scholarly journals On Tangency in Equisingular Families of Curves and Surfaces

2020 ◽  
Vol 71 (2) ◽  
pp. 485-505
Author(s):  
Arturo Giles Flores ◽  
O N Silva ◽  
J Snoussi
Keyword(s):  
Parameter Space ◽  
Tangent Cones ◽  
Curves And Surfaces ◽  
Families Of Curves ◽  
The Family ◽  
Surface Singularities ◽  
Whitney Equisingularity

Abstract We study the behavior of limits of tangents in topologically equivalent spaces. In the context of families of generically reduced curves, we introduce the $s$-invariant of a curve and we show that in a Whitney equisingular family with the property that the $s$-invariant is constant along the parameter space, the number of tangents of each curve of the family is constant. In the context of families of isolated surface singularities, we show through examples that Whitney equisingularity is not sufficient to ensure that the tangent cones of the family are homeomorphic. We explain how the existence of exceptional tangents is preserved by Whitney equisingularity but their number can change.

THE PARAMETER SPACE OF A QUADRATIC FAMILY OF POLYNOMIAL MAPS OF ℂ2

2008 ◽  
Vol 19 (05) ◽  
pp. 541-556
Author(s):  
ALINA ANDREI
Keyword(s):  
Lyapunov Exponents ◽  
Parameter Space ◽  
Mandelbrot Set ◽  
General Context ◽  
One Dimensional ◽  
Polynomial Maps ◽  
Regular Maps ◽  
The Family ◽  
The One ◽  
Polynomial Family

In this paper, we study the parameter space of the quadratic polynomial family fλ,μ(z, w) = (λz + w2, μw + z2), which exhibits interesting dynamics. Two distinct subsets of the parameter space are studied as appropriate analogs of the one-dimensional Mandelbrot set and some of their properties are proved by using Lyapunov exponents. In the more general context of holomorphic families of regular maps, we show that the sum of the Lyapunov exponents is a plurisubharmonic function of the parameter, and pluriharmonic on the set of expanding maps. Moreover, for the family fλ,μ, we prove that the sum of the Lyapunov exponents is continuous.

scholarly journals A compactification of the space of twisted cubics

2002 ◽  
Vol 91 (2) ◽  
pp. 221 ◽  
Author(s):  
I. Vainsencher ◽  
F. Xavier
Keyword(s):  
Parameter Space ◽  
Ruled Surfaces ◽  
Segre Varieties ◽  
The Family ◽  
Twisted Cubics ◽  
Nets Of Quadrics ◽  
Smooth Compactification

We give an elementary, explicit smooth compactification of a parameter space for the family of twisted cubics. The construction also applies to the family of subschemes defined by determinantal nets of quadrics, e.g., cubic ruled surfaces in $\boldsymbol P^4$, Segre varieties in $\boldsymbol P^5$. It is suitable for applications of Bott's formula to a few enumerative problems.

scholarly journals Level Algebras and $\mathbf {s}$-Lecture Hall Polytopes

10.37236/8626 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Florian Kohl ◽  
McCabe Olsen
Keyword(s):  
Tangent Cones ◽  
Lecture Hall ◽  
Ehrhart Theory ◽  
Lattice Polytopes ◽  
The Family ◽  
Lecture Hall Partitions ◽  
Level Algebras

Given a family of lattice polytopes, a common endeavor in Ehrhart theory is the classification of those polytopes in the family that are Gorenstein, or more generally level. In this article, we consider these questions for ${\boldsymbol s}$-lecture hall polytopes, which are a family of simplices arising from $\mathbf {s}$-lecture hall partitions. In particular, we provide concrete classifications for both of these properties purely in terms of ${\boldsymbol s}$-inversion sequences. Moreover, for a large subfamily of ${\boldsymbol s}$-lecture hall polytopes, we provide a more geometric classification of the Gorenstein property in terms of  its tangent cones. We then show how one can use the classification of level ${\boldsymbol s}$-lecture hall polytopes to construct infinite families of level ${\boldsymbol s}$-lecture hall polytopes, and to describe level ${\boldsymbol s}$-lecture hall polytopes in small dimensions.

HOW OFTEN DOES THE PARRONDO EFFECT APPEAR?

2007 ◽  
Vol 07 (02) ◽  
pp. C19-C25 ◽  
Author(s):  
GLORIA CERASELA CRISAN ◽  
ELENA NECHITA ◽  
MIHAI TALMACIU
Keyword(s):  
Parameter Space ◽  
The Family ◽  
Parrondo’S Paradox ◽  
Special Behaviour

It is already known that two simple losing games can derive a winning game when randomly combined. This special behaviour is known as Parrondo's Paradox, or the Parrondo effect. In this paper we estimate the volume of the parameter space that determines the Parrondo effect. In other words, how often two randomly chosen games are losing while their proper combination is leading to a winning game. Results for a typical class of relevant games indicate that the Parrondo effect is very unusual, because it appears with a probability of 0.0306%. By adding two more dimensions to the parameter space, the family of regions that exhibit the Parrondo effect is studied.

ON MULTICORNS AND UNICORNS I: ANTIHOLOMORPHIC DYNAMICS, HYPERBOLIC COMPONENTS AND REAL CUBIC POLYNOMIALS

2003 ◽  
Vol 13 (10) ◽  
pp. 2825-2844 ◽  
Author(s):  
SHIZUO NAKANE ◽  
DIERK SCHLEICHER
Keyword(s):  
Parameter Space ◽  
Bifurcation Diagrams ◽  
The Family ◽  
Real Analytic ◽  
Cubic Polynomials

We investigate the dynamics and the bifurcation diagrams of iterated antiholomorphic polynomials: These are complex conjugates of ordinary polynomials. Their second iterates are holomorphic polynomials, but dependence on parameters is only real-analytic. The structure of hyperbolic components of the family of unicritical antiholomorphic polynomials is revealed. In case of degree two, they arise naturally in the parameter space of real cubic (holomorphic) polynomials, which we investigate as well.

scholarly journals Contact Interactions in One-Dimensional Quantum Mechanics: a Family of Generalized б'-Potentials

2019 ◽  
Vol 64 (11) ◽  
pp. 1021 ◽  
Author(s):  
A. V. Zolotaryuk
Keyword(s):  
Parameter Space ◽  
Resonant Tunneling ◽  
Measure Zero ◽  
Contact Point ◽  
Delta Function ◽  
Point Interactions ◽  
Dimensional Parameter ◽  
One Dimensional ◽  
The Family ◽  
Delta Derivative

A “one-point” approximation is proposed to investigate the transmission of electrons through the extra thin heterostructures composed of two parallel plane layers. The typical example is the bilayer for which the squeezed potential profile is the derivative of Dirac’s delta function. The Schr¨odinger equation with this singular one-dimensional profile produces a family of contact (point) interactions each of which (called a “distributional” б′-potential) depends on the way of regularization. The discrepancies widely discussed so far in the literature regarding the family of delta derivative potentials are eliminated using a two-scale power-connecting parametrization of the bilayer potential that enables one to extend the family of distributional б′-potentials to a whole class of “generalized” б′-potentials. In a squeezed limit of the bilayer structure to zero thickness, the resonant tunneling through this structure is shown to occur in the form of sharp peaks located on the sets of Lebesgue’s measure zero (called resonance sets). A four-dimensional parameter space is introduced for the representation of these sets. The transmission on the complement sets in the parameter space is shown to be completely opaque.

Absolute instability of the Ekman layer and related rotating flows

1997 ◽  
Vol 331 ◽  
pp. 405-428 ◽  
Author(s):  
R. J. LINGWOOD
Keyword(s):  
Boundary Layer ◽  
Dispersion Relation ◽  
Parameter Space ◽  
Radial Direction ◽  
Ekman Layer ◽  
Rotating Flows ◽  
Absolute Instability ◽  
Boundary Layer Flows ◽  
Turbulent Transition ◽  
The Family

This paper is concerned with the theoretical behaviour of the laminar Ekman layer and the family of related rotating problems that includes both the Bödewadt and the von Kármán boundary-layer flows. Results from inviscid and viscous analyses are presented. In both cases, within specific regions of the parameter space, it is shown that the flows are absolutely unstable in the radial direction, i.e. disturbances grow in time at every radial location within these regions. Outside these regions, the flows are convectively unstable or stable. The absolute or convective nature of the flows is determined by examining the branch-point singularities of the dispersion relation. The onset of absolute instability is consistent with available experimental observations of the onset of laminar–turbulent transition in these flows.

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