Perfect 2-Colorings of the Grassmann Graph of Planes

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.

A family of formulas with reversal of high avoidability index

International Journal of Algebra and Computation ◽

10.1142/s0218196717500242 ◽

2017 ◽

Vol 27 (05) ◽

pp. 477-493 ◽

Cited By ~ 3

Author(s):

James Currie ◽

Lucas Mol ◽

Narad Rampersad

Keyword(s):

Simple Structure ◽

Complex Structure ◽

Infinite Family ◽

Index Formula ◽

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.

Linear programming analysis of VA/Q distributions: average distribution

Journal of Applied Physiology ◽

10.1152/jappl.1987.62.4.1356 ◽

1987 ◽

Vol 62 (4) ◽

pp. 1356-1362 ◽

Cited By ~ 6

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.

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

10.1142/s0218216501000743 ◽

2001 ◽

Vol 10 (01) ◽

pp. 97-107 ◽

Cited By ~ 1

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.

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

10.1142/s0218216520420080 ◽

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 ◽

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

10.1142/s0218216502001676 ◽

2002 ◽

Vol 11 (03) ◽

pp. 363-368 ◽

Cited By ~ 18

Author(s):

MIKAMI HIRASAWA ◽

YOSHIAKI UCHIDA

Keyword(s):

Simplicial Complex ◽

Infinite 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

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100011014 ◽

1933 ◽

Vol 29 (2) ◽

pp. 212-230 ◽

Cited By ~ 4

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.

A family of minimal and renormalizable rectangle exchange maps

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2019.77 ◽

2019 ◽

pp. 1-28

Author(s):

IAN ALEVY ◽

RICHARD KENYON ◽

REN YI

Keyword(s):

Dynamical System ◽

Jordan Domain ◽

Infinite Family ◽

Renormalization Scheme ◽

Galois Lattice ◽

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.

Biquandle virtual brackets

10.1142/s0218216519400030 ◽

2019 ◽

Vol 28 (11) ◽

pp. 1940003 ◽

Cited By ~ 1

Author(s):

Sam Nelson ◽

Kanako Oshiro ◽

Ayaka Shimizu ◽

Yoshiro Yaguchi

Keyword(s):

Commutative Ring ◽

Knot Theory ◽

Infinite Family ◽

Polynomial Form ◽

Link Diagrams ◽

Classical Definition ◽

Special Cases ◽

We introduce an infinite family of quantum enhancements of the biquandle counting invariant which we call biquandle virtual brackets. Defined in terms of skein invariants of biquandle colored oriented knot and link diagrams with values in a commutative ring [Formula: see text] using virtual crossings as smoothings, these invariants take the form of multisets of elements of [Formula: see text] and can be written in a “polynomial” form for convenience. The family of invariants defined herein includes as special cases all quandle and biquandle 2-cocycle invariants, all classical skein invariants (Alexander–Conway, Jones, HOMFLYPT and Kauffman polynomials) and all biquandle bracket invariants defined in [S. Nelson, M. E. Orrison and V. Rivera, Quantum enhancements and biquandle brackets, J. Knot Theory Ramifications 26(5) (2017) 1750034] as well as new invariants defined using virtual crossings in a fundamental way, without an obvious purely classical definition.

New AdS2 backgrounds and $$ \mathcal{N} $$ = 4 conformal quantum mechanics

Journal of High Energy Physics ◽

10.1007/jhep03(2021)277 ◽

2021 ◽

Vol 2021 (3) ◽

Author(s):

Yolanda Lozano ◽

Carlos Nunez ◽

Anayeli Ramirez ◽

Stefano Speziali

Keyword(s):

Quantum Mechanics ◽

Central Charge ◽

Infinite Family ◽

Dimensional System ◽

One Dimensional ◽

The Family ◽

The One ◽

Conformal Quantum Mechanics ◽

Chern Simons

Abstract We present a new infinite family of Type IIB backgrounds with an AdS2 factor, preserving $$ \mathcal{N} $$ N = 4 SUSY. For each member of the family we propose a precise dual Super Conformal Quantum Mechanics (SCQM). We provide holographic expressions for the number of vacua (the “central charge”), Chern-Simons terms and other non-perturbative aspects of the SCQM. We relate the “central charge” of the one-dimensional system with a combination of electric and magnetic fluxes in Type IIB. The Ramond-Ramond fluxes are used to propose an extremisation principle for the central charge. Other physical and geometrical aspects of these conformal quantum mechanics are analysed.