Determinantal representations of hyperbolic plane curves: An elementary approach

A smooth plane curve is said to admit a symmetric determinantal representation if it can be defined by the determinant of a symmetric matrix with entries in linear forms in three variables. We study the local-global principle for the existence of symmetric determinantal representations of smooth plane curves over a global field of characteristic different from two. When the degree of the plane curve is less than or equal to three, we relate the problem of finding symmetric determinantal representations to more familiar Diophantine problems on the Severi-Brauer varieties and mod 2 Galois representations, and prove that the local-global principle holds for conics and cubics. We also construct counterexamples to the local-global principle for quartics using the results of Mumford, Harris, and Shioda on theta characteristics.

${\cal U}$ntil now the representation (i.e. plotting) of curve in Parallel Coordinates is constructed from the point $\leftrightarrow$ line duality. The result is a ``line-curve'' which is seen as the envelope of it's tangents. Usually this gives an unclear image and is at the heart of the ``over-plotting'' problem; a barrier in the effective use of Parallel Coordinates. This problem is overcome by a transformation which provides directly the ``point-curve'' representation of a curve. Earlier this was applied to conics and their generalizations. Here the representation, also called dual, is extended to all planar algebraic curves. Specifically, it is shown that the dual of an algebraic curve of degree $n$ is an algebraic of degree at most $n(n - 1)$ in the absence of singular points. The result that conics map into conics follows as an easy special case. An algorithm, based on algebraic geometry using resultants and homogeneous polynomials, is obtained which constructs the dual image of the curve. This approach has potential generalizations to multi-dimensional algebraic surfaces and their approximation. The ``trade-off'' price then for obtaining {\em planar} representation of multidimensional algebraic curves and hyper-surfaces is the higher degree of the image's boundary which is also an algebraic curve in $\|$-coords.

For bivariate polynomials of degree we give fast numerical constructions of determinantal representations with matrices. Unlike some other existing constructions, our approach returns matrices of the smallest possible size for all (not just generic) polynomials of degree n and does not require any symbolic computation. We can apply these linearizations to numerically compute the roots of a system of two bivariate polynomials by using numerical methods for the two-parameter eigenvalue problems.

We study theory of curves in the complex hyperbola and show that special motions of curves are linked with the Burgers hierarchy, which also leads to a Hamiltonian formulation of the hierarchy and to the difference Burgers equation via their discretization. 1. Introduction It is widely recognized that a lot of differential equations in soliton theory arise from differential geometry especially theory of curves or surfaces ([1, 2, 3]). For example surfaces in the Euclidean 3-space with constant negative Gaussian or non-zero mean curvature are described by the sine or sinh Gordon equation respectively and proper affine spheres are described by the Tzitzeica equatio n. If we refer to theory of curves, the curvature of curves in the Euclidean 2-space evolves according to the mKdV equation under special motions. Pinkall ([4]) showed that the space of closed centroaffine curves in the centroaffine plane possesses a nat ural symplectic structure and the centroaffine curvature evolves according to the KdV e quation when the flow is generated by a Hamiltonian given by the total centroaffine curvature. Chou and Qu ([5, 6]) showed that many soliton equations arise from special motions of plane or space curves. Moreover, Hoffmann and Kutz ([7]) showed that special motions of curves in the complex 1 or 2-space or the complex projective line whose curvature evolves according to the mKdV or KdV equation can be discretized by use of cross ratios. In this paper we study theory of curves in the complex hyperbola which are determined by a certain curvature up to some symmetry. We shall show that special motions of curves are linked with the Burgers hierarchy, which can be formulated as a Hamiltonian system and also leads to the difference Burgers equation ([8]) via their discretization.

This paper presents an algorithm for approximating planar curves with conjugated circular arcs and straight-line segments. Approximation of digital planar curves using line segments and analytic curve pieces is a problem of fundamental importance in image analysis and pattern recognition. A specific variation of this problem is approximation by conjugated arcs and line segments, which also arises in robotics, particularly in motion planning. We propose an algorithmic solution to this problem, designed for the automatic generation of programs for robotic manipulators in the AS language. The originality of the method lies in employing a special representation of planar curves that is invariant under translation, rotation, and scaling. Such a representation enables the development of an optimal algorithm that automatically determines the number and placement of conjugated arcs and line segments approximating a given curve with a prescribed accuracy. The simplicity, flexibility with respect to input data, and low computational requirements of the proposed method make it suitable not only for image recognition tasks but also for robotic trajectory planning.

Cross-sectional profiles of medieval mouldings are often considered to be a vital form of data for art historians. Mouldings have often been used to highlight salient areas of buildings, and they provide a wealth of information. To date, little use of computer technology has been made in their acquisition, storage and analysis. The paper presents a representation for such planar curves, and an algorithm for its extraction from planar curves. The representation uses geometric primitives, specifically line and ore segments, which allow the curves to be represented in a compact form, approximating the circular sections more effectively than using linear segments alone. These idealised models provide templates which provide a means for the identification of specific features, within profiles, which can then serve as characteristics for use in case-based reasoning.

It is proved that every knot in the major subfamilies of J. Berge's lens space surgery (i.e., knots yielding a lens space by Dehn surgery) is presented by an L-shaped (real) plane curve as a "divide knot" defined by N. A'Campo in the context of singularity theory of complex curves. For each knot given by Berge's parameters, the corresponding plane curve is constructed. The surgery coefficients are also considered. Such presentations support us to study each knot itself, and the relationship among the knots in the set of lens space surgeries.

It is now well established that the negative emotions the child experiences for not understanding a mathematical topic mark their emotional memory associated with that topic. We’ve been investigating which tangible and accessible tools prevent the development of a pathological allergy to a fundamental concept as it is the Cartesian space, seeking for kid-friendly gates to the subject. When fear and pain for not understanding traces an escape pattern from this topic at a young age the child’s ability to relate to all its didactic applications can be seriously jeopardized, marking (when not identified) the school career and sub-sequent attitudes towards all the fields of theoretical and practical application of it.The elementary approach in explaining the Cartesian space principles to the children remains mainly linked to traditional visualization models of three-dimensional images on two-dimensional space, e.g., paper, blackboard, and screens. Only recently, augmented reality has been used as a teaching aid for visualizing objects in the actual three-dimensional space. Those systems are suitable for children naturally predisposed to mathematical and/or visual-cognitive intelligence, who are not suffering from any visual impairment. This is a non-inclusive system of access to understanding such fundamental topic as the Cartesian space. Topic which is later essential to an extended comprehension of geometry, mathematics, representation of objects, and concepts. The aim of the research was to find and test a support system to complement the standard two-dimensional and visual-only approach and to guarantee a complete and consistent sensorial experience of the definition of the Cartesian space through physical, material, and modular forms. We sought to create a bond between the concept and its real representation. This system should be extended to different ages of development and types of intelligence and backgrounds, transversal to environments and contexts of usage (family/school), also for visually impaired children. The developed tools pro-vide the child an early and positive emotional bond, prior to any traditional scholastic approach, with the fundamental principles of the Cartesian space through methods such as free play, trial and error, experimentation and share of the emotions while engaging in cooperative activities.

A set $S\subseteq\mathbb{R}^n$ is called semidefinite programming (SDP) representable or semidefinite representable if S equals the projection of a set in higher dimensional space which is describable by some linear matrix inequality (LMI). Clearly, if S is SDP representable, then S must be convex and semialgebraic (it is describable by conjunctions and disjunctions of polynomial equalities or inequalities). This paper proves sufficient conditions and necessary conditions for SDP representability of convex sets and convex hulls by proposing a new approach to constructing SDP representations. The contributions of this paper are: (i) For bounded SDP representable sets $W_1,\dots,W_m$, we give an explicit construction of an SDP representation for $\mathrm{conv}(\cup_{k=1}^mW_k)$. This provides a technique for building global SDP representations from the local ones. (ii) For the SDP representability of a compact convex semialgebraic set S, we prove sufficient: the boundary $\partial S$ is nonsingular and positively curved, while necessary is: $\partial S$ has nonnegative curvature at each nonsingular point. In terms of defining polynomials for S, nonsingular boundary amounts to them having nonvanishing gradient at each point on $\partial S$ and the curvature condition can be expressed as their strict versus nonstrict quasi-concavity at those points on $\partial S$ where they vanish. The gaps between them are $\partial S$ having or not having singular points either of the gradient or of the curvature's positivity. A sufficient condition bypassing the gaps is when some defining polynomials of S satisfy an algebraic condition called sos-concavity. (iii) For the SDP representability of the convex hull of a compact nonconvex semialgebraic set T, we find that the critical object is $\partial_c T$, the maximum subset of $\partial T$ contained in $\partial\mathrm{conv}(T)$. We prove sufficient for SDP representability: $\partial_c T$ is nonsingular and positively curved, and necessary is: $\partial_c T$ has nonnegative curvature at nonsingular points. The gaps between our sufficient and necessary conditions are similar to case (ii). The positive definite Lagrange Hessian (PDLH) condition, which meshes well with constructions, is also discussed.

Francis Sowerby Macaulay began his career working on Brill and Noether’s theory of algebraic plane curves and their interpretation of the Riemann–Roch and Cayley–Bacharach theorems; in fact it is Macaulay who first stated and proved the modern form of the Cayley–Bacharach theorem. Later in his career Macaulay developed ideas and results that have become important in modern commutative algebra, such as the notions of unmixedness, perfection (the Cohen–Macaulay property), and super-perfection (the Gorenstein property), work that was appreciated by Emmy Noether and the people around her. He also discovered results that are now fundamental in the theory of linkage, but this work was forgotten and independently rediscovered much later. The name of a computer algebra program (now Macaulay2) recognizes that much of his work is based on examples created by refined computation. Though he never spoke of the connection, the threads of Macaulay’s work lead directly from the problems on plane curves to his later theorems. In this paper we will explain what Macaulay did, and how his results are connected.

The dimensions of the graded quotients of the cohomology of a plane curve complement U = ℙ2 \ C with respect to the Hodge filtration are described in terms of simple geometrical invariants. The case of curves with ordinary singularities is discussed in detail. We also give a precise numerical estimate for the difference between the Hodge filtration and the pole order filtration on H2(U,ℂ).

The use of Hausdorff measures and dimension in the theory of Diophantine approximation dates back to the 1920s with the theorems of Jarník and Besicovitch regarding well-approximable and badly-approximable points.In this paper we consider three inhomogeneous problems that further develop these classical results.Firstly, we obtain a Jarník type theorem for the set S × 2 (ψ; θ) of multiplicatively approximable points in the plane R 2 .This Hausdorff measure statement does not reduce to Gallagher's Lebesgue measure statement as one might expect and is new even in the homogeneous setting (θ = 0).Next, we establish a Jarník type theorem for the set S × 2 (ψ; θ) ∩ C where C is a non-degenerate planar curve.This completes the Hausdorff theory for planar curves and clarifies a potential oversight in [2].Finally, we show that the set Bad(i, j; θ) of simultaneously inhomogeneously badly approximable points in R 2 is of full dimension.The underlying philosophy behind the proof has other applications; e.g.towards establishing the inhomogeneous version of Schmidt's Conjecture.The higher dimensional analogues of the planar results are also discussed.

Some remarks on the two-variable main conjecture of Iwasawa theory for elliptic curves without complex multiplication

Acting within the framework of geometric quantum mechanics, an interpretation of quantum uncertainty is discussed in terms of Jacobi fields, and a connection with the theory of elliptic curves is outlined, via classical integrability of Schrödinger's dynamics and the cross-ratio interpretation of quantum transition probabilities. Furthermore, a thoroughly geometrical construction of all special unitary representations of the 3-strand braid group on the quantum 1-qubit space is given, and the connection of one of them with elliptic curves admitting complex multiplication automorphisms — the physically relevant one corresponding to the anharmonic ratio — is shown. Also, contact is made with the Temperley–Lieb algebra theoretic constructions of Kauffman and Lomonaco, and it is shown that the standard trace relative to one of the above representations computes the Jones polynomial for particular values of the parameter, for knots arising as closures of 3-strand braids. Subsequently, a geometric entanglement criterion (in terms of Segre embeddings) is discussed, together with a projective geometrical portrait for quantum 2-gates. Finally, Aravind's idea of describing quantum states via knot theory is critically analyzed, and a geometrical picture — involving a blend of SU(2)-representation theory, classical projective geometry, binary trees and Brunnian and Hopf links — is set up in order to describe successive measurements made upon generalized GHZ states, close in spirit to the quantum knot picture again devised by Kauffman and Lomonaco.

Separation has always been an important task in the chemical industry. Even with the recent shift in emphasis and efforts to unify the physical and biological sciences, separation will remain an important workhorse in production and product/process design. The theory of simple distillation and residue curve maps has a long history—roughly 100 years—and has made a significant impact in the way many new separation processes are synthesized and designed. The key synthesis/design concept in this approach centers on understanding the ways in which constant boiling mixtures (or azeotropes) define separation boundaries. It is now well established that these azeotropes (or eutectics in melt or fractional crystallization and other solid–liquid separations) often define curved separation boundaries that place limitations on the degree of separation that can be achieved. Despite this, there remains no clear and exact understanding of separation boundaries and no straightforward way of accurately computing them in practice. A geometric methodology is presented that shows that exact separation boundaries can be defined through the use of differential geometry and dynamical systems theory and formulated as a constrained global optimization problem. Our novel approach is based on the observation that, for ternary homogeneous liquids, separation boundaries correspond to local maxima in the line integral within a given separation region. In addition, it is shown that these local maxima in the line integral correspond to one‐sided cusps and that global optimization is absolutely necessary, given that several local maxima can exist within any separation region. Several numerical examples are presented that show that the proposed geometric approach can accurately find separation boundaries for homogeneous and heterogeneous mixtures. Finally, it is shown that our new methodology is very general and readily extends to mixtures with four or more components, reactive separations, nonequilibrium models, and other processes such as crystallization and vapor degreasing. © 2005 American Institute of Chemical Engineers AIChE J, 2006