on the maslov index in this paper we give four definitions of maslov index and show that they all satisfy the same system of axioms and hence are equivalent to each other moreover relationships of several symplectic and differential geometric analytic and topological invariants including triple maslov indices eta invariants spectral flow and signatures of quadratic forms to the maslov index are developed and formulae relating them are given the broad presentation is designed with a view to applications both in geometry and in analysis 1994 john wiley sons inc

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

lower and upper bounds for the allocation problem and other nonlinear optimization problems we demonstrate the impossibility of strongly polynomial algorithms for the allocation problem in the comparison model and in the algebraic tree computation model that follow from lower bound results consequently there are no strongly polynomial algorithms for nonlinear concave separable optimization over a totally unimodular constraint matrix this is in contrast to the case when the objective is linear we present scaling based algorithms that use a greedy algorithm as a subroutine the algorithms are polynomial for the allocation problem and its extensions and are also optimal for the sample allocation problem and the generalized upper bounds allocation problem in that the complexity meets the lower bound derived from the comparison model for other extensions of the allocation problem the scaling based algorithms presented here are the fastest known these algorithms are also polynomial time algorithms for solving with e accuracy the allocation problem and its extension in continuous variables

combinatorial interior point methods for generalized network flow problems we present combinatorial interior point methods for the generalized minimum cost flow and the generalized circulation problems based on wallacher and zimmermann s combinatorial interior point method for the minimum cost network flow problem the algorithms have features of both a combinatorial algorithm and an interior point method they work towards optimality by iteratively reducing the value of a potential function while maintaining interior point solutions at each iteration flow is augmented along a generalized circulation which is computed by solving a tvpi two variables per inequality system the algorithms run in time where m and n are respectively the number of arcs and nodes in the graph and l is the length of the input data

convex separable optimization is not much harder than linear optimization the polynomiality of nonlinear separable convex concave optimization problems on linear constraints with a matrix with small subdeterminants and the polynomiality of such integer problems provided the inteter linear version of such problems ins polynomial is proven this paper presents a general purpose algorithm for converting procedures that solves linear programming problems the conversion is polynomial for constraint matrices with polynomially bounded subdeterminants among the important corollaries of the algorithm is the extension of the polynomial solvability of integer linear programming problems with totally unimodular constraint matrix to integer separable convex programming an algorithm for finding a e accurate optimal continuous solution to the nonlinear problem that is polynomial in log 1 e and the input size and the largest subdeterminant of the constraint matrix is also presented these developments are based on proximity results between the continuous and integral optimal solutions for problems with any nonlinear separable convex objective function the practical feature of our algorithm is that is does not demand an explicit representation of the nonlinear function only a polynomial number of function evaluations on a prespecified grid

towards a strongly polynomial algorithm for strictly convex quadratic programs an extension of tardos algorithm in a recent paper tardos described a polynomial algorithm for solving linear programming problems in which the number of arithmetic steps depends only on the size of the numbers in the constraint matrix and is independent of the size of the numbers in the right hand side and the cost coefficients in this paper we extend tardos results and present a polynomial algorithm for solving strictly convex quadratic programming problems in which the number of arithmetic steps is independent of the size of the numbers in the right hand side and the linear cost coefficients

strongly polynomial algorithms for the high multiplicity scheduling problem a high multiplicity scheduling problem consists of many jobs which can be partitioned into relatively few groups where all the jobs within each group are identical polynomial and even strongly polynomial algorithms for the standard scheduling problem in which all jobs are assumed to be distinct become exponential for the corresponding high multiplicity problem in this paper we study various high multiplicity problems of scheduling unit time jobs on a single machine we provide strongly polynomial algorithms for constructing optimal schedules with respect to several measures of efficiency completion time lateness tardiness the number of tardy jobs and their weighted counterparts the algorithms require a number of operations that are polynomial in the number of groups rather than in the total number of jobs as a by product we identify a new family of nxn transportation problems which are solvable in o n log n time by a simple greedy algorithm

integer programs for logic constraint satisfaction abstract logic constraint satisfaction problems are in general np hard and a general deterministic polynomial time algorithm is not known since several logic constraint problems can be reduced in polynomial time to the satisfaction of a conjunctive normal form cnf sat this case is very important we present here a technique to transform a cnf sat problem in an integer optimization problem that can be solved by linear programming the size of the obtained integer program has a polynomial growth in comparison with the original problem size

a two resource allocation problem solvable in linear time the allocation problem discussed here is as follows the processing time of a task is a linear function t ax by of amounts x y of resources allocated to it where t a b are positive constants given the amounts x and y of available resources we wish to allocate them among n independent tasks so as to minimize the maximal completion time we solve this problem in linear time

a strongly polynomial algorithm to solve combinatorial linear programs khachiyan and recently karmarkar gave polynomial algorithms to solve the linear programming problem these algorithms have a small theoretical drawback namely the number of arithmetic steps depends on the size of the input numbers we present a polynomial linear programming algorithm whose number of arithmetic steps depends only on the size of the numbers in the constraint matrix but is independent of the size of the numbers in the right hand side and objective vectors in particular it gives a polynomial algorithm for the minimum cost flow and multicommodity flow problems in which the number of arithmetic steps is independent of the size of the costs and capacities the algorithm makes use of an existing polynomial linear programming algorithm the problem of whether any algorithm has a running time that is independent even of the size of the numbers in the constraint matrix remains open

testing the necklace condition for shortest tours and optimal factors in the plane a tour of a finite set p of points is a necklace tour if there are disks with the points in p as centers such that two disks intersect if and only if their centers are adjacent in it has been observed by sanders that a necklace tour is an optimal traveling salesman tour

high multiplicity in earliness tardiness scheduling when a production shop has a large number of identical parts the parts are often recorded by a part description and quantity this differs from the type of description used by standard scheduling problems which assume that all parts or jobs are unique in high multiplicity scheduling problems identical jobs are encoded in an efficient format similar to that of the production shop the input describes one of the jobs and the number of such identical jobs we consider single machine high multiplicity problems with earliness and tardiness weights we investigate three categories of weights unit common and job specific for the unit and common weights problems a polynomial time algorithm is developed the algorithm takes advantage of identical jobs and finds solutions faster than by standard methods we provide a new method for creating a lower bound for the standard encoding of the job specific weights problem which is np complete we disaggregate each job into identical sub jobs with unit processing times then using high multiplicity encoding for this disaggregated problem we create a lower bound on the optimal objective function value of the original problem in polynomial time heuristic solutions are generated using a randomized rounding technique on the lower bound solution these results are used in a branch and bound solution method analytical and computational results are presented our combination of disaggregation and high multiplicity encoding provides a new method for creating lower bounds on the objective functions of np complete problems

spectral flow and bifurcation of critical points of strongly indefinite functionals abstract our main results here are as follows let x be a family of 2 periodic hamiltonian vectorfields that depend smoothly on a real parameter in a b and has a known trivial branch s of 2 periodic solutions let p be the poincare map of the linearization of x at s if the conley zehnder index of the path p does not vanish then any neighborhood of the trivial branch of periodic solutions contains 2 periodic solutions not on the branch moreover if each solution s is constant and each linearization a of x at s is time independent then bifurcation of 2 periodic orbits from the branch of equilibria arises whenever i a b i a b where i a is the index of the linear hamiltonian system ju au

polynomial methods for separable convex optimization in unimodular linear spaces with applications we consider the problem of minimizing a separable convex objective function over the linear space given by a system mx 0 with m a totally unimodular matrix in particular this generalizes the usual minimum linear cost circulation and cocirculation problems in a network and the problems of determining the euclidean distance from a point to the perfect bipartite matching polytope and the feasible flows polyhedron we first show that the idea of minimum mean cycle canceling originally worked out for linear cost circulations by goldberg and tarjan j assoc comput mach 36 1989 pp 873 886 and extended to some other problems t r ervolina and s t mccormick discrete appl math 46 1993 pp 133 165 a frank and a v karzanov technical report rr 895 m laboratoire artemis imag universite joseph fourier grenoble france 1992 t ibaraki a v karzanov and h nagamochi private communication 1993 m hadjiat technical report groupe intelligence artificielle faculte des sciences de luminy marseille france 1994 can be generalized to give a combinatorial method with geometric convergence for our problem we also generalize the computationally more efficient cancel and tighten method we then consider objective functions that are piecewise linear pure and piecewise quadratic or piecewise mixed linear and quadratic and we show how both methods can be implemented to find exact solutions in polynomial time strongly polynomial in the piecewise linear case these implementations are then further specialized for finding circulations and cocirculations in a network we finish by showing how to extend our methods to find optimal integer solutions to linear spaces of larger fractionality and to the case when the objective functions are given by approximate oracles

towards a genuinely polynomial algorithm for linear programming a linear programming algorithm is called genuinely polynomial if it requires no more than p m n arithmetic operations to solve problems of order m times n where p is a polynomial it is not known whether such an algorithm exists we present a genuinely polynomial algorithm for the simpler problem of solving linear inequalities with at most two variables per inequality the number of operations required is o mn 3 log text m the technique used was developed in a previous paper where a novel binary search idea was introduced

algorithms for the continuous nonlinear resource allocation problem new implementations and numerical studies patriksson 2008 provided a then up to date survey on the continuous separable differentiable and convex resource allocation problem with a single resource constraint since the publication of that paper the interest in the problem has grown several new applications have arisen where the problem at hand constitutes a subproblem and several new algorithms have been developed for its efficient solution this paper therefore serves three purposes first it provides an up to date extension of the survey of the literature of the field complementing the survey in patriksson 2008 with more then 20 books and articles second it contributes improvements of some of these algorithms in particular with an improvement of the pegging that is variable fixing process in the relaxation algorithm and an improved means to evaluate subsolutions third it numerically evaluates several relaxation primal and breakpoint dual algorithms incorporating a variety of pegging strategies as well as a quasi newton method our conclusion is that our modification of the relaxation algorithm performs the best at least for problem sizes up to 30 million variables the practical time complexity for the breakpoint and relaxation algorithms is linear

a bifurcation result for semi riemannian trajectories of the lorentz force equation we obtain a bifurcation result for solutions of the lorentz equation in a semi riemannian manifold such solutions are critical points of a certain strongly indefinite functionals defined in terms of the semi riemannian metric and the electromagnetic field the flow of the jacobi equation along each solution preserves the so called electromagnetic symplectic form and the corresponding curve in the symplectic group determines an integer valued homology class called the maslov index of the solution we study electromagnetic conjugate instants with symplectic techniques and we prove at first an analogous of the semi riemannian morse index theorem see calculus of variations prentice hall englewood cliffs nj usa 1963 by using this result together with recent results on the bifurcation for critical points of strongly indefinite functionals see j funct anal 162 1 1999 52 we are able to prove that each non degenerate and non null electromagnetic conjugate instant along a given solution of the semi riemannian lorentz force equation is a bifurcation point

on the complexity of quadratic programming in real number models of computation abstract the complexity of linearly constrained nonconvex quadratic programming is analyzed within the framework of real number models namely the one of blum shub and smale and its modification recently introduced by koiran weak bss model in particular we show that this problem is not np complete in the koiran setting applications to the full bss model are discussed

a survey on the continuous nonlinear resource allocation problem our problem of interest consists of minimizing a separable convex and differentiable function over a convex set defined by bounds on the variables and an explicit constraint described by a separable convex function applications are abundant and vary from equilibrium problems in the engineering and economic sciences through resource allocation and balancing problems in manufacturing statistics military operations research and production and financial economics to subproblems in algorithms for a variety of more complex optimization models this paper surveys the history and applications of the problem as well as algorithmic approaches to its solution the most common techniques are based on finding the optimal value of the lagrange multiplier for the explicit constraint most often through the use of a type of line search procedure we analyze the most relevant references especially regarding their originality and numerical findings summarizing with remarks on possible extensions and future research 2006 elsevier b v all rights reserved

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

spectral flow brouwer degree and hill s determinant formula abstract in 2005 a new topological invariant defined in terms of the brouwer degree of a determinant map was introduced by musso pejsachowicz and the first name author for counting the conjugate points along a semi riemannian geodesic this invariant was defined in terms of a suspension of a complexified family of linear second order dirichlet boundary value problems in this paper starting from this result we generalize this invariant to a general self adjoint morse sturm system and we prove a new spectral flow formula finally we discuss the relation between this spectral flow formula and the hill s determinant formula and we apply this invariant for detecting instability of periodic orbits of a hamiltonian system

a polynomial algorithm for an integer quadratic non separable transportation problem we study the problem of minimizing the total weighted tardiness when scheduling unti length jobs on a single machine in the presence of large sets of identical jobs previously known algorithms which do not exploit the set structure are at best pseudo polynomial and may be prohibitively inefficient when the set sizes are large we give a polynomial algorithm for the problem whose number of operations is independent of the set sizes the problem is reformulated as an integer program with a quadratic non separable objective and transportation constraints employing methods of real analysis we prove a tight proximity result between the integer solution to that problem and a fractional solution of a related problem the related problem is shown to be polynomially solvable and a rounding algorithm applied to its solution gives the optimal integer solution to the original problem

on bifurcation for semilinear elliptic dirichlet problems and the morse smale index theorem we study bifurcation from a branch of trivial solutions of semilinear elliptic dirichlet boundary value problems on star shaped domains where the bifurcation parameter is introduced by shrinking the domain in the proof of our main theorem we obtain in addition a special case of an index theorem due to s smale

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

polynomial algorithms for lp over a subring of the algebraic integers with applications to lp with circulant matrices we show that a modified variant of the interior point method can solve linear programs lps whose coefficients are real numbers from a subring of the algebraic integers by defining the encoding size of such numbers to be the bit size of the integers that represent them in the subring we prove the modified algorithm runs in time polynomial in the encoding size of the input coefficients the dimension of the problem and the order of the subring we then extend the tardos scheme to our case obtaining a running time which is independent of the objective and right hand side data as a consequence of these results we are able to show that lps with real circulant coefficient matrices can be solved in strongly polynomial time finally we show how the algorithm can be applied to lps whose coefficients belong to the extension of the integers by a fixed set of square roots

a strongly polynomial algorithm for a special class of linear programs we extend the list of linear programming problems that are known to be solvable in strongly polynomial time to include a class of lps which contains special cases of the generalized transshipment problem the result is facilitated by exploiting some special properties associated with leontief substitution systems and observing that a feasible solution to the system ax b x 0 in which no variable appears in more than two equations can be found in strongly polynomial time for b belonging to some set

deciding linear inequalities by computing loop residues v r pratt has shown that the real and integer feastbdlty of sets of linear mequallues of the form x y c can be decided quickly by examining the loops m certain graphs pratt s method is generahzed first to real feaslbdlty of mequahues m two variables and arbitrary coefficients and ultimately to real feaslbdlty of arbitrary sets of hnear mequahtles the method is well suited to apphcatlons m program verification

a strongly polynomial algorithm for minimum convex separable quadratic cost flow problems on two terminal series parallel networks we present strongly polynomial algorithms to find rational and integer flow vectors that minimize a convex separable quadratic cost function on two terminal series parallel graphs

a polynomial algorithm for minimum quadratic cost flow problems abstract network flow problems with quadratic separable costs appear in a number of important applications such as approximating input output matrices in economy projecting and forecasting traffic matrices in telecommunication networks solving nondifferentiable cost flow problems by subgradient algorithms it is shown that the scaling technique introduced by edmonds and karp 1972 in the case of linear cost flows for deriving a polynomial complexity bound for the out of kilter method may be extended to quadratic cost flows and leads to a polynomial algorithm for this class of problems the method may be applied to the solution of singly constrained quadratic programs and thus provides an alternative approach to the polynomial algorithm suggested by helgason kennington and lall 1980

some np complete problems in quadratic and nonlinear programming in continuous variable smooth nonconvex nonlinear programming we analyze the complexity of checking whether a a given feasible solution is not a local minimum and b the objective function is not bounded below on the set of feasible solutions we construct a special class of indefinite quadratic programs with simple constraints and integer data and show that checking a or b on this class is np complete as a corollary we show that checking whether a given integer square matrix is not copositive is np complete

ordinary differential operators in hilbert spaces and fredholm pairs let a t be a path of bounded operators on a real hilbert space hyperbolic at pm infty we study the fredholm theory of the operator f a d dt a t we relate the fredholm property of f a to the stable and unstable linear spaces of the associated system x prime a t x several examples are included to point out the differences with respect to the finite dimensional case in particular concerning the role of the spectral flow we define a general class of paths a for which many properties typical of the finite dimensional framework still hold our motivation is to develop the linear theory which is necessary for the set up of morse homology on hilbert manifolds

a signal dependent time frequency representation fast algorithm for optimal kernel design a time frequency representation based on an optimal signal dependent kernel has been previously been proposed in an attempt to overcome one of the primary limitations of bilinear time frequency distributions that the best kernel and distribution depend on the signal to be analyzed the optimization formulation for the signal dependent kernel results in a linear program with a unique feature a tree structure that summarizes a set of constraints on the kernel the authors present a fast algorithm based on sorting to solve a special class of linear programs that includes the problem of interest for a kernel with q variables the running time of the algorithm is o q log q which is several orders of magnitude less than any other known method for solving this class of linear program this efficiency enables the computation of the signal dependent optimal kernel time frequency representation at a cost that is on the same order as a fixed kernel distribution an important property of the optimal kernel is that it takes on essentially only the values of 1 and 0

the stationary thermistor problem with a current limiting device the electrical heating of a conductor connected with a current limiting device of total resistance r is studied under mixed boundary conditions a theorem of existence is given using a transformation which permits the reduction of the nonlinear system governing the problem to the classical mixed boundary value problem for the laplace equation

the traveling salesman problem with many visits to few cities we study the version of the traveling salesman problem in which a relatively small number of cities say six must be visited a huge number of times e g several hundred times each it costs to go

on certain doubly non uniformly and singular non uniformly elliptic systems abstract we consider the steady state of the thermistor problem consisting of a coupled set of nonlinear elliptic equations governing the temperature and the electric potential we study the existence of weak solutions under two kind of assumptions the first one considers the case in which the two diffusion coefficients are not bounded below far from zero arising to a doubly non uniformly elliptic system in the second one we assume in addition that the thermal conductivity blows up for a finite value of the temperature arising to a singular and non uniformly coupled system

a polynomial time primal dual affine scaling algorithm for linear and convex quadratic programming and its power series extension we describe an algorithm for linear and convex quadratic programming problems that uses power series approximation of the weighted barrier path that passes through the current iterate in order to find the next iterate if r 1 is the order of approximation used we show that our algorithm has time complexity o n 1 1 r l 1 1 r iterations and o n3 n2r arithmetic operations per iteration where n is the dimension of the problem and l is the size of the input data when r 1 we show that the algorithm can be interpreted as an affine scaling algorithm in the primal dual setup

about strongly polynomial time algorithms for quadratic optimization over submodular constraints we present new strongly polynomial algorithms for special cases of convex separable quadratic minimization over submodular constraints the main results are an o nm log n 2 m algorithm for the problemnetwork defined on a network onm arcs andn nodes an o n logn algorithm for thetree problem onn variables an o n logn algorithm for thenested problem and a linear time algorithm for thegeneralized upper bound problem these algorithms are the best known so far for these problems the status of the general problem and open questions are presented as well

efficient methods for selfish network design intuitively braess s paradox states that destroying a part of a network may improve the common latency of selfish flows at nash equilibrium such a paradox is a pervasive phenomenon in real world networks any administrator who wants to improve equilibrium delays in selfish networks is facing some basic questions is the network paradox ridden how can we delete some edges to optimize equilibrium flow delays how can we modify edge latencies to optimize equilibrium flow delays unfortunately such questions lead to np hard problems in general in this work we impose some natural restrictions on our networks e g we assume strictly increasing linear latencies our target is to formulate efficient algorithms for the three questions above we manage to provide a polynomial time algorithm that decides if a network is paradox ridden when latencies are linear and strictly increasing a reduction of the problem of deciding if a network with arbitrary linear latencies is paradox ridden to the problem of generating all optimal basic feasible solutions of a linear program that describes the optimal traffic allocations to the edges with constant latency an algorithm for finding a subnetwork that is almost optimal wrt equilibrium latency our algorithm is subexponential when the number of paths is polynomial and each path is of polylogarithmic length a polynomial time algorithm for the problem of finding the best subnetwork which outperforms any known approximation for the case of strictly increasing linear latencies a polynomial time method that turns the optimal flow into a nash flow by deleting the edges not used by the optimal flow and performing minimal modifications on the latencies of the remaining ones our results provide a deeper understanding of the computational complexity of recognizing the most severe manifestations of braess s paradox and our techniques show novel ways of using the probabilistic method and of exploiting convex separable quadratic programs

strongly polynomial algorithms for the quadratic transportation problem with a fixed number of sources the transportation problem with a linear objective function is known to be solvable in strongly polynomial time whereas instances with a convex nonquadratic objective function are not even for cases with the integrality constraints relaxed we present a linear time algorithm for the continuous quadratic transportation problem qtp with two source nodes further we show how problem instances with a fixed number of source or destination nodes k could be solved in strongly polynomial time onk 1 using an algebraic tree computation model the algorithms exploit a relationship between the transportation problem and the resource allocation problem the strong polynomiality of the algorithms presented here imply the existence of strongly polynomial algorithms for integer quadratic transportation problems with a fixed number of source nodes as well

spectral flow and bifurcation of critical points of strongly indefinite functionals part i general theory abstract spectral flow is a well known homotopy invariant of paths of selfadjoint fredholm operators we describe here a new construction of this invariant and prove the following theorem let i u rbe ac2function on the product of a real intervali a b with a neighborhood u of the origin of a real separable hilbert space h and such that for each in i 0is a critical point of the functional assume that the hessian l of at0is fredholm and moreover that laand lbare nonsingular if the spectral flow of the path l idoes not vanish then the interval i contains at least one bifurcation point from the trivial branch for solutions of the equation x x 0 equivalently every neighborhood of i 0 contains points of the form x wherex 0 is a critical point of

breakpoint searching algorithms for the continuous quadratic knapsack problem we give several linear time algorithms for the continuous quadratic knapsack problem in addition we report cycling and wrong convergence examples in a number of existing algorithms and give encouraging computational results for large scale problems

the notion of a rational convex program and an algorithm for the arrow debreu nash bargaining game we introduce the notion of a rational convex program rcp and we classify the known rcps into two classes quadratic and logarithmic the importance of rationality is that it opens up the possibility of computing an optimal solution to the program via an algorithm that is either combinatorial or uses an lp oracle next we define a new nash bargaining game called adnb which is derived from the linear case of the arrow debreu market model we show that the convex program for adnb is a logarithmic rcp but unlike other known members of this class it is nontotal our main result is a combinatorial polynomial time algorithm for adnb it turns out that the reason for infeasibility of logarithmic rcps is quite different from that for lps and quadratic rcps we believe that our ideas for surmounting the new difficulties will be useful for dealing with other nontotal rcps as well we give an application of our combinatorial algorithm for adnb to an important fair throughput allocation problem on a wireless channel finally we present a number of interesting questions that the new notion of rcp raises

a quasi linear algorithm for calculating the infimal convolution of convex quadratic functions in this paper we present an algorithm of quasi linear complexity to exactly calculate the infimal convolution of convex quadratic functions the algorithm exactly and simultaneously solves a separable uniparametric family of quadratic programming problems resulting from varying the equality constraint

on bifurcation for semilinear elliptic dirichlet problems on geodesic balls we study bifurcation from a branch of trivial solutions of semilinear elliptic dirichlet boundary value problems on a geodesic ball whose radius is used as the bifurcation parameter in the proof of our main theorem we obtain in addition a special case of an index theorem due to s smale

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