliptic curves.

scholarly journals Tropical extra-tropical thermocline water mass exchanges in the Community Climate Model v.3 Part I: the Atlantic Ocean

Ocean Science ◽  
2006 ◽  
Vol 2 (2) ◽  
pp. 137-146 ◽  
Author(s):  
I. Wainer ◽  
A. Lazar ◽  
A. Solomon
Keyword(s):  
Atlantic Ocean ◽  
Numerical Experiment ◽  
Numerical Experiments ◽  
Climate Model ◽  
Water Mass ◽  
Coupled Model ◽  
Atmospheric Component ◽  
The Third ◽  
Simulation Results ◽  
Fully Coupled

Abstract. The climatological annual mean tropical-extra-tropical pathways of thermocline waters in the Atlantic Ocean are investigated with the NCAR CCSM numerical coupled model. Results from three numerical experiments are analyzed: Two are fully coupled runs with different spatial resolution (T42 and T85) for the atmospheric component. The third numerical experiment is an ocean-only run forced by NCEP winds and fluxes. Results show that the different atmospheric resolutions have a significant impact on the subduction pathways in the Atlantic because of how the wind field is represented. These simulation results also show that the water subducted at the subtropics reaching the EUC is entirely from the South Atlantic. The coupled model ability to simulate the STCs is discussed.

scholarly journals CERTAIN VALUES OF GAUSSIAN HYPERGEOMETRIC SERIES AND A FAMILY OF ALGEBRAIC CURVES

2012 ◽  
Vol 08 (04) ◽  
pp. 945-961 ◽  
Author(s):  
RUPAM BARMAN ◽  
GAUTAM KALITA
Keyword(s):  
Finite Field ◽  
Elliptic Curves ◽  
Algebraic Curve ◽  
Hypergeometric Series ◽  
Algebraic Curves ◽  
Alternate Proof ◽  
Special Values ◽  
Gaussian Hypergeometric Series

Let λ ∈ ℚ\{0, -1} and l ≥ 2. Denote by Cl, λ the nonsingular projective algebraic curve over ℚ with affine equation given by [Formula: see text] In this paper, we give a relation between the number of points on Cl, λ over a finite field and Gaussian hypergeometric series. We also give an alternate proof of a result of [D. McCarthy, 3F2 Hypergeometric series and periods of elliptic curves, Int. J. Number Theory6(3) (2010) 461–470]. We find some special values of 3F2 and 2F1 Gaussian hypergeometric series. Finally we evaluate the value of 3F2(4) which extends a result of [K. Ono, Values of Gaussian hypergeometric series, Trans. Amer. Math. Soc.350(3) (1998) 1205–1223].

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