Arthur Cayley Research Articles

The Artistic, Algebraic, and Severe “Russian” Genius Arthur Cayley

The Artistic, Algebraic, and Severe “Russian” Genius Arthur Cayley

The $2$-Hessian and sextactic points on plane algebraic curves

In an article from 1865, Arthur Cayley claims that given a plane algebraic curve there exists an associated $2$-Hessian curve that intersects it in its sextactic points. In this paper we fix an error in Cayley's calculations and provide the correct defining polynomial for the $2$-Hessian. In addition, we present a formula for the number of sextactic points on cuspidal curves and tie this formula to the $2$-Hessian. Lastly, we consider the special case of rational curves, where the sextactic points appear as zeros of the Wronski determinant of the 2nd Veronese embedding of the curve.

The ontology of person-centered healthcare: Theoretical essentials to reground medicine within its humanistic framework. Part I - A centered concept of personhood: Some tools to conceptualise subjectivity

To address subjectivity, as a generally rooted phenomenon, other ways of visualisation must be applied than in conventional objectivistic approaches. Using ‘trees’ as operational metaphors, as employed in Arthur Cayley’s ‘theory of the analytical forms called trees’, one rooted ‘tree’ must be set beneath the other and, if such ‘trees’ are combined, the resulting ‘forest’ is nevertheless made up of individual ‘trees’ and not of a deconstructed mix of ‘roots’, ‘branches’, ‘leaves’ or further categories, each understood as addressable both jointly and individually. The reasons for why we have chosen a graph theory and corresponding discrete mathematics as an approach and application are set out in this first of our three articles. It combines two approaches that, in combination, are quite uncommon and which are therefore not immediately familiar to all readers. But as simple as it is to imagine a tree, or a forest, it is equally simple to imagine a child blowing soap bubbles with the aid of a blow ring. A little more challenging, perhaps, is the additional idea of arranging such blow rings in series, transforming the size of the soap bubble in one ring after the other. To finally combine both pictures, the one of trees and the other of blow rings, goes beyond simple imagination, especially when we prolong the imagined blow ring becoming a tunnel, with a specific inner shape. The inner shape of the blow ring and its expansion as a tunnel are understood as determined by discrete qualities, each forming an internal continuity, depicted as a scale, with the scales combined in the form of a glyph plot. The different positions on these scales determine their length and if the endpoints of the spines are connected with an enveloping line then this corresponds to the free space left open in the tunnel to go through it. Using so many visualisation techniques at once is testing. Nevertheless, this is what we propose here and to facilitate such a visualisation within the imagination, we do it step by step. As the intended result of this ‘juggling of three balls’, as it were, we end up with a concept of how living beings elaborate their principal structure to enable controlled outside-inside communication.

Counting of spanning trees of a complete graph

In 1889, Arthur Cayley published an article that contained a formula for counting the spanning trees of a complete graph. This theorem says that: Let n E N and Kn the complete graph with n vertices. Then the number of spanning trees of Kn is established by n n-2: The present work is constituted by a brief literary review about the basic concepts and results of the graph theory and detailed demonstration of the Cayley’s Formula, given by the meticulous construction of a bijection between the set of the spanning trees and a special set of numeric sequences. At the end we bring an algorithm that describes a precise construction of the spanning trees obtained of Kn from Cayley-Prufer sequences.

Arthur Cayley, Robert Harley and the quintic equation: newly discovered letters 1859–1863

Beginning in the 1840's, Arthur Cayley (1821–1895) led a vast invariant theory programme in algebra. After learning of results of James Cockle (1819–1895) and Robert Harley (1828–1910), he applied the techniques of invariant theory to the calculation of resolvents of quintic equations. Letters recently discovered reveal the priorities of Cayley and Harley with respect to the quintic, an approach which was at variance with that via the theory of groups. As another recently discovered manuscript reveals, Cayley returned to this subject in his final days.

Hurwitz’s counting isomers of alkanes

In a late entry in his mathematical diary from 1918 Adolf Hurwitz refined ideas due to Arthur Cayley in order to count trees of specific structure (namely isomers of alkanes). We shall analyze his reasoning in detail and discuss it in context with Cayley's original approach and George Pólya's later enumeration method.

In the footsteps of Euler and MacMahon: combinatorics, the mathematics that counts

This article is based on my presidential address, given at the BSHM Christmas meeting on 7 December 2013. It features the history of combinatorics, a much neglected area, with particular reference to the work of two mathematicians who have always interested me—Leonhard Euler and Major Percy MacMahon—and with further contributions by Arthur Cayley (on trees) and G H Hardy and S Ramanujan (on partitions).

Networks and Spanning Trees: The Juxtaposition of Prüfer and Borůvka

This paper outlines a method for teaching topics in undergraduate mathematics or computer science via historical curricular modules. The contents of one module, “Networks and Spanning Trees,” are discussed from the original work of Arthur Cayley, Heinz Prüfer, and Otakar Borůvka that motivates the enumeration and application of trees in graph theory. Cayley correctly identifies a pattern for the number of (labeled) trees on n fixed vertices. Prüfer’s paper provides a rigorous verification of this pattern, whereas Borůvka’s paper offers one of the first algorithms for finding a minimal spanning tree over the domain of labeled trees. These latter two papers in juxtaposition offer a pleasing confluence of concepts and applications, written verbally before the modern terminology of graph theory had been formulated.

Stably Cayley Groups in Characteristic Zero

A linear algebraic group G over a field k is called a Cayley group if it admits a Cayley map, that is, a G-equivariant birational isomorphism over k between the group variety G and its Lie algebra. A Cayley map can be thought of as a partial algebraic analog of the exponential map. A prototypical example is the classical “Cayley transform” for the special orthogonal group SOn defined by Arthur Cayley in 1846. A linear algebraic group G is called stably Cayley if is Cayley for some r≥0. Here denotes the split r-dimensional k-torus. These notions were introduced in 2006 by Lemire, Popov, and Reichstein, who classified Cayley and stably Cayley simple groups over an algebraically closed field of characteristic zero. In this paper, we study reductive Cayley groups over an arbitrary field k of characteristic zero. Our main results are a criterion for a reductive group G to be stably Cayley, formulated in terms of its character lattice, and a classification of stably Cayley simple groups.

A Review of Bayesian Networks and Structure Learning

This article reviews the topic of Bayesian networks. A Bayesian network is a factorisation of a probability distribution along a directed acyclic graph. The relation between graphical d-separation and independence is described. A short article by Arthur Cayley (1853) [7] is discussed, which laid ideas later used in Bayesian networks: factorisation, the noisy `or' gate, applications of algebraic geometry to Bayesian networks. The ideas behind Pearl's intervention calculus when the DAG represents a causal dependence structure; the relation between the work of Cayley and Pearl is commented on. Most of the discussion is about structure learning, outlining the two main approaches; search and score versus constraint based. Constraint based algorithms often rely on the assumption of faithfulness, that the data to which the algorithm is applied is generated from distributions satisfying a faithfulness assumption where graphical d- separation and independence are equivalent. The article presents some considerations for constraint based algorithms based on recent data analysis, indicating a variety of situations where the faithfulness assumption does not hold.

Editorial

Editorial

Arthur Cayley, Mathematician Laureate of the Victorian Age, by Tony CrillyJames Joseph Sylvester, Jewish Mathematician in a Victorian World, by Karen Parshall

<i>Arthur Cayley, Mathematician Laureate of the Victorian Age</i>, by Tony Crilly<i>James Joseph Sylvester, Jewish Mathematician in a Victorian World</i>, by Karen Parshall

Arthur Cayley mathematician laureate of the victorian age

Arthur Cayley mathematician laureate of the victorian age

Staircase rook polynomials and Cayley’s game of Mousetrap

First introduced by Arthur Cayley in the 1850’s, the game of Mousetrap involves removing cards from a deck according to a certain rule. In this paper we find the rook polynomial for the number of Mousetrap decks in which at least two specified cards are removed. We also find a new expression for the rook polynomial for the number of decks in which exactly one specified card is removed and give expressions for counts of two kinds of Mousetrap decks in terms of other known combinatorial numbers.

Equations from God: Pure Mathematics and Victorian Faith, by Daniel J. Cohen

Reviewed by: Equations from God: Pure Mathematics and Victorian Faith Tony Crilly (bio) Equations from God: Pure Mathematics and Victorian Faith, by Daniel J. Cohen; pp. x + 242. Baltimore and London: The Johns Hopkins University Press, 2007, $50.00, £33.50. History of science seems to follow St. Augustine's advice to Christians to beware of mathematicians—who have covenanted with the devil and have the effect of confining man to the bonds of Hell. Though the histories of mathematics and science do not mix, the currents between these two islands of thought are strong but deceptive. As suggested by John Hedley Brooke in his Science and Religion (1991), the reality of the relationship between religion and science is complex. We should expect the same reaction when mathematics is substituted for science. Daniel J. Cohen argues that the mathematicians of the early Victorian generation were inspired by religious concepts. He examines the lives of three prominent practitioners and uncovers their primary motivations. Implicit in his argument is that a very short time period in the nineteenth century saw a sharp break from mathematics infused with religion to secular mathematics. Pure mathematics is often regarded as the exemplar of a value-free subject. Bertrand Russell defined it in strict logical terms, as the class of all propositions of the form "[p] implies [q]. " Viewed thus, pure mathematics appears to be an abstract study divorced from any connection with application. Cohen's three pure mathematicians all preceded Russell: the American Benjamin Peirce and the English George Boole and Augustus De Morgan. But does his selection represent a more general trend? All Cohen's men are Christians, though they could also be described as mavericks. De Morgan was a non-joiner and claimed to be a "Christian unattached" (108). Boole was not a member of any church but leant towards the Unitarians. Peirce did not follow any organised creed but like the other two possessed Unitarian leanings. They were all seekers after Truth in a wider context in contrast to the more limited aim of the modern mathematician, who tends only to claim mathematical truth within the framework of mathematical theory. Boole was a dedicated seeker: "the great results of Science, and the primal truths of religion and morals, have an existence quite independent of our faculties and of our recognition. . . . It is given to us to discover Truth—we are permitted to comprehend it; but its sole origin is in the will or the character of the Creator; and this is the real connecting link between Science and Religion (Desmond MacHale, George Boole [1985] 43). Russell called him the "father of pure mathematics," but as Cohen shows, Boole was far removed from the "p implies q" definition of mathematics and the icy blast of mathematics as pure deduction. He was instead a "warm-blooded, religiously concerned idealist" (78). As a young man Peirce "maintained that the Bible embodied a higher form of truth than science. When there is doubt as to the relative positions of these streams of thought, science should bend to the Scriptures and not vice versa" (53), though this assertive viewpoint became less prominent as Peirce grew older. But God remained a focus. Whereas geniuses have the job of uncovering great mathematical laws, Peirce wrote that mathematics "vibrates in every soul" and that this ability of humankind to understand it is part of God's plan of giving "every individual the capacity to know of His existence and intelligence" (75). De Morgan has perhaps the strongest link with modernity. Much to his mother's consternation he moved away from established religion as a young man, and, by [End Page 336] the end of his career, concluded that religion and mathematics should not mix at all (136). Cohen's three subjects all left behind a wealth of evidence on these questions, and he skilfully assembles it into a lively and informative story. Though many mathematicians in Victorian Britain professed membership in the Anglican church, many treated the question of religious belief as a private matter. We may never know their true religious affinities or how these affected their motivations. A mathematical leader such as Arthur Cayley, for example, was a...

Geometric matrix algebra

Matrix multiplication was first introduced by Arthur Cayley in 1855 in agreement with the composition of linear transformations. We explore an underlying geometric framework in which matrix multiplication naturally arises from the product of numbers in a geometric (Clifford) algebra. Consequently, all invariants of a linear operator become geometric invariants of the multivectors that they represent. Two different kinds of bases for matrices emerge, a spectral basis of idempotents and nilpotents, and a standard basis of scalars, vectors, bivectors, and higher order k-vectors. The Kronecker product of matrices naturally arises when considering the block structure of a matrix. Conformal geometry of R3 is expressed in terms of the concept of an h-twistor, which is a generalization of a Penrose twistor.

George Boole and the origins of invariant theory

Historians have repeatedly asserted that invariant theory was born in two papers of George Boole (1841 and 1842). Although several themes and techniques of 19th-century invariant theory are enunciated in this work, in reacting to it (and thereby founding the British school of invariant theory), Arthur Cayley shifted Boole's research program.

Galton, Cayley, and the land across the river

The practical measurement of distance on land and sea has been of enduring interest for millennia. Here we outline a brief interaction between Francis Galton (1822-1911) and Arthur Cayley (1821-95) in which they too put the ‘geo’ back into geometry.Galton wanted to measure everything. In the adventurous life of this many-sided Victorian scientist, quantification was the hallmark of his work. In measuring geographical distances Galton used an idea suggested by Sir George Everest (1790-1866), the Surveyor-General of India in the 1830s. In the pages of the journal of the Royal Geographical Society, Everest showed how to calculate the distance from one point to an inaccessible point. One application would be to measure the distance from a point on one side of a river to a point on the other side.

The British development of the theory of invariants (1841–1895)

The two main British exponents of the theory of invariants, Arthur Cayley and James Joseph Sylvester, first encountered the idea of an “invariant” in an 1841 paper by George Boole. In the 1850s, Cayley, Sylvester, and the Irish mathematician, George Salmon, formulated the basic concepts, developed the key techniques, and set the research agenda for the field. As Cayley and Sylvester continued to extend the theory off and on through the 1880s, first Salmon in 1859 and later Edwin Bailey Elliott in 1895 codified it in high-level textbooks. This paper sketches the development of nineteenth-century invariant theory in British hands against a backdrop of personal, nationalistic, and internationalistic mathematical goals.

Quantum information and accounting information: Their salient features and conceptual applications

In an interdisciplinary effort, we explore the field of quantum information to search for promising conceptual applications to accounting. Salient features of quantum information are reviewed along with methods associated with the discipline. We then draw parallels to double-entry information through a discussion of the work of mathematician Arthur Cayley. We note that accounting information is a broader concept than double-entry information, and we identify some conceptual applications of quantum information to accounting, aiming to integrate the measurement process and its interactions with the environment.