Bounding Poincaré‐Hopf indices and Milnor numbers

We use Mather's finite determinacy theory and Baum‐Bott's theorem to give sharp bounds for the Poincaré‐Hopf index of a germ of homolorphic vector field with an isolated zero. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Proximal Point Method on Finslerian Manifolds and the “Effort–Accuracy” Trade-off

In this paper, we consider minimization problems with constraints. We show that, if the set of constraints is a Finslerian manifold of non-positive flag curvature, and the objective function is differentiable and satisfies the Kurdyka-Lojasiewicz property, then the proximal point method can be naturally extended to solve this class of problems. We prove that the sequence generated by our method is well defined and converges to a critical point. We show how tools of Finslerian geometry, specifically non-symmetrical metrics, can be used to solve non-convex constrained problems in Euclidean spaces. As an application, we give one result regarding decision-making speed and costs related to change.     

Local finiteness for Green’s relations in semigroup varieties

A semigroup variety is said to be locally 𝒦-finite, where 𝒦 stands for any of Green’s relations ?, ?, ?, 𝒟, or 𝒥, if every finitely generated semigroup in this variety has only finitely many 𝒦-classes. We characterize locally 𝒦-finite varieties of finite axiomatic rank in the language of “forbidden objects”.     

Zero-sets of complex homogeneous polynomials

In this paper we study the zero-sets of continuous n-homogeneous polynomials on complex nonseparable Banach spaces. We prove that the zero-set of any complex n-homogeneous polynomial P is a subspace if, and only if, there is a functional ? such that P(x)=? (x)ⁿ for every x. We give sufficient conditions on the Banach space to ensure that every continuous 2-homogeneous polynomial is identically zero on a nonseparable subspace. Also, we prove that, in the 2-homogeneous case, one of the following three properties holds: P ^?1(0) is a subspace; P ^?1(0) is the union of two different subspaces; and P ^?1(0) is the union of infinitely many different subspaces.     

Real Milnor Fibrations and (C)-Regularity

In this paper we study Milnor fibrations associated to real isolated singularities defined by map-germs f: (^m,0)(²,0). The main result relates the existence of the Milnor fibration with the (C)-regularity of the family of hypersurfaces with isolated singularity obtained by projecting f into the family L_– of all lines through the origin in the plane ².     

Effects of a Uniform Magnetic Field on Twisted Graphene Nanoribbons

In the present work, the relativistic quantum motion of massless fermions in a helicoidal graphene nanoribbon under the influence of a uniform magnetic field is investigated. Considering a uniform magnetic field (B) aligned along the axis of helicoid, this problem is explored in the context of Dirac equation in a curved space-time. As this system does not support exact solutions due to considered background, the bound-state solutions and local density of states (LDOS) are obtained numerically by means of the Numerov method. The combined effects of width of the nanoribbon (D), length of ribbon (L), twist parameter (ω), and B on the equations of motion and LDOS are analyzed and discussed. It is verified that the presence of B produces a constant minimum value of local density of state on the axis of helicoid, which is possible only for values large enough of ω, in contrast to the case for $B=0$ already studied in the literature.     

An experimental and numerical study on crystallization analysis fractionation (Crystaf)

Crystallization analysis fractionation (Crystaf) is a new technique used to estimate the chemical composition distribution (CCD) of semi-crystalline copolymers. In this study, the effect of chain microstructure and operation parameters on Crystaf profiles was investigated using a series of ethylene/1-hexene copolymers and their blends. The Crystaf profiles were also modeled via stochastic simulation based on the distribution of average ethylene sequence lengths.     

Molecular Weight and Long Chain Branch Distributions of Branch‐Block Olefinic Thermoplastic Elastomers

Olefinic thermoplastic elastomers can be prepared by incorporating semi‐crystalline macromonomers (e.g. isotactic or syndiotactic poly(propylene), high‐density polyethylene) onto amorphous backbones (e.g. atactic poly(propylene), ethylene/α‐olefin copolymers). The macromonomer incorporation reaction can be carried out in semi‐batch reactors by adding previously synthesized macromonomers to the reactor (ex situ approach), or by generating and incorporating the macromonomers in a single step (in situ approach). The differences in the microstructure of copolymers synthesized by in situ and ex situ techniques are explored herein through a mathematical model that can predict the concentration of linear and branched chains, their average molecular weights, polydispersity indices, and molecular weight distributions. In both cases linear chains predominate, but the ex situ approach produces a larger amount of branched chains with thermoplastic elastomer properties. Furthermore, for the in situ strategy, a significant amount of branched chains is only formed after the macromonomer concentration reaches a critical value.

Schematic representation of the polymerization mechanism.