Search Results (10)

During the last 15–20 years, the experimental methods for autonomous navigation and inter-satellite links have been developing rapidly in order to ensure navigation control and data processing without commands from Earth stations. Inter-satellite links are related to relative ranging between the satellites from one constellation or different constellations and measuring the distances between them with the precision of at least 1 $\mathsf{\mu }$m micrometer (=${10}^{-6}$ m), which should account for the bending of the light (radio or laser) signals due to the action of the Earth’s gravitational field. Thus, the theoretical calculation of the propagation time of a signal should be described in the framework of general relativity theory and the s.c. null cone equation. This review paper summarizes the latest achievements in calculating the propagation time of a signal, emitted by a GPS satellite, moving along a plane elliptical orbit or a space-oriented orbit, described by the full set of six Kepler parameters. It has been proved that for the case of plane elliptical orbit, the propagation time is expressed by a sum of elliptic integrals of the first, the second and the third kind, while for the second case (assuming that only the true anomaly angle is the dynamical parameter), the propagation time is expressed by a sum of elliptic integrals of the second and of the fourth order. For both cases, it has been proved that the propagation time represents a real-valued expression and not an imaginary one, as it should be. For the typical parameters of a GPS orbit, numerical calculations for the first case give acceptable values of the propagation time and, especially, the Shapiro delay term of the order of nanoseconds, thus confirming that this is a propagation time for the signal and not for the time of motion of the satellite. Theoretical arguments, related to general relativity and differential geometry have also been presented in favor of this conclusion. A new analytical method has been developed for transforming an elliptic integral in the Legendre form into an integral in the Weierstrass form. Two different representations have been found, one of them based on the method of four-dimensional uniformization, exposed in the monograph of Whittaker and Watson. The result of this approach is a new formulae for the Weierstrass invariants, depending in a complicated manner on the modulus parameter q of the elliptic integral in the Legendre form. Full article

We consider a series which combines two Dirichlet series constructed from the coefficients of a Laurent series and derive a general integral representation of the series as a Mellin transform. As an application, we obtain a family of Mellin integral identities involving the Weierstrass elliptic functions and some Lambert series. These identities are used to derive some of the properties of the Lambert series. Full article

In this paper, we introduce secure groups as a cryptographic scheme representing finite groups together with a range of operations, including the group operation, inversion, random sampling, and encoding/decoding maps. We construct secure groups from oblivious group representations combined with cryptographic protocols, implementing the operations securely. We present both generic and specific constructions, in the latter case specifically for number-theoretic groups commonly used in cryptography. These include Schnorr groups (with quadratic residues as a special case), Weierstrass and Edwards elliptic curve groups, and class groups of imaginary quadratic number fields. For concreteness, we develop our protocols in the setting of secure multiparty computation based on Shamir secret sharing over a finite field, abstracted away by formulating our solutions in terms of an arithmetic black box for secure finite field arithmetic or for secure integer arithmetic. Secure finite field arithmetic suffices for many groups, including Schnorr groups and elliptic curve groups. For class groups, we need secure integer arithmetic to implement Shanks’ classical algorithms for the composition of binary quadratic forms, which we will combine with our adaptation of a particular form reduction algorithm due to Agarwal and Frandsen. As a main result of independent interest, we also present an efficient protocol for the secure computation of the extended greatest common divisor. The protocol is based on Bernstein and Yang’s constant-time 2-adic algorithm, which we adapt to work purely over the integers. This yields a much better approach for multiparty computation but raises a new concern about the growth of the Bézout coefficients. By a careful analysis, we are able to prove that the Bézout coefficients in our protocol will never exceed $3max(a,b)$ in absolute value for inputs a and b. We have integrated secure groups in the Python package MPyC and have implemented threshold ElGamal and threshold DSA in terms of secure groups. We also mention how our results support verifiable multiparty computation, allowing parties to jointly create a publicly verifiable proof of correctness for the results accompanying the results of a secure computation. Full article

We introduce the real minimal surfaces family by using the Weierstrass data $({\zeta }^{-m},{\zeta }^m)$ for $\zeta \in \mathbb{C}$, $m\in {\mathbb{Z}}_{\ge 2},$ then compute the irreducible algebraic surfaces of the surfaces family in three-dimensional Euclidean space ${\mathbb{E}}^3$. In addition, we propose that family has a degree number (resp., class number) $2m(m+1)$ in the cartesian coordinates $x,y,z$ (resp., in the inhomogeneous tangential coordinates $a,b,c$). Full article

We consider the Enneper family of real maximal surfaces via Weierstrass data $(1,{\zeta }^m)$ for $\zeta \in \mathbb{C}$, $m\in {\mathbb{Z}}_{\ge 1}$. We obtain the irreducible surfaces of the family in the three dimensional Minkowski space ${\mathbb{E}}^{2,1}$. Moreover, we propose that the family has degree ${(2m+1)}^2$ (resp., class $2m(2m+1)$) in the cartesian coordinates $x,y,z$ (resp., in the inhomogeneous tangential coordinates $a,b,c$). Full article

In this work, we study the generalized 2D equal-width equation which arises in various fields of science. With the aid of numerous methods which includes Lie symmetry analysis, power series expansion and Weierstrass method, we produce closed-form solutions of this model. The exact solutions obtained are the snoidal wave, cnoidal wave, Weierstrass elliptic function, Jacobi elliptic cosine function, solitary wave and exponential function solutions. Moreover, we give a graphical representation of the obtained solutions using certain parametric values. Furthermore, the conserved vectors of the underlying equation are constructed by utilizing two approaches: the multiplier method and Noether’s theorem. The multiplier method provided us with four local conservation laws, whereas Noether’s theorem yielded five nonlocal conservation laws. The conservation laws that are constructed contain the conservation of energy and momentum. Full article

In this paper, fractal calculus, which is called $F^{\alpha }$-calculus, is reviewed. Fractal calculus is implemented on fractal interpolation functions and Weierstrass functions, which may be non-differentiable and non-integrable in the sense of ordinary calculus. Graphical representations of fractal calculus of fractal interpolation functions and Weierstrass functions are presented. Full article

We reveal the analytic relations between a matrix permanent and major nature’s complexities manifested in critical phenomena, fractal structures and chaos, quantum information processes in many-body physics, number-theoretic complexity in mathematics, and ♯P-complete problems in the theory of computational complexity. They follow from a reduction of the Ising model of critical phenomena to the permanent and four integral representations of the permanent based on (i) the fractal Weierstrass-like functions, (ii) polynomials of complex variables, (iii) Laplace integral, and (iv) MacMahon master theorem. Full article

We consider a family of higher degree Enneper minimal surface $E_m$ for positive integers m in the three-dimensional Euclidean space ${\mathbb{E}}^3$ . We compute algebraic equation, degree and integral free representation of Enneper minimal surface for $m=1,2,3$ . Finally, we give some results and relations for the family $E_m$ . Full article

Considering the Weierstrass data as $(\psi ,f,g)=(2,1-z^{-m},z^n)$ , we introduce a two-parameter family of Henneberg-type minimal surface that we call ${\mathfrak{H}}_{m,n}$ for positive integers $(m,n)$ by using the Weierstrass representation in the four-dimensional Euclidean space ${\mathbb{E}}^4$ . We define ${\mathfrak{H}}_{m,n}$ in $(r,\theta )$ coordinates for positive integers $(m,n)$ with $m\ne 1,n\ne -1,-m+n\ne -1$ , and also in $(u,v)$ coordinates, and then we obtain implicit algebraic equations of the Henneberg-type minimal surface of values $(4,2)$ . Full article