arXiv subjects

Edward Anderson

Publications and source records attributed to Edward Anderson.

At least 19 recordsLinked to original sources

arXiv2025

Poisson Inventory Models with Many Items: An Empirical Bayes Approach

We consider inventory decisions with many items, each of which has Poisson demand. The rate of demand for individual items is estimated on the basis of observations of past demand. The problem is to determine the items to hold in stock and the amount of each one. Our setting provides a natural framework for the application of the empirical Bayes methodology. We show how to do this in practice and demonstrate the importance of making posterior estimates of different demand levels, rather than just estimating the Poisson rate. We also address the question of when it is beneficial to separately analyse a group of items which are distinguished in some way. An example occurs when looking at inventory for a book retailer, who may find it advantageous to look separately at certain types of book (e.g. biographies). The empirical Bayes methodology is valuable when dealing with items having Poisson demand, and can be effective even with relatively small numbers of distinct items (e.g. 100). We discuss the best way to apply an empirical Bayes methodology in this context, and also show that doing this in the wrong way will reduce or eliminate the potential benefits.

stat.ME↗

arXiv2022

Minimax decision rules for planning under uncertainty

It is common to use minimax rules to make decisions for planning when there is great uncertainty on what will happen in the future. Minimax regret is one popular version of this. We give an analysis of the behaviour of minimax rules in the case with a finite set of possible future scenarios. The use of minimax rules avoids the need to determine probabilities for each scenario, which is an attractive feature in many public sector settings. However, minimax rules will have sensitivity to the choice of scenarios. In many cases using a minimax approach will mean the requirement for what may be regarded as arbitrary probabilities on scenarios is replaced by a similarly arbitrary choice of a very small number of specific scenarios. We investigate this phenomenon. When regret-based rules are used there are also problems arising since the independence of irrelevant alternatives property fails, which can lead to opportunities to game the process. Our analysis of these phenomena considers cases where the decision variables are chosen from a convex set in $R^n$, as well as cases with a finite set of decision choices.

math.OC↗

arXiv2020

A Local Resolution of the Problem of Time. VIII. Assignment of Observables

Given a state space, Assignment of Observables involves Taking Function spaces Thereover. At the classical level, the state space in question is phase space or configuration space. This assignment picks up nontrivialities when whichever combination of constraints and the quantum apply. For Finite Theories, weak observables equations are inhomogeneous-linear first-order PDE systems. Their general solution thus splits into complementary function plus particular integral: strong and nontrivially-weak observables respectively. We provide a PDE analysis for each of these. In the case of single observables equations - corresponding to single constraints - the Flow Method readily applies. Finding all the observables requires free characteristic problems. For systems, this method can be applied sequentially, due to integrability conferred by Frobenius' Theorem. In each case, the first part of this approach is Lie's Integral Theory of Geometrical Invariants, or the physical counterpart thereof. The second part finds the function space thereover, giving the entire space of (local) observables. We also outline the Field Theory counterpart. Here one has functional differential equations. Banach (or tame Fr\'{e}chet) Calculus is however sufficiently standard for the Flow Method and free characteristic problems to still apply. These calculi support the Lie-theoretic combination of machinery that our Local Resolution of the Problem of Time requires, by which Field Theory and GR are included.

gr-qc↗

arXiv2019

Comparative Theory of Background Independence

The Lie claw digraph has recently been shown to control Background Independence and thus both the Problem of Time and the nature of Physical Law. This is established for Flat and Differential Geometry with varying amounts of extra mathematical structure. This Lie claw digraph has Generator Closure at its centre, Relationalism at its root, and Assignment of Observables and Constructability from Less Structure Assumed (working if Deformation leads to Rigidity) on its other leaves. The centre is enabled by automorphisms and powered by the Lie Algorithm generalization of the Dirac Algorithm (itself holding for the canonical subcase, with constraints for generators). We now explain how such claws are universal over all levels of mathematical structure that could be considered to be Background Independent. This follows from automorphisms both being categorical and supplying the centre with enough machinery to control the 3 peripheral aspects. Such claws in general thus merit a new name: UBIC (Universal Background Independence Claws) including allusion to their ubiquity. For a given level, the Problem of Time facets are ordered into two UBIC: the spacetime primality copy and the space/dynamics/canonical primality copy. These may or may not have a Wheelerian 2-way route connecting them: Constructability of Spacetime from Space and Intermediary Object Independence. The 3 cases of Constructability furthermore take one up and down levels like the ramps in a multi-storey car park. Wheeler matrices and ramp matrices (or more generally digraphs) codify this information. These summarize the main features of each Background Independent Theory that can vary between levels, the Constructabilities and Intermediary Object Independence being selection principles rather than categorical. These selection principles indicate that Background Independence has reached maturity as a field of study.

gr-qc↗

arXiv2019

Lie Theory suffices to understand, and Locally Resolve, the Problem of Time

The Lie claw digraph controls Background Independence and thus the Problem of Time and indeed the Fundamental Nature of Physical Law. This has been established in the realms of Flat and Differential Geometry with varying amounts of extra mathematical structure. This Lie claw digraph has Generator Closure at its centre (Lie brackets), Relationalism at its root (implemented by Lie derivatives), and, as its leaves, Assignment of Observables (zero commutants under Lie brackets) and Constructability from Less Structure Assumed (working if generator Deformation leads to Lie brackets algebraic Rigidity). This centre is enabled by automorphisms and powered by the Generalized Lie Algorithm extension of the Dirac Algorithm (itself sufficing for the canonical subcase, for which generators are constraints). The Problem of Time's facet ordering problem is resolved.

gr-qc↗

arXiv2019

Nambu variant of Local Resolution of Problem of Time and Background Independence

A Local Resolution of the Problem of Time has recently been given, alongside reformulation as A Local Theory of Background Independence. The classical part of this can be viewed as requiring just Lie's Mathematics, albeit entrenched in subsequent topological and differential-geometric developments and extended to contemporary Physics' state spaces. We now widen this approach by mild recategorization to one based on Nambu's generalization of Lie's Mathematics, as follows. i) In this approach, the Lie derivative still suffices to encode Relationalism. ii) Closure is now assessed using the Nambu bracket - with $n$ slots rather than 2, so the first nontrivially Lie case has 3 slots - and a `Nambu Algorithm' analogue of the Dirac and Lie Algorithms. This produces a class of Nambu algebraic structures of generators or of first-class constraints. iii) Nambu observables are defined by Nambu brackets zero-commutation with generators or with first-class constraints; we use the Nambu analogue of the Jacobi identity to simplify this discussion relative to a previous treatment. These Nambu brackets relations can moreover be recast as explicit PDEs to be solved using the Flow Method. Nambu observables themselves form Nambu algebras. Lattices of Nambu constraint or generator algebraic substructures furthermore induce dual lattices of Nambu observables subalgebras. iv) Deformation of Nambu algebraic structures encountering Rigidity gives a means of Constructing more structure from less. v) Reallocation of Intermediary-Object Invariance gives the general Nambu algebraic structure's analogue of posing Refoliation Invariance for GR. We also draw some motivation from M-Theory's use of Nambu Mathematics along the lines of Bagger, Lambert and Gustavsson, finding some qualitative distinctions between this, GR and Supergravity as regards how Background Independence is realized.

gr-qc↗

arXiv2019

Nijenhuis-type variants of Local Theory of Background Independence

A local resolution of the Problem of Time has recently been given, alongside reformulation as a local theory of Background Independence. The classical part of this can be viewed as requiring just Lie's Mathematics, albeit entrenched in subsequent Topology and Differential Geometry developments and extended to the setting of contemporary Physics' state spaces. We now generalize this approach by mild recategorization to one based on Nijenhuis' generalization of Lie's Mathematics, as follows. 1) Relationalism is encoded using the Nijenhuis-Lie derivative. 2) Closure is assessed using the Schouten-Nijenhuis bracket, and a `Schouten-Nijenhuis Algorithm' analogue of the Dirac and Lie Algorithms. This produces a class of Gerstenhaber algebraic structures of generators or of constraints. 3) Observables are defined by a Schouten--Nijenhuis brackets relation, reformulating the constrained canonical case as explicit PDEs to be solved using the Flow Method, and forming their own Gerstenhaber algebras of observables. Lattices of Schouten-Nijenhuis-Gerstenhaber constraint or generator algebraic substructures furthermore induce dual lattices of Gerstenhaber observables subalgebras. 4) Deformation of Gerstenhaber algebraic structures of generators or constraints encountering Rigidity gives a means of Constructing more structure from less. 5) Reallocation of Intermediary-Object Invariance gives the general Schouten-Nijenhuis-Gerstenhaber algebraic structure's analogue of posing Refoliation Invariance for GR. We finally point to general Gerstenhaber bracket and Vinogradov bracket generalizations, with the former likely to play a significant role in Backgound-Independent Deformation Quantization and Quantum Operator Algebras.

gr-qc↗

arXiv2019

A Local Resolution of the Problem of Time. XIV. Grounding on Lie's Mathematics

In a major advance and simplification of this field, we show that A Local Resolution of the Problem of Time - also viewable as A Local Theory of Background Independence - can at the classical level be described solely by of Lie's Mathematics. This comprises i) Lie derivatives to encode Relationalism, including via solving the generalized Killing equation. ii) Lie brackets to formulate Closure, via Lie's Algorithm suitably extended to accommodate Dirac and topological insights, producing generator Lie algebraic structures: Lie algebras or algebroids. iii) Observables defined by Lie brackets relations, recastable as explicit PDE systems to be solved using the Flow Method, and constitute observables Lie algebras. iv) The `passing families of theories through the Dirac Algorithm' approach to Spacetime Construction from Space, and to obtaining more structure from less internally to each of space and spacetime separately, are identified as deformations that work selectively when Lie Algebraic Rigidity is encountered. v) Reallocation of Intermediary-Object (RIO) Invariance: the general Lie Theory's commuting-pentagon analogue of posing Refoliation Invariance for GR. i) to v) cover respectively the Relationalism, Closure, Observables, Deformations and RIO super-aspects of Background Independence, Lie Theory moreover already collates i) to iii) and the internal case of iv) as multiple interacting aspects. The Problem of Time's multiple interacting facets are then explained as, firstly, having 2 copies of this Lie collation, 1 for each of spacetime and space primalities. Secondly, a Wheelerian two-way route between these two primalities, comprising v) and iv)'s `spacetime from space' version. We further develop the Comparative Theory of Background Independence thus. We can even give a `pillars of the Foundations of Geometry' parallel of our Background Independence super-aspects.

gr-qc↗

arXiv2019

Problem of Time and Background Independence: classical version's higher Lie Theory

A local resolution of the Problem of Time has recently been given, alongside reformulation as a local theory of Background Independence. The classical part of this requires just Lie's Mathematics, much of which is basic: i) Lie derivatives to encode Relationalism. ii) Lie brackets for Closure giving Lie algebraic structures. iii) Observables defined by a Lie brackets relation, in the constrained canonical case as explicit PDEs to be solved using Lie's flow method, and themselves forming Lie algebras. iv) Lattices of constraint algebraic substructures induce dual lattices of observables subalgebras. The current Article focuses on two pieces of `higher Lie Theory' that are also required. Preliminarily, we extend Dirac's Algorithm for Constraint Closure to `Lie's Algorithm' for Generator Closure. 1) We then reinterpret `passing families of theories through the Dirac Algorithm' - a method used for Spacetime Construction (from space) and getting more structure from less structure assumed more generally - as the Dirac Rigidity subcase of Lie Rigidity. We also provide a Foundations of Geometry example of specifically Lie rather than Dirac Rigidity, to illustrate merit in extending from Dirac to Lie Algorithms. We point to such rigidity providing a partial cohomological (and thus global) selection principle for the Comparative Theory of Background Independence. 2) We finally pose the universal (theory-independent) analogue of GR's Refoliation Invariance for the general Lie Theory: Reallocation of Intermediary-Object Invariance. This is a commuting pentagon criterion: in evolving from an initial object to a final object, does switching which intermediary object one proceeds via amount to at most a difference by an automorphism of the final object? We argue for this to also be a selection principle in the Comparative Theory of Background Independence.

gr-qc↗

arXiv2019

A Local Resolution of the Problem of Time. V. Combining Temporal and Configurational Relationalism for Finite Theories

As we have known comprehensively since the early 1990's works of Isham and Kucha\v{r}, The Problem of Time mostly concerns interferences between its many facets. Having introduced the local facets in Articles I to IV, we now show how Article I's approach to Temporal Relationalism can be combined with Article II's to Configurational Relationalism. This requires reformulating some of the Principles of Dynamics to be Temporal Relationalism implementing, a strategy which we follow through in each subsequent Article. All in all, around half of the Principles of Dynamics needs to be rewritten to solve this noted, 50-year-old and hitherto unresolved foundational problem. This stands as sufficient reason to render new resultant Principles of Dynamics - `TRiPoD' - a significant and worthwhile development of the Principles of Dynamics. This amounts to taking Jacobi's action principle more seriously than Jacobi himself did, or any authors in between: to constitute a new starting point for the entirety of the Principles of Dynamics is to be reworked. This can moreover be viewed as a mild recategorization necessitated by the Problem of Time. While mathematically simple to carry out, it requires quite a lot of conceptual developments, by which it is both prudent and useful exposition to present this first for finite rather than Field Theoretic examples in the current Article. Article VI then extends this approach to Field Theory and GR.

gr-qc↗

arXiv2019

A Local Resolution of the Problem of Time. VI. Combining Temporal and Configurational Relationalism for Field Theories and GR

We next combine Temporal and Configurational Relationalism's resolution for Field Theory, including in particular for GR. The current Article also provides the finite-and-field theory portmanteau notation, by which the rest of this series' reworking of the Principles of Dynamics can be presented concurrently for Finite Theories and Field Theories. GR's Riem, superspace, and thin sandwich are also further outlined in support of the rest of this Series.

gr-qc↗

arXiv2019

A Local Resolution of the Problem of Time. VII. Constraint Closure

We now set up Constraint Closure in a manner consistent with Temporal and Configurational Relationalism. This requires modifying the Dirac Algorithm - which addresses the Constraint Closure Problem facet of the Problem of Time piecemeal - to the TRi-Dirac Algorithm. This is a member of the wider class of Dirac-type algorithms that enjoys the property of being Temporal Relationalism implementing (TRi). Constraint algebraic structures ensue. We include examples of types of constraint, outcomes of the Dirac Algorithm and different kinds of Constraint Closure Problems. Enough new Principles of Dynamics is required to support this venture that an Appendix on it is provided: differential Hamiltonians, anti-Routhians, and the brackets, state spaces and morphisms corresponding to these.

gr-qc↗

arXiv2019

A Local Resolution of the Problem of Time. IX. Spacetime Constructability

Assuming Temporal and Configurational Relationalism, GR as Geometrodynamic's DeWitt supermetric alongside local Lorentzian Relativity with its universal finite maximum propagation speed arises as one of very few options from Feeding Families through a Dirac-type Algorithm for Consistency. This amounts to Spacetime Construction from prior assumptions about space and dynamics alone. The other alternatives, arising as cofactors' likewise strongly vanishing roots are, firstly, Galileo-Riemann Geometrostatics with Galilean Relativity's infinte propagation speed. Secondly, Strong Gravity with Carrollian Relativity 's zero propagation speed. If none of these vanish, constant mean curvature of the spatial slice is enforced, paralleling previous work on decouping GR's constraints and addressing its initial-value problem. Assuming just Temporal Relationalism, spatial 3-diffeomorphism Configurational Relationalism is enforced as an integrability as one of very few options. The alternatives here are, once again, Galileo-Riemann Geometrostatics and Strong Gravity, for which spatial 3-diffeomorphisms are thereby optional, and local volume preserving diffeomorphisms. Options are few in each case above due to Rigidity kicking in. We furthermore demonstrate that such Rigidities are more generally Lie rather than specific to Dirac-type Algorithms. We do this by deriving the conformal versus projective ambiguity in top-geometry by feeding the general quadratic generator into the Lie Algorithm for Consistency. Both Feeding Families through a Brackets Consistency Algorithm and Rigidity are thus additionally of relevance to the Foundations of Geometry.

gr-qc↗

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.

cs.GT↗

arXiv2019

A Local Resolution of the Problem of Time. I. Introduction and Temporal Relationalism

This Series of Articles provides a local resolution of this major longstanding foundational problem between QM and GR, or, more generally, between Background Dependent and Background Independent Physics. We focus on the classical version; the concepts we use are moreover universal enough to admit quantum counterparts. This requires a series of articles to lay out because the Problem of Time is multi-faceted and is largely about interferences between facets, with traditional solutions to individual facets breaking down in attempted joint resolutions of facets. [98] already covered this at both the classical and semiclassical quantum levels. This Series serves, firstly, to isolate this resolution from [98]'s introductory and field-wide comparative material, passing from an 84-part work down to just a 14-part one. Secondly, to clarify that, at the usual differential-geometric level of structure used in physical theories, our classical local resolution is entirely catered for by Lie's mathematics. This renders our local classical part of the Problem of Time understandable by a very large proportion of Theoretical Physics or Mathematics majors. In this first article, we cover a first aspect: Temporal Relationalism. This generalizes the facet traditionally known as the Frozen Formalism Problem, that follows from the quantum form of GR's Hamiltonian constraint.

gr-qc↗

arXiv2019

A Local Resolution of the Problem of Time. II. Configurational Relationalism via a generalization of Group Averaging

In this article, we consider a second Problem of Time Facet. This started life as Wheeler's Thin Sandwich Problem, within the narrow context of 1) GR-as-Geometrodynamics, in particular its momentum constraint. 2) A Lagrangian variables level treatment. Conceiving in terms of Barbour's Best Matching is a freeing from 1), now in the context of first-class linear constraints. Conceiving in terms of the underlying Background Independence aspect - the titular Configurational Relationalism - serves moreover to remove 2) as well. This is implemented by the $G$-act, $G$-all method. I.e. given an object $O$ that is not $G$-invariant, we act on it with $G$ and then apply an operation involving the whole group. This has the effect of double-cancelling the introduction of our group action, thus yielding a $G$-independent version of $O$. A first example of whole-group operation is group averaging. This is a valuable prototype by its familiarity to a large proportion of Mathematics and Physics majors through featuring in elementary Group Theory and Representation Theory courses (and which can be traced back to Cauchy). In particular, this is much more familiar than Thin Sandwiches or Best Matching! Secondly, extremization over the group, of which Best Matching, and taking infs or sups over the group, are examples. Some further significant examples of this method in modern geometry and topology include those of Kendall, Younes, Hausdorff and Gromov. In this way, we populate this approach with examples well beyond the usual GR literature's by DeWitt, Barbour and Fischer (which we also outline).

gr-qc↗

arXiv2019

A Local Resolution of the Problem of Time. III. The other classical facets piecemeal

We introduce nine further local facets of the classical Problem of Time, and underlying Background Independence aspects, the previous two Articles having covered one further facet-and-aspect each for a total of eleven. I.e. 1) Constraint Closure, 2) Assignment of Observables, 3) Space Construction from Less Structured Space, A) Spacetime Construction from Space, 0') Spacetime Relationalism, 1') Spacetime Generator Closure, 2') Assignment of Spacetime Observables, 3') Spacetime Construction from Less Structured Spacetime, and B) Foliation Independence. These are classically implemented piecemeal by the Dirac Algorithm for 1) and the Lie Algorithm for 1'). Taking Function Spaces Thereover: over phase space for 2) and over the space of spacetimes for 2'). Feeding Deformed Families into the Dirac Algorithm for A) and into the Lie Algorithm for 3) and 3'), working out whenever Rigidity is encountered. Spacetime's diffeomorphism-invariance and perturbative Lie derivative for 0'). Finally, Refoliation Invariance following from the Dirac algebroid of the GR constraints for B). Each of these approaches is moreover grounded in Lie's Mathematics in accord with this Series' claim. The generalizations required to extend these resolutions to solve combined rather than just piecemeal facets is the main subject of Articles V to XIII. We include novel work on, firstly, Closure, including a blockwise breakdown into more specific Closure Problems. Secondly, on a coordinate-free Multi-tensor Calculus suitable for joint treatment of Background Independence aspects, presenting in particular the Dirac Algorithm, and ensuing theory of observables and of deformations. Finally, aspect 3') is new to the current Article.

gr-qc↗

arXiv2019

A Local Resolution of the Problem of Time. IV. Quantum outline and piecemeal Conclusion

In this final piecemeal treatment of local Problem of Time facets, and underlying Background Independence aspects, we first reconsider the ten local facets and aspects considered so far at the quantum level. This is essential both to appreciate past conceptualization and naming for these aspects and facets, and because the quantum Problem of Time is a more advanced goal than its classical counterpart. The case is made that each piecemeal aspect of Background Independence can be incorporated, i.e. each piecemeal facet of the Problem of Time is resolvable, in a local classical sense, using just Lie's Mathematics. This case is then extended in Articles V to XIII to a joint resolution, i.e. successfully handling the lion's share of the Problem of Time that resides in facet interferences. Relationalism, Closure, Observables and Constructions are argued to be super-aspects ordered to form a claw alias 3-star digraph with Closure as nexus. This is realized twice: faithfully for spacetime and with some modification in the split-Relationalism canonical case. There is finally a Wheelerian 2-way route between the two realizations, with Spacetime Construction from Space in one direction and Refoliation Invariance in the other.

gr-qc↗