Jacobian Curve Research Articles

Jacobian curve of singular foliations

Jacobian curve of singular foliations

Blockchain assisted node validation approach for security provisioning in WSN environment

Wireless sensor networks (WSNs) are a new technology that helps with a variety of practical uses, involving healthcare and monitoring the environment. In recent years, security has been considered as important topic in WSN since it is vulnerable to several security threats. Recent works uses cryptographic techniques to ensure security in WSN. In existing works, the security methodologies require high resources but still assure low level security. To resolve this issue, this paper proposes a node validation method which is lightweight as well as assures high level security. The main idea behind this work is to integrate Blockchain technology with WSN environment. We presented a novel Blockchain-assisted Node Validation (BlockNode) methodology for ensuring high level security. To maintain energy efficiency, the network is segregated into multiple clusters by Valid Cluster Formation (VCF) approach. In each cluster, optimum CH is selected by using type-II fuzzy algorithm. The VCF approach only allows the valid nodes which are authorized by Blockchain validation. Then, the data transmission is secured by Jacobian Curve Encryption (JCE) algorithm. For optimal route selection, Energy-aware Reinforcement Learning (ERL) algorithm is proposed. Overall, the proposed work high level security with minimum resource consumption. The experimental results obtained from NS-3.25 simulation tool confirms that the proposed work achieves better performance in security level, encryption & decryption time, delay, energy consumption, delivery ratio and throughput.

Invertibility of the dual energy x-ray data transform.

The Alvarez-Macovski method extracts the x-ray energy-dependent information by expanding the attenuation coefficient as a linear combination of functions of energy multiplied by basis set coefficients. Since the basis functions are known a priori, the coefficients represent all the energy-dependent information. The method then computes the line integrals of these coefficients, summarized as a vector A, from measurements with multiple x-ray spectra, summarized as a vector L. The purpose of this paper is to determine the factors that affect the invertibility of the L(A) transformation with a two function basis set and two spectral measurements, the dual energy transformation. A general invertibility theorem is applied that requires testing for zero values of the Jacobian of the transformation in its input domain. General conditions for invertibility are proved. It is shown that the generalized A vector noise variance is proportional to the generalized measurement noise variance divided by the square of the Jacobian. The relationship between the zero Jacobian values and ambiguous sets of A vector points with the same L values is determined. The effect of zero Jacobian values on an iterative algorithm that inverts L(A) is simulated. The choice of a particular valid basis set does not affect invertibility. Nonoverlapping measurement spectra such as those from photon counting detectors with perfect pulse height analysis are invertible. The widely used x-ray tube spectra with different voltages are shown to be invertible. Spectra with the same maximum energy, such as those from layered detectors, approach noninvertibility with small absolute value Jacobian for large object thicknesses. The zero Jacobian values fall on curves in A vector space that, except for a simple artificial case, are close to but not exactly straight lines. With noninvertible spectra, pairs of ambiguous points are located on opposite sides of the zero Jacobian curve. The iterative algorithm has large convergence errors near zero Jacobian curves and converges to the closest ambiguous point to the initial estimate for other points. The invertibility of dual energy systems is determined by the presence of zero values of the Jacobian of the dual x-ray energy data transformation L(A) in the input domain.

Computing the intersections of three conics according to their Jacobian curve

In this study, we describe the intersection of three conics based on the singularities of their corresponding Jacobian curve. In particular, we show that certain singular points and sub-lines of the Jacobian curve are the precise common points and common tangent lines of the conics, respectively. Based on our results, these points or the tangent line can be computed as the singularities of the Jacobian curve. These results facilitate investigations of the relationships between a net of conics and their Jacobian curve.

Special Fibres and Critical Locus for a Pencil of Plane Curve Singularities

We will analyze the relationships between the special fibres of a pencil Λ of plane curve singularities and the Jacobian curve J of Λ (defined by the zero locus of the Jacobian determinant for any fixed basis ′ ∈ Λ). From the results, we find decompositions of J (and of any special fibre of the pencil) in terms of the minimal resolution of Λ. Using these decompositions and the topological type of any generic pair of curves of Λ, we obtain some topological information about J. More precise decompositions for J can be deduced from the minimal embedded resolution of any pair of fibres (not necessarily generic) or from the minimal embedded resolution of all the special fibres.

Exceptional curves on rational numerically elliptic surfaces

A numerically elliptic surface is a complete smooth algebraic surface X over an algebraically closed field k with a proper morphism f: X-* C to a smooth curve C such that the general fiber off is an integral curve of arithmetic genus 1. If the generic fiber is smooth the surface is called elliptic; otherwise it is called quasi-elliptic. The latter only can occur if k has characteristic 2 or 3, in which case the general fiber is a rational curve with an ordinary cusp [BM]. If no fiber off contains an exceptional cwue (i.e., a smooth irreducible curve isomorphic to [Fp’ and having self-intersection l), then X is said to be minimal; all elliptic surfaces will hereafter be assumed to be minimal. One says that an elliptic surface X is Jacobian if the smooth points of the generic fiber X, comprise the Jacobian curve of X,,. It is equivalent for X, to have a rational point, or for f to have a section. For a rational Jacobian numerically elliptic surface, the exceptional curves are precisely the sections of the libration, which provides a tool by which an enumeration of the exceptional curves on a rational Jacobian numerically elliptic surface can be carried out (cf. [MP, HL, MOP]). Whether Jacobian or not, a rational

Orders of the torsion of points of curves of genus 1

Let K be an algebraic number field of degree n; let be the number of divisor classes of the field K; y: v2=u4+au2+B is the Jacobian curve over $$K; B\left( {a^2 - 48} \right) = c^2 \prod\limits_{i - 1}^N {q_i } $$ where C is an integral divisor, q1, ..., qN are distinct prime divisors. One proves that there exists an effectively computable constant c=c(n, h(K), N), such that the order m of the torsion of any primitive K-point on is bounded by it: m⩽C.

Local factorization of the coordinates of torsion points

Let ℐ be a Jacobian curve, defined over the global field K, and let g be the group of K-points on ℐ of finite order. In this note we give the local factorization of the coordinates of points from g.

Notes on a Net of Quadric Surfaces: III. The Scroll of Trisecants of the Jacobian Curve

Notes on a Net of Quadric Surfaces: III. The Scroll of Trisecants of the Jacobian Curve

A special net of quadrics

It is well-known that a general net of quadric surfaces cannot be obtained as the net of polar quadrics of the points of a plane in regard to a cubic surface; in order that it may be so obtained it must have various properties that a general net of quadrics does not have. The locus of the vertices of the cones which belong to the net of quadrics is a curve ϑ –the Jacobian curve of the net of quadrics, and the trisecants of ϑ generate a scroll. Any plane which contains two trisecants of ϑ is a bitangent plane of the scroll and, for a general net of quadrics, there are eighteen of these bitangent planes passing through an arbitrary point. When however the net of quadrics is a net of polar quadrics it is found that any plane which contains two trisecants of ϑ contains two other trisecants also; it thus contains four trisecants in all and counts six times as a bitangent plane of the scroll. The bitangent developable of the scroll, which is, for a general net of quadrics, of class eighteen, degenerates, in the special case when the net of quadrics is a net of polar quadrics, into a developable of class three counted six times; the planes of the developable are therefore the osculating planes of a twisted cubic γ. The plane which, together with a cubic surface, gives rise to the net of polar quadrics must be one of the osculating planes of γ. It is also found, further, that the osculating planes of γ are grouped into pentahedra, the vertices of all these pentahedra lying on ϑ.

Certain contact properties of linear systems of hypersurfaces

where the ('s are homogeneous parameters and thef's are homogeneous functions of degree n in the variables x0, xl, , xr, represents an o P-system, I VI, of hypersurfaces of order n in an r-space, Sr. Interpreting the i's as the homogeneous coordinates of a point of a p-space, 2p, we have a one-to-one correspondence between the points of 2p and the hypersurfaces of VI . In this paper we propose to deal with certain contact properties of the system j Vj and to investigate some of the properties and relations to one another of the corresponding loci in 2p. Our results will be generalizations of certain results which W. L. Edget has, in connection with his study of octadic surfaces, described for the case r = 3, p =2, n = 2. We should mention that T. R. Hollcroft has derived: the properties of the curve in the plane 12 Of the parameters which corresponds to the Jacobian curve of a net of hypersurfaces in Sr, and in another paper? those of the surface in the 3-dimensional space 13 of the parameters which corresponds to the Jacobian surface of a web of surfaces in S3. In neither of these papers, however, has the author touched upon the results which we are going to derive. In the following we shall, first, describe the general case. This is done in ?1. Then we shall consider the case r=p=2 in ?2 and the case n=2, r=p general in ?3. Finally, in ?4, we shall conclude the paper with a description of the results for n = 2, r =p = 3. 1. The general case. Let n, r, p be general. Since for a hypersurface to acquire a conical point or hypernode is equivalent to one condition, there are ooPhypersurfaces of the system I VI each possessing a hypernode. The

The Jacobian Curve of a Net of Quadrics

Much has been written, from the algebraical as well as from the geometrical standpoint, on the subject of pencils of quadrics: algebraically the problem consists of the canonical reduction of a pencil of quadratic forms, and the classical paper on the subject is by Weierstrass. But among the different kinds of pencils of quadratic forms there is the “singular pencil,” in which the discriminant of every form belonging to the pencil is zero; interpreted geometrically this means that every quadric belonging to the pencil is a cone. This case was expressly excluded from consideration by Weierstrass, and the canonical reduction was only accomplished later by Kronecker. But, although Weierstrass and Kronecker together solved completely the problem of the canonical reduction of a pencil of quadratic forms, a much clearer insight into the nature of the problem was gained when Segre gave the geometrical solution. He published two papers, one dealing with the non-singular pencils and the other with the singular pencil.

X. On hyperjacobian surfaces and curves

In a paper published in the ‘Mathematische Annalen’ (vol. iii. p. 459), Brill has discussed the question of curves having three-point contact with a doubly infinite pencil of curves; and in particular he has investigated some of the properties of the curve passing through all the points of contact with the individual curves of the pencil. In the same Journal (vol. x. p. 221) Krey of Kiel has applied a method similar to that of Brill with partial success to the question of curves having four-point contact with a triply infinite pencil. Some formulæ, however, given in my paper “On the Sextactic Points of a Plane Curve” (Phil. Trans. 1865, p. 657) have proved to be directly applicable to both questions. An application of them to Brill’s problem will be found in the ‘Comptes Rendus’ for 1876 (2nd semestre, p. 627), and a solution of Krey’s problem in the 'Proceedings of the London Mathematical Society for the same year (vol. viii. p. 29). The present subject was in the first instance suggested by the foregoing papers; and from one point of view it may be regarded as an attempt to extend the question to the case of surfaces; viz. to determine a curve which shall pass through the points of contact of a given surface U with certain surfaces belonging to a pencil V, and to investigate some of its properties. From a slightly different point of view, however, it may be considered as an extension of two ideas, viz. first, that of the Jacobian surface, or locus of the points whose polar planes with regard to four surfaces meet in a point; and secondly, that of the Jacobian curve, or locus of points whose polar planes with regard to three surfaces have a right line in common. More particularly, commencing with the facts, first, that if a surface of the form a ϕ + b ψ + c Z touch a surface U, the point of contact is a point on the Jacobian, and secondly, that if a surface of the form aϕ + bψ touch a surface U, the point of contact is a point on the Jacobian curve, I have endeavoured to extend them to higher degrees of contact.