Memory distribution in complex fitness landscapes

In a co-evolutionary context, the survive probability of individual elements of a system depends on their relation with their neighbors. The natural selection process depends on the whole population, which is determined by local events between individuals. Particular characteristics assigned to each individual, as larger memory, usually improve the individual fitness, but an agent possess also endogenous characteristics that induce to re-evaluate her fitness landscape and choose the best-suited kind of interaction, inducing a non absolute value of the outcomes of the interaction. In this work, a novel model with agents combining memory and rational choice is introduced, where individual choices in a complex fitness landscape induce changes in the distribution of the number of agents as a function of the time. In particular, the tail of this distribution is fat compared with distributions for agents interacting only with memory.Comment: 6 pages, 3 figures, submited to Physica

Burgers-like equation for spontaneous breakdown of the chiral symmetry in QCD

We link the spontaneous breakdown of chiral symmetry in Euclidean QCD to the collision of spectral shock waves in the vicinity of zero eigenvalue of Dirac operator. The mechanism, originating from complex Burger's-like equation for viscid, pressureless, one-dimensional flow of eigenvalues, is similar to recently observed weak-strong coupling phase transition in large $N_c$ Yang-Mills theory. The spectral viscosity is proportional to the inverse of the size of the random matrix that replaces the Dirac operator in the universal (ergodic) regime. We obtain the exact scaling function and critical exponents of the chiral phase transition for the averaged characteristic polynomial for $N_c \ge3$ QCD. We reinterpret our results in terms of known properties of chiral random matrix models and lattice data.Comment: 12 page

Universal shocks in the Wishart random-matrix ensemble - a sequel

We study the diffusion of complex Wishart matrices and derive a partial differential equation governing the behavior of the associated averaged characteristic polynomial. In the limit of large size matrices, the inverse Cole-Hopf transform of this polynomial obeys a nonlinear partial differential equation whose solutions exhibit shocks at the evolving edges of the eigenvalue spectrum. In a particular scenario one of those shocks hits the origin that plays the role of an impassable wall. To investigate the universal behavior in the vicinity of this wall, a critical point, we derive an integral representation for the averaged characteristic polynomial and study its asymptotic behavior. The result is a Bessoid function.Comment: 7 pages, 2 figure

Universal shocks in the Wishart random matrix ensemble - 1

We show that the derivative of the logarithm of the average characteristic polynomial of a diffusing Wishart matrix obeys an exact partial differential equation valid for an arbitrary value of N, the size of the matrix. In the large N limit, this equation generalizes the simple Burgers equation that has been obtained earlier for Hermitian or unitary matrices. The solution through the method of characteristics presents singularities that we relate to the precursors of shock formation in fluid dynamical equations. The 1/N corrections may be viewed as viscous corrections, with the role of the viscosity being played by the inverse of the doubled dimension of the matrix. These corrections are studied through a scaling analysis in the vicinity of the shocks, and one recovers in a simple way the universal Bessel oscillations (so-called hard edge singularities) familiar in random matrix theory.Comment: 9 page

M74 Motorway, Glasgow – Geotechnical Aspects of Design and Construction

The 7.8 kilometre long M74 Completion project forms the final part of the motorway box around the city of Glasgow. Construction commenced in May 2008 and was completed in June 2011. The urban route corridor presented many geotechnical challenges to the design and construction teams. It is underlain by Recent, lightly over-consolidated Clyde Alluvium of maximum 35 metres thickness over Glacial Till and Carboniferous Coal Measure Sandstone bedrock. Additionally, it was the location of industries over the 19th and 20th centuries which deposited waste over the natural ground containing chromium, steel works slag and hydrocarbons. The route corridor is also underlain by historical coal mining. This paper details the geotechnical design and construction to overcome the challenges for earthworks and structure foundations posed by th

Large N_c confinement and turbulence

We suggest that the transition that occurs at large $N_c$ in the eigenvalue distribution of a Wilson loop may have a turbulent origin. We arrived at this conclusion by studying the complex-valued inviscid Burgers-Hopf equation that corresponds to the Makeenko-Migdal loop equation, and we demonstrate the appearance of a shock in the spectral flow of the Wilson loop eigenvalues. This picture supplements that of the Durhuus-Olesen transition with a particular realization of disorder. The critical behavior at the formation of the shock allows us to infer exponents that have been measured recently in lattice simulations by Narayanan and Neuberger in $d=2$ and $d=3$. Our analysis leads us to speculate that the universal behavior observed in these lattice simulations might be a generic feature of confinement, also in $d=4$ Yang-Mills theory.Comment: 4 pages, no figures- Some rewriting - Typos corrected - References completed and some correcte

Diffusion in the space of complex Hermitian matrices - microscopic properties of the averaged characteristic polynomial and the averaged inverse characteristic polynomial

We show that the averaged characteristic polynomial and the averaged inverse characteristic polynomial, associated with Hermitian matrices whose elements perform a random walk in the space of complex numbers, satisfy certain partial differential, diffusion-like, equations. These equations are valid for matrices of arbitrary size. Their solutions can be given an integral representation that allows for a simple study of their asymptotic behaviors for a broad range of initial conditions.Comment: 26 pages, 4 figure