This has remarkable implications. Since this quantity never changes sign, only one of R and G can... more This has remarkable implications. Since this quantity never changes sign, only one of R and G can ever be zero. If the initial value of (3) is positive, for example, only G can equal zero, and so only red can win. So this quantity is key in determining who will win the battle, and a side’s fighting strength – we shall call rR and gG the red and green forces’ respective fighting strengths – varies as the units’ fighting effectiveness times the square of their numbers.
The Battle of Britain is the focus of much historical controversy. We show here how the statistic... more The Battle of Britain is the focus of much historical controversy. We show here how the statistical technique of weighted bootstrapping can be used to create a new quantitative basis to help address such controversies. Bootstrapping facilitates the exploration of alternative campaign possibilities with different tactics. This results in comparative probabilities of “victory” for the actual and various counterfactual campaigns, providing a quantified assessment of the likelihood of German achievement of air superiority, thereby facilitating invasion. We find this more likely had the Luftwaffe targeted airfields more heavily, and greatly more likely had Germany brought forward its air campaign.
We consider integrable field theories with achiral boundary conditions and uncover the underlying... more We consider integrable field theories with achiral boundary conditions and uncover the underlying achiral twisted Yangian algebra. This construction arises from old work on the bosonic principal chiral model on a half-line, but finds a modern realization as the hidden symmetry in the planar limit of the scattering of worldsheet excitations of the AdS/CFT light-cone superstring off a D5-brane.
We present the `Heisenberg picture' of the reflection algebra by explicitly constructing the ... more We present the `Heisenberg picture' of the reflection algebra by explicitly constructing the boundary Yangian symmetry of an AdS/CFT superstring which ends on a boundary with non-trivial degrees of freedom and which preserves the full bulk Lie symmetry algebra. We also consider the spectrum of bulk and boundary states and some automorphisms of the underlying algebras.
We present a nested algebraic Bethe ansatz for a one dimensional open spin chain whose boundary q... more We present a nested algebraic Bethe ansatz for a one dimensional open spin chain whose boundary quantum spaces are irreducible so_2n- or sp_2n-representations and the monodromy matrix satisfies the defining relations of the Olshanskii twisted Yangian Y^±(gl_2n). We use a generalization of the Bethe ansatz introduced by De Vega and Karowski which allows us to relate the spectral problem of a so_2n- or sp_2n-symmetric open spin chain to that of a gl_n-symmetric periodic spin chain. We explicitly derive the structure of the Bethe vectors and the nested Bethe equations.
We calculate the first quantum corrections to the masses of solitons in imaginary-coupling affine... more We calculate the first quantum corrections to the masses of solitons in imaginary-coupling affine Toda theories using the semi-classical method of Dashen, Hasslacher and Neveu. The theories divide naturally into those based on the simply-laced, the twisted and the untwisted non-simply-laced algebras. We find that the classical relationships between soliton and particle masses found by Olive et al. persist for the first two classes, but do not appear to do so naively for the third.
We explore the house price distributions for the English cities of London, Manchester, Bristol, N... more We explore the house price distributions for the English cities of London, Manchester, Bristol, Newcastle, Birmingham and Leeds. We find Pareto (power law) behaviour in their upper tails, which is clearly distinct from lognormal and gamma distributions in the cases of London, Manchester and Newcastle. For London, the city with the lowest power, this is a striking match with that found in the wealth distribution of the super-rich. We propose an index of Housing Wealth Inequality based on the Pareto exponent and analogous to the Gini coefficient, and comment on its possible uses.
By placing theories with Yangian charges on the lattice in the analogue of the St Petersburg scho... more By placing theories with Yangian charges on the lattice in the analogue of the St Petersburg school's approach to the sine-Gordon system, we exhibit the Yangian structure of the auxiliary algebra, and explain how the two Yangians are related.
Pioneers in Arts, Humanities, Science, Engineering, Practice
Simple dynamical systems, in the spirit of Richardson’s arms race, can be used to investigate the... more Simple dynamical systems, in the spirit of Richardson’s arms race, can be used to investigate the core dynamics of various models of insurgent and multilateral war. This chapter describes two such models. The first combines Richardson’s two-nation arms race with Lanchester’s attrition model and Deitchman’s guerrilla variant of it to create a model in which the typical long-term outcome is neither annihilation nor escalation but rather a stable fixed point, a stalemate. The scaling it implies for the force required to defeat an insurgency matches that which has been observed. The second model is of multilateral attritional war, in the spirit of Richardson’s multinational arms race. We describe the case of three antagonists, whose objective is to win but, if they cannot win, to minimize their remaining opponents. In contrast to truels and triads in which the objective is survival, and the weakest actor often emerges in a position of surprising strength, here the outcome is mutual anni...
We show that, in the open Hubbard model with integrable boundary conditions, the bulk Yangian sym... more We show that, in the open Hubbard model with integrable boundary conditions, the bulk Yangian symmetry is broken to a twisted Yangian. We prove that the associated charges commute with the Hamiltonian and the reflection matrix, and that they form a coideal subalgebra.
A Yang-Baxter relation-based formalism for generalized quantum affine algebras with the structure... more A Yang-Baxter relation-based formalism for generalized quantum affine algebras with the structure of an infinite Hopf family of (super-) algebras is proposed. The structure of the infinite Hopf family is given explicitly on the level of L matrices. The relation with the Drinfeld current realization is established in the case of 4×4 R-matrices by studying the analogue of the Ding-Frenkel theorem. By use of the concept of algebra “comorphisms ” (which generalize the notion of algebra comodules for standard Hopf algebras), a possible way of constructing infinitely many commuting operators out of the generalized RLL algebras is given. Finally some examples of the generalized RLL algebras are briefly discussed. 1
We generalize Deitchman’s guerrilla warfare model to account for trade-off between intelligence (... more We generalize Deitchman’s guerrilla warfare model to account for trade-off between intelligence (‘bits’) and firepower (‘shots’). Intelligent targeting leads to aimed fire; absence of intelligence leads to unaimed fire, dependent on targets’ density. We propose a new Lanchester-type model that mixes aimed and unaimed fire, the balance between these being determined by quality of information. We derive the model’s conserved quantity, and use it to analyze the trade-off between investments in intelligence and in firepower—for example, in counterinsurgency operations.
A Yang-Baxter relation-based formalism for generalized quantum affine algebras with the structure... more A Yang-Baxter relation-based formalism for generalized quantum affine algebras with the structure of an infinite Hopf family of (super-) algebras is proposed. The structure of the infinite Hopf family is given explicitly on the level of $L$ matrices. The relation with the Drinfeld current realization is established in the case of $4\times4$ $R$-matrices by studying the analogue of the Ding-Frenkel theorem. By use of the concept of algebra ``comorphisms'' (which generalize the notion of algebra comodules for standard Hopf algebras), a possible way of constructing infinitely many commuting operators out of the generalized $RLL$ algebras is given. Finally some examples of the generalized $RLL$ algebras are briefly discussed.
This has remarkable implications. Since this quantity never changes sign, only one of R and G can... more This has remarkable implications. Since this quantity never changes sign, only one of R and G can ever be zero. If the initial value of (3) is positive, for example, only G can equal zero, and so only red can win. So this quantity is key in determining who will win the battle, and a side’s fighting strength – we shall call rR and gG the red and green forces’ respective fighting strengths – varies as the units’ fighting effectiveness times the square of their numbers.
The Battle of Britain is the focus of much historical controversy. We show here how the statistic... more The Battle of Britain is the focus of much historical controversy. We show here how the statistical technique of weighted bootstrapping can be used to create a new quantitative basis to help address such controversies. Bootstrapping facilitates the exploration of alternative campaign possibilities with different tactics. This results in comparative probabilities of “victory” for the actual and various counterfactual campaigns, providing a quantified assessment of the likelihood of German achievement of air superiority, thereby facilitating invasion. We find this more likely had the Luftwaffe targeted airfields more heavily, and greatly more likely had Germany brought forward its air campaign.
We consider integrable field theories with achiral boundary conditions and uncover the underlying... more We consider integrable field theories with achiral boundary conditions and uncover the underlying achiral twisted Yangian algebra. This construction arises from old work on the bosonic principal chiral model on a half-line, but finds a modern realization as the hidden symmetry in the planar limit of the scattering of worldsheet excitations of the AdS/CFT light-cone superstring off a D5-brane.
We present the `Heisenberg picture' of the reflection algebra by explicitly constructing the ... more We present the `Heisenberg picture' of the reflection algebra by explicitly constructing the boundary Yangian symmetry of an AdS/CFT superstring which ends on a boundary with non-trivial degrees of freedom and which preserves the full bulk Lie symmetry algebra. We also consider the spectrum of bulk and boundary states and some automorphisms of the underlying algebras.
We present a nested algebraic Bethe ansatz for a one dimensional open spin chain whose boundary q... more We present a nested algebraic Bethe ansatz for a one dimensional open spin chain whose boundary quantum spaces are irreducible so_2n- or sp_2n-representations and the monodromy matrix satisfies the defining relations of the Olshanskii twisted Yangian Y^±(gl_2n). We use a generalization of the Bethe ansatz introduced by De Vega and Karowski which allows us to relate the spectral problem of a so_2n- or sp_2n-symmetric open spin chain to that of a gl_n-symmetric periodic spin chain. We explicitly derive the structure of the Bethe vectors and the nested Bethe equations.
We calculate the first quantum corrections to the masses of solitons in imaginary-coupling affine... more We calculate the first quantum corrections to the masses of solitons in imaginary-coupling affine Toda theories using the semi-classical method of Dashen, Hasslacher and Neveu. The theories divide naturally into those based on the simply-laced, the twisted and the untwisted non-simply-laced algebras. We find that the classical relationships between soliton and particle masses found by Olive et al. persist for the first two classes, but do not appear to do so naively for the third.
We explore the house price distributions for the English cities of London, Manchester, Bristol, N... more We explore the house price distributions for the English cities of London, Manchester, Bristol, Newcastle, Birmingham and Leeds. We find Pareto (power law) behaviour in their upper tails, which is clearly distinct from lognormal and gamma distributions in the cases of London, Manchester and Newcastle. For London, the city with the lowest power, this is a striking match with that found in the wealth distribution of the super-rich. We propose an index of Housing Wealth Inequality based on the Pareto exponent and analogous to the Gini coefficient, and comment on its possible uses.
By placing theories with Yangian charges on the lattice in the analogue of the St Petersburg scho... more By placing theories with Yangian charges on the lattice in the analogue of the St Petersburg school's approach to the sine-Gordon system, we exhibit the Yangian structure of the auxiliary algebra, and explain how the two Yangians are related.
Pioneers in Arts, Humanities, Science, Engineering, Practice
Simple dynamical systems, in the spirit of Richardson’s arms race, can be used to investigate the... more Simple dynamical systems, in the spirit of Richardson’s arms race, can be used to investigate the core dynamics of various models of insurgent and multilateral war. This chapter describes two such models. The first combines Richardson’s two-nation arms race with Lanchester’s attrition model and Deitchman’s guerrilla variant of it to create a model in which the typical long-term outcome is neither annihilation nor escalation but rather a stable fixed point, a stalemate. The scaling it implies for the force required to defeat an insurgency matches that which has been observed. The second model is of multilateral attritional war, in the spirit of Richardson’s multinational arms race. We describe the case of three antagonists, whose objective is to win but, if they cannot win, to minimize their remaining opponents. In contrast to truels and triads in which the objective is survival, and the weakest actor often emerges in a position of surprising strength, here the outcome is mutual anni...
We show that, in the open Hubbard model with integrable boundary conditions, the bulk Yangian sym... more We show that, in the open Hubbard model with integrable boundary conditions, the bulk Yangian symmetry is broken to a twisted Yangian. We prove that the associated charges commute with the Hamiltonian and the reflection matrix, and that they form a coideal subalgebra.
A Yang-Baxter relation-based formalism for generalized quantum affine algebras with the structure... more A Yang-Baxter relation-based formalism for generalized quantum affine algebras with the structure of an infinite Hopf family of (super-) algebras is proposed. The structure of the infinite Hopf family is given explicitly on the level of L matrices. The relation with the Drinfeld current realization is established in the case of 4×4 R-matrices by studying the analogue of the Ding-Frenkel theorem. By use of the concept of algebra “comorphisms ” (which generalize the notion of algebra comodules for standard Hopf algebras), a possible way of constructing infinitely many commuting operators out of the generalized RLL algebras is given. Finally some examples of the generalized RLL algebras are briefly discussed. 1
We generalize Deitchman’s guerrilla warfare model to account for trade-off between intelligence (... more We generalize Deitchman’s guerrilla warfare model to account for trade-off between intelligence (‘bits’) and firepower (‘shots’). Intelligent targeting leads to aimed fire; absence of intelligence leads to unaimed fire, dependent on targets’ density. We propose a new Lanchester-type model that mixes aimed and unaimed fire, the balance between these being determined by quality of information. We derive the model’s conserved quantity, and use it to analyze the trade-off between investments in intelligence and in firepower—for example, in counterinsurgency operations.
A Yang-Baxter relation-based formalism for generalized quantum affine algebras with the structure... more A Yang-Baxter relation-based formalism for generalized quantum affine algebras with the structure of an infinite Hopf family of (super-) algebras is proposed. The structure of the infinite Hopf family is given explicitly on the level of $L$ matrices. The relation with the Drinfeld current realization is established in the case of $4\times4$ $R$-matrices by studying the analogue of the Ding-Frenkel theorem. By use of the concept of algebra ``comorphisms'' (which generalize the notion of algebra comodules for standard Hopf algebras), a possible way of constructing infinitely many commuting operators out of the generalized $RLL$ algebras is given. Finally some examples of the generalized $RLL$ algebras are briefly discussed.
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