Singular Curves Research Articles

Construction of a solution to a velocity problem in the case of violation of the smoothness of the curvature of the target set boundary

For the development of analytical and numerical algorithms for constructing nonsmooth solutions of optimal control problems, procedures are proposed for constructing scattering curves for a single class of control velocity problems. We consider the reduction problems for a minimal time of solutions of a dynamical system with a circular velocity vectogram for the case where the target set is generally nonconvex, and its boundary has points at which the curvature smoothness is violated. These points are referred to as pseudovertices, the characteristic points of the target set, which are responsible for the occurrence of the singularity of the optimal result function. When forming a proper reparameterization (in this case, taking into account the geometry of the velocity vector diagram) of the arc of the boundary of the target set containing a pseudovertex, the scattering curve is constructed as an integral curve. Moreover, the initial conditions of the corresponding Cauchy problem are determined by the properties of the pseudovertex. One of the numerical characteristics of the pseudovertex, the pseudovertex marker, determines the initial velocity of the material point describing a smooth portion of the scattering curve. This approach to the identification and construction (in analytical or numerical form) of singular curves was previously substantiated for a number of cases of a target boundary that are different in the order of smoothness. It should be emphasized that the case considered in this paper is the most specific, in particular, because of the revealed connection between the dynamic problem and the problem of polynomial algebra. It is proved that the pseudovertex marker is the nonpositive root of some third-order polynomial whose coefficients are determined by the one-sided derivatives of curvatures of the pseudovertex of the target set. The effectiveness of the developed theoretical methods and numerical procedures is illustrated by specific examples.

Fractal and Multifractal Analysis of Pore Size Distribution in Low Permeability Reservoirs Based on Mercury Intrusion Porosimetry

To quantitatively evaluate the complexities and heterogeneities of pore structures in sandstone reservoirs, we apply single fractal theory and multifractal theory to explore the fractal characteristics of pore size distributions based on mercury intrusion porosimetry. The fractal parameters were calculated and the relationships between the petrophysical parameters (permeability and entry pressure) and the fractal parameters were investigated. The results show that the single fractal curves exhibit two-stage characteristics and the corresponding fractal dimensions D1 and D2 can characterize the complexity of pore structure in different sizes. Favorable linear relationships between log(ε) and log(μ,(ε)) indicate that the samples satisfy multifractal characteristics and ε is the sub-intervals with size ε = J × 2−k. The multifractal singularity curves used in this study exhibit a right shape, indicating that the heterogeneity of the reservoir is mainly affected by pore size distributions in sparse regions. Multifractal parameters, D(0), D(1), and Δf, are positively correlated with permeability and entry pressure, while D(0), D(1), and Δf are negatively correlated with permeability and entry pressure. The ratio of larger pores volumes to total pore volumes acts as a control on the fractal dimension over a specific pore size range, while the range of the pore size distribution has a definite impact on the multifractal parameters. Results indicate that fractal analysis and multifractal analysis are feasible methods for characterizing the heterogeneity of pore structures in a reservoir. However, the single fractal models ignore the influence of microfractures, which could result in abnormal values for calculated fractal dimension. Compared to single fractal analysis, multifractal theory can better quantitatively characterize the heterogeneity of pore structure and establish favorable relationships with reservoir physical property parameters.

Are algebraic links in the Poincaré sphere determined by their Alexander polynomials?

The Alexander polynomial in several variables is defined for links in three-dimensional homology spheres, in particular, in the Poincare sphere: the intersection of the surface $$S=\{(z_1,z_2,z_3)\in {\mathbb {C}}^3: z_1^5+z_2^3+z_3^2=0\}$$ with the 5-dimensional sphere $${\mathbb {S}}_{\varepsilon }^5=\{(z_1,z_2,z_3)\in {\mathbb {C}}^3: \vert z_1\vert ^2+\vert z_2\vert ^2+\vert z_3\vert ^2=\varepsilon ^2\}$$. An algebraic link in the Poincare sphere is the intersection of a germ of a complex analytic curve in (S, 0) with the sphere $${\mathbb S}^5_\varepsilon $$ of radius $$\varepsilon $$ small enough. Here we discuss to which extent the Alexander polynomial in several variables of an algebraic link in the Poincare sphere determines the topology of the link. We show that, if the strict transform of a curve in (S, 0) does not intersect the component of the exceptional divisor corresponding to the end of the longest tail in the corresponding $$E_8$$-diagram, then its Alexander polynomial determines the combinatorial type of the minimal resolution of the curve and therefore the topology of the corresponding link. The Alexander polynomial of an algebraic link in the Poincare sphere is determined by the Poincare series of the filtration defined by the corresponding curve valuations. (They coincide with each other for a reducible curve singularity and differ by the factor $$(1-t)$$ for an irreducible one.) We show that, under conditions similar to those for curves, the Poincare series of a collection of divisorial valuations determines the combinatorial type of the minimal resolution of the collection.

Some Singular Curves and Surfaces Arising from Invariants of Complex Reflection Groups

We construct highly singular projective curves and surfaces defined by invariants of primitive complex reflection groups. As a result, we improve some lower bounds on the number of singularities of a given type that a plane curve or a surface in of a given degree might have.

Triviality properties of principal bundles on singular curves

For $G$ a split semi-simple group scheme and $P$ a principal $G$-bundle on a relative curve $X\to S$, we study a natural obstruction for the triviality of $P$ on the complement of a relatively ample Cartier divisor $D \subset X$. We show, by constructing explicit examples, that the obstruction is nontrivial if $G$ is not simply connected but it can be made to vanish, if $S$ is the spectrum of a dvr (and some other hypotheses), by a faithfully flat base change. The vanishing of this obstruction is shown to be a sufficient condition for etale local triviality if $S$ is a smooth curve, and the singular locus of $X-D$ is finite over $S$.

Introduction to Heegaard Floer homology

Heegaard Floer homology is an invariant of knots, links, and 3-manifolds introduced by Ozsváth and Szabó about 15 years ago. This survey defines Heegaard Floer homology and describes its basic properties. Also discussed is the relation between Heegaard Floer homology and invariants of singularities of curves and surfaces. Bibliography: 72 titles.

Duality on value semigroups

We establish a combinatorial counterpart of the Cohen-Macaulay duality on a class of curve singularities which includes algebroid curves. For such singularities the value semigroup and the value semigroup ideals of all fractional ideals satisfy axioms that define so-called good semigroups and good semigroup ideals. We prove that each good semigroup admits a canonical good semigroup ideal which gives rise to a duality on good semigroup ideals. We show that the Cohen-Macaulay duality and our good semigroup duality are compatible under taking values.

A note on a question of Dimca and Greuel

In this note, we give a positive answer to a question of Dimca and Greuel about the quotient between the Milnor and Tjurina numbers of an isolated plane curve singularity in the cases of one Puiseux pair and semi-quasi-homogeneous singularities.

The auto Igusa-zeta function of a plane curve singularity is rational

It is shown that the auto-Igusa zeta function of a plane curve singularity is rational. This gives a new criterion for a plane curve over an algebraically closed field of characteristic zero to be smooth at a point.

Defining Curvature as a Measure via Gauss–Bonnet on Certain Singular Surfaces

We show how to define curvature as a measure using the Gauss–Bonnet Theorem on a family of singular surfaces obtained by gluing together smooth surfaces along boundary curves. We find an explicit formula for the curvature measure as a sum of three types of measures: absolutely continuous measures, measures supported on singular curves, and discrete measures supported on singular points. We discuss the spectral asymptotics of the Laplacian on these surfaces.

Generalized Calogero and Toda Models

A Calogero–Sutherland system with two types of interacting spin variables has been described using the Hitchin approach and quasicompact structure. Complete integrability has been established by means of the Lax equation specified on a singular curve and the classical r-matrix depending on the spectral parameter. Generalized Toda systems have also been considered. Their phase portraits have been described.

Echo Signal Extraction Based on Improved Singular Spectrum Analysis and Compressed Sensing in Wavelet Domain

A novel framework of signal extraction is proposed based on improved singular spectrum analysis (SSA) and compressed sensing (CS) for laser radar. The improved signal selection method, which locates a cut-off point on the singular spectrum curve, is fused into the SSA to reduce the noise. Afterward, the CS in the wavelet domain is employed to acquire the observation vector, and the improved reconstruction algorithm with a hard threshold denoising method reduces the noise furthermore. The simulations and experiments are performed to verify the superiority of the proposed algorithm compared with other algorithms. The proposed algorithm can enhance a signal-to-noise ratio while reducing the distortion of the received signal.

Mazur's conjecture and an unexpected rational curve on Kummer surfaces and their superelliptic generalisations

We prove the following special case of Mazur's conjecture on the topology of rational points. Let $E$ be an elliptic curve over $\mathbb{Q}$ with $j$-invariant $1728$. For a class of elliptic pencils which are quadratic twists of $E$ by quartic polynomials, the rational points on the projective line with positive rank fibres are dense in the real topology. This extends results obtained by Rohrlich and Kuwata-Wang for quadratic and cubic polynomials. For the proof, we investigate a highly singular rational curve on the Kummer surface $K$ associated to a product of two elliptic curves over $\mathbb{Q}$, which previously appeared in publications by Mestre, Kuwata-Wang and Satge. We produce this curve in a simpler manner by finding algebraic equations which give a direct proof of rationality. We find that the same equations give rise to rational curves on a class of more general surfaces extending the Kummer construction. This leads to further applications apart from Mazur's conjecture, for example the existence of rational points on simultaneous twists of superelliptic curves. Finally, we give a proof of Mazur's conjecture for the Kummer surface $K$ without any restrictions on the $j$-invariants of the two elliptic curves.

Asymptotic expansions and summability with respect to an analytic germ

In a previous article [CMS], monomial asymptotic expansions, Gevrey asymptotic expansions, and monomial summability were introduced and applied to certain systems of singularly perturbed differential equations. In the present work, we extend this concept, introducing (Gevrey) asymptotic expansions and summability with respect to a germ of an analytic function in several variables – this includes polynomials. The reduction theory of singularities of curves and monomialization of germs of analytic functions are crucial to establish properties of the new notions, for example a generalization of the Ramis–Sibuya theorem for the existence of Gevrey asymptotic expansions. Two examples of singular differential equations are presented for which the formal solutions are shown to be summable with respect to a polynomial: one ordinary and one partial differential equation.

On the multiplicity of tangent cones of monomial curves

Let $\Lambda$ be a numerical semigroup, $\mathcal{C}\subseteq \mathbb{A}^n$ the monomial curve singularity associated to $\Lambda$, and $\mathcal{T}$ its tangent cone. In this paper we provide a sharp upper bound for the least positive integer in $\Lambda$ in terms of the codimension and the maximum degree of the equations of $\mathcal{T}$, when $\mathcal{T}$ is not a complete intersection. A special case of this result settles a question of J. Herzog and D. Stamate.

Bifurcations of Traveling Wave Solutions for a Modified Novikov’s Cubic Equation

Exact traveling solutions for a modified Novikov’s cubic equation is considered based on the bifurcation method of dynamical systems in this paper. The two-dimensional system of modified Novikov’s cubic equation exists singular curve \(\phi ^2+a^2y^2=R\) when \(R>0\) and then it is proved that the corresponding traveling wave system of the modified Novikov’s cubic equation exists smooth solitary wave solutions and breaking wave solutions.

An adaptive subdivision method for root finding of univariate polynomials

The most common methods for root finding of polynomials are iterative. This type of methods shows some drawbacks, like the impossibility of parallelization, the unpredictable nature of the iterations, that prevents to search roots inside a pre-specified region of complex plane, and the moderated degree of the polynomials that they can handle. This work describes a recursive root finding method. It is based on the winding number of plane curves, that is related to the number of zeros of a polynomial in a plane region. By our previous work (García Zapata and Díaz Martín, 2014) we find the winding number at reasonable computational cost, so we can approximate the roots by recursive subdivision of the search region. This subdivision approach is parallelizable, with a known bound of the computational cost, and the search can be restrained, in contrast to the iterative methods. We use error returns to identify and avoid singular curves, adapting the subdivision to the roots. This allows us to handle high degree polynomials without resorting to the expensive multiprecision arithmetics of other recursive methods. A significant contribution is that we formally prove its correctness.

The Maslov Complex Germ and Semiclassical Spectral Series Corresponding to Singular Invariant Curves of Partially Integrable Hamiltonian Systems

We study semiclassical eigenvalues of the Schroedinger operator, corresponding to singular invariant curve of the corresponding classical system. The latter system is assumed to be partially integrable. We describe geometric object corresponding to the eigenvalues (comlex vector bundle over a graph) and compute semiclassical eigenvalues in terms of the corresponding holonomy group.

Evolutes of fronts in the Minkowski plane

We consider the differential geometry of evolutes of singular curves and give the definitions of spacelike fronts and timelike fronts in the Minkowski plane. We also give the notions of moving frames along the non‐lightlike fronts in the Minkowski plane. By using the moving frames, we define the evolutes of non‐lightlike fronts and investigate the geometric properties of these evolutes. We obtain that the evolute of a spacelike front is a timelike front and the evolute of a timelike front is a spacelike front. Since the evolute of a non‐lightlike front is also a non‐lightlike front, we can take evolute again. We study the Minkowski Zigzag number of non‐lightlike fronts and give the n‐th evolute of the non‐lightlike front. Finally, we give an example to illustrate our results.

Modules over plane curve singularities in any ranks and DAHA

We generalize the construction of geometric superpolynomials for unibranch plane curve singularities from our prior paper from rank one to any ranks; explicit formulas are obtained for torus knots. The new feature is the definition of counterparts of Jacobian factors (directly related to compactified Jacobians) for higher ranks, which is parallel to the classical passage from invertible sheaves to vector bundles over algebraic curves. This is an entirely local theory, connected with affine Springer fibers for non-reduced (germs of) spectral curves. We conjecture and justify numerically the connection of our geometric polynomials in arbitrary ranks with the corresponding DAHA superpolynomials for any algebraic knots colored by columns.