scholarly journals A family of minimal and renormalizable rectangle exchange maps

2019 ◽  
pp. 1-28
Author(s):  
IAN ALEVY ◽  
RICHARD KENYON ◽  
REN YI
Keyword(s):  
Dynamical System ◽  
Jordan Domain ◽  
Infinite Family ◽  
Renormalization Scheme ◽  
Galois Lattice ◽  
The Family

A domain exchange map (DEM) is a dynamical system defined on a smooth Jordan domain which is a piecewise translation. We explain how to use cut-and-project sets to construct minimal DEMs. Specializing to the case in which the domain is a square and the cut-and-project set is associated to a Galois lattice, we construct an infinite family of DEMs in which each map is associated to a Pisot–Vijayaraghavan (PV) number. We develop a renormalization scheme for these DEMs. Certain DEMs in the family can be composed to create multistage, renormalizable DEMs.

From the Outside In

2015 ◽  
Author(s):  
Derek Smith
Keyword(s):  
Positive Integer ◽  
Infinite Family ◽  
Open Problems ◽  
The Family

This chapter discusses Slothouber–Graatsma–Conway puzzle, which asks one to assemble six 1 × 2 × 2 pieces and three 1 × 1 × 1 pieces into the shape of a 3 × 3 × 3 cube. The puzzle has been generalized to larger cubes, and there is an infinite family of such puzzles. The chapter's primary argument is that, for any odd positive integer n = 2k + 1, there is exactly one way, up to symmetry, to make an n × n × n cube out of n tiny 1 × 1 × 1 cubes and six of each of a set of rectangular blocks. The chapter describes a way to solve each puzzle in the family and explains why there are no other solutions. It then presents several related open problems.

scholarly journals A family of formulas with reversal of high avoidability index

2017 ◽  
Vol 27 (05) ◽  
pp. 477-493 ◽  
Author(s):  
James Currie ◽  
Lucas Mol ◽  
Narad Rampersad
Keyword(s):  
Simple Structure ◽  
Complex Structure ◽  
Infinite Family ◽  
Index Formula ◽  
The Family

We present an infinite family of formulas with reversal whose avoidability index is bounded between [Formula: see text] and [Formula: see text], and we show that several members of the family have avoidability index [Formula: see text]. This family is particularly interesting due to its size and the simple structure of its members. For each [Formula: see text] there are several previously known avoidable formulas (without reversal) of avoidability index [Formula: see text] but they are small in number and they all have rather complex structure.

scholarly journals Perfect 2-Colorings of the Grassmann Graph of Planes

10.37236/8672 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Stefaan De Winter ◽  
Klaus Metsch
Keyword(s):  
Infinite Family ◽  
Grassmann Graph ◽  
The Family

We construct an infinite family of intriguing sets, or equivalently perfect 2-colorings, that are not tight in the Grassmann graph of planes of PG$(n,q)$, $n\ge 5$ odd, and show that the members of the family are the smallest possible examples if $n\ge 9$ or $q\ge 25$.

Linear programming analysis of VA/Q distributions: average distribution

1987 ◽  
Vol 62 (4) ◽  
pp. 1356-1362 ◽  
Author(s):  
K. S. Kapitan ◽  
P. D. Wagner
Keyword(s):  
Infinite Family ◽  
Inert Gas ◽  
Structural Detail ◽  
The Family ◽  
Single Exception ◽  
Average Distribution ◽  
Log Normal ◽  
Programming Analysis ◽  
Ventilation Perfusion Ratio ◽  
Simple Average

The defining equations of the multiple inert gas elimination technique are underdetermined, and an infinite number of VA/Q ratio distributions exists that fit the same inert gas data. Conventional least-squares analysis with enforced smoothing chooses a single member of this infinite family whose features are assumed to be representative of the family as a whole. To test this assumption, the average of all ventilation-perfusion ratio (VA/Q) distributions that are compatible with given data was calculated using a linear program. The average distribution so obtained was then compared with that recovered using enforced smoothing. Six typical sets of inert gas data were studied. In all sets but one, the distribution recovered with conventional enforced smoothing closely matched the structure of the average distribution. The single exception was associated with the broad log-normal VA/Q distribution, which is rarely observed using the technique. We conclude that the VA/Q distribution conventionally recovered approximates a simple average of all compatible distributions. It therefore displays average features and only that degree of fine structural detail that is typical of the family as a whole.

Subharmonic Orbits of a Strongly Nonlinear Oscillator Forced by Closely Spaced Harmonics

2010 ◽  
Vol 6 (1) ◽  
Author(s):  
Themistoklis P. Sapsis ◽  
Alexander F. Vakakis
Keyword(s):  
Dynamical System ◽  
Nonlinear Oscillator ◽  
Resonance Condition ◽  
Asymptotic Results ◽  
Strongly Nonlinear ◽  
Singular Perturbation Analysis ◽  
Local Bifurcation Analysis ◽  
The Family ◽  
Subharmonic Orbits ◽  
Strongly Nonlinear Oscillator

We study asymptotically the family of subharmonic responses of an essentially nonlinear oscillator forced by two closely spaced harmonics. By expressing the original oscillator in action-angle form, we reduce it to a dynamical system with three frequencies (two fast and one slow), which is amenable to a singular perturbation analysis. We then restrict the dynamics in neighborhoods of resonance manifolds and perform local bifurcation analysis of the forced subharmonic orbits. We find increased complexity in the dynamics as the frequency detuning between the forcing harmonics decreases or as the order of a secondary resonance condition increases. Moreover, we validate our asymptotic results by comparing them to direct numerical simulations of the original dynamical system. The method developed in this work can be applied to study the dynamics of strongly nonlinear (nonlinearizable) oscillators forced by multiple closely spaced harmonics; in addition, the formulation can be extended to the case of transient excitations.

STATISTICAL LAWS FOR COMPUTATIONAL COLLAPSE OF DISCRETIZED CHAOTIC MAPPINGS

1996 ◽  
Vol 06 (12a) ◽  
pp. 2389-2399 ◽  
Author(s):  
PHIL DIAMOND ◽  
ALEXEI POKROVSKII
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Dynamical Systems ◽  
Close Agreement ◽  
Computer Experiments ◽  
Continuum State ◽  
Chaotic Dynamical Systems ◽  
The Family ◽  
Random Mappings ◽  
Statistical Laws

When a dynamical system is realized on a computer, the computation is of a discretization, where finite machine arithmetic replaces continuum state space. For chaotic dynamical systems, the discretizations often have collapsing effects to a fixed point or to short cycles. Statistical properties of this phenomenon can be modeled by random mappings with an absorbing center. The model gives results which are very much in line with computational experiments and there appears to be a type of universality summarised by an Arcsine Law. The effects are discussed with special reference to the family of mappings fl(x)=1−|1−2x|l,x∈[0, 1], 1<l≤2. Computer experiments show close agreement with predictions of the model.

A FAMILY OF BRUNNIAN LINKS BASED ON EDWARDS' CONSTRUCTION OF VENN DIAGRAMS

2001 ◽  
Vol 10 (01) ◽  
pp. 97-107 ◽  
Author(s):  
ARNAUD MAES ◽  
CORINNE CERF
Keyword(s):  
Infinite Family ◽  
Venn Diagrams ◽  
The Family ◽  
Brunnian Links

We construct an infinite family of brunnian links whose projections give the family of Venn diagrams for many sets constructed by Edwards.

scholarly journals Gordian complexes of knots and virtual knots given by region crossing changes and arc shift moves

2020 ◽  
Vol 29 (10) ◽  
pp. 2042008
Author(s):  
Amrendra Gill ◽  
Madeti Prabhakar ◽  
Andrei Vesnin
Keyword(s):  
Simplicial Complex ◽  
Infinite Family ◽  
High Dimensional ◽  
Dimensional Simplex ◽  
Virtual Knot ◽  
Virtual Knots ◽  
The Family

Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in [Formula: see text]. Later Horiuchi and Ohyama defined Gordian complex of virtual knots using [Formula: see text]-move and forbidden moves. In this paper, we discuss Gordian complex of knots by region crossing change and Gordian complex of virtual knots by arc shift move. Arc shift move is a local move in the virtual knot diagram which results in reversing orientation locally between two consecutive crossings. We show the existence of an arbitrarily high-dimensional simplex in both the Gordian complexes, i.e. by region crossing change and by the arc shift move. For any given knot (respectively, virtual knot) diagram we construct an infinite family of knots (respectively, virtual knots) such that any two distinct members of the family have distance one by region crossing change (respectively, arc shift move). We show that the constructed virtual knots have the same affine index polynomial.

THE GORDIAN COMPLEX OF KNOTS

2002 ◽  
Vol 11 (03) ◽  
pp. 363-368 ◽  
Author(s):  
MIKAMI HIRASAWA ◽  
YOSHIAKI UCHIDA
Keyword(s):  
Simplicial Complex ◽  
Infinite Family ◽  
The Family

In this paper, we define the Gordian complex of knots, which is a simplicial complex whose vertices consist of all oriented knot types in the 3-sphere. We show that for any knot K, there exists an infinite family of distinct knots containing K such that any pair (Ki, Kj) of the member of the family, the Gordian distance dG(Ki, Kj) = 1.

Isolated singular points in the theory of algebraic surfaces

1933 ◽  
Vol 29 (2) ◽  
pp. 212-230 ◽  
Author(s):  
D. W. Babbage
Keyword(s):  
Infinite Family ◽  
Singular Points ◽  
Algebraic Surfaces ◽  
Three Dimensions ◽  
Linearly Independent ◽  
The Family ◽  
Image Position

If F(x0, x1, x2, x3) = 0 is the equation of a surface in space of three dimensions which has an ordinary isolated s-ple point O, then by means of the substitutionswhere Φ0 = 0, Φ1 = 0, …, Φr = 0 are the equations of r + 1 linearly independent surfaces passing simply through O, F is transformed into a surface F′ in [r], on which to the point O of F there corresponds a simple curve γ. The points of γ arise from the points of F in the first neighbourhood of O, and in this simple case the genus of γ is ½ (s − 1) (s − 2). In the study of properties which are common to all members of an infinite family of birationally equivalent surfaces no distinction is made between O and γ, O being regarded as a curve which has become infinitesimal on the particular surface of the family in question.

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