existence and bifurcation of solutions for an elliptic degenerate problem abstract we investigate the existence multiplicity and bifurcation of solutions of a model nonlinear degenerate elliptic differential equation x 2 u u u p 1 u in 0 1 u 0 u 1 0 this model is related to a simplified version of the nonlinear wheeler dewitt equation as it appears in quantum cosmological models we prove the existence of multiple positive solutions more precisely we show that there exists an infinite number of connected branches of solutions which bifurcate from the bottom of the essential spectrum of the corresponding linear operator nous etudions ici l existence multiplicite et proprietes de bifurcation des solutions d un probleme elliptique degenere x 2 u u u p 1 u in 0 1 u 0 u 1 0 ce probleme modele est proche d une version simplifiee et non lineaire de l equation de wheeler dewitt utilisee dans des modeles de cosmologie quantique nous prouvons l existence d une infinite de branches de solutions qui bifurquent a partir de l infimum du spectre continu de l operateur lineaire correspondant

critical storage capacity of the j 1 neural network for neural networks in which the coupling jij are allowed to take on the values jij 1 or jij 1 the authors determine numerically the critical storage capacity for random unbiased patterns as a function of the stability they use an exact enumeration scheme based on the gray code and a continuous distribution for the patterns to control finite size effects results are presented for n 25 they indicate an optimal storage capacity of c 0 82 n

sufficient conditions for strong local minima the case of c 1 extremals in this paper we settle a conjecture of ball that uniform quasiconvexity and uniform positivity of the second variation are sufficient for a c 1 extremal to be a strong local minimizer our result holds for a class of variational functionals with a power law behavior at infinity the proof is based on the decomposition of an arbitrary variation of the dependent variable into its purely strong and weak parts we show that these two parts act independently on the functional the action of the weak part can be described in terms of the second variation whose uniform positivity prevents the weak part from decreasing the functional the strong part localizes i e its action can be represented as a superposition of weierstrass needles which cannot decrease the functional either due to the uniform quasiconvexity conditions

asymptotic analysis of a random walk with a history dependent step length we study an unbiased discrete time random walk on the nonnegative integers with the origin absorbing and a history dependent step length letting y denote the maximum distance the walker has ever been from the origin steps that do not change y have length v while those that increase y taking the walker to a site that has never been visited have length n the process serves as a simplified model of spreading in systems with an infinite number of absorbing configurations asymptotic analysis of the probability generating function shows that for large t the survival probability decays as s t t with v 2n our expression for the decay exponent is in agreement with results obtained via numerical iteration of the transition matrix

error estimation and adaptive meshing in strongly nonlinear dynamic problems we present work aimed at developing a general framework for mesh adaption in strongly nonlinear possibly dynamic problems we begin by showing that the solutions of the incremental boundary value problem for a wide class of materials including nonlinear elastic materials compressible newtonian fluids and viscoplastic solids obey a minimum principle provided that the constitutive updates are formulated appropriately this minimum principle can be taken as a basis for asymptotic error estimation in particular we chose to monitor the error of a lower order projection of the finite element solution the optimal mesh size distribution then follows from a posteriori error indicators which are purely local i e can be computed element by element we demonstrate the robustness and versatility of the computational framework with the aid of convergence studies and selected examples of application

partial regularity of strong local minimizers in the multi dimensional calculus of variations let n be a bounded domain and f a given strongly quasiconvex integrand of class c2 satisfying the growth condition f xi le c 1 xi p for some c 0 and 2 p consider the multiple integral i u int omega f nabla u where u w1 p n the main result of the paper is the proof that any strong local minimizer of i is of class c1 loc for any 0 1 on an open set of full n dimensional measure in the case of weak local minimizers we establish the same result under the extra assumption that the oscillations in the gradient of the minimizer are not too large without such an assumption weak local minimizers need not be partially regular as we show by a class of examples we also briefly discuss the question of existence of strong local minimizers for i and connections of our results to extensions of weierstrass sufficiency theorem to the multi dimensional setting

bifurcation points of a singular boundary value problem on 0 1 abstract following earlier work on some special cases 11 17 and on the analogous problem in higher dimensions 10 20 we make a more thorough investigation of the bifurcation points for a nonlinear boundary value problem of the form a x u x f x u x u x for 0 x 1 0 1 a x u x 2 d x and u 1 0 where for all r and x 0 1 f x 0 0 0 and 3 f x 0 0 v x and 4 f x 0 0 0 so that the formal linearization about a trivial solution u 0 is a x v x v x v x v x for 0 x 1 even when f is a smooth function of all its variables standard bifurcation theory does not apply to the problem and the results differ from the usual conclusions this is because we deal with the case where the coefficient a has a critical degeneracy as x 0 in the sense that a c 0 1 with a x 0 for x 0 1 and lim x 0 a x x 2 a 0 it was observed in 11 17 that if the exponent 2 is replaced by a value less than 2 then classical bifurcation theory can be used to treat the problem the paper 17 deals with the case f x s t sin s whereas 11 covers the more general form f x s t f s here we admit a much broader class of nonlinearities and some new phenomena appear in particular we encounter situations where bifurcation does not occur at a simple eigenvalue of the linearization lying below the essential spectrum

unilateral contact of two solids subject to large deformations formulation and existence results the approach that is currently the most widely used for formulating the unilateral frictionless contact between two deformable solids involves two key elements the orthogonal projection of one potential contact material surface on another and the action reaction principle of contact forces in the present paper we investigate this approach in the general case of large deformations and study unilateral frictionless boundary value problems of elasticity formulated by means of it the conditions for the orthogonal projection to be unique or further a c 1 diffeomorphism are established an extended action reaction principle is explicitly given for the forces acting on a pair of potential contact surfaces assuming the solids under consideration can be described by polyconvex hyperelastic laws we formulate the associated unilateral frictionless boundary value problem as a minimum principle and show the existence of a solution to it

storage capacity of neural networks effect of the fluctuations of the number of active neurons per memory the storage capacity in an attractor neural network with excitatory couplings is shown to depend not only on the fraction of active neurons per pattern or coding rate but also on the fluctuations around this value in the thermodynamical limit the capacity is calculated in the case of exactly the same number of active neurons in every pattern for every coding level the capacity is increased with respect to the case of random patterns results are supported by numerical simulations done with an exhaustive search algorithm and partly solve in the sparse coding limit the paradox of the discrepancy of the capacity of the willshaw model with optimal capacity

w1 p quasiconvexity and variational problems for multiple integrals abstract variational problems for the multiple integral i u g u x dx where r m and u r n are studied a new condition on g called w1 p quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of c b morrey it being shown in particular to be necessary for sequential weak lower semicontinuity of i in w 1 p r n and for the existence of minimizers for certain related integrals counterexamples are given concerning the weak continuity properties of jacobians in w 1 p r n p n m an existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses

hamiltonian structures and stability for rigid bodies with flexible attachments the dynamics of a rigid body with flexible attachments is studied a general framework for problems of this type is established in the context of poisson manifolds and reduction a simple model for a rigid body with an attached linear extensible shear beam is worked out for illustration second the energy casimir method for proving nonlinear stability is recalled and specific stability criteria for our model example are worked out the poisson structure and stability results take into account vibrations of the string rotations of the rigid body their coupling at the point of attachment and centrifugal and coriolis forces

stability of poisson equilibria and hamiltonian relative equilibria by energy methods we develop a general stability theory for equilibrium points of poisson dynamical systems and relative equilibria of hamiltonian systems with symmetries including several generalisations of the energy casimir and energy momentum methods using a topological generalisation of lyapunov s result that an extremal critical point of a conserved quantity is stable we show that a poisson equilibrium is stable if it is an isolated point in the intersection of a level set of a conserved function with a subset of the phase space that is related to the topology of the symplectic leaf space at that point this criterion is applied to generalise the energy momentum method to hamiltonian systems which are invariant under non compact symmetry groups for which the coadjoint orbit space is not hausdorff we also show that a g stable relative equilibrium satisfies the stronger condition of being a stable where a is a specific group theoretically defined subset of g which contains the momentum isotropy subgroup of the relative equilibrium the results are illustrated by an application to the stability of a rigid body in an ideal irrotational fluid

nonlinear stability of circular vortex patches this paper proves that circular vortex patches in the plane are stable for the nonlinear dynamical system generated by the euler equations of incompressible fluids this is achieved by establishing a relative variational principle in terms of either energy or angular momentum thus we exploit and extend arnold s idea in 1965 1969 to a nonsmooth setting as well

local minimizers and the schmid law in corner shaped domains this paper focuses on the mathematical analysis of biaxial loading experiments in martensite more particularly on how hysteresis relates to metastability these experiments were carried out by chu and james and their mathematical treatment was initiated by ball chu and james experimentally it is observed that a homogeneous deformation y 1 is the stable state for small loads while y 2 is stable for large loads a model was proposed by ball chu and james which for a certain intermediate range of loads predicts crucially that y 1 remains metastable that is a local as opposed to global minimiser of the energy this result explains convincingly the hysteresis that is observed experimentally it is easy to get an upper bound on the load at which metastability finishes however it was also noticed that this bound the schmid law may not be sharp though this required some geometric conditions on the sample in this research we rigorously justify the ball chu james model by means of de giorgi s convergence establish some properties of local minimisers of the limiting energy and prove the metastability result mentioned above an important part of the paper is then devoted to establishing which geometric conditions are necessary and sufficient for the counter example to the schmid law to apply namely the presence of sharp corners in the sample

the discrete diffusive coagulation fragmentation equations with scattering we consider the discrete coagulation fragmentation equations with diffusion presupposing a maximal cluster size such a feature requires a new interaction mechanism in order to prevent occurrence of too large clusters being produced by coagulation existence of a unique solution for this model is proven and long time behaviour is studied in situations where equilibria are explicitly known

bifurcation at isolated singular points of the hadamard derivative for banach spaces x and y we consider bifurcation from the line of trivial solutions for the equation f lambda u 0 where f r x x y with f lambda 0 0 for all lambda is an element of r the focus is on the situation where f lambda center dot is only hadamard differentiable at 0 and lipschitz continuous on some open neighbourhood of 0 without requiring any frechet differentiability applications of the results obtained here to some problems involving nonlinear elliptic equations on r n where frechet differentiability is not available are presented in some related papers which shed light on the relevance of our hypotheses

quasiconvexity at the boundary positivity of the second variation and elastic stability for one dimensional problems of the calculus of variations there is a well developed theory relating the positivity of the second variation at a critical point i e linearized stability to strong relative minima these results largely due to weierstrass are usually proved by the method of fields of extremals as is described in bolza 1904 and morrey 1966 or by an argument involving a careful use of taylor series as in hestenes 1966 chapter 3 14

homotopy linearization and bifurcation on presente une prise en compte essentiellement auto contenue de la relation entre bifurcation locale et la determination a l aide de la linearisation de la non trivialite de certaines classes d homotopie d operateurs non lineaires

asymptotic linearity and hadamard differentiability for a mapping between banach spaces two weaker variants of the usual notion of asymptotic linearity are defined and explored it is shown that under inversion through the unit sphere they correspond to hadamard and weak hadamard differentiability at the origin of the inversion nemytskii operators from sobolev spaces to lebesgue spaces over rn share these weaker properties but they are not asymptotically linear in the usual sense c 2011 elsevier ltd all rights reserved

criteria for the existence of a potential well abstract we consider a critical point u 0 of a functional f c 1 h r where h is a real hilbert space and formulate criteria ensuring that u 0 lies in a potential well of f without supposing that f is frechet differentiable at u 0 the derivative is required to be g teaux differentiable at u 0 but positive definiteness of f u 0 does not even ensure that f has a local minimum at u 0 when f is not frechet differentiable at u 0 this issue is also discussed in the context of the energy functional for a parameter dependent nonlinear eigenvalue problem and then for a particular case involving a degenerate elliptic dirichlet problem on a bounded domain in r n

sufficiency theorems for local minimizers of the multiple integrals of the calculus of variations let omega subset of r n be a bounded domain and let f omega x r n x r nxn r consider the functional i u f omega f x u du dx over the class of sobolev functions w 1 w q omega r n 1 less than or equal to q less than or equal to infinity for which the integral on the right is well defined in this paper we establish sufficient conditions on a given function u 0 and f to ensure that u 0 provides an l r local minimizer for i where 1 less than or equal to r less than or equal to infinity the case r infinity is somewhat known and there is a considerable literature on the subject treating the case min n n 1 mostly based on the field theory of the calculus of variations the main contribution here is to present a set of sufficient conditions for the case 1 less than or equal to r less than or equal to infinity our proof is based on an indirect approach and is largely motivated by an argument of hestenes relying on the concept of directional convergence

nonlinear stability for saddle solutions of ideal flows and symmetry breaking in this paper we generalize arnol d s method for nonlinear stability of the two dimensional incompressible euler equation s stationary solutions the method is extended so as to apply for saddle solutions which include many cases of practical interest in particular the case of solutions which do not share the symmetry of the domain the extension is made possible by our technique of supporting functionals and the use of additional flow invariants

a new identity for the surface impedance matrix and its application to the determination of surface wave speeds it is well known in surface wave theory that the secular equation for the surface wave speed can be written as det m 0 in terms of the surface impedance matrix m it is shown in this paper that m satisfies the simple identity m i r t 1 m i r t q 2 i 0 in the usual notation in the stroh formalism this identity provides an efficient method for calculating the wave speed of surface waves in unstressed or pre stressed elastic half spaces the method is explained and illustrated by examples it is also shown that the buckling wrinkling pre stress for a pre stressed elastic half space can be calculated using the same procedure but with pre stress playing the role of

slow relaxation in weakly open rational polygons the interplay between the regular piecewise linear and irregular vertex angle boundary effects in nonintegrable rational polygonal billiards of m equal sides is discussed decay dynamics in polygons of perimeter p m and small opening delta is analyzed through the late time survival probability s m approximately equal t delta two distinct slow relaxation channels are established the primary universal channel exhibits relaxation of regular sliding orbits with delta 1 the secondary channel is given by delta 1 and becomes open when m p m delta it originates from vertex order disorder dual effects and is due to relaxation of chaoticlike excitations

bifurcation for some non frechet differentiable problems abstract we study the bifurcation points of an equation of the form f u u in a real hilbert space since f is only required to be hadamard but not frechet differentiable at u 0 bifurcation points need not belong to the spectrum of f 0 the abstract results are illustrated in the case of a nonlinear schrodinger equation

null lagrangians weak continuity and variational problems of arbitrary order abstract we consider the problem of minimizing integral functionals of the form i u f x k u x dx where r p u r and k u denotes the set of all partial derivatives of u with orders k the method is based on a characterization of null lagrangians l ku depending only on derivatives of order k applications to elasticity and other theories of mechanics are given

nonlinear stability of a rotating planar liquid drop incompressible inviscid free boundary fluid flows are hamiltonian systems i e the evolution of the flow is determined by a poisson bracket and a hamiltonian function the poisson structure for such flows generalizes a previous structure of zakharov for irrotational free boundary flow and a structure of arnold for flow of an incompressible fluid in a region with fixed boundary the poisson bracket is determined by reduction from canonical variables in the lagrangian material description

the space of interactions in neural network models the typical fraction of the space of interactions between each pair of n ising spins which solve the problem of storing a given set of p random patterns as n bit spin configurations is considered the volume is calculated explicitly as a function of the storage ratio alpha p n of the value kappa 0 of the product of the spin and the magnetic field at each site and of the magnetisation m here m may vary between 0 no correlation and 1 completely correlated the capacity increases with the correlation between patterns from alpha 2 for correlated patterns with kappa 0 and tends to infinity as m tends to 1 the calculations use a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is shown to be locally stable a local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist

three dimensional adaptive meshing by subdivision and edge collapse in finite deformation dynamic plasticity problems with application to adiabatic shear banding this paper is concerned with the development of a general framework for adaptive mesh refinement and coarsening in three dimensional finite deformation dynamic plasticity problems mesh adaption is driven by a posteriori global error bounds derived on the basis of a variational formulation of the incremental problem the particular mesh refinement strategy adopted is based on rivara s longest edge propagation path lepp bisection algorithm our strategy for mesh coarsening or unrefinement is based on the elimination of elements by edge collapse the convergence characteristics of the method in the presence of strong elastic singularities are tested numerically an application to the three dimensional simulation of adiabatic shear bands in dynamically loaded tantalum is also presented which demonstrates the robustness and versatility of the method copyright c 2001 john wiley sons ltd

elastic surface substrate interactions a theory for three dimensional finite deformations of elastic solids with conforming elastic films attached to their bounding surfaces is described the gurtin murdoch theory incorporating elastic resistance of the film to strain is generalized to account for the effects of intrinsic flexural resistance this modification yields a model that can be used to describe equilibrium deformations in the presence of compressive surface stress fields an associated variational theory is given and material symmetry considerations are discussed the theory is illustrated by examples

three unfinished works on the optimal storage capacity of networks the optimal storage properties of three different neural network models are studied for two of these models the architecture of the network is a perceptron with or j interactions whereas for the third model the output can be an arbitrary function of the inputs analytic bounds and numerical estimates of the optimal capacities and of the minimal fraction of errors are obtained for the first two models the third model can be solved exactly and the exact solution is compared to the bounds and to the results of numerical simulations used for the two other models

incompatible sets of gradients and metastability we give a mathematical analysis of a concept of metastability induced by incompatibility the physical setting is a single parent phase just about to undergo transformation to a product phase of lower energy density under certain conditions of incompatibility of the energy wells of this energy density we show that the parent phase is metastable in a strong sense namely it is a local minimizer of the free energy in an l1 neighbourhood of its deformation the reason behind this result is that due to the incompatibility of the energy wells a small nucleus of the product phase is necessarily accompanied by a stressed transition layer whose energetic cost exceeds the energy lowering capacity of the nucleus we define and characterize incompatible sets of matrices in terms of which the transition layer estimate at the heart of the proof of metastability is expressed finally we discuss connections with experiments and place this concept of metastability in the wider context of recent theoretical and experimental research on metastability and hysteresis

nonlinear dynamical theory of the elastica the dynamical behaviour of a slender rod is analyzed here in terms of a generalization of euler s elastica theory the model includes a linear stress strain relation but nonlinear geometric terms properties of the rod may vary along its length and various boundary conditions are considered a rotational inertia term that is neglected in many theories is retained and is essential to the analysis by use of the equivalence of an energy and a sobolev norm and by reformulation of the equations as a semilinear system global existence of solutions is proved for any smooth initial data equilibrium solutions that are stable in the static sense of minimizing the potential energy are then proved to be stable in the dynamic sense due to liapounov

existence of minimizers and microstructure in nonlinear elasticity in this paper the existence of a solution in the form of a minimizer or microstructure is established for the boundary value problems of nonlinear elasticity with certain nonconvex stored energy functions such as those of st venant kirchhoff type materials necessary and sufficient conditions for minimizing sequences of the potential energy to converge to a minimizer or to microstructure are given

capacity of neural networks with discrete synaptic couplings the authors study the optimal storage capacity of neural networks with discrete local constraints on the synaptic couplings jij models with such constraints include those with binary couplings jij or 1 or jij 0 1 quantised couplings with larger synaptic range e g jij or 1 l or 2 l or 1 and in the limit continuous couplings confined to the hypercube mod jij mod or 1 box confinement they find that the optimal storage capacity alpha kappa is best determined by the vanishing of a suitably defined entropy as calculated in the replica symmetric approximation they also extend their results to cases with biased memories and make contact with sparse coding models

a variational rod model with a singular nonlocal potential the classical theory of elastic rods does not account for the possibility that large deformations may involve distinct points along the rod occupying the same physical space we develop an elastic rod model with a pairwise repulsive potential such that if two non adjacent points along the rod are close in physical space there is an energy barrier which prevents contact while for points nearby along the rod the potential is describable classically this framework is developed to prove the existence of minimizers within each homotopy class where the idea of topological homotopy of a curve is generalized to elastic rods as framed curves finally the relevant first order optimality conditions are derived and used to investigate the regularity of minimizers

boundary regularity and sufficient conditions for strong local minimizers abstract in this paper we present a new proof of the sufficiency theorem for strong local minimizers concerning c 1 extremals at which the second variation is strictly positive the results are presented in the quasiconvex setting in accordance with the original statement by grabovsky and mengesha 31 the strategy that we follow relies on a decomposition theorem that allows to split a sequence of variations into its oscillating and its concentrating parts as well as on a sufficiency result according to which smooth extremals are spatially local minimizers furthermore we prove partial regularity up to the boundary for strong local minimizers in the non homogeneous case and a full regularity result for lipschitz extremals with gradient of vanishing mean oscillation as a consequence we also establish a sufficiency result for this class of extremals the regularity results are established via a blow up argument

nonlinear stability of intermediate baroclinic flow on a sloping bottom the governing equations describing the dynamics of mesoscale gravity currents or coupled density fronts and steadily travelling coherent cold eddies on a sloping bottom are shown to possess a non canonical hamiltonian structure we exploit the hamiltonian formalism to obtain a variational principle that describes arbitrary steady solutions in terms of a suitably constrained hamiltonian two arnol d like stability theorems are obtained which can establish the linear stability in the sense of liapunov of these steady solutions based on this analyses two a priori estimates are derived which bound the disturbance energy and the liapunov norm with respect to the initial disturbance potential enstrophy and energy in the limit of parallel shear flow solutions corresponding to a current flowing along isobaths the first formal stability theorem reduces to a previously established normal mode stability result based on the formal stability analysis convexity conditions are given for the constrained hamiltonian that can rigorously establish nonlinear stability in the sense of liapunov for the steady current solutions a variational principle is also presented which can describe steadily travelling isolated cold eddy solutions of the model the principle is based on constraining the hamiltonian with appropriately chosen casimir and momentum invariants it is shown that a suitably extended form of andrews theorem holds for our model equations therefore the stability of the steadily travelling isolated eddy solutions cannot be established using the energy casimir analysis developed here

associative memory on the puzzling sparse coding limit recent studies of the information capacity in a sparsely coded memory net has led to some contradictory results in the willshaw model where the couplings are binary 0 or 1 the maximal quantity of information that can be stored is 1n 2 approximately 0 69 bits per synapse on the other hand a calculation a la gardner 1988 for 0 1 couplings gives an upper bound for the maximal capacity of about 0 29 bits per synapse in this study the author considers two possible sources for this discrepancy the first one is that the criteria for defining the maximal capacity are different with or without a constraint of perfect errorless storage the second one is a difference in the choice of the probability distribution of the random patterns used to compute this capacity this analysis shows in particular that for the willshaw model the maximal information capacity is much larger when the number of active neurons is exactly the same in every stored pattern than when it is given only in average in addition he gives an argument showing that this result may be generic e g valid for any activity level and independent of the learning rule

quasiconvexity and partial regularity in the calculus of variations we prove partial regularity of minimizers of certain functionals in the calculus of variations under the principal assumption that the integrands be uniformly strictly quasiconvex this is of interest since quasiconvexity is known in many circumstances to be necessary and sufficient for the weak sequential lower semicontinuity of these functionals on appropriate sobolev spaces examples covered by the regularity theory include functionals with integrands which are convex in the determinants of various submatrices of the gradient matrix

scaling behavior for random walks with memory of the largest distance from the origin we study a one dimensional random walk with memory the behavior of the walker is modified with respect to the simple symmetric random walk only when he or she is at the maximum distance ever reached from his or her starting point home in this case having the choice to move farther or to move closer the walker decides with different probabilities if the probability of a forward step is higher then the probability of a backward step the walker is bold otherwise he or she is timorous we investigate the asymptotic properties of this bold timorous random walk showing that the scaling behavior varies continuously from subdiffusive timorous to superdiffusive bold the scaling exponents are fully determined with a new mathematical approach based on a decomposition of the dynamics in active journeys the walker is at the maximum distance and lazy journeys the walker is not at the maximum distance

bifurcation at isolated singular points for a degenerate elliptic eigenvalue problem we consider bifurcation from the line of trivial solutions for a nonlinear eigenvalue problem on a bounded open subset omega of r n with n 3 containing 0 the leading term is a degenerate elliptic operator of the form l u del a del u where a is an element of c omega over bar with a 0 on omega over bar 0 and lim x 0 a x vertical bar x vertical bar 2 is an element of 0 infinity solutions should satisfy u 0 on partial derivative omega and the energy associated with l should be finite integral omega a vertical bar del u vertical bar 2 dx infinity the nonlinear terms are of lower order depending only on u and del u under our hypotheses the associated nemytskii operators are not frechet differentiable at the trivial solution u 0 c 2014 elsevier ltd all rights reserved

the becker doring cluster equations basic properties and asymptotic behaviour of solutions existence and uniqueness results are established for solutions to the becker doring cluster equations the density is shown to be a conserved quantity under hypotheses applying to a model of a quenched binary alloy the asymptotic behaviour of solutions with rapidly decaying initial data is determined denoting the set of equilibrium solutions byc 0 s the principal result is that if the initial density 0 s then the solution converges strongly toc o while if 0 s the solution converges weak toc s in the latter case the excess density 0 s corresponds to the formation of larger and larger clusters i e condensation the main tools for studying the asymptotic behaviour are the use of a lyapunov function with desirable continuity properties obtained from a known lyapunov function by the addition of a special multiple of the density and a maximum principle for solutions

parity and generalized multiplicity assuming that x and y are banach spaces and a a b 2 x y is a path of linear fredholm operators with invertible endpoints in f pl we defined a homotopy invariant of a a a i g z2 the parity of a on i the parity plays a fundamental role in bifurcation problems and in degree theory for nonlinear fredholm type mappings here we prove a that generically the parity is a mod 2 count of the number of transversal intersec tions of a i with the set of singular operators b that if a0 is an isolated singular point of a then the local parity a a x0 lim o a x0 a0 remains invariant under lyapunov schmidt reduction and c that a a a0 1 i where mg x0 is any one of the various concepts of generalized multiplicity which have been defined in the context of linearized bifurcation data