scholarly journals LEGENDRIAN GRAPHS GENERATED BY TANGENTIAL FAMILY GERMS

2006 ◽  
Vol 49 (1) ◽  
pp. 29-37 ◽  
Author(s):  
Gianmarco Capitanio
Keyword(s):  
Cotangent Bundle ◽  
Normal Forms ◽  
Plane Curve ◽  
Parameter Family ◽  
Apparent Contour ◽  
The Family ◽  
Whitney Umbrella ◽  
Envelope Theory

AbstractWe construct a Legendrian version of envelope theory. A tangential family is a one-parameter family of rays emanating tangentially from a regular plane curve. The Legendrian graph of the family is the union of the Legendrian lifts of the family curves in the projectivized cotangent bundle $PT^*\mathbb{R}^2$. We study the singularities of Legendrian graphs and their stability under small tangential deformations. We also find normal forms of their projections into the plane. This allows us to interpret the beak-to-beak perestroika as the apparent contour of a deformation of the double Whitney umbrella singularity $A_1^\pm$.

scholarly journals Centre Symmetry Sets of Families of Plane Curves

2015 ◽  
Vol 48 (2) ◽  
Author(s):  
Peter Giblin ◽  
Graham Reeve
Keyword(s):  
Normal Forms ◽  
Direct Calculation ◽  
Parameter Family ◽  
Plane Curves ◽  
The Family ◽  
Study Centre

AbstractWe study centre symmetry sets and equidistants for a I-parameter family of plane curves where, for a special member of the family, there exist two inflexions with parallel tangents. Some results can be obtained by reducing a generating family to normal forms, but others require direct calculation from the generating family.

scholarly journals On the Dynamics of the Unified Chaotic System Between Lorenz and Chen Systems

2015 ◽  
Vol 25 (09) ◽  
pp. 1550122 ◽  
Author(s):  
Jaume Llibre ◽  
Ana Rodrigues
Keyword(s):  
Chaotic System ◽  
Parameter Family ◽  
Algebraic Surfaces ◽  
Hopf Bifurcations ◽  
Global Dynamics ◽  
Differential Systems ◽  
Special Values ◽  
Unified Chaotic System ◽  
Invariant Algebraic Surfaces ◽  
The Family

A one-parameter family of differential systems that bridges the gap between the Lorenz and the Chen systems was proposed by Lu, Chen, Cheng and Celikovsy. The goal of this paper is to analyze what we can say using analytic tools about the dynamics of this one-parameter family of differential systems. We shall describe its global dynamics at infinity, and for two special values of the parameter a we can also describe the global dynamics in the whole ℝ3using the invariant algebraic surfaces of the family. Additionally we characterize the Hopf bifurcations of this family.

Double points in families of map germs from ℝ2 to ℝ3

2020 ◽  
pp. 1-18
Author(s):  
J. A. Moya-Pérez ◽  
J. J. Nuño-Ballesteros
Keyword(s):  
Double Point ◽  
Parameter Family ◽  
Milnor Number ◽  
Double Points ◽  
The Family ◽  
Number Formula ◽  
Real Analytic

We show that a 1-parameter family of real analytic map germs [Formula: see text] with isolated instability is topologically trivial if it is excellent and the family of double point curves [Formula: see text] in [Formula: see text] is topologically trivial. In particular, we deduce that [Formula: see text] is topologically trivial when the Milnor number [Formula: see text] is constant.

scholarly journals SYMMETRIC ITINERARY SETS

2019 ◽  
Vol 100 (1) ◽  
pp. 97-108
Author(s):  
MICHAEL F. BARNSLEY ◽  
NICOLAE MIHALACHE
Keyword(s):  
Dynamical Systems ◽  
Parameter Family ◽  
The Family

We consider a one-parameter family of dynamical systems $W:[0,1]\rightarrow [0,1]$ constructed from a pair of monotone increasing diffeomorphisms $W_{i}$ such that $W_{i}^{-1}:$$[0,1]\rightarrow [0,1]$$(i=0,1)$. We characterise the set of symbolic itineraries of $W$ using an attractor $\overline{\unicode[STIX]{x1D6FA}}$ of an iterated closed relation, in the terminology of McGehee, and prove that there is a member of the family for which $\overline{\unicode[STIX]{x1D6FA}}$ is symmetrical.

scholarly journals Parameterizations of 1-Bridge Torus Knots

2003 ◽  
Vol 12 (04) ◽  
pp. 463-491 ◽  
Author(s):  
Doo Ho Choi ◽  
Ki Hyoung Ko
Keyword(s):  
Normal Form ◽  
Fundamental Group ◽  
Explicit Formula ◽  
Normal Forms ◽  
Double Cover ◽  
Torus Knots ◽  
Number Of Components ◽  
Torus Knot ◽  
Branched Cover ◽  
The Family

A 1-bridge torus knot in a 3-manifold of genus ≤ 1 is a knot drawn on a Heegaard torus with one bridge. We give two types of normal forms to parameterize the family of 1-bridge torus knots that are similar to the Schubert's normal form and the Conway's normal form for 2-bridge knots. For a given Schubert's normal form we give algorithms to determine the number of components and to compute the fundamental group of the complement when the normal form determines a knot. We also give a description of the double branched cover of an ambient 3-manifold branched along a 1-bridge torus knot by using its Conway's normal form and obtain an explicit formula for the first homology of the double cover.

scholarly journals Absolute stability properties of the explicit linear 3-step formulae

1980 ◽  
Vol 9 (124) ◽  
Author(s):  
Zahari Zlatev ◽  
Ole Østerby
Keyword(s):  
Numerical Solution ◽  
Absolute Stability ◽  
Numerical Experiments ◽  
Parameter Family ◽  
Stability Properties ◽  
The Family ◽  
The Absolute

A three-parameter family of explicit linear 3-step formulae is derived. The conditions which ensure zero-stability of the formulae in the family are formulated. The absolute stability properties of the zero-stable formulae in the family are investigated both for p = 3 and p = 2 where p is the order of the formulae under consideration. Some numerical experiments are carried out in order to illustrate that formulae with good absolute stability properties can efficiently be used in the numerical solution of problems in which the absolute stability properties are dominant.

APPLYING TWO-PARAMETER ENVELOPE THEORY TO DETERMINING SPHERICAL CAM PROFILE WITH CYLINDERICAL FOLLOWERS

2000 ◽  
Vol 24 (2) ◽  
pp. 415-435 ◽  
Author(s):  
Shyue-Cheng Yang ◽  
Cha’o-Kuang Chen
Keyword(s):  
Geometric Model ◽  
Geometric Characteristic ◽  
Parameter Family ◽  
Geometric Models ◽  
Numerical Example ◽  
Pressure Angle ◽  
Cam Profile ◽  
Cutting Path ◽  
Two Parameter ◽  
Envelope Theory

Using the envelope theory of two-parameter family of ball surfaces, two geometric models of spherical cam can be easily obtained when the follower-motion program has been given. The results of the envelope theory are used to determine an optimal spherical cam profile with an oscillating cylindrical follower. Some investigations of geometric characteristic, such as pressure angle and cutting path, are determined using the obtained geometric model. The principle curvatures are analyzed to avoid undercutting. Finally, a numerical example is given to illustrate the application of the procedure.

A mathematical model of the rotor profile of the single-screw compressor

2002 ◽  
Vol 216 (3) ◽  
pp. 343-351 ◽  
Author(s):  
S-C Yang
Keyword(s):  
Mathematical Model ◽  
Parameter Family ◽  
Cutting Edge ◽  
Screw Compressor ◽  
Single Screw ◽  
Rotor Profile ◽  
The Family ◽  
Screw Rotor ◽  
The One ◽  
Screw Rotors

This paper presents a method for determining the basic profile of a single-screw compressor including a gate rotor and a screw rotor. The inverse envelope concept for determining the cutting-edge curve of the gate rotor is presented. Based on this concept, the required cutter for machining the screw rotor can be obtained by an envelope of the one-parameter family of obtained screw rotors. The obtained screw rotor is an envelope of the family of gate rotor surfaces. Let the obtained envelope of the one-parameter family of gate rotor surfaces become the generating surface. The inverse envelope can be used to obtain the envelope of the family of generating surfaces. Then, the profile of the gate rotor with the cutting-edge curve can be easily obtained. The proposed method shows that the gate rotor and the screw rotor are engaged along the contact line at every instant. This is essential to reduce the effect of leakage on compressor performance. In this paper, a mathematical model of the meshing principle of the screw rotor with the gate rotor is established. As an example, the single-screw compressor for a compressor ratio of 11:6 is determined with the aid of the proposed mathematical model. Results from these mathematical models should have applications in the design of single-screw compressors.

scholarly journals Growth of positive words and lower bounds of the growth rate for Thompson’s groups F(p)

2017 ◽  
Vol 27 (01) ◽  
pp. 1-21
Author(s):  
José Burillo ◽  
Victor Guba
Keyword(s):  
Growth Rate ◽  
Lower Bound ◽  
Lower Bounds ◽  
Normal Forms ◽  
New Normal ◽  
Thompson's Groups ◽  
The Family ◽  
Thompson’S Group

Let [Formula: see text], [Formula: see text] be the family of generalized Thompson’s groups. Here, [Formula: see text] is the famous Richard Thompson’s group usually denoted by [Formula: see text]. We find the growth rate of the monoid of positive words in [Formula: see text] and show that it does not exceed [Formula: see text]. Also, we describe new normal forms for elements of [Formula: see text] and, using these forms, we find a lower bound for the growth rate of [Formula: see text] in its natural generators. This lower bound asymptotically equals [Formula: see text] for large values of [Formula: see text].

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