On the algebraic structure of rational discrete dynamical systems

We show how singularities shape the evolution of rational discrete dynamical systems. The stabilisation of the form of the iterates suggests a description providing among other things generalised Hirota form, exact evaluation of the algebraic entropy as well as remarkable polynomial factorisation properties. We illustrate the phenomenon explicitly with examples covering a wide range of models

The Contribution of Douglass North to New Institutional Economics

Douglass North, along with Ronald Coase and Oliver Williamson, transformed the early intuitions of new institutional economics into powerful conceptual and analytical tools that spawned a robust base of empirical research. NIE arose in response to questions not well explained by standard neoclassical models, such as make or buy and why rich or poor? Today NIE is a success story by many measures: four Nobel laureates in under 20 years, increasing penetration of mainstream journals, and significant impact on major policy debates from anti-trust law to development aid. This paper provides a succinct overview of North's evolving ideas about institutions and explains how North's work shaped the emerging field of new institutional economics and had a potent impact on economics and the social sciences more broadly. North provides a powerful example of how persistent and well placed confidence and hard work can productively transform the status quo. North's influence continues strong and his enthusiasm for exploring new frontiers and cooperating across artificial academic boundaries has never waned.New Institutional Economics, institutions, transaction costs, development and growth

Algebraic entropy for differential-delay equations

We extend the definition of algebraic entropy to a class of differential-delay equations. The vanishing of the entropy, as a structural property of an equation, signals its integrability. We suggest a simple way to produce differential-delay equations with vanishing entropy from known integrable differential-difference equations

Duality relations in the auxiliary field method

The eigenenergies $\epsilon^{(N)}(m;\{n_i,l_i\})$ of a system of $N$ identical particles with a mass $m$ are functions of the various radial quantum numbers $n_i$ and orbital quantum numbers $l_i$. Approximations $E^{(N)}(m;Q)$ of these eigenenergies, depending on a principal quantum number $Q(\{n_i,l_i\})$, can be obtained in the framework of the auxiliary field method. We demonstrate the existence of numerous exact duality relations linking quantities $E^{(N)}(m;Q)$ and $E^{(p)}(m';Q')$ for various forms of the potentials (independent of $m$ and $N$) and for both nonrelativistic and semirelativistic kinematics. As the approximations computed with the auxiliary field method can be very close to the exact results, we show with several examples that these duality relations still hold, with sometimes a good accuracy, for the exact eigenenergies $\epsilon^{(N)}(m;\{n_i,l_i\})$

Permutations preserving divisibility

We give a proof of a theorem on the common divisibility of polynomials and permuted polynomials (over GF(2)) by a polynomial g(x)

Nonequilibrium Stationary States and Phase Transitions in Directed Ising Models

We study the nonequilibrium properties of directed Ising models with non conserved dynamics, in which each spin is influenced by only a subset of its nearest neighbours. We treat the following models: (i) the one-dimensional chain; (ii) the two-dimensional square lattice; (iii) the two-dimensional triangular lattice; (iv) the three-dimensional cubic lattice. We raise and answer the question: (a) Under what conditions is the stationary state described by the equilibrium Boltzmann-Gibbs distribution? We show that for models (i), (ii), and (iii), in which each spin "sees" only half of its neighbours, there is a unique set of transition rates, namely with exponential dependence in the local field, for which this is the case. For model (iv), we find that any rates satisfying the constraints required for the stationary measure to be Gibbsian should satisfy detailed balance, ruling out the possibility of directed dynamics. We finally show that directed models on lattices of coordination number $z\ge8$ with exponential rates cannot accommodate a Gibbsian stationary state. We conjecture that this property extends to any form of the rates. We are thus led to the conclusion that directed models with Gibbsian stationary states only exist in dimension one and two. We then raise the question: (b) Do directed Ising models, augmented by Glauber dynamics, exhibit a phase transition to a ferromagnetic state? For the models considered above, the answers are open problems, to the exception of the simple cases (i) and (ii). For Cayley trees, where each spin sees only the spins further from the root, we show that there is a phase transition provided the branching ratio, $q$, satisfies $q \ge 3$

Rationale for tau aggregation inhibitor therapy in Alzheimer's disease and other tauopathies

Preprin

Light baryon masses in different large-NcN_c limits

We investigate the behavior of light baryon masses in three inequivalent large-$N_c$ limits: 't~Hooft, QCD$_{{\rm AS}}$ and Corrigan-Ramond. Our framework is a constituent quark model with relativistic-type kinetic energy, stringlike confinement and one-gluon-exchange term, thus leading to well-defined results even for massless quarks. We analytically prove that the light baryon masses scale as $N_c$, $N_c^2$ and $1$ in the 't~Hooft, QCD$_{{\rm AS}}$ and Corrigan-Ramond limits respectively. Those results confirm previous ones obtained by using either diagrammatic methods or constituent approaches, mostly valid for heavy quarks.Comment: Final version to appear in Phys. Rev.