Legendre Form Research Articles

DIRECTIONAL DIFFERENTIABILITY OF THE OPTIMAL VALUE IN QUADRATIC PROGRAMMING PROBLEMS UNDER LINEAR CONSTRAINTS ON HILBERT SPACES

We investigate the first-order directional differentiability of the optimal value function in parametric quadratic programming problems under linear constraints in Hilbert spaces. We derive an explicit formula for computing the directional derivative of the optimal value function in cases where the quadratic part of the objective function is in Legendre form.

Algebraic Geometry and Elliptic Integrals Approach for Calculation of the Propagation Time of a Signal Between GPS Satellites with Account of General Relativity Theory

The approach for calculating the propagation time of a signal between GPS satellites will be summarized, based on the proposed new theoretical approach in several previous publications, as well as the perspectives for future development of the theory. Topics include: 1. Basic notions of inter-satellite communications. 2. Shapiro delay formulae in General Relativity Theory - basic formalism and the necessity to extend the formalism by taking into account the satellite motion on a plane elliptic or space-distributed elliptic orbit. 3. Basic facts about the disturbed motion in celestial mechanics and the necessity to incorporate it in the theory of inter-satellite communications, accounting for General Relativity Effects. 4. Propagation time of a signal, emitted by a satellite on a plane and also space-distributed elliptical orbit in terms of zero-order elliptic integrals and respectively of higher order integrals. Proof of the real-valuedness of the propagation time for all cases as one of the criteria for the correctness of the theoretical approach. 5. New analytical algorithms for calculation of zero-order elliptic integrals in the Legendre form. Relation to two representations in the Weierstrass form. 6. The new formalism of intersecting four-dimensional null cones and the resulting physical notions of the (intersecting)) space-time interval (with the property of being positive, negative or equal to zero) and the (intersecting) geodesic distance (being only positive, because is related to the distance, travelled by light or radio signals). Proof of these properties in the general case and in some partial cases. New numerical estimate E lim > 45.002510943228 [deg], above which the space-time interval is positive and thus inter-satellite communications between satellites on one plane elliptical orbit are possible. The angular distance of 45[ deg] is typical for the disposition of 8 satellites on one orbit in the Russian satellite constellation GLON ASS, so it might be claimed that such a configuration is favourable from the point of view of inter-satellite communications (with account of GRT effects).

On elements of large order on elliptic curves and multiplicative dependent images of rational functions over finite fields

Let E1 and E2 be elliptic curves in Legendre form with integer parameters. We show there exists a constant C such that for almost all primes, for all but at most C pairs of points on the reduction of E1×E2 modulo p having equal x coordinate, at least one among P1 and P2 has a large group order. We also show similar abundance over finite fields of elements whose images under the reduction modulo p of a finite set of rational functions have large multiplicative orders.

Exact analysis of the nonlinear vibration of an archetypal non-natural oscillator

The exact periodic solution of an archetypal non-natural oscillator with quadratic inertia and cubic nonlinear static stiffness has been derived in terms of the elliptic integral of the third kind. The periodic solution was derived based on a new form of elliptic integral that arose naturally from the model of the archetypal non-natural oscillator. It was shown that this new form of elliptic integral is an alternate form of the elliptic integral of the third kind and it has several advantages over the classical Legendre form. Hence, some new properties of elliptic integrals were derived based on the new elliptic integral. A bifurcation analysis of the present oscillator revealed the possible combinations of static stiffness parameters that can produce periodic solutions and the corresponding initial amplitude range where these periodic solutions exist. The bifurcation analysis informed investigations on the frequency–amplitude response which included the effect of the different static stiffness parameters. It was observed that the frequency increased for an increase in the positive cubic nonlinear stiffness parameter and vice versa. Additionally, the large-amplitude displacement history showed the same profile notwithstanding the combination and magnitudes of the stiffness parameters.

On Schwarzschild’s Interior Solution and Perfect Fluid Star Model

We solve the boundary value problem for Einstein’s gravitational field equations in the presence of matter in the form of an incompressible perfect fluid of density ρ and pressure field p(r) located in a ball r≤r0. We find a 1-parameter family of time-independent and radially symmetric solutions ga,ρa,pa:−2m<a<a1 satisfying the boundary conditions g=gS and p=0 on r=r0, where gS is the exterior Schwarzschild solution (solving the gravitational field equations for a point mass M concentrated at r=0) and containing (for a=0) the interior Schwarzschild solution, i.e., the classical perfect fluid star model. We show that Schwarzschild’s requirement r0>9κM/(4c2) identifies the “physical” (i.e., such that pa(r)≥0 and pa(r) is bounded in 0≤r≤r0) solutions {pa:a∈U0} for some neighbourhood U0⊂(−2m,+∞) of a=0. For every star model {ga:a0<a<a1}, we compute the volume V(a) of the region r≤r0 in terms of abelian integrals of the first, second, and third kind in Legendre form.

Semistable models of elliptic curves over residue characteristic 2

AbstractGiven an elliptic curve E in Legendre form $y^2 = x(x - 1)(x - \lambda )$ over the fraction field of a Henselian ring R of mixed characteristic $(0, 2)$, we present an algorithm for determining a semistable model of E over R that depends only on the valuation of $\lambda $. We provide several examples along with an easy corollary concerning $2$-torsion.

Kummer for Genus One Over Prime-Order Fields

This work considers the problem of fast and secure scalar multiplication using curves of genus one defined over a field of prime order. Previous work by Gaudry and Lubicz (Finite Fields Appl 15(2):246–260, 2009) had suggested the use of the associated Kummer line to speed up scalar multiplication. In the present work, we explore this idea in detail. The first task is to obtain an elliptic curve in Legendre form which satisfies necessary security conditions such that the associated Kummer line has small parameters and a base point with small coordinates. It turns out that the ladder step on the Kummer line supports parallelism and can be implemented very efficiently in constant time using the single-instruction multiple-data (SIMD) operations available in modern processors. For the 128-bit security level, this work presents three Kummer lines denoted as \(K_1:=\mathsf{KL2519(81,20)}\), \(K_2:=\mathsf{KL25519(82,77)}\) and \(K_3:=\mathsf{KL2663(260,139)}\) over the three primes \(2^{251}-9\), \(2^{255}-19\) and \(2^{266}-3\), respectively. Implementations of scalar multiplications for all three Kummer lines using Intel intrinsics have been done, and the code is publicly available. Timing results on the Skylake and the Haswell processors of Intel indicate that both fixed base and variable base scalar multiplications for \(K_1\) and \(K_2\) are faster than those achieved by Sandy2x, which is a highly optimised SIMD implementation in assembly of the well-known Curve25519. On Skylake, both fixed base and variable base scalar multiplications for \(K_3\) are faster than Sandy2x, whereas on Haswell, fixed base scalar multiplication for \(K_3\) is faster than Sandy2x while variable base scalar multiplication for both \(K_3\) and Sandy2x takes roughly the same time. In practical terms, the particular Kummer lines that are introduced in this work are serious candidates for deployment and standardisation. We further illustrate the usefulness of the proposed Kummer lines by instantiating the quotient Digital Signature Algorithm on all the three Kummer lines.

Connecting Legendre with Kummer and Edwards

Scalar multiplication on suitable Legendre form elliptic curves can be speeded up in two ways. One can perform the bulk of the computation either on the associated Kummer line or on an appropriate twisted Edwards form elliptic curve. This paper provides details of moving to and from between Legendre form elliptic curves and associated Kummer line and moving to and from between Legendre form elliptic curves and related twisted Edwards form elliptic curves. Further, concrete twisted Edwards form elliptic curves are identified which correspond to known Kummer lines at the 128-bit security level which provide very fast scalar multiplication on modern architectures supporting SIMD operations.

Legendre Forms in Reflexive Banach Spaces

Legendre forms are used in the literature for second-order sufficient optimality conditions of optimization problems in (reflexive) Banach spaces. We show that if a Legendre form exists on a reflexive Banach space, then this space is already isomorphic to a Hilbert space.

On the Solution Existence of Nonconvex Quadratic Programming Problems in Hilbert Spaces

We propose conditions for the existence of solutions for nonconvex quadratic programming problems whose constraint set is defined by finitely many convex linear-quadratic inequalities in Hilbert spaces. In order to obtain our results, we use either properties of the Legendre form or properties of compact operators with closed range. The results are established without requesting the convexity of the objective function or the compactness of the constraint set. As a special case, we obtain some on the existence of solutions results for the quadratic programming problems under linear constraints in Hilbert spaces.

On the Solution Existence of Convex Quadratic Programming Problems in Hilbert Spaces

We provide solution existence results for the convex quadratic programming problems in Hilbert spaces, which the constraint set is defined by finitely many convex quadratic inequalities. In order to obtain our results, we shall use either the properties of the Legendre form or the properties of the finite-rank operator. The existence results are established without requesting neither coercivity of the objective function nor compactness of the constraint set.

Six unlikely intersection problems in search of effectivity

AbstractWe investigate four properties related to an elliptic curveEtin Legendre form with parametert: the curveEthas complex multiplication,E−thas complex multiplication, a point onEtwith abscissa 2 is of finite order, andtis a root of unity. Combining all pairs of properties leads to six problems on unlikely intersections. Using a variety of techniques we solve these problems with varying degrees of effectivity (and for three of them we even present the list of all possiblet).

Matrix Arithmetic-Geometric Mean and the Computation of the Logarithm

We investigate the stability of the matrix arithmetic-geometric mean (AGM) iteration. We show that the classical formulation of this iteration may be not stable (a necessary and sufficient condition for its stability is given) and investigate the numerical properties of alternative formulations. It turns out that the so-called Legendre form is the right choice for matrices. Due to its fast convergence and good numerical properties, our AGM formulation has the potential to play an important role in the computation of matrix functions. In fact, we developed an algorithm, whose main block is an optimized AGM scheme, for the computation of the logarithm of a matrix, which is shown to be competitive, in terms of accuracy, with the state-of-the-art methods. Methods that do not require an initial reduction to the Schur form are potentially more efficient on parallel computers. For this reason, our current implementation does not include such reduction and operates with full matrices till the end. As compared to ...

Characterization of local quadratic growth for strong minima in the optimal control of semi-linear elliptic equations

In this article we consider an optimal control problem of a semi-linear elliptic equation, with bound constraints on the control. Our aim is to characterize local quadratic growth for the cost function J J in the sense of strong solutions. This means that the function J J grows quadratically over all feasible controls whose associated state is close enough to the nominal one, in the uniform topology. The study of strong solutions, classical in the Calculus of Variations, seems to be new in the context of PDE optimization. Our analysis, based on a decomposition result for the variation of the cost, combines Pontryagin’s principle and second order conditions. While these two ingredients are known, we use them in such a way that we do not need to assume that the Hessian of the Lagrangian of the problem is a Legendre form, or that it is uniformly positive on an extended set of critical directions.

Infinite rank of elliptic curves over Q\mathrmab

If $E$ is an elliptic curve defined over a quadratic field $K$, and the $j$-invariant of $E$ is not $0$ or $1728$, then $E(\mathbb{Q}^{\mathrm{ab}})$ has infinite rank. If $E$ is an elliptic curve in Legendre form, $y^2 = x(x-1)(x-\lambda)$, where $\math

Positive rank quadratic twists of four elliptic curves

Let K be a number field and E i / K an elliptic curve defined over K for i = 1 , 2 , 3 , 4 . We prove that there exists a number field L containing K such that there are infinitely many d k ∈ L × / ( L × ) 2 such that E i d k ( L ) has positive rank, equivalently all four elliptic curves E i have growth of the rank over each of quadratic extensions L k : = L ( d k ) , more strongly, for any i 1 , i 2 , … , i m , rank ( E i ( L i 1 ⋯ L i m ) ) > rank ( E i ( L i 1 ⋯ L i m − 1 ) ) > ⋯ > rank ( E i ( L i 1 ) ) > rank ( E i ( L ) ) . We also prove that if each elliptic curve E i for i = 1 , 2 , 3 can be written in Legendre form over a cubic extension K of a number field k , then there are infinitely many d ∈ k × / ( k × ) 2 such that E i d ( K ) for i = 1 , 2 , 3 is of positive rank.

Mesures d'indépendance linéaire de carrés de périodes et quasi-périodes de courbes elliptiques

A linear independence measure over Q is obtained for values of some generalized hypergeometric functions at rational points and for the squares of real periods and quasi-periods of elliptic curves in Legendre form y 2 = x ( x − 1 ) ( x − 1 / q ) for almost every q ∈ Z .

A [formula omitted]-adic point counting algorithm for elliptic curves on legendre form

In 2000 T. Satoh gave the first p - adic point counting algorithm for elliptic curves over finite fields. Satoh' s algorithm was followed by the Satoh–Skjernaa–Taguchi algorithm and furthermore by the arithmetic–geometric mean and modified SST algorithms for characteristic two only. All four algorithms are important to Elliptic Curve Cryptography. In this paper, we present the general framework for p -adic point counting and we apply it to elliptic curves on Legendre form. We show how the λ - modular polynomial can be used for lifting the curve and Frobenius isogeny to characteristic zero and we show how the associated multiplier gives the action of the lifted Frobenius isogeny on the invariant differential. The result is a point counting algorithm for elliptic curves on Legendre form. The algorithm runs in a time complexity of O ( n 2 μ + 1 / ( μ + 1 ) ) for fixed p and a space complexity of O ( n 2 ) where p n is the field size. We include results from experimeriments in characteristic p = 3 , 5 , … , 19 .

Legendre Elliptic Curves over Finite Fields

We show that every elliptic curve over a finite field of odd characteristic whose number of rational points is divisible by 4 is isogenous to an elliptic curve in Legendre form, with the sole exception of a minimal respectively maximal elliptic curve. We also collect some results concerning the supersingular Legendre parameters.

Legendre Elliptic Curves over Finite Fields

We show that every elliptic curve over a finite field of odd characteristic whose number of rational points is divisible by 4 is isogenous to an elliptic curve in Legendre form, with the sole exception of a minimal respectively maximal elliptic curve. We also collect some results concerning the supersingular Legendre parameters.