A Combinatorial Distinction Between Unit Circles and Straight Lines: How Many Coincidences Can they Have?

2009 ◽  
Vol 18 (5) ◽  
pp. 691-705 ◽  
Author(s):  
GYÖRGY ELEKES ◽  
MIKLÓS SIMONOVITS ◽  
ENDRE SZABÓ
Keyword(s):  
Parameter Family ◽  
Sufficient Condition ◽  
Simple Application ◽  
Quadratic Number ◽  
General Sufficient Condition ◽  
Family Of Curves ◽  
Unit Circles ◽  
Triple Points ◽  
Straight Lines ◽  
Special Case

We give a very general sufficient condition for a one-parameter family of curves not to have n members with ‘too many’ (i.e., a near-quadratic number of) triple points of intersections. As a special case, a combinatorial distinction between straight lines and unit circles will be shown. (Actually, this is more than just a simple application; originally this motivated our results.)

scholarly journals Nets of Lines with the Combinatorics of the Square Grid and with Touching Inscribed Conics

2021 ◽  
Author(s):  
Alexander I. Bobenko ◽  
Alexander Y. Fairley
Keyword(s):  
Projective Plane ◽  
Parameter Family ◽  
Common Edge ◽  
Square Grid ◽  
Polygonal Chains ◽  
Straight Lines ◽  
Special Case

AbstractIn the projective plane, we consider congruences of straight lines with the combinatorics of the square grid and with all elementary quadrilaterals possessing touching inscribed conics. The inscribed conics of two combinatorially neighbouring quadrilaterals have the same touching point on their common edge-line. We suggest that these nets are a natural projective generalisation of incircular nets. It is shown that these nets are planar Koenigs nets. Moreover, we show that general Koenigs nets are characterised by the existence of a 1-parameter family of touching inscribed conics. It is shown that the lines of any grid of quadrilaterals with touching inscribed conics are tangent to a common conic. These grids can be constructed via polygonal chains that are inscribed in conics. The special case of billiards in conics corresponds to incircular nets.

scholarly journals A Uniqueness Result for a Simple Superlinear Eigenvalue Problem

2021 ◽  
Vol 31 (2) ◽  
Author(s):  
Michael Herrmann ◽  
Karsten Matthies
Keyword(s):  
Eigenvalue Problem ◽  
Numerical Simulations ◽  
Convolution Operator ◽  
Existence And Uniqueness ◽  
Uniqueness Result ◽  
Constitutive Laws ◽  
Parameter Family ◽  
Shape Constraint ◽  
Special Case ◽  
Nonlinear Eigenfunctions

AbstractWe study the eigenvalue problem for a superlinear convolution operator in the special case of bilinear constitutive laws and establish the existence and uniqueness of a one-parameter family of nonlinear eigenfunctions under a topological shape constraint. Our proof uses a nonlinear change of scalar parameters and applies Krein–Rutman arguments to a linear substitute problem. We also present numerical simulations and discuss the asymptotics of two limiting cases.

A Classification and Closure Properties of Languages for Describing Concurrent System Behaviours

1981 ◽  
Vol 4 (3) ◽  
pp. 531-549 ◽  
Author(s):  
Miklós Szijártó
Keyword(s):  
Formal Languages ◽  
Parallel Program ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Concurrent System ◽  
Closure Properties ◽  
Sequential Program ◽  
Necessary And Sufficient ◽  
Special Case ◽  
Independence Relations

The correspondence between sequential program schemes and formal languages is well known (Blikle and Mazurkiewicz (1972), Engelfriet (1974)). The situation is more complicated in the case of parallel program schemes, and trace languages (Mazurkiewicz (1977)) have been introduced to describe them. We introduce the concept of the closure of a language on a so called independence relation on the alphabet of the language, and formulate several theorems about them and the trace languages. We investigate the closedness properties of Chomsky classes under closure on independence relations, and as a special case we derive a new necessary and sufficient condition for the regularity of the commutative closure of a language.

Existence of Periodic Solution for Equation of Motion of Simple Beams With Harmonically Variable Length

1995 ◽  
Author(s):  
Ebrahim Esmailzadeh ◽  
Gholamreza Nakhaie-Jazar ◽  
Bahman Mehri
Keyword(s):  
Periodic Solution ◽  
Fixed Point ◽  
Nonlinear Function ◽  
Axial Direction ◽  
Equation Of Motion ◽  
The Other ◽  
Sufficient Condition ◽  
Vibrating String ◽  
The Stability ◽  
Special Case

Abstract The transverse vibrating motion of a simple beam with one end fixed while driven harmonically along its axial direction from the other end is investigated. For a special case of zero value for the rigidity of the beam, the system reduces to that of a vibrating string with the corresponding equation of its motion. The sufficient condition for the periodic solution of the beam is then derived by means of the Green’s function and Schauder’s fixed point theorem. The criteria for the stability of the system is well defined and the condition for which the performance of the beam behaves as a nonlinear function is stated.

scholarly journals Curvature Properties of Generalized pp-Wave Metrics

2021 ◽  
Vol 45 (02) ◽  
pp. 237-258
Author(s):  
ABSOS ALI SHAIKH ◽  
TRAN QUOC BINH ◽  
HARADHAN KUNDU
Keyword(s):  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Sufficient Condition ◽  
Type Condition ◽  
Pure Radiation ◽  
Projective Curvature ◽  
Pp Wave ◽  
Codazzi Type ◽  
Curvature Tensors ◽  
Special Case

The main objective of the present paper is to investigate the curvature properties of generalized pp-wave metrics. It is shown that a generalized pp-wave spacetime is Ricci generalized pseudosymmetric, 2-quasi-Einstein and generalized quasi-Einstein in the sense of Chaki. As a special case it is shown that pp-wave spacetime is semisymmetric, semisymmetric due to conformal and projective curvature tensors, R-space by Venzi and satisfies the pseudosymmetric type condition P ⋅ P = −13Q(S,P). Again we investigate the sufficient condition for which a generalized pp-wave spacetime turns into pp-wave spacetime, pure radiation spacetime, locally symmetric and recurrent. Finally, it is shown that the energy-momentum tensor of pp-wave spacetime is parallel if and only if it is cyclic parallel. Again the energy momentum tensor is Codazzi type if it is cyclic parallel but the converse is not true as shown by an example. Finally, we make a comparison between the curvature properties of the Robinson-Trautman metric and generalized pp-wave metric.

A sufficient condition for instability of BGK type waves in both unmagnetized and magnetized plasmas

1977 ◽  
Vol 17 (1) ◽  
pp. 57-68 ◽  
Author(s):  
E. Infeld ◽  
G. Rowlands
Keyword(s):  
Magnetic Field ◽  
Linear Theory ◽  
General Problem ◽  
Uniform Magnetic Field ◽  
Magnetized Plasmas ◽  
Sufficient Condition ◽  
General Sufficient Condition ◽  
Special Cases

This paper investigates the general problem of stability of Bernstein—Greene— Kruskal type waves. By investigating perturbations perpendicular to the wave, we obtain a general sufficient condition for instability. This is then extended to the case of magnetized plasmas with a uniform magnetic field in the direction of the BGK wave. New perturbed modes, having no counterpart in linear theory, are also found. Various special cases are considered and previous, more particular results confirmed.

scholarly journals Convergence of the Chern–Moser–Beloshapka normal forms

2020 ◽  
Vol 2020 (765) ◽  
pp. 205-247
Author(s):  
Bernhard Lamel ◽  
Laurent Stolovitch
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Direct Proof ◽  
Sufficient Condition ◽  
Normalizing Transformation ◽  
General Sufficient Condition ◽  
Real Analytic ◽  
Moser Normal Form

AbstractIn this article, we give a normal form for real-analytic, Levi-nondegenerate submanifolds of{\mathbb{C}^{N}}of codimension{d\geq 1}under the action of formal biholomorphisms. We find a very general sufficient condition on the formal normal form that ensures that the normalizing transformation to this normal form is holomorphic. In the case{d=1}our methods in particular allow us to obtain a new and direct proof of the convergence of the Chern–Moser normal form.

scholarly journals On composition of formal power series

2002 ◽  
Vol 30 (12) ◽  
pp. 761-770 ◽  
Author(s):  
Xiao-Xiong Gan ◽  
Nathaniel Knox
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Open Problem ◽  
Constant Term ◽  
Radius Of Convergence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Formal Power ◽  
Special Case

Given a formal power seriesg(x)=b0+b1x+b2x2+⋯and a nonunitf(x)=a1x+a2x2+⋯, it is well known that the composition ofgwithf,g(f(x)), is a formal power series. If the formal power seriesfabove is not a nonunit, that is, the constant term offis not zero, the existence of the compositiong(f(x))has been an open problem for many years. The recent development investigated the radius of convergence of a composed formal power series likefabove and obtained some very good results. This note gives a necessary and sufficient condition for the existence of the composition of some formal power series. By means of the theorems established in this note, the existence of the composition of a nonunit formal power series is a special case.

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