on the stability of the quenched state in mean field spin glass models while the gibbs states of spin glass models have been noted to have an erratic dependence on temperature one may expect the mean over the disorder to produce a continuously varying quenched state the assumption of such continuity in temperature implies that in the infinite volume limit the state is stable under a class of deformations of the gibbs measure the condition is satisfied by the parisi ansatz along with an even broader stationarity property the stability conditions have equivalent expressions as marginal additivity of the quenched free energy implications of the continuity assumption include constraints on the overlap distribution which are expressed as the vanishing of the expectation value for an infinite collection of multi overlap polynomials the polynomials can be computed with the aid of a real replica calculation in which the number of replicas is taken to zero

existence of solutions to some singular initial value problems on etudie l existence des solutions et leurs domaines de definition pour des problemes aux valeurs initiales pour des equations differentielles d ordre 1 et 2 qui sont singulieres soit dans les variables dependantes soit dans les variables independantes

an eulerian lagrangian approach to the navier stokes equations we present a formulation of the incompressible viscous navier stokes equation based on a generalization of the inviscid weber formula in terms of a diffusive back to labels map and a virtual velocity we derive a generalization of the inviscid cauchy formula and obtain certain bounds for the objects introduced

a geometrical study of 3d incompressible euler flows with clebsch potentials a long lived euler flow and its power law energy spectrum abstract a simple initial condition for vorticity sin y z sin z x sin x y which has clebsch potentials has been identified to lead to a flow evolution with a very weak energy transfer this allows us to integrate the euler equations in time longer than commonly expected to reach a stage at which the total enstrophy attains its peak for the corresponding navier stokes flow it thereby enables us to study the relationship between the inviscid limit and totally inviscid behaviours numerically in spite of small energy dissipation rate the navier stokes flow shows a power law spectrum whose exponent is around 5 3 and 2 a similar behaviour is also observed for the euler flow in physical space this flow has groups of vorticity layers which hesitate to roll up

existence of positive solution to a singular second order initial value problem with impulses this paper deals with the existence of positive solutions to singular initial value problem with impulse effects the right hand side of the differential equation can be singular in its dependent variable

stochastic symmetry breaking in a gaussian hopfield model we study a two pattern hopfield model with gaussian disorder we find that there are infinitely many pure states at low temperatures in this model and that the metastate is supported on an infinity of symmetric pairs of pure states the origin of this phenomenon is the random breaking of a rotation symmetry of the distribution of the disorder variables

about the ergodic regime in the analogical hopfield neural networks moments of the partition function in this paper we introduce and exploit the real replica approach for a minimal generalization of the hopfield model by assuming the learned patterns to be distributed according to a standard unit gaussian we consider the high storage case when the number of patterns linearly diverges with the number of neurons we study the infinite volume behavior of the normalized momenta of the partition function we find a region in the parameter space where the free energy density in the infinite volume limit self averages around its annealed approximation as well as the entropy and the internal energy density moreover we evaluate the corrections to their extensive counterparts with respect to their annealed expressions the fluctuations of properly introduced overlaps which act as order parameters are also discussed

singularities of euler flow not out of the blue does three dimensional incompressible euler flow with smooth initial conditions develop a singularity with infinite vorticity after a finite time this blowup problem is still open after briefly reviewing what is known and pointing out some of the difficulties we propose to tackle this issue for the class of flows having analytic initial data for which hypothetical real singularities are preceded by singularities at complex locations we present some results concerning the nature of complex space singularities in two dimensions and propose a new strategy for the numerical investigation of blowup

an eulerian lagrangian approach for incompressible fluids local theory we study a formulation of the incompressible euler equations in terms of the inverse lagrangian map in this formulation the equations become a first order advective nonlinear system of partial differential equations

the thermodynamic limit in mean field spin glass models we present a simple strategy in order to show the existence and uniqueness of the infinite volume limit of thermodynamic quantities for a large class of mean field disordered models as for example the sherrington kirkpatrick model and the derrida p spin model the main argument is based on a smooth interpolation between a large system made of n spin sites and two similar but independent subsystems made of n 1 and n 2 sites respectively with n 1 n 2 n the quenched average of the free energy turns out to be subadditive with respect to the size of the system this gives immediately convergence of the free energy per site in the infinite volume limit moreover a simple argument based on concentration of measure gives the almost sure convergence with respect to the external noise similar results hold also for the ground state energy per site

finite difference schemes for incompressible flows in the velocity impulse density formulation we consider finite difference schemes based on the impulse density variable we show that the original velocity impulse density formulation of oseledets is marginally ill posed for the inviscid flow and this has the consequence that some ordinarily stable numerical methods in other formulations become unstable in the velocity impulse density formulation we present numerical evidence of this instability we then discuss the construction of stable finite difference schemes by requiring that at the numerical level the nonlinear terms be convertible to similar terms in the primitive variable formulation finally we give a simplified velocity impulse density formulation which is free of these complications and yet retains the nice features of the original velocity impulse density formulation with regard to the treatment of boundary we present numerical results on this simplified formulation for the driven cavity flow on both the staggered and non staggered grids

an unconstrained hamiltonian formulation for incompressible fluid flow the equations governing the time evolution of an ideal fluid in material coordinates are expressed as an unconstrained canonical hamiltonian system the incompressibility of the flow is consequent upon certain first integrals of the motion the variable conjugate to the configuration field is not the usual linear momentum but is instead a quantity that is related to linear momentum through an auxiliary scalar field whose time derivative is the pressure the definition of the hamiltonian involves a minimization with respect to this auxiliary field the method of derivation may be generally applied to obtain unconstrained hamiltonian descriptions of lagrangian field equations subject to pointwise constraints

vortex flows and related numerical methods many important phenomena in fluid motion are evident in vortex flow i e flows in which vortical structures are significant in determining the whole flow this book which consists of lectures given at a nato arw held in grenoble france in june 1992 provides an up to date account of current research in the study of these phenomena by means of numerical methods and mathematical modelling such methods include eulerian methods finite difference spectral and wavelet methods as well as lagrangian methods contour dynamics vortex methods and are used to study such topics as 2 or 3 dimensional turbulence vorticity generation by solid bodies shear layers and vortex sheets and vortex reconnection for researchers and graduate students in computational fluid dynamics numerical analysis and applied mathematics

filtered viscous fluid equations abstract we analyze an eulerian lagrangian description of filtered incompressible fluid equations a viscous cauchy formula is derived for such equations the models discussed are related to vortex methods and to filtered kuzmin oseledets equations

percolation on correlated random networks we consider a class of random weighted networks obtained through a redefinition of patterns in an hopfield like model and by performing percolation processes we get information about topology and resilience properties of the networks themselves given the weighted nature of the graphs different kinds of bond percolation can be studied stochastic deleting links randomly and deterministic deleting links based on rank weights each mimicking a different physical process the evolution of the network is accordingly different as evidenced by the behavior of the largest component size and of the distribution of cluster sizes in particular we can derive that weak ties are crucial in order to maintain the graph connected and that when they are the most prone to failure the giant component typically shrinks without abruptly breaking apart these results have been recently evidenced in several kinds of social networks

analytic solution of attractor neural networks on scale free graphs we study the influence of network topology on retrieval properties of recurrent neural networks using replica techniques for dilute systems the theory is presented for a network with an arbitrary degree distribution p k and applied to power law distributions p k k i e to neural networks on scale free graphs a bifurcation analysis identifies phase boundaries between the paramagnetic phase and either a retrieval phase or a spin glass phase using a population dynamics algorithm the retrieval overlap and spin glass order parameters may be calculated throughout the phase diagram it is shown that there is an enhancement of the retrieval properties compared with a poissonian random graph we compare our findings with simulations

boundary layer for the navier stokes alpha model of fluid turbulence we study boundary layer turbulence using the navier stokes alpha model obtaining an extension of the prandtl equations for the averaged flow in a turbulent boundary layer in the case of a zero pressure gradient flow along a flat plate we derive a nonlinear fifth order ordinary differential equation which is an extension of the blasius equation we study it analytically and prove the existence of a two parameter family of solutions satisfying physical boundary conditions matching these parameters with the skin friction coefficient and the reynolds number based on momentum thickness we get an agreement of the solutions with experimental data in the laminar and transitional boundary layers as well as in the turbulent boundary layer for moderately large reynolds numbers

finite connectivity attractor neural networks we study a family of diluted attractor neural networks with a finite average number of symmetric connections per neuron as in finite connectivity spin glasses their equilibrium properties are described by order parameter functions for which we derive an integral equation in replica symmetric approximation a bifurcation analysis of this equation reveals the locations of the paramagnetic to recall and paramagnetic to spin glass transition lines in the phase diagram the line separating the retrieval phase from the spin glass phase is calculated at zero temperature all phase transitions are found to be continuous

singular dirichlet second order bvps with impulses abstract the paper deals with the existence of solutions to singular second order differential equations with impulse effects and with the dirichlet boundary conditions the right hand side of the differential equation can be singular in its phase variable

geometric statistics in turbulence the author presents results regarding certain average properties of incompressible fluids derived from the equations of motion the author estimates the average dissipation rate the average dimension of level sets the role played by the field of direction of vorticity in the three dimensional euler and navier stokes equations is discussed and a class of two dimensional equations that are useful models of incompressible dynamics is described the author presents results concerning scaling exponents in turbulence

liouville type theorems for the forced euler equations and the navier stokes equations in this paper we study the liouville type properties for solutions to the steady incompressible euler equations with forces in mathbb r n if we assume single signedness condition on the force then we can show that a c 1 mathbb r n solution v p with v 2 p in l frac q 2 mathbb r n q in frac 3n n 1 infty is trivial v 0 for the solution of the steady navier stokes equations satisfying v x to 0 as x to infty the condition int mathbb r 3 delta v frac 6 5 dx infty which is stronger than the important d condition int mathbb r 3 nabla v 2 dx infty but both having the same scaling property implies that v 0 in the appendix we reprove theorem 1 1 chae commun math phys 273 203 215 2007 using the self similar euler equations directly

on the euler equations of incompressible fluids euler equations of incompressible fluids use and enrich many branches of mathematics from integrable systems to geometric analysis they present important open physical and mathematical problems examples include the stable statistical behavior of ill posed free surface problems such as the rayleightaylor and kelvin helmholtz instabilities the paper describes some of the open problems related to the incompressible euler equations with emphasis on the blowup problem the inviscid limit and anomalous dissipation some of the recent results on the quasigeostrophic model are also mentioned

a spherical hopfield model a spherical hopfield type neural network is introduced involving neurons and patterns that are continuous variables both the thermodynamics and dynamics of this model are studied in order to have a retrieval phase a quartic term is added to the hamiltonian the thermodynamics of the model is exactly solvable and the results are replica symmetric a langevin dynamics leads to a closed set of equations for the order parameters and effective correlation and response function typical for neural networks the stationary limit corresponds to the thermodynamic results numerical calculations illustrate these findings

confinement of vorticity for the 2d euler α equations abstract in this article we consider weak solutions of the euler equations in the full plane we take as initial unfiltered vorticity an arbitrary nonnegative compactly supported bounded radon measure global well posedness for the corresponding initial value problem is due m oliver and s shkoller we show that for all time the support of the unfiltered vorticity is contained in a disk whose radius grows no faster than o t log t 1 4 this result is an adaptation of the corresponding result for the incompressible 2d euler equations with initial vorticity compactly supported nonnegative and p th power integrable p 2 due to d iftimie t sideris and p gamblin and independently to ph serfati

nonexistence of self similar singularities for the 3d incompressible euler equations we prove that there exists no self similar finite time blowing up solution to the 3d incompressible euler equations if the vorticity decays sufficiently fast near infinity in mathbb r 3 by a similar method we also show nonexistence of self similar blowing up solutions to the divergence free transport equation in mathbb r n this result has direct applications to the density dependent euler equations the boussinesq system and the quasi geostrophic equations for which we also show nonexistence of self similar blowing up solutions

nonexistence of self similar singularities in the viscous magnetohydrodynamics with zero resistivity abstract we are concerned on the possibility of finite time singularity in a partially viscous magnetohydrodynamic equations in r n n 2 3 namely the mhd with positive viscosity and zero resistivity in the special case of zero magnetic field the system reduces to the navier stokes equations in r n in this paper we exclude the scenario of finite time singularity in the form of self similarity under suitable integrability conditions on the velocity and the magnetic field we also prove the nonexistence of asymptotically self similar singularity this provides us information on the behavior of solutions near possible singularity of general type as described in corollary 1 1

global regularity for vortex patches we present a proof of chemin s 4 result which states that the boundary of a vortex patch remains smooth for all time if it is initially smooth

organization and evolution of synthetic idiotypic networks we introduce a class of weighted graphs whose properties are meant to mimic the topological features of idiotypic networks namely the interaction networks involving the b core of the immune system each node is endowed with a bit string representing the idiotypic specificity of the corresponding b cell and the proper distance between any couple of bit strings provides the coupling strength between the two nodes we show that a biased distribution of the entries in bit strings can yield fringes in the weighted degree distribution small world features and scaling laws in agreement with experimental findings we also investigate the role of aging thought of as a progressive increase in the degree of bias in bit strings and we show that it can possibly induce mild percolation phenomena which are investigated too

on a model for the navier stokes equations using magnetization variables abstract it is known that in a classical setting the navier stokes equations can be reformulated in terms of so called magnetization variables w that satisfy 1 t w p w w p w w w 0 and relate to the velocity u via a leray projection u p w we will prove the equivalence of these formulations in the setting of weak solutions that are also in l 0 t h 1 2 l 2 0 t h 3 2 on the 3 dimensional torus our main focus is the proof of global well posedness in h 1 2 for a new variant of 1 where p w is replaced by w in the second nonlinear term 2 t w p w w 1 2 w 2 w 0 this is based on a maximum principle analogous to a similar property of the burgers equations

quaternions and ideal flows after a review of some of the recent works by holm and gibbon on quaternions and their application to lagrangian flows particularly the incompressible euler equations and the equations of ideal mhd this paper investigates the compressible and relativistic euler equations using these methods

a stochastic perturbation of inviscid flows we prove existence and regularity of the stochastic flows used in the stochastic lagrangian formulation of the incompressible navier stokes equations with periodic boundary conditions and consequently obtain a ck local existence result for the navier stokes equations our estimates are independent of viscosity allowing us to consider the inviscid limit we show that as 0 solutions of the stochastic lagrangian formulation with periodic boundary conditions converge to solutions of the euler equations at the rate of o sqrt nu t

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

existence theory for single and multiple solutions to singular positone boundary value problems abstract in this paper we present some old and new existence results for singular boundary value problems our nonlinearity may be singular in its dependent variable

analogue neural networks on correlated random graphs we consider a generalization of the hopfield model where the entries of patterns are gaussian and diluted we focus on the high storage regime and we investigate analytically the topological properties of the emergent network as well as the thermodynamic properties of the model we find that by properly tuning the dilution in the pattern entries the network can recover different topological regimes characterized by peculiar scalings of the average coordination number with respect to the system size the structure is also shown to exhibit a large degree of cliquishness even when very sparse moreover we obtain explicitly the replica symmetric free energy and the self consistency equations for the overlaps order parameters of the theory which turn out to be classical weighted sums of sub overlaps defined on all possible sub graphs finally a study of criticality is performed through a small overlap expansion of the self consistencies and through a whole fluctuation theory developed for their rescaled correlations both approaches show that the net effect of dilution in pattern entries is to rescale the critical noise level at which ergodicity breaks down

geometric gradient flow dynamics with singular solutions the gradient flow dynamics of an arbitrary geometric quantity is derived using a generalization of darcy s law we consider flows in both lagrangian and eulerian formulations the lagrangian formulation includes a dissipative modification of fluid mechanics eulerian equations for self organization of scalars 1 forms and 2 forms are shown to reduce to nonlocal characteristic equations we identify singular solutions of these equations corresponding to collapsed clumped states and discuss their evolution

nonlinear superlinear singular and nonsingular second order boundary value problems abstract many new existence theorems are presented for singular and nonsingular second order boundary value problems singularities at y 0 t 0 and t 1 will be discussed