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Edward Anderson

Publications and source records attributed to Edward Anderson.

At least 37 records · Page 2Linked to original sources
arXiv2019

Capacity Games with Supply Function Competition

This paper studies a setting in which multiple suppliers compete for a buyer's procurement business. The buyer faces uncertain demand and there is a requirement to reserve capacity in advance of knowing the demand. Each supplier has costs that are two dimensional, with some costs incurred before demand is realized in order to reserve capacity and some costs incurred after demand is realized at the time of delivery. A distinctive feature of our model is that the marginal costs may not be constants, and this naturally leads us to a supply function competition framework in which each supplier offers a schedule of prices and quantities. We treat this problem as an example of a general class of capacity games and show that, when the optimal supply chain profit is submodular, in equilibrium the buyer makes a reservation choice that maximizes the overall supply chain profit, each supplier makes a profit equal to their marginal contribution to the supply chain, and the buyer takes the remaining profit. We further prove that this submodularity property holds under two commonly studied settings: (1) there are only two suppliers; and (2) in the case of more than two suppliers, the marginal two-dimension costs of each supplier are non-decreasing and constant, respectively.

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arXiv2019

Shape Theories. I. Their Diversity is Killing-Based and thus Nongeneric

Kendall's Shape Theory covers shapes formed by $N$ points in $\mathbb{R}^d$ upon quotienting out the similarity transformations. This theory is based on the geometry and topology of the corresponding configuration space: shape space. Kendall studied this to build a widely useful Shape Statistics thereupon. The corresponding Shape-and-Scale Theory -- quotienting out the Euclidean transformations -- is useful in Classical Dynamics and Molecular Physics, as well as for the relational `Leibnizian' side of the Absolute versus Relational Motion Debate. Kendall's shape spaces moreover recur withing this `Leibnizian' Shape-and-Scale Theory. There has recently been a large expansion in diversity of Kendall-type Shape(-and-Scale) Theories. The current article outlines this variety, and furthermore roots it in solving the poset of generalized Killing equations. This moreover also places a first great bound on how many more Shape(-and-Scale) Theories there can be. For it is nongeneric for geometrically-equipped manifolds -- replacements for Kendall's $\mathbb{R}^d$ carrier space (absolute space to physicists) - to possess any generalized Killing vectors. Article II places a second great bound, now at the topological level and in terms of which Shape(-and-Scale) Theories are technically tractable. Finally Article III explains how the diversity of Shape(-and-Scale) Theories - from varying which carrier space and quotiented-out geometrical automorphism group are in use - constitutes a theory of Comparative Background Independence: a topic of fundamental interest in Dynamics, Gravitation and Theoretical Physics more generally. Article I and II's great bounds moreover have significant consequences for Comparative Background Independence.

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arXiv2019

Shape Theories. II. Compactness Selection Principles

Shape(-and-scale) spaces - configuration spaces for generalized Kendall-type Shape(-and-Scale) Theories - are usually not manifolds but stratified manifolds. While in Kendall's own case - similarity shapes - the shape spaces are analytically nice - Hausdorff - for the Image Analysis and Computer Vision cases - affine and projective shapes - they are not: merely Kolmogorov. We now furthermore characterize these results in terms of whether one is staying within, or straying outside of, some compactness conditions which provide protection for nice analytic behaviour. We furthermore list which of the recent wealth of proposed shape theories lie within these topological-level selection principles for technical tractability. Most cases are not protected, by which the merely-Kolmogorov behaviour may be endemic and the range of technically tractable Shape(-and-Scale) Theories very limited. This is the second of two great bounds on Shape(-and-Scale) Theories, each of which moreover have major implications for the Comparative Theory of Background Independence as per Article III.

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arXiv2019

Shape Theory. III. Comparative Theory of Backgound Independence

Background Independence is the modern form of the relational side of the Absolute versus Relational Debate. Difficulties with its implementation form the Problem of Time. Its 9 facets - Isham and Kuchař's conceptual classification - correspond to 9 aspects of Background Independence as per the Author's classical-or-quantum theory-independent upgrade. 8 are local. The most significant arena for these is brackets algebra, the first 5 involving canonical constraints as follows. 1) Handling instantaneous gauge invariance. 2) Resolving its apparent timelessness. 3) Closure of 1-2)'s constraints. 4) Expression in terms of observables: commutants with constraints. 5) Reconstructing spacetime from constraint algebra rigidity. 6) is the spacetime counterpart of 2-4), and 7) is spacetime's foliation independence. 8) handles nonuniqueness. 9) renders 1-8) globally sound. We show how Shape(-and-Scale) Theory's mastery of 1) for N-point-particle models extends by placing a mechanics over shape(-and-scale) space to model 1-4). For flat-space Euclidean and similarity models, this gives a local resolution of the Problem of Time. This is moreover consistent within a global treatment if its reduced spaces are Hausdorff paracompact, which admit a Shrinking Lemma. GR's superspace is also Hausdorff paracompact. While 1-9) are poseable for all relativistic theories, and 1-4), 8), 9) for all theories - to all levels of mathematical structure: affine, projective, conformal, topological manifold, topological space... - resolution is on a case-by-case basis. Among N-point-particle theories, then, Article II's Hausdorff paracompact reduced space guarantee selects a very small subset. In particular, affine and projective shapes are precluded and conceiving in terms of shapes in space is preferred over doing so in spacetime. A substantial Selection Principle for Comparative Background Independence is thus born.

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arXiv2018

Explicit partial and functional differential equations for beables or observables

We provide explicit partial differential equations - in finite cases - and functional differential equations - in field-theoretic cases - which determine observables or beables in the senses of Kuchař and of Dirac. These cover a wide range of relational mechanics models as well as Electromagnetism, Yang--Mills Theory and General Relativity. We give an underlying reason why pure-configuration Kuchař observables are already well-known: various types of shape, E-fields, B-fields, loops and 3-geometries. The partial differential equations or functional differential equations for pure-momentum observables are also posed, as are those for observables which have a mixture of configuration and momentum functional dependence.

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arXiv2018

Shape Derivatives

Shape Theory, together with Shape-and-Scale Theory, comprise Relational Theory. This consists of $N$-point models on a manifold $M$, for which some geometrical automorphism group $G$ is regarded as meaningless and is thus quotiented out from the $N$-point model's product space $\times_{I = 1}^N M$. Each such model has an associated function space of preserved quantities, solving the PDE system for zero brackets with the sums over $N$ of each of $G$'s generators. These are smooth functions of the $N$-point geometrical invariants. Each $(M, G)$ pair has moreover a `minimal nontrivially relational unit' value of $N$; we now show that relationally-invariant derivatives can be defined on these, yielding the titular notions of shape(-and-scale) derivatives. We obtain each by Taylor-expanding a functional version of the underlying geometrical invariant, and isolating a shape-independent derivative factor in the nontrivial leading-order term. We do this for translational, dilational, dilatational and projective geometries in 1-$d$, the last of which gives a shape-theoretic rederivation of the Schwarzian derivative. We next phrase and solve the ODEs for zero and constant values of each derivative. We then consider translational, dilational, rotational, rotational-and-dilational, Euclidean and equi-top-form (alias unimodular affine) cases in $\geq 2$-$d$. We finally pose the PDEs for zero and constant values of each of our $\geq 2$-$d$ derivatives, and solve a subset of these geometrically-motivated PDEs. This work is significant for Relational Motion and Background Independence in Theoretical Physics, and foundational for both Flat and Differential Geometry.

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arXiv2018

Quadrilaterals in Shape Theory. II. Alternative Derivations of Shape Space: Successes and Limitations

We show that the recent derivation that triangleland's topology and geometry is $S^2$ from Heron's formula does not extend to quadrilaterals by considering Brahmagupta, Bretschneider and Coolidge's area formulae. That $N$-a-gonland is more generally $CP^{N - 2}$ (with $CP^1 = S^2$ recovering the triangleland sphere) follows from Kendall's extremization that is habitually used in Shape Theory, or the generalized Hopf map. We further explain our observation of non-extension in terms of total area not providing a shape quantity for quadrilaterals. It is rather the square root of of sums of squares of subsystem areas that provides a shape quantity; we clarify this further in representation-theoretic terms. The triangleland $S^2$ moreover also generalizes to $d$-simplexlands being $S^{d(d + 1)/2 - 1}$ topologically by Casson's observation. For the 3-simplex - alias tetrahaedron - while volume provides a shape quantity and is specified by the della Francesca-Tartaglia formula, the analogue of finding Heron eigenvectors is undefined. $d$-volume moreover provides a shape quantity for the $d$-simplex, specified by the Cayley-Menger formula generalization of the Heron and della Francesca-Tartaglia formulae. While eigenvectors can be defined for the even-$d$ Cayley-Menger formulae, the dimension count does not however work out for these to provide on-sphere conditions. We finally point out the multiple dimensional coincidences behind the derivation of the space of triangles from Heron's formula. This article is a useful check on how far the least technically involved derivation of the smallest nontrivial shape space can be taken. This is significant since Shape Theory is a futuristic branch of mathematics, with substantial applications in both Statistics (Shape Statistics) and Theoretical Physics (Background Independence: of major relevance to Classical and Quantum Gravitational Theory).

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arXiv2018

Spaces of Observables from Solving PDEs. I. Translation-Invariant Theory

Finding classical canonical observables consists of taking a function space over phase space. For constrained theories, these functions must form zero brackets with a closed algebraic structure of first-class constraints. This brackets condition can moreover be recast as a first-order PDE system, to be treated as a free characteristic problem. We explore explicit observables equations and their concrete solutions for a translation- and reparametrization-invariant action, thereby populating the following variety of notions of observables with examples. 1) The brackets can be strongly or weakly zero in Dirac's sense, i.e.\ a linear combination of constraints. 2) Observables can admit pure-configuration and pure-momentum restrictions. 3) Our model provides the translation constraint P$_i$ encoding zero total momentum of the model universe and depending homogeneous-linearly on momenta, and the Chronos constraint equation of time reinterpretation of the `constant-energy condition' and depending quadratically on momenta. These are mechanical analogues of GR's momentum M$_i$ and Hamiltonian H constraints respectively. P$_i$ and Chronos moreover algebraically close separately (only M$_i$ does for GR). Our model thus supports translation gauge-observables $G$ and Chronos observables $C$, as well as unrestricted observables $U$ and Dirac observables $D$ brackets-commuting with neither and both respectively. We relate the strong to properly-weak split of weak observables to the complementary-function to particular-integral split of the complete solution, with the properly-weak rendered of measure-0 relative to the strong by the latter's free characteristicness. The closed algebraic structures form a bounded lattice and the corresponding notions of observables a dual lattice, with the observables themselves forming a presheaf of function spaces thereover.

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arXiv2018

Isotropy Groups and Kinematic Orbits for 1 and 2-$d$ $N$-Body Problems

Mitchell and Littlejohn showed that isotropy groups and orbits for $N$-body problems attain a sense of genericity for $N = 5$. The author recently showed that the arbitrary-$d$ generalization of this 3-$d$ result is that genericity in this sense occurs for $N = d + 2$. The author also showed that a second sense of genericity -- now order-theoretic rather than a matter of counting -- occurs for $N = 2 d + 1$, excepting $d = 3$, for which it is not 7 but 8. Applications of this work include 1) that some of the increase in complexity in passing from 3 to 4 and 5 body problems in 3-$d$ is already present in the more-well known setting of passing from intervals to triangles and then to quadrilaterals in 2-$d$. 2) That not $(d, N) = (3, 6)$ but $(4, 6)$ is a natural theoretical successor of $(3, 5)$. 3) Such consideration isotropy groups and orbits is moreover a model for a larger case of interest, namely that of GR's reduced configuration spaces. The current Article presents the lower-$d$ cases explicitly: 0, 1 and 2-$d$, including also the topological and geometrical form of the corresponding isotropy groups and orbits.

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arXiv2018

Specific PDEs for Preserved Quantities in Geometry. I. Similarities and Subgroups

We provide specific PDEs for preserved quantities $Q$ in Geometry, as well as a bridge between this and specific PDEs for observables $O$ in Physics. We furthermore prove versions of four other theorems either side of this bridge: the below enumerated sentences. For the generic geometry - in the sense of it possessing no generalized Killing vectors, i.e.\ continuous geometrical automorphisms - the $P$ form a smooth space of free functions over said geometry. If a geometry possesses the corresponding type of Killing vectors, the $P$ must Lie-brackets commute with `sums-over-points of the automorphism generators', $S$. The observables counterpart of this is that in the presence of first-class constraints $F$, the $O$ must Poisson-brackets commute with these. Then 1) defining $Q$, $O$ requires closed subalgebras of $S$, $F$. 2) The $Q$, and the $O$, themselves form closed algebras. 3) The subalgebras of $Q$, $O$ form bounded lattices dual to those of $S$, $F$ respectively. Both $S$, $Q$ and $F$, $O$ commutations can moreover be reformulated as first-order linear PDEs, treated free-characteristically. The secondmost generic case has just one $S$ or $F$, and so just one PDE, which standardly reduces to an ODE system. The more highly nongeneric case of multiple $S$ or $F$, however, returns an over-determined PDE system. 4) We prove that nonetheless these are always integrable. This is significant by being mostly-opposite to how the more familiar generalized Killing equations themselves behave. We finally solve for the preserved quantities of similarity geometry and its subgroups; companion papers extend this program to affine, projective and conformal geometries.

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arXiv2018

Specific PDEs for Preserved Quantities in Geometry. III. 1-d Projective Transformations and Subgroups

We extend finding geometrically-significant preserved quantities by solving specific PDEs to 1-$d$ projective transformations and subgroups. This can be viewed not only as a purely geometrical problem but also as a subcase of finding physical observables, and furthermore as part of extending the comparative study of Background Independence level-by-level in mathematical structure to include projective structure. Full 1-$d$ projective invariants are well-known to be cross-ratios. We moreover rederive this fact as the unique solution of 1-$d$ projective geometry's preserved equation PDE system. We also provide the preserved quantities for the 1-$d$ geometries whose only transformations are 1) special-projective transformations $Q$, giving differences of reciprocals. 2) $Q$ alongside dilations $D$, now giving ratios of difference of reciprocals. This analysis moreover firstly points to a new interpretation of cross-ratio: those ratios of differences that are concurrently differences of reciprocals, and secondly motivates 1) and 2) as corresponding to bona fide and distinctive Geometries.

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arXiv2018

Specific PDEs for Preserved Quantities in Geometry. II. Affine Transformations and Subgroups

We extend finding geometrically-significant preserved quantities by solving specific PDEs to the affine transformations and subgroups. This can be viewed not only as a purely geometrical problem but also as a subcase of finding physical observables, and furthermore as part of the comparative study of Background Independence level-by-level in mathematical structure. While cross and scalar-triple products (combined with differences and ratios) suffice to formulate these preserved quantities in 2- and 3-$d$ respectively, the arbitrary-dimensional generalization evokes the theory of forms. The affine preserved quantities are ratios of $d$-volume forms of differences, $d$-volume forms being the `top forms' supported by dimension $d$, and referring moreover to $d$-volumes of relationally-defined subsystems.

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arXiv2018

$N$-Body Problem: Minimal $N$'s for Qualitative Nontrivialities

We review the $N$-Body Problem in arbitrary dimension $d$ at the kinematical level, with modelling Background Independence in mind. In particular, we give a structural analysis of its reduced configuration spaces, decomposing this subject matter into basic Topology, Geometry, Group Theory, Linear Algebra, Graph Theory and Order Theory. At the metric level, these configuration spaces of shapes form basic geometric series for 1- and 2-$d$: $\mathbb{S}^{N - 2}$ and $\mathbb{CP}^{N - 2}$, though there are no more such series for $d \geq 3$. $d \geq 3$ also sees an onset of stratification. Casson's diagonal, for which $N = d + 1$, plays a critical role which we explain in simple Linear Algebra terms; these are moreover topologically spheres. $N = d$ and $N = d + 2$ have further significance as well, the latter as regards a counting notion of genericity of the isotropy groups and kinematical orbits realized. These observations twin the $N$ = (3, 4, 5) progression in qualitative complexity that almost all $N$-Body Problem work for concrete $N$ concentrates on with the much better-known $N$ = (2, 3, 4) progression in 2-$d$: from intervals to triangles to quadrilaterals: a large source of intuitions and mathematical analogies. We furthermore provide an Order-Theoretic genericity criterion, which is almost always bounded by the double-slope $N = 2 d + 1$ line, though an accidental relation pushes $N$ up by 1 to 8 in 3-$d$. We finally consider rubber shapes, for which the configuration spaces are graphs: much simpler than stratified manifolds, and yet containing quite a few of metric-level shapes' qualitative features. This singles out $N = 5$ and 6 in 1-$d$ and $N = 6$ and 8 in $\geq$ 2-$d$ for the onsets of various graph-theoretical nontrivialities.

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arXiv2018

Absolute versus Relational Debate: a Modern Global Version

Suppose one seeks to free oneself from a symmetric absolute space by quotienting out its symmetry group. This in general however fails to erase all memory of this absolute space's symmetry properties. Stratification is one major reason for this, which is present in both a) Kendall-type Shape Theory and subsequent Relational Mechanics, and b) General Relativity configuration spaces. We consider the alternative starting point with a generic absolute space, meaning with no nontrivial generalized Killing vectors whatsoever. In this approach, generically Shape-and-Scale Theory is but trivially realized, there is no separate Shape Theory and indeed no stratification. While the GR configuration space version of these considerations was already expounded in 1996 by Fischer and Moncrief, the Kendall-type shape theory version is new to the current article. In each case, this amounts to admitting some small deformation by which symmetry's hard consequences at the level of reduced configuration spaces are warded off.We end by discussing the senses in which each of the above two strategies retain absolutist features, each's main known technical advantages and disadvantages, and the desirability of replacing Kendall-type Shape Theory with a Local-and-Approximate Shape Theory. This article is in honour of Prof. Niall ó Murchadha, on the occasion of his Festschrift.

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arXiv2018

Rubber Relationalism: Smallest Graph-Theoretically Nontrivial Leibniz Spaces

Kendall's Similarity Shape Theory for constellations of N points in the carrier space $\mathbb{R}^d$ as quotiented by the similarity group was developed for use in Probability and Statistics. It was subsequently shown to reside within Mechanics' Shape-and-Scale Theory, in which points are interpreted as particles, carrier space plays the role of absolute space, and the Euclidean group is quotiented out. Let us jointly refer to Shape(-and-Scale) Theory as Relational Theory, and to its reduced configuration spaces as relational spaces. We now consider a less structured version: the Topological Relational Theory of `rubber configurations'. This already encodes some features of the much more diverse Geometrical Relational Theories. In contrast with the latter's (stratified) manifold relational spaces, the former's are graphs: much simpler to treat; their edges encode topological adjacency. We concentrate on Leibniz spaces, corresponding to indistinguishable points and mirror-image identification. These are moreover the building blocks of the distinguishable and (where possible) mirror-image distinct cases' relational spaces. For connected manifold without boundary carrier spaces, there are just 3 'rubber relationalisms: $\mathbb{R}$, $\mathbb{S}^1$, and a joint one for all carrier spaces with $d \geq 2$. For $d \geq 2$, rubber configurations are in 1:1 correspondence with partitions, with $\mathbb{S}^1$ and $\mathbb{R}$ giving successive refinements. We find that generic and maximal configurations are universally present as cone points, as are binaries in the first 2 cases. Deconing leaves us with residue graphs containing the N-specific information. We provide graph-theoretical nontriviality criteria for which N = 6, 6 and 5 are minimal across these models, and stronger such for which N = 8, 8 and 6 are minimal, and outline GR topology-change analogue-model and N-body problem applications.

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arXiv2018

Background Independence: $\mathbb{S}^1$ and $\mathbb{R}$ absolute spaces differ greatly in Shape-and-Scale Theory

Kendall-type Shape(-and-Scale) Theory on $\mathbb{R}^d$ involves $N$ point configurations therein quotiented by some geometrically meaningful automorphism group. This occurs in Shape Statistics, the Classical and Quantum $N$-body Problem and as a model for many aspects of Generally Relativistic theories' Background Independence. Shape-and-Scale theory on the circle $\mathbb{S}^1$ is significant at the level of `rubber shapes' as 1 of only 3 classes of connected-without-boundary absolute spaces. It is also the first $\mathbb{T^d}$ and $\mathbb{RP}^d$ as well as the first sphere; spheres and tori are motivated by spatially-closed GR and $\mathbb{RP}^d$ by Image Analysis and Computer Vision. We now investigate the $\mathbb{S}^1$ case at the geometrical level. With $Isom(\mathbb{S}^1) = SO(2)$ itself a $\mathbb{S}^1$, the shape-and-scale $N$-body configuration spaces are systematically $\mathbb{T}^{N - 1}$. We show moreover that 3 points on the circle already suffices for major differences to occur relative to on the line $\mathbb{R}$. Scale is now obligatory. Totally antipodal configurations are as significant as the maximal collision. Topologically, partially antipodal configurations play an equivalent role to right angles: specifically a $d \geq 2$ notion on $\mathbb{R}^d$. Using up less and more arc than an antipodal configuration are the respective topological analogues of acute and obtuse triangles. The idea that quotienting out geometrical automorphisms banishes an incipient notion of absolute space is dead. Such indirect modelling is, rather, well capable of remembering the incipient absolute space's topology. Thus topological considerations of Background Independence have become indispensible even in mechanics models. In General Relativity, this corresponds to passing from Wheeler's Superspace to Fischer's Big Superspace.

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arXiv2018

Topological Shape Theory

Kendall's Similarity Shape Theory for constellations of points in the carrier space $\mathbb{R}^n$ was developed for use in Probability and Statistics. It was subsequently shown to reside within (Classical and Quantum) Mechanics' Shape-and-Scale Theory, in which the points are interpreted as particles and the carrier space plays the role of absolute space. In other more recent work, Kendall's Similarity Shape Theory has been generalized to affine, projective, conformal and supersymmetric versions, as well as to $\mathbb{T}^n$, $\mathbb{S}^n$, $\mathbb{RP}^n$ and Minkowski spacetime carrier spaces. This has created a sizeable field of study: generalized Kendall-type Geometrical Shape(-and-Scale) Theory. Aside from offering a wider range of shape-statistical applications, this field of study is an exposition of models of Background Independence of relevance to the Absolute versus Relational Motion Debate, and the Foundations and Dynamics of General Relativity and Quantum Gravity. In the current article, we consider a simpler type of Shape Theory, comprising relatively few types of behaviour: the Topological Shape Theory of rubber shapes. This underlies the above much greater diversity of more structured Shape Theories; in contrast with the latter's (stratified) manifolds shape spaces, the former's are just graphs. We give examples of these graphs for the smallest nontrivial point-or-particle numbers, and outline how these feature within a wider range of Geometric Shape Theories' shape spaces, and are furthermore straightforward to do Statistics, Dynamics and Quantization with.

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