Some properties of Malgrange isomonodromic deformations of linear 2 × 2 systems

We study movable singularities of the Malgrange isomonodromic deformation of a linear differential 2 × 2 system with two irregular singularities of Poincaré rank 1 and with an arbitrary number of Fuchsian singular points.     

Global Structure of Planar Quadratic Semi-Quasi-Homogeneous Polynomial Systems

This paper study the planar quadratic semi-quasi-homogeneous polynomial systems(short for PQSQHPS). By using the nilpotent singular points theorem, blow-up technique, Poincaré index formula, and Poincaré compaction method, the global phase portraits of such systems in canonical forms are discussed. Furthermore, we show that all the global phase portraits of PQSQHPS can be-classed into six topological equivalence classes.     

A rearrangement formula for a singular Cauchy-Szegö integral in a ball from ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

We obtain an analog of the Poincaré-Bertrand formula for a singular Cauchy-Szegö integral in a multidimensional ball. We understand the principal value of the integral in the Cauchy sense. The obtained formula differs from that of Poincaré-Bertrand for the Cauchy integral in a complex plane.     

Chaos in singular systems

In this paper we consider singular systems of differential equations and we show that, under right conditions, the Poincaré map associated to those systems, and not just a suitable iterate, behaves chaotically. We use the notion of exponential dichotomies to prove the existence of a transverse homoclinic orbit of our system and after use the shadow lemma to show that the Poincaré map associated to its topologically conjugate to the Bernouilli shift on a set of two symbols. Entrata in Redazione il 3 aprile 1997 e, in versione riveduta, il 30 ottobre 1997.     

On the algebraic fundamentals of singular perturbation theory

We suggest an algebraic approach to singular perturbation theory and present a generalization of the Poincaré expansion theorem.     

A singular integral of the composite operator

We establish the Poincaré-type inequalities for the composition of the homotopy operator and the projection operator. We also obtain some estimates for the integral of the composite operator with a singular density.     

Topological recursion for the Poincaré polynomial of the combinatorial moduli space of curves

We show that the Poincaré polynomial associated with the orbifold cell decomposition of the moduli space of smooth algebraic curves with distinct marked points satisfies a topological recursion formula of the Eynard–Orantin type. The recursion uniquely determines the Poincaré polynomials from the initial data. Our key discovery is that the Poincaré polynomial is the Laplace transform of the number of Grothendieck’s dessins d’enfants.     

Ridge directional singular points for fingerprint recognition and matching

In this paper, a new approach to extract singular points in a fingerprint image is presented. It is usually difficult to locate the exact position of a core or a delta due to the noisy nature of fingerprint images. These points are the most widely used for fingerprint classification and matching. Image enhancement, thinning, cropping, and alignment are used for minutiae extraction. Based on the Poincaré curve obtained from the directional image, our algorithm extracts the singular points in a fingerprint with high accuracy. It examines ridge directions when singular points are missing. The algorithm has been tested for classification performance on the NIST‐4 fingerprint database and found to give better results than the neural networks algorithm. Copyright © 2005 John Wiley & Sons, Ltd.     

On the center-focus problem for systems with a degenerate singular point

For an analytic system with nonzero cubic part and with a degenerate monodromic singular point, we indicate an algorithm for constructing the asymptotic expansion of the Poincaré map. We present closed-form expressions for the first four focus quantities and find two necessary conditions for a center.     

Multidimensional Poincaré construction and singularities of lifted fields for implicit differential equations

This paper is devoted to singular points of the so-called lifted vector fields, which arise in studying systems of implicit differential equations by using the method of lifting the equation to a surface, a generalization of the construction used by Poincaré for a single implicit equation. The author studies the phase portraits and renormal forms of such fields in a neighborhood of their singular points. In conclusion, this paper considers the lifted vectors fields generated by Euler-Lagrange and Euler-Poisson equations and fast-slow systems. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 19, Optimal Control, 2006.     

A disjoint disks property for 3-manifolds

We show that the map separation property (MSP), a concept due to H.W. Lambert and R.B. Sher, is an appropriate analogue of J.W. Cannon’s disjoint disks property (DDP) for the class $\text{C}$ of compact generalized 3-manifolds with zero-dimensional singular set, modulo the Poincaré conjecture. Our main result is that the Poincaré conjecture (in dimension three) is equivalent to the conjecture that every X?$\text{C}$ with the MSP is a topological 3-manifold.     

Singular hyperbolicity for transitive attractors with singular points of 3-dimensional <Emphasis Type="Italic">C</Emphasis><Superscript>2</Superscript>-flows

In the context of C^r-flows on 3-manifolds (r ≥ 1), the notion of singular hyperbolicity, inspired on the Lorenz Attractor, is the right generalization of hyperbolicity (in the sense of Smale) for C¹-robustly transitive sets with singularities. We estabish conditions (on the associated linear Poincaré flow and on the nature of the singular set) under which a transitive attractor with singularities of a C²-flow on a 3-manifold is singular hyperbolic.     

H?lder continuity for Trudinger’s equation in measure spaces

We complete the study of the regularity for Trudinger’s equation by proving that weak solutions are H?lder continuous also in the singular case. The setting is that of a measure space with a doubling non-trivial Borel measure supporting a Poincaré inequality. The proof uses the Harnack inequality and intrinsic scaling.     

Modulus and Poincaré Inequalities on Non-Self-Similar Sierpiński Carpets

A carpet is a metric space homeomorphic to the Sierpiński carpet. We characterize, within a certain class of examples, non-self-similar carpets supporting curve families of nontrivial modulus and supporting Poincaré inequalities. Our results yield new examples of compact doubling metric measure spaces supporting Poincaré inequalities: these examples have no manifold points, yet embed isometrically as subsets of Euclidean space.     

Poincar and Weak Poincar Inequalities for the Mixed Poisson Measure

By using the Mecke identity, we study a class of birth-death type Dirichlet forms associated with the mixed Poisson measure. Both Poincar and weak Poincar inequalities are established, while another Poincar type inequality is disproved under some reasonable assumptions.     

Arithmetic properties of coefficients of half-integral weight Maass–Poincaré series

Zagier [23] proved that the generating functions for the traces of level 1 singular moduli are weight 3/2 modular forms. He also obtained generalizations for “twisted traces”, and for traces of special non-holomorphic modular functions. Using properties of Kloosterman-Salié sums, and a well known reformulation of Salié sums in terms of orbits of CM points, we systematically show that such results hold for arbitrary weakly holomorphic and cuspidal half-integral weight Poincaré series in Kohnen’s Γ₀(4) plus-space. These results imply the aforementioned results of Zagier, and they provide exact formulas for such traces.     

Dirichlet problem for semilinear edge-degenerate elliptic equations with singular potential term

In this paper, we introduce weighted p-Sobolev spaces on manifolds with edge singularities. We give the proof for the corresponding edge type Sobolev inequality, Poincaré inequality and Hardy inequality. As an application of these inequalities, we prove the existence of nontrivial weak solutions for the Dirichlet problem of semilinear elliptic equations with singular potentials on manifolds with edge singularities.     

Singular integral operators in Morrey spaces and interior regularity of solutions to systems of linear PDE’s

We obtain boundedness in Morrey spaces of singular integral operators with Calderón-Zygmund type kernel of mixed homogeneity. These estimates are used for the study of the interior regularity of the solutions of linear elliptic/parabolic systems. The proved Poincaré-type inequality permits to describe the Hölder, Morrey, and BMO regularity of the lower-order derivatives of the solutions.     

Optimization of Poincaré sections for discriminating between stochastic and deterministic behavior of dynamical systems

This paper studies the problem of finding optimal parameters for a Poincaré section used for determining the type of behavior of a time series: a deterministic or stochastic one. To reach that goal optimization algorithms are coupled with the Poincaré & Higuchi (P&H) method, which calculates the Higuchi dimension using points obtained by performing a Poincaré section of a certain attractor. The P&H method generates distinctive patterns that can be used for determining if a given attractor is produced by a deterministic or a stochastic system, but this method is sensitive to the parameters of the Poincaré section. Patterns generated by the P&H method can be characterized using numerical measures which in turn can be used for finding such parameters for the Poincaré section for which the patterns produced by the P&H method are the most prominent. This paper studies several approaches to parameterization of the Poincaré section. Proposed approaches are tested on twelve time series, six produced by deterministic chaotic systems and six generated randomly. The obtained results show, that finding good parameters of the Poincaré section is important for determining the type of behavior of a time series. Among the tested methods the evolutionary algorithm was able to find the best Poincaré sections for use with the P&H method.