on the hughes model for pedestrian flow the one dimensional case in this paper we investigate the mathematical theory of hughes model for the ow of pedestrians cf 17 consisting of a nonlinear conservation law for the density of pedestrians coupled with an eikonal equation for a potential modelling the common sense of the task we rst consider an approximation of the original model in which the eikonal equation is replaced by an elliptic approximation for such an approximated system we prove existence and uniqueness of entropy solutions in one space dimension in the sense of kru zkov 21 in which the boundary conditions are posed following the approach of bardos et al 7 we use bv estimates on the density and stability estimates on the potential in order to prove uniqueness furthermore we analyse the evolution of characteristics for the original hughes model in one space dimension and study the behaviour of simple solutions in order to reproduce interesting phenomena related to the formation of shocks and rarefaction waves the characteristic calculus is supported by numerical simulations

a k theoretical invariant and bifurcation for a parameterized family of functionals abstract let f f x x x be a family of functionals defined on a hilbert manifold e and smoothly parameterized by a compact connected orientable n dimensional manifold x and let x e be a smooth section of critical points of f the aim of this paper is to give a sufficient topological condition on the parameter space x which detects bifurcation of critical points for f from the trivial branch finally we are able to give some quantitative properties of the bifurcation set for perturbed geodesics on semi riemannian manifolds

boundary behavior of diffusion approximations to markov jump processes we show that diffusion approximations including modified diffusion approximations can be problematic since the proper choice of local boundary conditions if any exist is not obvious for a class of markov processes in one dimension we show that to leading order it is proper to use a diffusion fokker planck approximation to compute mean exit times with a simple absorbing boundary condition however this is only true for the leading term in the asymptotic expansion of the mean exit time higher order correction terms do not in general satisfy simple absorbing boundary conditions in addition the diffusion approximation for the calculation of mean exit times is shown to break down as the initial point approaches the boundary and leads to an increasing relative error by introducing a boundary layer we show how to correct the diffusion approximation to obtain a uniform approximation of the mean exit time we illustrate these considerations with a number of examples including a jump process which leads to kramers diffusion model this example represents an extension to a multivariate process

discretization of nonlinear models by sinc collocation interpolation methods abstract this paper deals with the solution of initial boundary value problems for nonlinear evolution equations in one and two space dimensions the solution technique is based on collocation interpolation methods which use sinc functions the paper aims to provide a guideline towards a large number of nonlinear problems of interest in applied sciences

general population systems macroscopic limit of a class of stochastic semigroups relationships between three possible descriptions microscopic mesoscopic and macroscopic that are related to a large population composed of several subpopulations are formulated the systems describing the evolution of a large number of individuals undergoing stochastic interactions a description at microscopic level in terms of stochastic semigroups are considered the solutions of the general lotka volterra type equations without or with weak diffusion descriptions at macroscopic level are approximated by solutions of stochastic systems when the number of individuals n tends to the rate of approximation is controlled

population kinetic models for social dynamics dependence on structural parameters abstract a kinetic equation of the boltzmann kind is adopted to approach the analysis of a population of individuals subject to social interactions the state variable referred to a dominant social feature such as the individual wealth is defined on the whole real axis different kinds of interactions are allowed both on a stochastic and a deterministic basis structural parameters are varied and the related system sensitivity analysed

compactications defined by arrangements ii locally symmetric varieties of type iv we define a new class of completions of locally symmetric varieties of type iv which interpolates between the baily borel compactification and mumford s toric compactifications an arithmetic arrangement in a locally symmetric variety of type iv determines such a completion canonically this completion admits a natural contraction that leaves the complement of the arrangement untouched the resulting completion of the arrangement complement is very much like a baily borel compactification it is the proj of an algebra of meromorphic automorphic forms when that complement has a moduli space interpretation then what we get is often a compactification obtained by means of geometric invariant theory we illustrate this with several examples moduli spaces of polarized k3 and enriques surfaces and the semi universal deformation of a triangle singularity we also discuss the question when a type iv arrangement is definable by an automorphic form

orientability of fredholm families and topological degree for orientable nonlinear fredholm mappings abstract we construct a degree theory for oriented fredholm mappings of index zero between open subsets of banach spaces and between banach manifolds our approach is based on the orientation of fredholm mappings it does not use fredholm structures on the domain and target spaces we provide a computable formula for the change in degree through an admissible homotopy that is necessary for applications to global bifurcation the notion of orientation enables us to establish rather precise relationships between our degree and many other degree theories for particular classes of fredholm maps including the elworthy tromba degree which have appeared in the literature in a seemingly unrelated manner

algebraic shift equivalence and primitive matrices motivated by symbolic dynamics we study the problem given a unital subring s of the reals when is a matrix a algebraically shift equivalent over s to a primitive matrix we conjecture that simple necessary conditions on the nonzero spectrum of a are sufficient and establish the conjecture in many cases if s is the integers we give some lower bounds on sizes of realizing primitive matrices for dedekind domains we prove that algebraic shift equivalence implies algebraic strong shift equivalence

on the distribution of dominance in populations of social organisms bumble bees may possess a scalar character called dominance which changes according to certain rules as a result of encounters between pairs of organisms an equation for the distribution of dominance in a population is derived based on a set of plausible axioms the resulting boltzmann like integrodifferential equation is analyzed analytically and or numerically for certain important special cases

on the modelling of crowd dynamics by generalized kinetic models abstract we introduce a new approach for modelling the behavior of pedestrians which borrows techniques from the kinetic theory of rarefied gazes this approach is based on concepts from the kinetic theory for active particles in our modelling of crowd movements pedestrians are regarded as entities whose microscopic states include a geometric variable to denote position a mechanical variable to denote speed and a variable to describe the rate of development of individual strategies that influence the collective dynamics indeed the third variable is introduced to account for the ability of individuals to modify their dynamics according to specific objectives and strategies the equations eventually obtained are a set of nonlinear integro differential equations of vlasov or boltzmann type

harmonic wavelets towards the solution of nonlinear pde in this paper harmonic wavelets which are analytically defined and band limited are studied together with their differentiable properties harmonic wavelets were recently applied to the solution of evolution problems and more generally to describe evolution operators in order to consider the evolution of a solitary profile and to focus on the localization property of wavelets it seems to be more expedient to make use of functions with limited compact support either in space or in frequency the connection coefficients of harmonic wavelets are explicitly computed in the following at any order and characterized by some recursive formulas in particular they are functionally and finitely defined by a simple formula for any order of the basis derivatives

convergence of numerical schemes to a nonlinear kinetic model of population dynamics with nonlocal boundary conditions we examine existence and uniqueness of a global solution and some basic mathematical issues associated with the development of a numerical scheme for a model of a tumor immune system interaction the system consists of nonlinear transport equations with a bilinear operator like a boltzmann type and with a nonlocal boundary condition we construct a numerical scheme as a combination of a s t operator semigroup associated with the continuous problem and p delta projection operator on the discrete space the three estimates 1 l infty bound 2 a uniform total variation bound and 3 l 1 continuity in time of the approximate solution are established with a self contained treatment of the stability and convergence properties numerical calculations are reported

partially reflected diffusion the radiation reactive or robin boundary condition for the diffusion equation is widely used in chemical and biological applications to express reactive boundaries the underlying trajectories of the diffusing particles are believed to be partially absorbed and partially reflected at the reactive boundary however the relation between the reaction constant in the robin boundary condition and the reflection probability is not well defined in this paper we define the partially reflected process as a limit of the markovian jump process generated by the euler scheme for the underlying ito dynamics with partial boundary reflection trajectories that cross the boundary are terminated with probability p sqrt delta t and otherwise are reflected in a normal or oblique direction we use boundary layer analysis of the corresponding master equation to resolve the nonuniform convergence of the probability density function of the numerical scheme to the solution of the fokker planck equation in a half space wi

an asymptotic theory of large deviations for markov jump processes we present new asymptotic methods for the analysis of markov jump processes the methods based on the wkb and other singular perturbation techniques are applied directly to the kolmogorov equations and not to approximate equations that come e g from diffusion approximations for time homogeneous processes we construct approximations to the stationary density function and the mean first passage time from a given domain examples involving a random walk and a problem in queueing theory are presented to illustrate our methods for a class of time inhomogeneous processes we construct long time approximations to the transition probability density function and the probability of large deviations from a stable state the law of large numbers is obtained as a special case

on the numerical solutions of second order macroscopic models of pedestrian flows the main target of this paper is focused on the numerical simulation of macroscopic models two dimensional hyperbolic conservation law of pedestrian flows therefore finite volume methods can be used to discretize the equations actually the algorithms that have been used are particularly suited for solving hyperbolic problems moreover simulations using first order accurate numerical solvers and first godunov type schemes s k godunov a finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics mathematik sbornik 47 1959 271 290 have been developed this article is motivated by recent research activity focused on the problem of modelling systems of the living matter

global several parameter bifurcation and continuation theorems a unified approach via complementing maps we use the concept of complementing map in order to present a unified approach to the continuation principle global implicit function theorem and global bifurcation results for compact vector fields depending on several parameters

a stochastic system of particles modelling the euler equations we consider a system of n spheres interacting through elastic collisions at a stochastic distance in the limit n for a suitable rescaling of the interaction parameters we prove that the one particle distribution function converges to a local maxwellian whose gross density velocity and temperature satisfy the euler equation

recovery of the absorption coefficient from diffused reflected light using a discrete diffusive model an algorithm based on a discrete model for photon migration is proposed for recovering the absorption coefficient in a semi infinite slab a pulsed light source is modeled and only reflected signals are used in the reconstruction the diffusion coefficient and speed of light are assumed constant and known the algorithm is based on the result that the earliest reflected signals obtained from a particular voxel depend either in a linear or in a quadratical fashion on the square of the absorption coefficient in that voxel by solving the discrete model in the forward direction in a region where the absorption coefficient has been calculated already the absorption coefficients in the voxels on the boundary of the region are calculated using interpolation results show that simple absorptive regions can be accurately recovered

asymptotic analysis of a state dependent m g 1 queueing system we present new asymptotic methods for the analysis of queueing systems these methods are applied to a state dependent m g 1 queue we formulate problems for and compute approximations to i the stationary density of the unfinished work ii the mean length of time until the end of a busy period iii the mean length of a busy period and iv the mean time until the unfinished work reaches or exceeds a specified capacity the methods are applied to the full kolmogorov equations scaled so that the arrival rate is rapid and the mean service is small thus we do not truncate equations as in diffusion approximations for state independent m g 1 queues our results are shown to agree with the known exact solutions we include comparisons both analytic and numerical between our results and those obtained from diffusion approximations

strong shift equivalence and shear adjacency of nonnegative square integer matrices abstract for two square matrices a b of possibly different sizes with nonnegative integer entries write a 1 b if a rs and b sr for some two nonnegative integer matrices r s the transitive closure of this relation is called strong shift equivalence and is important in symbolic dynamics where it has been shown by r f williams to characterize the isomorphism of two topological markov chains with transition matrices a and b one invariant is the characteristic polynomial up to factors of however no procedure for deciding strong shift equivalence is known even for 2 2 matrices a b in fact for n n matrices with n 2 no nontrivial sufficient condition is known this paper presents such a sufficient condition that a and b are in the same component of a directed graph whose vertices are all n n nonnegative integer matrices sharing a fixed characteristic polynomial and whose edges correspond to certain elementary similarities for n 2 this result gives confirmation of strong shift equivalences that previously could not be verified for n 2 previous results are strengthened and the structure of the directed graph is determined

nonorientability of the index bundle and several parameter bifurcation the universal parameter space for sturm liouville boundary value problems is a two torus we show that the index bundle of the universal family of linear s l boundary value problems is non orientable using this we find a one dimensional set of bifurcation points for nonlinear bvp of sturm liouville type

solution of nonlinear initial boundary value problems by sinc collocation interpolation methods abstract this paper deals with the solution of initial boundary value problems for nonlinear evolution equations in one and two space dimensions the solution technique is based on collocation interpolation methods which use sinc functions the application refers to some direct and inverse type problems

entropy at a weight per symbol and embeddings of markov chains in 11 krieger gave a remarkable embedding result for shifts of finite type he proved that two simple necessary conditions strict inequality of the topological entropies of the shifts and the existence of an embedding for their periodic points are in fact sufficient for markov chains when the appropriate definitions of embedding and subsystem were supplied in 10 analogous necessary conditions strict inequality of functions and the existence of a weight preserving embedding for periodic points became apparent however an example in 10 also revealed these conditions to not be sufficient in this paper we bring to bear on embeddings of markov chains the ideas we introduced in 16 in fact we work in the closely related setting of triples z a a a f where 2 a aa is a shift of finite type and f is a function assigning weights to the edges of the graph associated with sa aa the func t ionf takes values in ir or more generally in n q for some qen the weight of a periodic orbit which corresponds to a cycle in the graph is then the sum of the weights of the edges lying on the cycle division by the length of the cycle gives the weight per symbol in 16 we continued the rational linear span of the set wps i of weights per symbol is a finite dimensional vector space vs the convex hull of wps i in v i is a polytope wpsi called the weight per symbol polytope consideration of the faces of wps i gives subsystems of r a aa f which are ordered according to the ordering of the faces the resulting nesting of subsystems is the scaffold of a o a f an embedding of a triple i a a a f in another 2 b ab g is a continuous shift commuting injection q 2 a 12 which preserves weights of periodic orbits the two necessary conditions mentioned above imply w p s i c w when the boundary c wpsi meets wpsg that is when a face f of wps i meets a face g of wpsg any embedding must carry the subsystem corresponding to f into that corresponding to g and this gives rise to further necessary conditons a discussion of these complications and a description of the example of 10 in these terms will be given in sect 7 before that the main embedding

compact moduli of plane curves we construct a compactication md of the moduli space of plane curves of degreed we regard a plane curvec p 2 as a surface divisor pair p 2 c and dene md as a moduli space of pairs x d where x is a degeneration of the plane we show that if d is not divisible by 3 the stackmd is smooth and the degenerate surfaces x can be described explicitly msc2000 14h10 14j10 14e30

on multisymplecticity of sinc gauss legendre collocation discretizations for some hamiltonian pdes abstract in this paper a new multisymplectic scheme based on sinc collocation discretization in space and gauss legendre collocation discretization in time for some hamiltonian pdes is developed the scheme preserves the multisymplectic geometry structure exactly by satisfying the discrete multisymplectic conservation law moreover the discrete energy and momentum conservative properties of the multisymplectic integrator are also discussed in order to testify the superiority of the multisymplectic method it is applied to nonlinear dirac equation numerical experiments are given to illustrate the accuracy of the approximation and the conserved quantities

bifurcation of heteroclinic orbits via an index theory heteroclinic orbits for one parameter families of nonautonomous vectorfields appear in a very natural way in many physical applications inspired by a recent bifurcation result for homoclinic trajectories of nonautonomous vectorfield proved by author in 13 we define a new mathbf z 2 index and we construct a index theory for heteroclinic orbits of nonautonomous vectorfield we prove an index theorem by showing that under some standard transversality assumptions the mathbf z 2 index is equal to the parity a homotopy invariant for paths of fredholm operators of index 0 as a direct consequence of the index theory developed in this paper we get a new bifurcation result for heteroclinic orbits

on the existence and nonexistence of finitary codings for a class of random fields we study the existence of finitary codings also called finitary homomorphisms or finitary factor maps from a finite valued i i d process to certain random fields for markov random fields we show using ideas of marton and shields that the presence of a phase transition is an obstruction for the existence of the above coding this yields a large class of bernoulli shifts for which no such coding exists conversely we show that for the stationary distribution of a monotone exponentially ergodic probabilistic cellular automaton such a coding does exist the construction of the coding is partially inspired by the propp wilson algorithm for exact simulation in particular combining our results with a theorem of martinelli and olivieri we obtain the fact that for the plus state for the ferromagnetic ising model on mathbf z d d geq 2 there is such a coding when the interaction parameter is below its critical value and there is no such coding when the interaction parameter is above its critical value

williams s conjecture is false for reducible subshifts we show that for two subshifts of finite type having exactly two irreducible components strong shift equivalence is not the same as shift equiv alence this refutes the williams conjecture w in the reducible case the irreducible case remains an open problem mathematics research group alabama state university montgomery alabama 36101 this content downloaded from 207 46 13 180 on thu 08 sep 2016 04 22 09 utc all use subject to http about jstor org terms

domain decomposition and discretization of continuous mathematical models abstract this paper deals with the development of decomposition of domains methods related to the discretization by collocation interpolation methods of continuous models described by nonlinear partial differential equations the objective of this paper is to show how generalized collocation and domain decomposition methods can be applied for problems where different models are used within the same domain

completions branched covers artin groups and singularity theory we study the curvature of metric spaces and branched covers of riemannian manifolds with applications in topology and algebraic geometry here curvature bounds are expressed in terms of the cat inequality we prove a general cat extension theorem giving sufficient conditions on and near the boundary of a locally cat metric space for the completion to be cat we use this to prove that a branched cover of a complete riemannian manifold is locally cat if and only if all tangent spaces are cat 0 and the base has sectional curvature bounded above by we also show that the branched cover is a geodesic space using our curvature bound and a local asphericity assumption we give a sufficient condition for the branched cover to be globally cat and the complement of the branch locus to be contractible we conjecture that the universal branched cover of cn over the mirrors of a finite coxeter group is cat 0 conditionally on this conjecture we use our machinery to prove the arnoldpham thom conjecture on k 1 spaces for artin groups also conditionally we prove the asphericity of moduli spaces of amply lattice polarized k3 surfaces and of the discriminant complements of all the unimodal hypersurface singularities in arnold s hierar

the moduli of curves of genus six and k3 surfaces we prove that the coarse moduli space of curves of genus six is birational to an arithmetic quotient of a bounded symmetric domain of type iv by giving a period map to the moduli space of some lattice polarized k3 surfaces

on the moduli space of pairs consisting of a cubic threefold and a hyperplane abstract we study the moduli space of pairs x h consisting of a cubic threefold x and a hyperplane h in p 4 the interest in this moduli comes from two sources the study of certain weighted hypersurfaces whose middle cohomology admit hodge structures of k3 type and on the other hand the study of the singularity o 16 the cone over a cubic surface in this paper we give a hodge theoretic construction of the moduli space of cubic pairs by relating x h to certain lattice polarized cubic fourfolds y a period map for the pairs x h is then defined using the periods of the cubic fourfolds y the main result is that the period map induces an isomorphism between a git model for the pairs x h and the baily borel compactification of some locally symmetric domain of type iv

monotone difference schemes for equations with mixed derivatives abstract there are considered elliptic and parabolic equations of arbitrary dimension with alternating coefficients at mixed derivatives for such equations monotone difference schemes of the second order of local approximation are constructed schemes suggested satisfy the principle of maximum a priori estimates of stability in the norm c without limitation on the grid steps and h 1 2 p are obtained unconditional stability

existence and uniqueness for a kinetic model of population dynamics with character dependence and spatial structure this work we consider a one species population dynamics model with character dependence spatial structure and a nonlocal renewal process arising as a boundary condition the individual interaction are based on boltzmann kinetic type modeling using fixed point arguments and the div rot lemma we prove that our model admits a unique global nonnegative solution

proof of dynamical scaling in smoluchowski s coagulation equation with constant kernel smoluchowski s coagulation equation for irreversible aggregation with constant kernel is considered in its discrete version open image in new window wherect c1 t is the concentration ofl particle clusters at timet we prove that for initial data satisfyingc1 0 0 and the condition 0 cl 0 0 the solutions behave asymptotically likec1 t t 2 c lt 1 ast withlt 1 kept fixed the scaling function c is 1 gr where rho sum l lc l 0 a conserved quantity is the initial number of particles per unit volume an analous result is obtained for the continuous version of smoluchowski s coagulation equation frac partial partial t c v text t int 0 v du text c v u text t text c u text t 2c v text t int 0 infty du text c u text t wherec v t is the oncentration of clusters of sizev

solution of nonlinear evolution problems by parallelized collocation interpolation methods abstract this paper deals with the solution of initial boundary value problems for nonlinear evolution equations the solution technique is based on collocation interpolation methods which are improved in order to reduce computational errors at fixed discretizations of the independent variables the method consists in parallelizing the approximation of the space derivatives so that the same approximation is reached by a lower number of collocation points and hence by a lower computation time the analysis includes a theoretical and computational estimate both of the approximation error and of the computational time

first order dynamics driven by rapid markovian jumps the effect of rapid random jumps on the behavior of deterministically stable dynamics is considered the random driving force noise is taken to be a state dependent markov jump process with either finite or infinite state space while the jump rate is high o 1 varepsilon the jump size is assumed to be o 1 for a particle that is deterministically confined to a potential well the stationary joint probability density function of the state and noise variables the probability distribution of the exit points and the mean first passage time from the well are computed in contrast to the case of white noise where these quantities depend on a barrier height determined solely by the potential here they depend on an effective barrier height determined by both the potential and the noise process the method used here is to introduce the small parameter varepsilon where 1 varepsilon is a measure of the rapid jump rate and to employ singular perturbation methods to solve the forward and

on the cauchy problem for macroscopic model of pedestrian flows abstract in this paper we discuss the limit behavior of hyperbolic systems of conservation laws with stiff relaxation terms to the local systems as the relaxation time tends to zero the prototype is crowd models derived from crowd dynamics according to macroscopic scaling when the flow of crowds is supposed to satisfy the paradigms of continuum mechanics under an appropriate structural stability condition the asymptotic expansion is obtained when one assumes the existence of a smooth solution to the equilibrium system in this case the local existence of a classical solution is also shown

degree of locally condensing perturbations of fredholm maps with positive index and applications abstract we give a construction of the topological degree for locally condensing perturbations of c 1 fredholm maps of positive index with values in the g l c framed cobordism group the construction of the degree extends constructions of the degree for fredholm maps of positive index and also its finite dimensional compact and compactly restricted perturbations it based on the finite dimensional reduction method we apply the constructed degree to prove the existence theorem for a system of ordinary differential equations with hopf s boundary conditions

numerical methods based on whittaker cardinal or sinc functions this paper summarizes the results known to date for using sine functions composed with other functions as bases for approximations in numerical analysis described in this paper are methods of interpolation and approximation of functions and their derivatives quadrature the approximate evaluation of transforms hilbert fourier laplace hankel and mellin and the approximate solution of differential and integral equations the methods have many advantages over classical methods which use polynomials as bases in addition all of the methods converge at an optimal rate if singularities on the boundary of approximation are ignored

on the derivation of reaction diffusion equations as limit dynamics of systems of moderately interacting stochastic processes we consider systems of moderately interacting particles which are divided into a finite number of different subpopulations and show that in the limit as the population size tends to infinity the empirical processes of the subpopulations converge to the solution of a system of reaction diffusion equations

the automorphism group of a shift of finite type let xt at be a shift of finite type and g aut vt denote the group of homeomorphisms of xt commuting with t we investigate the algebraic properties of the countable group g and the dynamics of its action on xt and associated spaces using marker constructions we show g contains many groups such as the free group on two generators however g is residually finite so does not contain divisible groups or the infinite symmetric group the doubly exponential growth rate of the number of automorphisms depending on n coordinates leads to a new and nontrivial topological invariant of crt whose exact value is not known we prove that modulo a few points of low period g acts transitively on the set of points with least at period n using padic analysis we generalize to most finite type shifts a result of boyle and krieger that the gyration function of a full shift has infinite order the action of g on the dimension group of at is investigated we show there are no proper infinite compact g invariant sets we give a complete characterization of the g orbit closure of a continuous probability measure and deduce that the only continuous g invariant measure is that of maximal entropy examples questions and problems complement our analysis and we conclude with a brief survey of some remaining open problems

rice s theorem for the limit sets of cellular automata abstract rice s theorem is a well known result in the theory of recursive functions a corresponding theorem for cellular automata limit sets is proved all nontrivial properties of limit sets of cellular automata cas are shown undecidable the theorem remains valid even if only one dimensional cas are considered

first passage time problems for a class of master equations with separable kernels abstract we discuss first passage time problems for a class of one dimensional master equations with separable kernels for this class of master equations the integral equation for first passage time moments can be transformed exactly into ordinary differential equations when the separable kernel has only a single term the equation for the mean first passage time obtained is exactly that for simple diffusion the boundary conditions however differ from those appropriate to simple diffusion the equations for higher moments differ slightly from those for simple diffusion analysis is presented of a generalization of a model of a random walk with long range jumps first investigated by lindenberg and shuler since the equations can be solved exactly one can study the behavior of boundary conditions in the continuum limit the generalization to a larger number of terms in the separable kernel leads to higher order equations for the first passage time moments in each case boundary conditions can be found directly from the original master equation

deformations of singularities and variation of git quotients we study the deformations of the minimally elliptic surface sin gularity n16 a standard argument reduces the study of the deformations of n16 to the study of the moduli space of pairs c l consisting of a plane quintic curve and a line we construct this moduli space in two ways via the periods of k3 surfaces and by using geometric invariant theory git the git construction depends on the choice of the linearization in particular for one choice of linearization we recover the space constructed via k3 surfaces and for another we obtain the full deformation space of n16 the two spaces are related by a series of explicit flips in conclusion by using the flexibil ity given by git and the standard tools of hodge theory we obtain a good understanding of the deformations of n16

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