A class of random fields on complete graphs with tractable partition function

The aim of this short note is to draw attention to a method by which the partition function and marginal probabilities for a certain class of random fields on complete graphs can be computed in polynomial time. This class includes Ising models with homogeneous pairwise potentials but arbitrary (inhomogeneous) unary potentials. Similarly, the partition function and marginal probabilities can be computed in polynomial time for random fields on complete bipartite graphs, provided they have homogeneous pairwise potentials. We expect that these tractable classes of large scale random fields can be very useful for the evaluation of approximation algorithms by providing exact error estimates.Comment: accepted for publication in IEEE TPAMI (short paper

Isochronism and tangent bifurcation of band edge modes in Hamiltonian lattices

In {\em Physica D} {\bf 91}, 223 (1996), results were obtained regarding the tangent bifurcation of the band edge modes ($q=0,\pi$) of nonlinear Hamiltonian lattices made of $N$ coupled oscillators. Introducing the concept of {\em partial isochronism} which characterises the way the frequency of a mode, $\omega$, depends on its energy, $\epsilon$, we generalize these results and show how the bifurcation energies of these modes are intimately connected to their degree of isochronism. In particular we prove that in a lattice of coupled purely isochronous oscillators ($\omega(\epsilon)$ strictly constant), the in-phase mode ($q=0$) never undergoes a tangent bifurcation whereas the out-of-phase mode ($q=\pi$) does, provided the strength of the nonlinearity in the coupling is sufficient. We derive a discrete nonlinear Schr\"odinger equation governing the slow modulations of small-amplitude band edge modes and show that its nonlinear exponent is proportional to the degree of isochronism of the corresponding orbits. This equation may be seen as a link between the tangent bifurcation of band edge modes and the possible emergence of localized modes such as discrete breathers.Comment: 23 pages, 1 figur

Breathers on lattices with long range interaction

We analyze the properties of breathers (time periodic spatially localized solutions) on chains in the presence of algebraically decaying interactions $1/r^s$. We find that the spatial decay of a breather shows a crossover from exponential (short distances) to algebraic (large distances) decay. We calculate the crossover distance as a function of $s$ and the energy of the breather. Next we show that the results on energy thresholds obtained for short range interactions remain valid for $s>3$ and that for $s < 3$ (anomalous dispersion at the band edge) nonzero thresholds occur for cases where the short range interaction system would yield zero threshold values.Comment: 4 pages, 2 figures, PRB Rapid Comm. October 199

Discrete breathers in thermal equilibrium: distributions and energy gaps

We study a discrete two-dimensional nonlinear system that allows for discrete breather solutions. We perform a spectral analysis of the lattice dynamics at thermal equilibrium and use a cooling technique to measure the amount of breathers at thermal equilibrium. Our results confirm the existence of an energy threshold for discrete breathers. The cooling method provides with a novel computational technique of measuring and analyzing discrete breather distribution properties in thermal equilibrium.Comment: 20 pages, 14 figure

On the Equivariant Tamagawa Number Conjecture for Tate Motives, Part II

Let K be any finite abelian extension of Q, k any subfield of K and r any integer. We complete the proof of the equivariant Tamagawa Number Conjecture for the pair (h⁰(Spec(K))(r),Z[Gal(K/k)])

Obtaining Breathers in Nonlinear Hamiltonian Lattices

We present a numerical method for obtaining high-accuracy numerical solutions of spatially localized time-periodic excitations on a nonlinear Hamiltonian lattice. We compare these results with analytical considerations of the spatial decay. We show that nonlinear contributions have to be considered, and obtain very good agreement between the latter and the numerical results. We discuss further applications of the method and results.Comment: 21 pages (LaTeX), 8 figures in ps-files, tar-compressed uuencoded file, Physical Review E, in pres

LIMEtree: Interactively Customisable Explanations Based on Local Surrogate Multi-output Regression Trees

Systems based on artificial intelligence and machine learning models should be transparent, in the sense of being capable of explaining their decisions to gain humans' approval and trust. While there are a number of explainability techniques that can be used to this end, many of them are only capable of outputting a single one-size-fits-all explanation that simply cannot address all of the explainees' diverse needs. In this work we introduce a model-agnostic and post-hoc local explainability technique for black-box predictions called LIMEtree, which employs surrogate multi-output regression trees. We validate our algorithm on a deep neural network trained for object detection in images and compare it against Local Interpretable Model-agnostic Explanations (LIME). Our method comes with local fidelity guarantees and can produce a range of diverse explanation types, including contrastive and counterfactual explanations praised in the literature. Some of these explanations can be interactively personalised to create bespoke, meaningful and actionable insights into the model's behaviour. While other methods may give an illusion of customisability by wrapping, otherwise static, explanations in an interactive interface, our explanations are truly interactive, in the sense of allowing the user to "interrogate" a black-box model. LIMEtree can therefore produce consistent explanations on which an interactive exploratory process can be built

On the local Tamagawa number conjecture for Tate motives over tamely ramified fields

The local Tamagawa number conjecure, first formulated by Fontaine and Perrin-Riou, expresses the compatibility of the (global) Tamagawa number conjecture on motivic $L$-functions with the functional equation. The local conjecture was proven for Tate motives over finite unramified extensions $K/\mathbb{Q}_p$ by Bloch and Kato. We use the theory of $(\phi, \Gamma_K)$-modules and a reciprocity law due to Cherbonnier and Colmez to provide a new proof in the case of unramified extensions, and to prove the conjecture for the motive $\mathbb{Q}_p(2)$ over certain tamely ramified extensions.Comment: 45 pages, LaTeX; extensive revisions and clarifications based on feedback; to appear in Algebra & Number Theor