Martingales, nonstationary increments, and the efficient market hypothesis

We discuss the deep connection between nonstationary increments, martingales, and the efficient market hypothesis for stochastic processes x(t) with arbitrary diffusion coefficients D(x,t). We explain why a test for a martingale is generally a test for uncorrelated increments. We explain why martingales look Markovian at the level of both simple averages and 2-point correlations. But while a Markovian market has no memory to exploit and cannot be beaten systematically, a martingale admits memory that might be exploitable in higher order correlations. We also use the analysis of this paper to correct a misstatement of the ‘fair game’ condition in terms of serial correlations in Fama’s paper on the EMH. We emphasize that the use of the log increment as a variable in data analysis generates spurious fat tails and spurious Hurst exponents.     

Statistical independence in nonlinear maps coupled to non-invertible transformations

We investigate the connections between functions of type xn=p(θTzn) and nonlinear maps coupled to non-invertible transformations. These systems can produce unpredictable dynamics. We study the higher-order correlations in the generated sequences. We show that (theoretically) it is possible to construct systems that can generate sequences that constitute a set of statistically independent random variables. We apply the results in the improvement of a two-dimensional coupled map system that has been used in practical applications as e.g. cryptosystems and data compression.     

Sur la determination des constantes optiques du calcium a partir de couches epaisses

We investigate how it is possible to determine the best the optical constants of calcium from measurements of the transmittance T and reflectance at the interfaces metal-vacuum R and metal substrate R′ on thin films, using a method of successive approximations. We compare the results obtained using the experimental values of (R, R′, T) or (R, T) or (R′, T), for two films of different thicknesses. We show that a satisfactory result is obtained with the values of (R′, T). We determine, in this manner, the optical constants of calcium in the energy range 2 –5.6 eV, and we compare the results obtained with those determined by other authors.     

Can the quark mass matrix be real?

We study the possibility of solving the strong CP problem in a theory where CP is broken spontaneously and the quark mass matrix, M_ij, not just its determinant, remains real to one part in 10⁸. We find that it is difficult to build natural models consistent with the known CP violation phenomenology. We find that such models require new fermions and possess a hierarchy problem. We present an SO(10) example which succeeds at the one-loop level, however, two-loop effects will introduce phases into M_ij.     

F-theory,T-duality on K3 surfaces,and N = 2 supersymmetric gauge theories in four dimensions

We construct T-duality on K3 surfaces. The T-duality exchanges a 4-brane RR charge and a 0-brane RR charge. We study the action of the T-duality on the moduli space of 0-branes located at points of K3 and 4-branes wrapping it. We apply the construction to F-theory compactified on a Calabi-Yau 4-fold and study the duality of N = 2 SU(N_c) gauge theories in four dimensions. We discuss the generalization to the N = 1 duality scenario.     

Massive yang-mills approach to Skyrmions with vector mesons⋆

We construct an effective Lagrangian involving pions,ρ andω mesons. We calculate the bulk properties of hedgehog Skyrmions in this model. Canonical quantization is performed to project onto states of good spin and isospin, and various approximations going into the calculation of the moment of inertia are discussed in extensio. We find that theω-excitations are clearly dominating the excitations of theρ field.     

Comparison of the π and Δ operator approximations for the two-dimensional t–J model

We compare the spectra of the new π operator of the SO(5) theory and the conventional Δ operator for the two-dimensional t–J model. We also calculate the weight transferred to the two-hole ground state from half-filling by these operators. We find that the spectra of these operators are quite similar and the weight for the π operator is smaller than the weight for the Δ operator. We argue that the two-dimensional t–J model does not have a good approximate SO(5) symmetry claimed in Ref. [1].     

Exclusive semileptonic heavy meson decays including lepton mass effects

We discuss the exclusive semi-leptonic (s.l.) bottom meson decaysB→D(D ^*)+l+v where we include non-zero lepton mass effects in the kinematics and dynamics. We develop the general formalism for the non-zero lepton mass case. We then look at how rates, spectra and angular correlations are affected by non-zero lepton masses in the context of a specific spectator quark model. Numerical results are presented for s.l. decays involving thee-, μ- and τ-leptons. We also discuss the s.l. decaysB→π(ρ),D→K(K ^*) and the free quark decay model.     

On the number of solutions for the two-point boundary value problem on Riemannian manifolds

We study the solutions joining two fixed points of a time-independent dynamical system on a Riemannian manifold (M,g) from an enumerative point of view. We prove a finiteness result for solutions joining two points p,q∈M that are non-conjugate in a suitable sense, under the assumption that (M,g) admits a non-trivial convex function. We discuss in some detail the notion of conjugacy induced by a general dynamical system on a Riemannian manifold. Using techniques of infinite dimensional Morse theory on Hilbert manifolds we also prove that, under generic circumstances, the number of solutions joining two fixed points is odd. We present some examples where our theory applies.     

Phase transitions in one dimension and less

Phase transitions can occur in one-dimensional classical statistical mechanics at non-zero temperature when the number of components N of the spin is infinite. We show how to solve such magnets in one dimension for any N, and how the phase transition develops at N=∞. We discuss SU(N) and Sp(N) magnets, where the transition is second-order. In the new high-temperature phase, the correlation length is zero. We also show that for the SU(N) magnet on exactly three sites with periodic boundary conditions the transition becomes first order.     

Duality diagrams and the decays of the ψ and ψ′

We apply duality diagrams to the ψ and ψ′ decays. We propose that the ψ → 3π decay is described by a Virasoro formula, and show that the low mass ππ suppression in ψ′ → ψ2π is a consequence of the threshold factor contained in the Pomeron-2π coupling. We stress the importance of the ψ and ψ′ decays for the study of heavy mesonic resonances.     

Duality Theorems in Ergodic Transport

We analyze several problems of Optimal Transport Theory in the setting of Ergodic Theory. In a certain class of problems we consider questions in Ergodic Transport which are generalizations of the ones in Ergodic Optimization. Another class of problems is the following: suppose ?? is the shift acting on Bernoulli space X={1,2,??,d}^?, and, consider a fixed continuous cost function c:X×X???. Denote by ?? the set of all Borel probabilities ?? on X×X, such that, both its x and y marginals are ??-invariant probabilities. We are interested in the optimal plan ?? which minimizes ??c? d?? among the probabilities in ??. We show, among other things, the analogous Kantorovich Duality Theorem. We also analyze uniqueness of the optimal plan under generic assumptions on c. We investigate the existence of a dual pair of Lipschitz functions which realizes the present dual Kantorovich problem under the assumption that the cost is Lipschitz continuous. For continuous costs c the corresponding results in the Classical Transport Theory and in Ergodic Transport Theory can be, eventually, different. We also consider the problem of approximating the optimal plan ?? by convex combinations of plans such that the support projects in periodic orbits.     

Energy exchange in fast optical soliton collisions as a random cascade model

We study the dynamics of a probe soliton propagating in an optical fiber and exchanging energy in fast collisions with a random sequence of pump solitons. The energy exchange is induced by Raman scattering or by cubic nonlinear loss/gain. We show that the equation describing the dynamics of the probe soliton's amplitude has the same form as the equation for the local space average of energy dissipation in random cascade models in turbulence. We characterize the statistics of the probe soliton's amplitude by the τq exponents from multifractal theory and by the Cramér function S(x). We find that the nth moment of the two-time correlation function and the bit-error-rate contribution from amplitude decay exhibit power-law behavior as functions of propagation distance, where the exponents can be expressed in terms of τq or S(x).     

Ecological interaction and phylogeny, studying functionality on composed networks

We study a class of composed networks that are formed by two tree networks, T_P and T_A, whose end points touch each other through a bipartite network B_PA. We explore this network using a functional approach. We are interested in how much the topology, or the structure, of T_X (X=A or P) determines the links of B_PA. This composed structure is a useful model in evolutionary biology, where T_P and T_A are the phylogenetic trees of plants and animals that interact in an ecological community. We make use of ecological networks of dispersion of fruits, which are formed by frugivorous animals and plants with fruits; the animals, usually birds, eat fruits and disperse their seeds. We analyse how the phylogeny of T_X determines or is correlated with B_PA using a Monte Carlo approach. We use the phylogenetic distance among elements that interact with a given species to construct an index κ that quantifies the influence of T_X over B_PA. The algorithm is based on the assumption that interaction matrices that follows a phylogeny of T_X have a total phylogenetic distance smaller than the average distance of an ensemble of Monte Carlo realisations. We find that the effect of phylogeny of animal species is more pronounced in the ecological matrix than plant phylogeny.     

Compressible effect algebras

We define a special type of additive map J on an effect algebra E called a compression. We call J(1) the focus of J and if p is the focus of a compression then p is called a projection. The set of projections in E is denoted by P(E). A compression J is direct if J(a) ≤ a for all a ε E. We show that direct compressions are equivalent to projections onto components of cartesian products. An effect algebra E is said to be compressible if every compression on E is uniquely determined by its focus and every compression on E has a supplement. We define and characterize the commutant C(p) of a projection p and show that a compression with focus p is direct if and only if C(p) = E. We show that P(E) is an orthomodular poset. It is proved that the cartesian product of effect algebras is compressible if and only if each component is compressible. We then consider compressible sequential effect algebras, Lüders maps and conditional probabilities.     

An Infinite Family of Adsorption Models and Restricted Lukasiewicz Paths

We define (k,?)-restricted Lukasiewicz paths, k??????₀, and use these paths as models of polymer adsorption. We write down a polynomial expression satisfied by the generating function for arbitrary values of (k,?). The resulting polynomial is of degree ?+1 and hence cannot be solved explicitly for sufficiently large ?. We provide two different approaches to obtain the phase diagram. In addition to a more conventional analysis, we also develop a new mathematical characterisation of the phase diagram in terms of the discriminant of the polynomial and a zero of its highest degree coefficient. We then give a bijection between (k,?)-restricted Lukasiewicz paths and ??rise??-restricted Dyck paths, identifying another family of path models which share the same critical behaviour. For (k,?)=(1,??) we provide a new bijection to Motzkin paths. We also consider the area-weighted generating function and show that it is a q-deformed algebraic function. We determine the generating function explicitly in particular cases of (k,?)-restricted Lukasiewicz paths, and for (k,?)=(0,??) we provide a bijection to Dyck paths.     

On the measure of gaps and spectra for discrete 1D Schrödinger operators

We study the lebesgue measure of gaps and spectra, of ergodic Jacobi matrices. We show that: |σ/A|+|G|≥v, where: σ is the spectrum,G is the union of the gaps,A is the set of energies where the Lyaponov exponent vanishes andv is an appropriate seminorm of the potential. We also study in more detail periodic Jacobi matrices, and obtain a lower bound and large coupling asymptotics for the measure of the spectrum. We apply the results of the periodic case, to limit periodic Jacobi matrices, and obtain sufficient conditions for |G|≥v and for |σ|>0.     

Factorization at higher twist in hadron-hadron scattering

We study one-loop soft gluon effects in a scalar quark QCD model of the Drell-Yan process. We find that factorization of the type suggested by Politzer holds at order m²/s, but fails at order m⁴/s². We suggest that this pattern holds to higher orders, and that the non-factoring terms are not suppressed by form factors. We compute the non-factoring contributions, and give a classical interpretation of our results.     

Doped graphene as a superconductor

We study s-wave superconductivity state in doped graphene within the extended attractive Hubbard model and BCS theory. We use the Green's function approach and coherent potential approximation. We obtain critical temperature of graphene, Tc, as a function of the impurity concentration, c, as well as impurity strength, δ. The results show that when c and δ, are increased, Tc remains finite and doped graphene can be a superconductor.     

Some tests for the helicity structure of the Pomeron in ep collisions

We discuss the reaction ep→epp?X in the onephoton exchange approximation, where it is in essence the reaction γ*p→p?X. A large rapidity gap is required between the particle or particles of the proton remnant pJ and those of X. We define a suitable azimuthal angle between a leptonic and a hadronic plane. The dependence of the cross section on is given explicitly and can be used to extract cross sections and interference terms for the reactionγ*p→p?X corresponding to the various helicities of the virtual photon γ*. The interference terms can be used to test models for the large rapidity gap events in a sensitive way. We discuss in detail models with factorizing Pomeron exchange and in particular the Donnachie- Landshoff Pomeron model. We make some remarks on soft colour exchange models and on possible effects of QCD background vacuum fields. We conclude with a suggestion to look for Odderon exchange in exclusive deep inelastic high energy reactions like γ*p→,p?π⁰and ?p→p?η.