Singular Curves Research Articles

Detection of the effect of the Percentage of Items with DIF on the strength and efficacy of the singular characteristic curve test in the three-parameter model: الكشف عن أثر نسبة المفردات ذات الأداء التفاضلي على قوة وفاعلية اختبار منحنى خصائص المفردة في النموذج ثلاثي المعلم

The current research is following the simulation studies in IRT to experiment with the effect and power of the ICC's method on DIF items. To achieve the experiment, data were generated for three tests consisting of 100 items to each one containing (10% -20%-30%) items with DIF. These three tests were applied through the simulation program to 3000 individuals with normal abilities are distributed in average (0) and standard deviation (1). The results of the study indicated the effectiveness of the ICC's test by using the Rajo equation (1988), which measuring the area below the curve between the Reference groups and the Focal groups in detection items of (10% -20% -30%) amount of DIF and counting them in each of the three tests. The results showed that the strength of the test was (0, 619) for the DIF of 10% and the test strength of the DIF of 20% is amount (0.703) and the strength of the test was (0786) for the DIF of 30%, all of Power Values are higher than the average rating by Cohen, D test to derive the size of the impact. And we recommend taking the results in instruments containing differential items.

Singular curve and critical curve for doubly nonlinear Lane–Emden type equations

This paper is concerned with singular curve and critical curve for the periodic doubly nonlinear Lane–Emden type equation . In 2010, under a convex assumption on the domain Ω, Wang et al. (2010) considered a partial case of . While, there are no results about the cases of and before the present work. In this paper, we fill the gap for these two cases and give a total classification for the exponents. Furthermore, by constructing a special blow‐up sequence, we remove the convex assumption on the domain Ω, not only for the partial case considered in Wang et al. (2010) but also for other remaining cases.

Partition of the complement of good semigroup ideals and Apéry sets

Good semigroups form a class of submonoids of containing the value semigroups of curve singularities. In this article, we describe a partition of the complement of good semigroup ideals. As main application, we describe the Apéry sets of good semigroups with respect to arbitrary elements. This generalizes to any the results in the study of D’Anna et al., which are proved in the case d = 2 and only for the standard Apéry set with respect to the smallest nonzero element. Several new results describing good semigroups in are also provided.

Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles

Quantum curves were introduced in the physics literature. We develop a mathematical framework for the case associated with Hitchin spectral curves. In this context, a quantum curve is a Rees $\mathcal{D}$-module on a smooth projective algebraic curve, whose semi-classical limit produces the Hitchin spectral curve of a Higgs bundle. We give a method of quantization of Hitchin spectral curves by concretely constructing one-parameter deformation families of opers. We propose a variant of the topological recursion of Eynard--Orantin and Mirzakhani for the context of singular Hitchin spectral curves. We show that a PDE version of topological recursion provides all-order WKB analysis for the Rees $\mathcal{D}$-modules, defined as the quantization of Hitchin spectral curves associated with meromorphic $SL(2,\mathbb{C})$-Higgs bundles. Topological recursion can be considered as a process of quantization of Hitchin spectral curves. We prove that these two quantizations, one via the construction of families of opers, and the other via the PDE recursion of topological type, agree for holomorphic and meromorphic $SL(2,\mathbb{C})$-Higgs bundles. Classical differential equations such as the Airy differential equation provides a typical example. Through these classical examples, we see that quantum curves relate Higgs bundles, opers, a conjecture of Gaiotto, and quantum invariants, such as Gromov--Witten invariants

KASITS: A Graphical User Interface for Kinematic Analysis and Synthesis of Five-Bar Linkage with Prismatic Joint

In this paper, a novel graphical user interface is developed for kinematic analysis and synthesis of five-bar linkage with prismatic joint, named KASITS. This interface has two menus that the users can freely select, namely for analysis and synthesis. In the analysis menu, the direct kinematics are derived to visually depict the overall workspace of the mechanism. Within this workspace, the singularity curves are plotted. In the synthesis menu, the value of design parameters is obtained for a given trajectory. An optimization is employed based on Pareto optimal solutions. The demonstration is provided to guide the users better.

On resolving singularities of plane curves via a theorem attributed to Alfred Clebsch

This paper discusses a theorem in birational geometry that J.L. Coolidge attributed to Alfred Clebsch. The background is reconstructed from letters Felix Klein exchanged with Max Noether in 1894, when Noether was completing work on a lengthy report with Alexander Brill on the history of algebraic functions. Noether was deeply troubled to learn that Klein had informed him back in 1869 about relevant results that Clebsch and Leopold Kronecker had discussed in Berlin. These exchanges with Klein led to revisions in the Brill-Noether report, made in part to ensure Noether's own priority rights and larger intellectual legacy.

Milnor numbers in deformations of homogeneous singularities

Let f0 be a plane curve singularity. We study the Minor numbers of singularities in deformations of f0. We completely describe the set of these Milnor numbers for homogeneous singularities f0 in the case of non-degenerate deformations and obtain some partial results on this set in the general case.

Space Curves and Solitons of the KP Hierarchy. I. The l-th Generalized KdV Hierarchy

It is well known that algebro-geometric solutions of the KdV hierarchy are constructed from the Riemann theta functions associated with hyperelliptic curves, and that soliton solutions can be obtained by rational (singular) limits of the corresponding curves. In this paper, we discuss a class of KP solitons in connections with space curves, which are labeled by certain types of numerical semigroups. In particular, we show that some class of the (singular and complex) KP solitons of the $l$-th generalized KdV hierarchy with $l\ge 2$ is related to the rational space curves associated with the numerical semigroup $\langle l,lm+1,\dots, lm+k\rangle$, where $m\ge 1$ and $1\le k\le l-1$. We also calculate the Schur polynomial expansions of the $\tau$-functions for those KP solitons. Moreover, we construct smooth curves by deforming the singular curves associated with the soliton solutions. For these KP solitons, we also construct the space curve from a commutative ring of differential operators in the sense of the well-known Burchnall-Chaundy theory.

Geometric Algebra of Singular Ruled Surfaces

Singular ruled surface is an interesting research object and is the breakthrough point of exploring new problems. However, because of singularity, it’s difficult to study the properties of singular ruled surfaces. In this paper, we combine singularity theory and Clifford algebra to study singular ruled surfaces. We take advantage of the dual number of Clifford algebra to make the singular ruled surfaces transform into the dual singular curves on the dual unit sphere. By using the research method on the singular curves, we give the definition of the dual evolute of the dual front in the dual unit sphere, we further provide the k-th dual evolute of the dual front. Moreover, we consider the ruled surface corresponding to the dual evolute and k-th dual evolute and provide the developable conditions of these ruled surfaces.

Factoring polynomials over finite fields

In this paper, we describe a new polynomial factorization algorithm over finite fields with odd characteristics. The main ingredient of the algorithm is special singular curves. The algorithm relies on the extension of the Mumford representation and Cantor’s algorithm to these special singular curves.

Duality between wavefronts and caustics

There is a classical optical system: a planar curvilinear mirror illuminated by light rays coming from a coplanar point light source. An interesting and meaningful problem is to determine the focal points on the mirror corresponding to the cusps on the caustic. Due to the restriction of research on singular curves, this problem is still not very clear. In our last paper, we obtain a one-to-one correspondence between cusps on the orthotomic and cusps on the caustic. However, we know that vertexes on the orthotomic also induce cusps on the caustic. In this paper, we give their explicit correspondence and study the relation among mirror, orthotomic and caustic. This completes the work of the correspondence among mirror, wavefront (orthotomic) and caustic when the mirror has cusps and inflection points.

Birational models of 𝓜2,2 arising as moduli of curves with nonspecial divisors

Abstract We study birational projective models of 𝓜2,2 obtained from the moduli space of curves with nonspecial divisors. We describe geometrically which singular curves appear in these models and show that one of them is obtained by blowing down the Weierstrass divisor in the moduli stack of 𝓩-stable curves 𝓜 2,2(𝓩) defined by Smyth. As a corollary, we prove projectivity of the coarse moduli space M 2,2(𝓩).

Normal forms for the endpoint map near nice singular curves for rank-two distributions

Given a rank-two sub-Riemannian structure (M, Δ) and a point x0 ∈ M, a singular curve is a critical point of the endpoint map F : γ ↦ γ (1) defined on the space of horizontal curves starting at x0. The typical least degenerate singular curves of these structures are called regular singular curves; they are nice if their endpoint is not conjugate along γ. The main goal of this paper is to show that locally around a nice singular curve γ, once we choose a suitable topology on the control space we can find a normal form for the endpoint map, in which F writes essentially as a sum of a linear map and a quadratic form. This is a preparation for a forthcoming generalization of the Morse theory to rank-two sub-Riemannian structures.

On the analytic classification of irreducible plane curve singularities

On the analytic classification of irreducible plane curve singularities

The minimal Tjurina number of irreducible germs of plane curve singularities

In this paper we give a positive answer to a question of Dimca and Greuel about the quotient between the Milnor and the Tjurina numbers for any irreducible germ of plane curve singularity. This result is based on a closed formula for the minimal Tjurina number of an equisingularity class in terms of the sequence of multiplicities of the strict transform along a resolution. The key points for the proof are previous results by Genzmer, Wall and Mattei.

Biometric authenticator algorithm based on multiresolution analysis

Features extraction is an essential process in identifying person biometrics because the effectiveness of the system depends on it. Multiresolution Analysis success can be used in the system of a person’s identification and pattern recognition. In this paper, we present a feature extraction method for two-dimensional face and iris authentication. Our approach is a combination of principal component analysis (PCA) and curvelet transform as an improved fusion approach for feature extraction. The proposed fusion approach involves image denoising using 2D-Curvelet transform to achieve compact representations of curves singularities. This is followed by the application of PCA as a fusion rule to improve upon the spatial resolution. The limitations of the only PCA algorithm are a poor recognition speed and complex mathematical calculating load, to reduce these limitations, we are applying the curvelet transform. <br /> To assess the performance of the presented method, we have employed three classification techniques: Neural networks (NN), K-Nearest Neighbor (KNN) and Support Vector machines (SVM).<br />The results reveal that the extraction of image features is more efficient using Curvelet/PCA.

Monotonic invariants under blowups

We prove that the numerical invariant [Formula: see text] of a reduced irreducible plane curve singularity germ is non-negative, non-decreasing under blowups and strictly increasing unless the curve is non-singular. This provides a new perspective to understand the question posed by Dimca and Greuel. Moreover, our work can be put in the general framework of discovering monotonic invariants under blowups.

Analytic torsion for surfaces with cusps II. Regularity, asymptotics and curvature theorem

We study the Quillen metric on the determinant line bundle associated with a family of complex singular curves with hyperbolic cusp singularities.More precisely, we fix a family of complex curves, which admit at most double-point singularities. We endow the fibers of this family with Kähler metrics, which are defined away from a finite set of points, a divisor on the total space of the family, in the vicinity of which the metric has Poincaré-type singularities. We fix a holomorphic vector bundle over the total space of the family and endow it with a Hermitian metric with at most logarithmic singularities on the divisor, coming from the power of the relative canonical line bundle twisted by the divisor. An important example is the family of stable surfaces with marked points endowed with constant scalar curvature metric.The associated determinant line bundle is naturally endowed with the Quillen metric defined using the analytic torsion from the first paper of this series, generalizing the classical definition of Ray-Singer. We study the regularity of this Quillen metric near the locus of singular curves. The singularities turn out to be reasonable enough, so that the curvature of the Chern connection of the determinant line bundle endowed with the Quillen norm is well-defined as a current. We derive the explicit formula for this current, which gives a refinement of Riemann-Roch-Grothendieck theorem at the level of currents. This generalizes the curvature theorems of Takhtajan-Zograf and Bismut-Bost.

An instability criterion for volume-preserving area-stationary surfaces with singular curves in sub-Riemannian 3-space forms

We study stable surfaces, i.e., second order minima of the area for variations of fixed volume, in sub-Riemannian space forms of dimension 3. We prove a stability inequality and provide sufficient conditions ensuring instability of volume-preserving area-stationary $$C^2$$ surfaces with a non-empty singular set of curves. Combined with previous results, this allows to describe any complete, orientable, embedded and stable $$C^2$$ surface $$\Sigma $$ in the Heisenberg group $${\mathbb {H}}^1$$ and the sub-Riemannian sphere $${\mathbb {S}}^3$$ of constant curvature 1. In $${\mathbb {H}}^1$$ we conclude that $$\Sigma $$ is a Euclidean plane, a Pansu sphere or congruent to the hyperbolic paraboloid $$t=xy$$ . In $${\mathbb {S}}^3$$ we deduce that $$\Sigma $$ is one of the Pansu spherical surfaces discovered in Hurtado and Rosales (Math Ann 340(3):675–708, 2008). As a consequence, such spheres are the unique $$C^2$$ solutions to the sub-Riemannian isoperimetric problem in $${\mathbb {S}}^3$$ .

Computing the equisingularity type of a pseudo-irreducible polynomial

Germs of plane curve singularities can be classified accordingly to their equisingularity type. For singularities over $$\mathbb {C}$$ , this important data coincides with the topological class. In this paper, we characterise a family of singularities, containing irreducible ones, whose equisingularity type can be computed in an expected quasi-linear time with respect to the discriminant valuation of a Weierstrass equation.