scholarly journals Algebraic Properties of Chromatic Roots

10.37236/6578 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Peter J. Cameron ◽  
Kerri Morgan
Keyword(s):  
Algebraic Integer ◽  
Tutte Polynomial ◽  
Chromatic Polynomial ◽  
Galois Groups ◽  
Algebraic Numbers ◽  
Isaac Newton ◽  
Isaac Newton Institute ◽  
Chromatic Root ◽  
The Subject ◽  
Chromatic Roots

A chromatic root is a root of the chromatic polynomial of a graph.  Any chromatic root is an algebraic integer. Much is known about the location of chromatic roots in the real and complex numbers, but rather less about their properties as algebraic numbers. This question was the subject of a seminar at the Isaac Newton Institute in late 2008.  The purpose of this paper is to report on the seminar and subsequent developments.We conjecture that, for every algebraic integer $\alpha$, there is a natural number n such that $\alpha+n$ is a chromatic root. This is proved for quadratic integers; an extension to cubic integers has been found by Adam Bohn. The idea is to consider certain special classes of graphs for which the chromatic polynomial is a product of linear factors and one "interesting" factor of larger degree. We also report computational results on the Galois groups of irreducible factors of the chromatic polynomial for some special graphs. Finally, extensions to the Tutte polynomial are mentioned briefly.

scholarly journals Chromatic roots as algebraic integers

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Adam Bohn
Keyword(s):  
Algebraic Integer ◽  
Algebraic Integers ◽  
Chromatic Polynomials ◽  
Integer Multiple ◽  
Chromatic Root ◽  
Il Y A ◽  
International Audience ◽  
The Subject ◽  
Large Families ◽  
Chromatic Roots

International audience A chromatic root is a zero of the chromatic polynomial of a graph. At a Newton Institute workshop on Combinatorics and Statistical Mechanics in 2008, two conjectures were proposed on the subject of which algebraic integers can be chromatic roots, known as the ``$α +n$ conjecture'' and the ``$nα$ conjecture''. These say, respectively, that given any algebraic integer α there is a natural number $n$ such that $α +n$ is a chromatic root, and that any positive integer multiple of a chromatic root is also a chromatic root. By computing the chromatic polynomials of two large families of graphs, we prove the $α +n$ conjecture for quadratic and cubic integers, and show that the set of chromatic roots satisfying the nα conjecture is dense in the complex plane. Une racine chromatique est un zéro du polynôme chromatique d'un graphe. A un atelier au Newton Institute sur la combinatoire et la mécanique statistique en 2008, deux conjectures ont été proposées dont le sujet des entiers algébriques peut être racines chromatiques, connus sous le nom ``la conjecture $α + n$'' et ``la conjecture $n α$ ''. Les conjectures veulent dire, respectivement, que pour chaque entier algébrique $α$ il y a un nombre entier naturel $n$, tel que $α + n$ est une racine chromatique, et que chaque multiple entier positif d'une racine chromatique est aussi une racine chromatique . En calculant les polynômes chromatiques de deux grandes familles de graphes, on prouve la conjecture $α + n$ pour les entiers quadratiques et cubiques, et montre que l'ensemble des racines chromatiques qui confirme la conjecture $nα$ est dense dans le plan complexe.

The Curious Eye

2020 ◽  
Author(s):  
Erin Webster
Keyword(s):  
Seventeenth Century ◽  
Human Vision ◽  
Francis Bacon ◽  
Robert Hooke ◽  
Robert Boyle ◽  
Isaac Newton ◽  
Johannes Kepler ◽  
The Social ◽  
The Subject ◽  
Historical Moment

The Curious Eye explores early modern debates over two related questions: what are the limits of human vision, and to what extent can these limits be overcome by technological enhancement? Today, in our everyday lives we rely on optical technology to provide us with information about visually remote spaces even as we question the efficacy and ethics of such pursuits. But the debates surrounding the subject of technologically mediated vision have their roots in a much older literary tradition in which the ability to see beyond the limits of natural human vision is associated with philosophical and spiritual insight as well as social and political control. The Curious Eye provides insight into the subject of optically mediated vision by returning to the literature of the seventeenth century, the historical moment in which human visual capacity in the West was first extended through the application of optical technologies to the eye. Bringing imaginative literary works by Francis Bacon, John Milton, Margaret Cavendish, and Aphra Behn together with optical and philosophical treatises by Johannes Kepler, René Descartes, Robert Hooke, Robert Boyle, and Isaac Newton, The Curious Eye explores the social and intellectual impact of the new optical technologies of the seventeenth century on its literature. At the same time, it demonstrates that social, political, and literary concerns are not peripheral to the optical science of the period but rather an integral part of it, the legacy of which we continue to experience.

scholarly journals V. A letter to Dr. Jurin, Coll. Med. Lond. Soc. & Secr. R. S. concerning the abovementioned appearance in the rainbow, with some other reflections on the same subject

1723 ◽  
Vol 32 (375) ◽  
pp. 245-261
Keyword(s):  
Isaac Newton ◽  
The Subject ◽  
Made In

Sir, Upon your communicating to me the curious Observations, your Friend Dr. Langwith had made on the Rainbow, I inform'd you those Appearances might, I thought, be explain'd by the Discoveries, the Great Sir Isaac Newton had made in the Subject of Light and Colours, in his Wonderful Treatise of Optics .

Time: A Very Short Introduction

2021 ◽  
Author(s):  
Jenann Ismael
Keyword(s):  
History Of Physics ◽  
Philosophical Thinking ◽  
Short Introduction ◽  
Isaac Newton ◽  
Experience Of Time ◽  
The Everyday ◽  
History Of ◽  
The Subject ◽  
Nature Of Time ◽  
Everyday Living

Time: A Very Short Introduction explores questions about the nature of time that have been at the heart of philosophical thinking since its beginnings: questions like whether time has a beginning or end, whether and in what sense time passes, how time is different from space, whether time has a direction, and whether it is possible to travel in time. These questions passed into the hands of scientists with the work of Isaac Newton when the structure of space and time became connected to motion and included the subject matter of physics. This VSI charts the way that the history of physics, from Isaac Newton through Albert Einstein’s two revolutions, wrought changes to the conception of time. There are parts of physics that are in a state of confusion, but this strand of development is a story of philosophical illumination and conceptual beauty. The discussion here provides an opportunity to see what distinguishes the methods of physics from those of philosophy. It brings together physics, cognitive science, and phenomenology in the service of reconciling what modern theories tell us about the nature of time with the everyday living experience of time.

scholarly journals At the Anniversary Meeting, November 30, 1844

1851 ◽  
Vol 5 ◽  
pp. 519-538
Keyword(s):  
Royal Society ◽  
Sir Isaac Newton ◽  
Isaac Newton ◽  
Legal Proceedings ◽  
The Public ◽  
Foreign Countries ◽  
The Subject ◽  
The Government ◽  
The Antarctic

Gentlemen, The time has again come round for my addressing you, and for ex­pressing my own gratitude, as well as yours, to your Council for their constant and zealous attention to the interests of the Royal Society. We have been compelled during several late years to have recourse to legal proceedings on the subject of the great tithes of Mablethorp, a portion of the Society’s property, and I rejoice to say with success. In my last address, I was required to give our thanks to Mr. Watt and to Mr. Dollond for the valuable busts which they had kindly presented to us. That of Mr. Dollond is placed at the commence­ment of the staircase leading to our apartments, and serves to indi­cate that his valuable improvements in the construction of our tele­scopes have been so many steps to the acquisition of higher and higher knowledge of the great universe of which this globe forms so insignificant a part. By the liberality of Mr. Watt we shall soon be furnished with handsome pedestals for the busts of his father and of Sir Isaac Newton, the two great lights of British mechanical genius and British philosophical science. Mr. Gilbert has kindly undertaken to furnish a similar pedestal for the bust of his father, and we have thought it right to provide one for that of Sir Joseph Banks. These will shortly form a conspicuous ornament of our place of meeting. The magnetical observatories are still carrying on their observa­tions, both in Her Majesty’s dominions and in foreign countries, and another naval officer, Lieut. Moore, has proceeded to the Antarctic Seas to complete a portion of the survey of Captain Sir James Ross, which was interrupted by stress of weather. That gallant and enter­ prising officer will, I hope, ere long give to us and to the public his own narrative of his important discoveries. Detailed accounts of the botany and zoology of the regions visited by him are preparing under the patronage of the Government, while Colonel Sabine is proceeding with the raagnetical observations, which were the more immediate objects of this, one of the most important voyages of discovery ever undertaken.

A generalization of the Whitney rank generating function

1993 ◽  
Vol 113 (2) ◽  
pp. 267-280 ◽  
Author(s):  
G. E. Farr
Keyword(s):  
Generating Function ◽  
Linear Code ◽  
Tutte Polynomial ◽  
Natural Generalization ◽  
Chromatic Polynomial ◽  
Weight Enumerator ◽  
Hadamard Transform ◽  
Percolation Probability ◽  
Special Cases

AbstractThe Whitney quasi-rank generating function, which generalizes the Whitney rank generating function (or Tutte polynomial) of a graph, is introduced. It is found to include as special cases the weight enumerator of a (not necessarily linear) code, the percolation probability of an arbitrary clutter and a natural generalization of the chromatic polynomial. The crucial construction, essentially equivalent to one of Kung, is a means of associating, to any function, a rank-like function with suitable properties. Some of these properties, including connections with the Hadamard transform, are discussed.

A Tutte Polynomial for Maps

2018 ◽  
Vol 27 (6) ◽  
pp. 913-945 ◽  
Author(s):  
ANDREW GOODALL ◽  
THOMAS KRAJEWSKI ◽  
GUUS REGTS ◽  
LLUÍS VENA
Keyword(s):  
Finite Group ◽  
Tutte Polynomial ◽  
Chromatic Polynomial ◽  
Polynomial Invariant ◽  
Flow Polynomial ◽  
Orientable Surfaces

We follow the example of Tutte in his construction of the dichromate of a graph (i.e. the Tutte polynomial) as a unification of the chromatic polynomial and the flow polynomial in order to construct a new polynomial invariant of maps (graphs embedded in orientable surfaces). We call this the surface Tutte polynomial. The surface Tutte polynomial of a map contains the Las Vergnas polynomial, the Bollobás–Riordan polynomial and the Krushkal polynomial as specializations. By construction, the surface Tutte polynomial includes among its evaluations the number of local tensions and local flows taking values in any given finite group. Other evaluations include the number of quasi-forests.

Brighter than how many suns? Sir Isaac Newton’s burning mirror

1989 ◽  
Vol 43 (1) ◽  
pp. 31-51 ◽  
Keyword(s):  
Royal Society ◽  
Scientific Literature ◽  
Radiant Heat ◽  
The Other ◽  
Sir Isaac Newton ◽  
Isaac Newton ◽  
Other Hand ◽  
The Press ◽  
The Subject ◽  
Hearsay Evidence

There are a number of references in the scientific literature to a burning mirror designed by Sir Isaac Newton (1). Together, they record that it was made from seven separate concave glasses, each about a foot in diameter, that Newton demonstrated its effects at several meetings of the Royal Society and that he presented it to the Society. Nonetheless, neither the earliest published list of instruments possessed by the Royal Society nor the most recent one mentions the burning mirror; the latest compiler does not even include it amongst those items, once owned, now lost. No reference to the instrument apparently survives in the Society’s main records. It is not listed by the author of the recent compendium on Newton’s life and work (2). There is, however, some contemporary information still extant (Appendix 1). Notes of the principles of its design and some of its effects are to be found in the Society’s Journal Book for 1704; some of the dimensions and the arrangement of the mirrors are given in a Lexicon published by John Harris which he donated to the Royal Society at the same meeting, 12 July 1704, at which Newton gave the Society the speculum. The last reference in the Journal Book is dated 15 November that year, when Mr Halley, the then secretary to the Society, was desired to draw up an account of the speculum and its effects (3). No such account appears to have been presented to the Royal Society. There is no reference in Newton’s published papers and letters of his chasing Halley to complete the task, nor is there any mention of it in the general references to Halley. The latter was, of course, quite accustomed to performing odd jobs for Newton; that same year he was to help the Opticks through the press. The only other contemporary reference to the burning mirror, though only hearsay evidence since Flamsteed was not present at the meeting, is in a letter the latter wrote to James Pound; this confirms that there were seven mirrors and that the aperture of each was near a foot in diameter (4). Because John Harris gave his Dictionary to the Royal Society in Newton’s presence, it is reasonable to assume that his description is accurate. As Newton would hardly have left an inaccurate one unchallenged, then, belatedly, the account desired of Mr Halley can be presented. In some respects, the delay is advantageous, since the subject of radiant heat and its effects, although already by Newton’s period an ancient one, is today rather better understood. On the other hand, some data has to be inferred, that could have been measured, and some assumptions made about Newton’s procedures and understanding that could have been checked (5).

scholarly journals Developments in turbulence research: a review based on the 1999 Programme of the Isaac Newton Institute, Cambridge

2001 ◽  
Vol 436 ◽  
pp. 353-391 ◽  
Author(s):  
J. C. R. HUNT ◽  
N. D. SANDHAM ◽  
J. C. VASSILICOS ◽  
B. E. LAUNDER ◽  
P. A. MONKEWITZ ◽  
...  
Keyword(s):  
Reynolds Number ◽  
Large Eddy Simulation ◽  
Numerical Simulations ◽  
Turbulent Flows ◽  
Small Scale ◽  
Isaac Newton ◽  
Eddy Simulation ◽  
Isaac Newton Institute ◽  
Large Eddy ◽  
Eddy Structures

Recent research is making progress in framing more precisely the basic dynamical and statistical questions about turbulence and in answering them. It is helping both to define the likely limits to current methods for modelling industrial and environmental turbulent flows, and to suggest new approaches to overcome these limitations. Our selective review is based on the themes and new results that emerged from more than 300 presentations during the Programme held in 1999 at the Isaac Newton Institute, Cambridge, UK, and on research reported elsewhere. A general conclusion is that, although turbulence is not a universal state of nature, there are certain statistical measures and kinematic features of the small-scale flow field that occur in most turbulent flows, while the large-scale eddy motions have qualitative similarities within particular types of turbulence defined by the mean flow, initial or boundary conditions, and in some cases, the range of Reynolds numbers involved. The forced transition to turbulence of laminar flows caused by strong external disturbances was shown to be highly dependent on their amplitude, location, and the type of flow. Global and elliptical instabilities explain much of the three-dimensional and sudden nature of the transition phenomena. A review of experimental results shows how the structure of turbulence, especially in shear flows, continues to change as the Reynolds number of the turbulence increases well above about 104 in ways that current numerical simulations cannot reproduce. Studies of the dynamics of small eddy structures and their mutual interactions indicate that there is a set of characteristic mechanisms in which vortices develop (vortex stretching, roll-up of instability sheets, formation of vortex tubes) and another set in which they break up (through instabilities and self- destructive interactions). Numerical simulations and theoretical arguments suggest that these often occur sequentially in randomly occurring cycles. The factors that determine the overall spectrum of turbulence were reviewed. For a narrow distribution of eddy scales, the form of the spectrum can be defined by characteristic forms of individual eddies. However, if the distribution covers a wide range of scales (as in elongated eddies in the ‘wall’ layer of turbulent boundary layers), they collectively determine the spectra (as assumed in classical theory). Mathematical analyses of the Navier–Stokes and Euler equations applied to eddy structures lead to certain limits being defined regarding the tendencies of the vorticity field to become infinitely large locally. Approximate solutions for eigen modes and Fourier components reveal striking features of the temporal, near-wall structure such as bursting, and of the very elongated, spatial spectra of sheared inhomogeneous turbulence; but other kinds of eddy concepts are needed in less structured parts of the turbulence. Renormalized perturbation methods can now calculate consistently, and in good agreement with experiment, the evolution of second- and third-order spectra of homogeneous and isotropic turbulence. The fact that these calculations do not explicitly include high-order moments and extreme events, suggests that they may play a minor role in the basic dynamics. New methods of approximate numerical simulations of the larger scales of turbulence or ‘very large eddy simulation’ (VLES) based on using statistical models for the smaller scales (as is common in meteorological modelling) enable some turbulent flows with a non-local and non-equilibrium structure, such as impinging or convective flows, to be calculated more efficiently than by using large eddy simulation (LES), and more accurately than by using ‘engineering’ models for statistics at a single point. Generally it is shown that where the turbulence in a fluid volume is changing rapidly and is very inhomogeneous there are flows where even the most complex ‘engineering’ Reynolds stress transport models are only satisfactory with some special adaptation; this may entail the use of transport equations for the third moments or non-universal modelling methods designed explicitly for particular types of flow. LES methods may also need flow-specific corrections for accurate modelling of different types of very high Reynolds number turbulent flow including those near rigid surfaces.This paper is dedicated to the memory of George Batchelor who was the inspiration of so much research in turbulence and who died on 30th March 2000. These results were presented at the last fluid mechanics seminar in DAMTP Cambridge that he attended in November 1999.

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