John Ceres Amson's Papers

John Amson wrote papers on a wide variety of topics. We hope to illustrate this with a list of titles of some of his papers, many with additional information such as the Abstract.

John C Amson, An analysis of the gas-shielded consumable metal arc welding system, British Welding Journal 41 (4) (1962), 232-249.

Equations characterising drop detachment in the gas-shielded consumable-metal-arc welding system are derived, using basic physical principles together with certain physical assumptions and approximations deduced from observations. ....


The consumable-metal-arc system is well known with regard to its metal transfer features and will not be described here.

The system is physically very complicated, and no complete analysis is yet possible or foreseeable. This is due largely to ignorance of the detailed physical behaviour of the important parts of the system. Nothing precise is known of the behaviour of material at the interface between the molten drops on the end of the wire electrode and the arc plasma supported there; nor of the precise way in which the wire electrode material is reduced into the rapid succession of detaching drops in spray transfer, and the effect of this reduction on the siting of the arc plasma as each drop is formed and detached; nor of the location of current conduction paths and current density between the ephemeral drops' surfaces and the constantly disturbed arc plasma. Many mechanisms of detachment have been proposed and counter-proposed and until recently experimental evidence tended to exclude only the most unlikely. The production of high energy streams within and arc has been shown to constitute a sufficient mechanism for the transport of drops co-axially within the arc, when once they become detached from the parent electrode, but their effect on such detachment is uncertain. However, it will be shown here that on making many simplifying assumptions, estimates of drop detachment frequency, and drop size at detachment , as functions of either the arc current or the related wire electrode feed speed, can be deduced; and that these estimates compare favourably for the most part with observed values.

John C Amson and G R Salter, Analysis of the gas-shielded consumable metal-arc welding system - Effect of ambient gas pressure, British Welding Journal 10 (9) (1963), 472-483.

Simulated weld beads have been carried out in argon with Al wire on Al plate using the MIG welding process, the gas pressure ranging between 100 and 1520 torr; changes in drop detachment frequency, arc current, voltage, bead penetration etc.

John C Amson, Lorentz force in the molten tip of an arc electrode, British Journal of Applied Physics 16 (8) (1965), 1169-1179.


A general expression for the Lorentz force in the molten tip of an arc electrode is derived from the surface integral of Maxwell stress. The Lorentz force in typical situations is calculated and its variation and effect during the formation and detachment of the molten tip are discussed. Values much larger than those customarily associated with molten tips are seen to appear towards the end of the tip detachment cycle.


Consider an electric arc in a gas atmosphere, one of whose electrodes (of either polarity) is a thin rod or wire. Heat from the arc process and the flow of current through the wire can cause the tip of the wire to melt. The molten tip thus formed is often detached as a droplet from the electrode and transferred through the arc by a combination of electro-magnetic, fluid-dynamic, and gravitational forces. If the electrode wire as it melts is advanced steadily to maintain a quasi-steady system in which the arc length and current etc. fluctuate only slightly during the cycle of melting and detachment of the electrode tip, the process of tip detachment can become regular. Such a well regulated system is the basis, for example, of the automatic consumable-electrode welding process (see Amson 1962 for a short bibliography of the process). (Typical rates of droplet detachment from the tip of an aluminium wire, 116\large\frac{1}{16}\normalsize in. diameter, in an argon arc, vary between 1 per second at 30 A to over 1000 per second at 350 A, the electrode feed speeds being 1.2 and 14 cm sec1^{-1} respectively.)

Gravity is not always a predominant force in tip detachment, since detachment can take place coaxially with the electrode wire, regardless of the system's orientation, providing the current is high enough. Forces other than gravity arise from the interaction of the electric current and its own magnetic field. The direct electromagnetic forces are the Lorentz forces within the body of the molten tip prior to detachment; the indirect ones are those due to the flow of arc plasma and the gas about the tip of the wire, induced by the Lorentz forces in the conducting arc plasma itself. These flows are often violent, and plasma velocities of the order of 1 km sec1^{-1}} are not unusual, whilst detached droplets with a mass of about 10 mg, in free flight along the arc axis, have often been observed cinematically to receive accelerations of over 100 g from such plasma jet streams (Amson 1962, Maecker 1955, Wells 1962, Amson and Salter 1963).

In this paper we study the contribution to the forces of tip detachment arising from the axial component of Lorentz force in the molten tip. For simplicity we only consider a symmetrical arrangement, i.e. one in which the current, the magnetic field, the droplet at the tip, and the arc are all disposed symmetrically with respect to the axis of the cylindrical wire electrode. This restriction also ensures that whatever torques appear within the molten tip shall have axial symmetry, and hence that the resulting force acting on the droplet is torque-free. When the system is not symmetric, the combination of thrust and torque in the detaching tip may well explain the very different motions of droplets observed in this situation (Needham 1963, British Welding Research Association Report A1/38/63). Whilst the radial components of Lorentz force produce an increase of pressure throughout a conductor in general, this contributes nothing to the axial force on the conductor in the absence of fluid motion in the latter. But when fluid motion is present, and especially for example in the compound conductor formed by the interior of the molten tip of an arc electrode and the arc plasma about its exterior, the situation can alter radically and additional axial forces, the so-called magnetokinetic forces, can appear (Amson and Salter 1963, Serdjuk 1962). Both these two further aspects of Lorentz force in the molten tip of an arc electrode lie beyond the scope of the present study, and must be reserved for a later investigation.


This paper is condensed from Report A1/35/63 prepared for the A1 Committee of the British Welding Research Association, whose permission to publish is gratefully recorded.

John C Amson and A F Parker-Rhodes, Essentially finite chains, Scientific Report, Cambridge Language Research Unit (1965).


A sequence whose terms begin with a specific term the originator and whose subsequent terms are determined by a specific iteration procedure the formulator such that each inherits the structure of its predecessor, is called a chain. Examples are given to clarify the meaning of essentially. The paper examines a number of instances in which an essentially infinite chain becomes essentially finite, and gives the necessary conditions on the dimensionality of the originator for the chains to be essentially finite and specifies the member at which the chain terminates. Where GG is a group and FF is the process of selecting any proper subgroup of GG, the modification to FF seems largely independent of the nature of GG, except that GG should have finite dimension, which permits explication in a setting of finite abelian groups and their endomorphisms.

John C Amson, Proc. 2nd Commonwealth Welding Conference (The Welding Institute, London, 1965), P2.1-P2.8.

John C Amson, Equilibrium Models of Cities: 1. An Axiomatic Theory, Environment and Planning A: Economy and Space 4 (4) (1972), 429-444.


A study is undertaken of the concept of a city as an 'urban gravitational plasma' consisting of one or more species of civic matter (populations, activity rates, and so on) interacting on themselves and each other, and, at the same time, responding to relocation coercions induced by satisfaction potentials of various kinds (housing rentals, amenity levels, and so on). The latter are assumed to be coupled to the territorial densities of the individual species of civic matter through equations of state, for which the housing rental-population density relation in market equilibrium theory is a prototype.

The study is divided into four parts. The first part (presented here) approaches the problem from a formal axiomatic viewpoint, and the axioms and definitions are discussed in relation to the real urban situations from which they are abstracted. The notion of equilibrium configurations for a city is introduced, and the general equilibrium equations necessary for their existence are developed. Three particular illustrations of these equations are offered: that of a single species city, and of a two species city - both with an ideal (polytropic) state equation - and that of a single species city with an imperfect (van der Waals) state equation. These illustrations will be examined in detail in the subsequent three parts of this study.

John C Amson, The Dependence of Population Distribution on Location Costs, Environment and Planning A: Economy and Space 4 (2) (1972), 163-181.


Part of a household's constrained budget is assumed to be comprised of a general cost of location. By using a simple generalisation of Muth's economic theory of the spatial pattern of urban housing, the population density distribution in a plane city is shown to be dependent on the distribution of this generalised cost of location. If housing demand is elastic, then this dependence is shown to be exponential, a conclusion which agrees with a familiar result obtained by maximising the entropy of a population system subject to a cost constraint. If the housing demand is less or more elastic, then the dependence is binomial. Implications of the theory are discussed and illustrations given.

John C Amson, Electrode voltage in the consumable-electrode arc system, Journal of Physics D: Applied Physics 5 (1) (1972), 89-96.


In a consumable-electrode arc system the variable portion of the wire electrode between the current pick-up point and the electrode-arc interface (the 'electrode stickout') forms an important resistive part of the arc circuit. An approximate relation for the voltage fall along the stickout is calculated in terms of the arc current, the electrode feed speed and the stickout length, on the assumption that the electrode resistivity varies linearly with temperature. Some numerical illustrations are given for stainless steel wire electrodes.

John C Amson, Representations of Multilinear and Polynomial Operators on Vector Spaces, Journal of the London Mathematical Society (2) 4 (3) (1972), 394-400.


That linear functions (operators) on vector spaces are representable by matrices relative to chosen bases, is familiar. By generalising the notion of a matrix to that of a "matroid" we can extend the representation theory of linear functions to multilinear functions and polynomial functions. In Section 1, I explain what I mean by a "matroid" and a "pensymmetric matroid"; in Section 2 the representation of multilinear and symmetric multilinear functions is described in terms of matroids and pensymmetric matroids; these notions are specialised in Section 3 for the case of polynomial functions, and a simple illustration is given in the last section.

John C Amson, Mutual spaces and mutuality, Mathematical Proceedings of the Cambridge Philosophical Society 71 (2) (1972), 203-210.


The theory of dual spaces and duality is extended from a pair of vector spaces in duality to a list of more than two vector spaces in mutuality. The notions of a canonical bilinear functional and of compatible topologies for a dual pair are extended to those of a canonical multilinear functional and of compatible topologies for a mutual list. Compatible topologies are characterised by means of an extended version of the Mackey-Arens Theorem. Directions for further work are noted.

John C Amson, Multimatrix Polyalgebra Representations of the Polar Composition of Polynomial Operators, Proceedings of the London Mathematical Society (3) 25 (3) (1972), 465-485.


In this paper I shall show how the familiar linear representation theory of the composition of linear operators on vector spaces in terms of a linear algebra of matrices can be generalised to a non-linear representation theory of the polar composition of polynomial operators on vector spaces in terms of a non-linear algebra of generalised matrices. In §1, I discuss vector spaces of multimatrices (generalised matrices) of degree nNn \in N, their subspaces of pensymmetric and penantisymmetric multimatrices, and the corresponding vector spaces of multilinear functions, symmetric and antisymmetric multilinear functions. New notions of null multimatrices and axi-null multilinear functions are introduced and discussed - every multimatrix is the unique sum of a pensymmetric and a null multimatrix. In §2, a nonlinear algebraic structure - a polyalgebra - is defined, and concrete examples are given in terms of direct sums of vector spaces of multimatrices and generalised matrix multiplications. The direct sum subspace of null multimatrices is shown to be a right polyideal but not a left polyideal, and hence not a polyideal in the polyalgebra of multimatrices. Consequently null multimatrices cannot be factorised out by passing to a quotient polyalgebra of pensymmetric multimatrices. Since the direct sum subspace of pensymmetric multimatrices is also shown to be not stable for polymultiplication, it is not a subpolyalgebra of the polyalgebra of multimatrices. However, pensymmetrised polymultiplications are shown to give to that subspace the structure of a polyalgebra of pensymmetric multimatrices. In §3, I turn to polynomial operators on vector spaces and show that they possess a more general composition (polar composition) than ordinary function composition. By suitably formalising this composition, the vector space of all polynomial operators on a vector space is shown to be a polyalgebra with respect to polar compositions. Finally, this polyalgebra of polynomial operators is shown to be isomorphic to a corresponding polyalgebra of pensymmetric multimatrices. With this structural theorem we achieve our goal of generalising the matrix linear algebra representation theory of linear operators to polynomial operators.

John C Amson, Multilinear and Polynomial Transformations of Bounded-Sequence Spaces, Journal of the London Mathematical Society (2) 4 (4) (1972), 643-646.

John C Amson, Some Continuous, and Compact, Polynomial Operators on Function Spaces, Journal of the London Mathematical Society (2) 5 (2) (1972), 223-230.


In a previous paper homogeneous polynomial operators on vector spaces were shown to be representable by suitable "matroid operators"... A matroid operator is an obvious generalisation of a matrix operator - a "matroid of degree nn" being simply a matrix of type (In;J)I^{n}; J) over a scalar field KK ( = R\mathbb{R} or C\mathbb{C}). The index sets II and JJ for the matroid in the operator have the same cardinality as the vector space bases in the operator's domain and codomain, respectively. In that algebraic situation the representation theory is complete: a vector space of homogeneous polynomial operators of arbitrary degree nn is isomorphic to a vector space of (pensymmetrlc) matroids of the same degree. This result contains as a special case (of degree n=1n = 1) the familiar representation theory for linear operators on vector spaces in terms of matrix operators. A corresponding representation theory for continuous and compact polynomial operators on normed sequence spaces in terms of normed matroid operators, is to be published shortly.

In this present paper, I undertake the first steps towards a representation theory for continuous and compact polynomial operators on spaces of measurable functions, in terms of integral operators determined by "kernel" functions generalising the role of matroids in the algebraic theory. It appears that the class of such integral operators, though forming only a small part of the class of integral operators at large, is still very wide. In particular, it includes the important class of non-linear integral operators of what we shall call Lyapunov-Schmidt type, which were the first to appear in the protypes of all non-linear integral equations - those discussed independently by Lyapunov and by Schmidt at the beginning of this century ...

John C Amson, Equilibrium Models of Cities: 2. Single-Species Cities, Environment and Planning A: Economy and Space 5 (3) (1973), 295-338.


This second part of a study of a city as an 'urban gravitational plasma' investigates in detail the case where the city consists of only one species of civic matter, and is circularly symmetric. To increase the relevance of the theory to actual urban situations, this civic matter is assumed throughout to be a citizen population, though the theory would apply just as well if other illustrations, such as floor space or traffic flows, etc., were to be chosen instead. The population is assumed to attract itself in a way which tends to increase its density in high density regions and to decrease it in low density regions. This 'clumping' effect is offset by another inducement on the population to relocate itself in places where some 'dissatisfaction potential' is less. Again, for illustration, it is assumed throughout that the dissatisfaction has the form of a housing rental, that is, the price of the composite bundle of 'housing' commodities and utilities.

It is shown that the competition between the two civic forces of attraction and dispersal can lead to equilibrium distributions of the population in which the forces are everywhere in balance. The forms of these distributions depend greatly on the extent to which the housing rental is proportional to the local population density. Different degrees of this dependence are shown to give rise to many different forms of the equilibrium configurations available to a city. These are classified according to a regular scheme, and their properties explored and illustrated in detail. The manner in which one equilibrium configuration may grow into another, with or without any change in the total population in the city, leads to the idea of an 'equilibrium growth' in a city. Again their different possible types are examined in detail.

Finally, certain classes of the equilibrium configurations are shown to resemble closely the familiar negative exponential and gaussian distributions of population density. The resemblance can be so close as to make it extremely likely that many actual cities, that have been shown elsewhere to exhibit population density distributions of those forms, may in fact be exhibiting equilibrium distributions of the kind deduced in this study.

John C Amson, Equilibrium and catastrophic modes of urban growth, in E L Cripps (ed.) London Papers in Regional Science 4. Space-Time Concepts in Urban and Regional Models (Pion, London, 1974), 108-128.


Let CC denote a plane abstract city having only a single species PP of civic matter [see Amson (1972a) for a formal account of these ideas]. For definiteness we may take CC to have circular symmetry in the city's civic space XX ( = R2\mathbb{R}^{2}, the euclidean plane), and PP to be a population of 'citizens' distributed smoothly at a density τ(r)(r0)\tau (r) (r ≥ 0) throughout XX. I have shown (1972b) that if the population is exposed to two kinds of competing coercions (civic forces) - namely (a) a cohesive coercion (civic attraction) of gravity type with the coercion law m1.m2d\large\frac{m_1 . m_2}{d} of the two-dimensional (note, not three-dimensional) Newtonian form having a logarithmic potential function klog(1r)k \log (\large\frac{1}{r}\normalsize ), and (b) a relocation inducement of potential type given by the gradient gradp(=dpdrgrad p (= -\large\frac{dp}{dr}\normalsize in the outward radial direction) of a civic pressure function (potential) p(r)(r0)p(r) (r ≥ 0), for which the price ...

John C Amson, Catastrophe Theory: A Contribution to the Study of Urban Systems?, Environment and Planning B: Planning and Design 2 (2) (1975), 177-221.


The possibility of applying Thorn's catastrophe theory to the study of urban systems is discussed in broad terms, the emphasis being laid on the importance it attaches to the idea of morphogenesis - changes in structural form. The theory is introduced through the idea of a system being composed of a control manifold and a behaviour manifold which, together, form the system space; the system's possible behaviours lie in a behaviour subset of this space. Critical regions occur in the behaviour set where the kind of behaviour observed undergoes a catastrophic change of form. Control points associated with these changes form the catastrophe set for the system, and the control manifold is partitioned into various regions where the different behaviours are located. The system's behaviour is governed by families of regulating (potential) functions and supplementary conventions such as Maxwell's, the Perfect Delay, etc. There follows an extensive and illustrated description of twelve elementary catastrophes (including Thom's 'magnificent seven'). The article concludes with a note of the few existing applications in urban studies and a brief summary of some technical details.

John C Amson, A regional plasma theory of land use, in G J Papageorgiou (ed.) Mathematical land use theory (D C Heath, Lexington, MA, 1976), 99-116.

The Present Theory and Some Sketches for Its Later Development.

By a regional plasma we mean a system of distinct populations distributed throughout a region in which each population interacts not only with itself but with each other population. Each population has associated with it a spatial density function, and it is the relative proportion of the different densities at any location that determines the land use at that location. The population interactions we envisage arise through two effects: gravity attractions (coercions) and density dependent location costs. Two kinds of plasmas will be considered: those in which interactions of the gravity type alone re present ("first" kind) and those in which both types of interactions are present ("second" kind).

While the particular natures of the populations in our plasma need not be specified at this stage, it is perhaps helpful to think of then as being populations of, say, different socioeconomic classes, or job places, workers' residences, traffic flow rates, and so on. We shall restrict ourselves in this essay to plasmas having at most three populations in order not to obscure the main  ideas. Each population in the plasma is distributed as a spatial measure ...

John C Amson, On Polynomial Operator Equations in Banach Space, Journal of the London Mathematical Society (2) 10 (1975), 171-174.

John C Amson, A Note on Civic State Equations. Environment and Planning A: Economy and Space 9 (1) (1977), 105-110.


A more general relationship ('state equation') between housing costs and population density is deduced by an improved version of an earlier argument (Amson, 1972a). The relative importance of land factors in housing production, and the consumer's personal housing-cost index, can be used as parameters in the relationship.

T Bastin, H P Noyes, John C Amson and C W Kilmister, On the physical interpretation and the mathematical structure of the combinatorial hierarchy, International Journal of Theoretical Physics 18 (7) (1979), 445-488.

From a review by Slawomir Bugajski.

The basic mathematical object of the paper is an abelian group SS isomorphic to the direct sum of NN copies of the cyclic two-element group. Given such a group, the set E(S)E(S) of all its endomorphisms equipped with the natural point-wise addition forms an abelian group, isomorphic to the direct sum of N2N^{2} copies of the cyclic two-element group. Thus any SS determines iteratively a sequence E0,E1,E2,...,Em,...E^{0}, E^{1}, E^{2}, ... , E^{m}, ... where E0=SE^{0} = S and Em=E(Em1)E^{m} = E(E^{m−1}) for any natural mm. Some special embeddings Em1EmE^{m-1} \hookrightarrow E^m are constructed for min HH, with HH depending on NN, leading to a "hierarchy" E0E1EHE^{0} \subset E^{1} \subset ⋯ \subset E^{H}. The only nontrivial such hierarchy exists for N=2,H=3N = 2, H = 3. A physical meaning is attached to the four sets E0,E1,E2,E3E^{0}, E^{1}, E^{2}, E^{3}: they are to describe baryons, mesons, leptons, and photons, respectively. Starting with this interpretation, the authors are able to calculate the ratio of proton and electron masses as well as the binding energy of the ground state of hydrogen. Some cosmological implications are also considered.

John C Amson, Discrimination Systems and Projective Geometries, in Discrete and Combinatorial Physics, Proceedings of Alternative Natural Philosophy Association 9 (1988), 158-189.

John C Amson and N Gopal Reddy, A Hilbert algebra of Hilbert-Schmidt quadratic operators, Bulletin of the Australian Mathematical Society 41 (1) (1990), 123-134.

From a review by Hermann König.

Quadratic operators of Hilbert-Schmidt class are studied and shown to be uniquely representable by a sequence of linear, self-adjoint Hilbert-Schmidt operators. This leads to a multiplication of such quadratic operators by component-wise composition of the corresponding sequences of operators. As such, the class of quadratic Hilbert-Schmidt operators becomes a linear HH^{∗}-algebra containing all nuclear quadratic operators. A block diagonal representation is given in the finite-dimensional case.

A F Parker-Rhodes and John C Amson, Hierarchies of Descriptive Levels in Physical Theory, International Journal of General Systems 27 (1-3) (1998), 57-80.

It is shown that if we attempt to build up a mathematical system for the description of physical phenomena, on the assumption that the essential observations are two-valued, so that the appropriate algebra is the field of characteristic two, we are led to formulate a hierarchical system. The principal entities in this system are matrices over the field J2J_{2} whose orders increase in successive levels. A rule for constructing the various levels of the hierarchy is given, and it is shown that in non-trivial cases the number of levels is always finite. But of these finite systems there is one which appears complex enough to serve as a basis for physical theory on the lines laid out by Bastin and Kilmister (1954).

John C Amson and A F Parker-Rhodes, Essentially Finite Chains, International Journal of General Systems 27 (1-3) (1998), 81-92,

A chain is defined to be a sequence of structured bets (e.g. groups) constructed from a first member (the originator) by a structure preserving mapping (the formulator). Depending on the precise nature of the originator and the formulator the characteristic evolution of the chain may be finite, infinite or cyclic and in an essential or non-essential way. Some specific instances of chains of groups are examined and their characteristic evolution determined in terms of necessary conditions on the dimensions of the originator, the member of the chain at which the evolution terminates being determined in cases where the chain is essentially finite. An important illustration is given of an essentially finite chain which arose in a study of an information structure hierarchy of descriptive levels in physical theory.

John C Amson and K Bowden, A Combinatoric Bit-Hoop System, Proceedings of Alternative Natural Philosophy Association, Cambridge, "Against Bull" (2005), 407-440.

John C Amson, Gregory's meridian line of 1673-74: a St. Andrews detective story, BSHM Bulletin: Journal of the British Society for the History of Mathematics 23 (2) (2008), 58-72.

Hidden beneath a bland carpet held down by bookcases and tables in the Upper Parliament Hall in South Street, a delicate strip of wood is inlaid in the wooden floorboards. It runs under the carpet from the north side of the room to beneath a window on the south side. Outside the window a curious metal bracket is bolted to the stonework, camouflaged to the point of invisibility in greyish paint. The outlook from the window shows the blank, empty, wall of a University building erected in the 1890s. From the next window to the right some nondescript trees on the Scoonie Hill skyline can be seen two miles away. Amongst the trees, should one care to see for oneself, is a washing-line post: a pillar made of stone with a sad, bent, bronze trident a-top of it.

What is remarkable is that the two Longitudes, of the bracket and the pillar, are almost exactly the same. The difference in the last decimal place translates into an East-West difference on the ground of under one metre, at that Latitude. To all intents the two points have the same longitude. They lie on the same geographical meridian line running from the North to the South Pole of the Earth, and are separated by a north-south horizontal distance of 2.410 kilometres.

An inlaid wooden strip in a floor, a window bracket, a trident on a stone pillar two miles away due south: these are all we have left to remember one of the most amazing scientific achievements in an age of great scientific advances over three centuries ago - the conception and laying-out in the 1670s of an Astronomical Meridian Line by James Gregory.

L H Kauffman and John C Amson (eds.), Scientific Essays In Honor Of H Pierre Noyes On The Occasion Of His 90th Birthday (World Scientific, 2013).

This book is a Festschrift for the 90th birthday of the physicist Pierre Noyes. The book is a representative selection of papers on the topics that have been central to the meetings over the last three decades of ANPA, the Alternative Natural Philosophy Association. ANPA was founded by Pierre Noyes and his colleagues the philosopher-linguist-physicist Frederick Parker-Rhodes, the physicist Ted Bastin, and the mathematicians Clive Kilmister, John Amson. Many of the topics in the book centre on the combinatorial hierarchy discovered by the originators of ANPA. Other topics explore geometrical, cosmological and biological aspects of those ideas, and foundational aspects related to discrete physics and emergent quantum mechanics. The book will be useful to readers interested in fundamental physics, and particularly to readers looking for new and important viewpoints in Science that contain the seeds of futurity.

John C Amson, Unital Homogeneous Polynomial Operators on Hilbert Space with applications to nonlinear theoretical physics, in L H Kauffman and John C Amson (eds.), Scientific Essays In Honor Of H Pierre Noyes On The Occasion Of His 90th Birthday (World Scientific, 2013), 1-27.


The familiar notion of the linear unit operator on a Hilbert space is extended to its homogeneous polynomial operator analogues of arbitrary degree: 'unital' homogeneous polynomial operators, whose multimatrix representations are identified with generalised Kronecker symbols (tensors). Some properties are lost, many other novel structural properties are established.

The set of unital operators of all degrees on a Hilbert space constitutes a unique object, a 'Unital System'. It is unit-normed, closed and commutative under functional composition, and closed under functional polar composition. It is a multiplicatively and additively graded set, graded by the positive integer subset of the integer ring. It is the disjoint union of its homogeneous component 'grade' subsets each comprised of an equivalence class (under unitary similarity) of unital operators of same degree 1, the first grade subset being the only one itself closed under composition.

Some operator equations of Fredholm and Characteristic type involving unital homogeneous polynomial operators are illustrated, their solubility described, and connections with an emerging theory of the spectrum of tensors and homogeneous polynomial operators are noted.

Possible applications to the theory of nonlinear phenomena in the presence of very high field strengths are indicated.

Last Updated September 2023