on global diffeomorphisms of euclidian space as is well known the first attempt at characterizing global diffeomorphisms of ir goes back to the 1906 work of hadamard l since that time progress in the global diffeomorphism problem has been rather slow notable contributions are those of levy 2 who generalized hadamard s theorem to banach spaces and of banach and mazur 3 and caccioppoli 4 who proved the so called properness criterion saying that a proper local homeomorphism of a banach space is a global homeomorphism the standard proofs of the theorems of hadamard and levy and of banach mazur and caccioppoli rely upon ideas of what is now called covering space theory via the important homotopy lifting property of coverings see e g 5 this approach makes it clear that it is only lack of compactness of ir that allows for the existence of noninjective local homeomorphisms when n l 2 the complex exponential when n 2 being the most famous example the theorems mentioned above were found again by various authors sometimes in special cases and sometimes in a different framework but always through some variant of the same argument a synthesis was made by rheinboldt 6 in 1969 part of his results were rediscovered five years later by plastock 7 and given a somewhat simpler exposition for the special case of banach spaces other contributions were made by meisters and olech 8 global homeomorphisms of bounded domains in i meyer 9 complementing the work of levy and radulescu and radulescu lo expanding upon plastock s results many other articles have been devoted to special cases where extra hypotheses are introduced consistently with the problem at hand for instance polynomial mappings only are relevant to the jacobian conjecture of algebraic geometry and hypotheses about the spectrum of the derivative at each point come up in a natural way in global stability questions for differential equations the new results presented in this paper have been obtained through arguments from critical point theory and differential topology here the central notion is that of admissible flow given a mapping f ii ir the main property of what we call an f admissible flow q ir x ii f o i f o is that for every x e ir f o the function t jf q t x l where 1 1 is the euclidean norm is strictly increasing and tends to 0 resp 00 as t tends to co resp 00 the basic feature we prove is that a local diffeomorphism f is a global one if and only if f o zi and there is an f admissible flow this statement makes it clear that 0 plays a special role regarding f admissibility and the condition f o 0 cannot be eliminated as examples will show the introduction of f admissibility followed the author s remark that in hilbert space hadamard s theorem could be derived from the mountain pass theorem of ambrosetti and

on the construction of global coordinate systems in euclidean spaces abstract several techniques for constructing global coordinate systems in r n adapted to functions are given some of these techniques are based on geometrical assumptions and other ones on analytical assumptions

global subjectivity of submersions via contractibility of the fibers abstract we give sufficient condition for a c 1 submersion f x y x y real banach spaces to be surjective with contractible fibers f 1 y roughly speaking this condition interpolates two well known but unrelated hypotheses coresponding to the two extreme cases hadanard s criterion when y x and f is a local diffeomorphism and the palais smale condition when y r these results may be viewed as a global variant of the implicit function theorem which unlike the local one does not require split kernels they are derived form a deformation theorem tailored to fit functionals with a norm like nondifferentiability

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

a class of hamilton jacobi equations on banach finsler manifolds abstract the concept of subdifferentiability is studied in the context of c 1 finsler manifolds modeled on a banach space with a lipschitz c 1 bump function a class of hamilton jacobi equations defined on c 1 finsler manifolds is studied and several results related to the existence and uniqueness of viscosity solutions are obtained

global optimization of polynomials using gradient tentacles and sums of squares we consider the problem of computing the global infimum of a real polynomial f on mathbb r n every global minimizer of f lies on its gradient variety i e the algebraic subset of mathbb r n where the gradient of f vanishes if f attains a minimum on mathbb r n it is therefore equivalent to look for the greatest lower bound of f on its gradient variety nie demmel and sturmfels proved recently a theorem about the existence of sums of squares certificates for such lower bounds based on these certificates they find arbitrarily tight relaxations of the original problem that can be formulated as semidefinite programs and thus be solved efficiently we deal here with the more general case when f is bounded from below but does not necessarily attain a minimum in this case the method of nie demmel and sturmfels might yield completely wrong results in order to overcome this problem we replace the gradient variety by larger semialgebraic subsets of mathbb r n which we call gradient tentacles it now gets substantially harder to prove the existence of the necessary sums of squares certificates

ehresmann fibrations and palais smale conditions for morphisms of finsler manifolds the central result of this paper is a generalization of the ehresmann fibration theorem to the infinite dimensional and or non proper setting with this aim we introduce the concepts of strong submersion and of mapping with uniformly split kernels first applications include a global implicit function theorem and a necessary and sufficient version of hadamard s global invertibility criterion in the setting of finsler manifolds another major application makes use of the idea of asymptotic critical value not critical point which helps formulate various generalizations of the palais smale condition for morphisms of finsler manifolds not merely functionals we obtain a critical point theory for morphisms of finsler manifolds extending many results known only for functionals notably ekeland s variational principle these results are used to discuss a new approach to lagrange multiplier and nonlinear eigenvalue problems and to develop an intrinsic critical point theory for complex analytic functionals the latter reveals an intimate connection between conditions of palais smale type and the structure of polynomial automorphisms jacobian conjecture

brouwer s degree without properness a degree deg f y is defined for every continuous function f rn rn which possesses all the properties of brouwer s degree provided that y is restricted to the complement of some closed set a f of asymptotic values sufficient conditions are given for a f to be nowhere dense it is also shown that in the opposite direction a f having nonempty interior has a direct impact on the solutions of f x y which cannot be discovered by degree arguments

exponential decay of the solutions of quasilinear second order equations and pohozaev identities keywords pohozaev identities decay nonlinear elliptic equation reference ana article 2000 002doi 10 1006 jdeq 1999 3749view record in web of science record created on 2008 12 10 modified on 2016 08 08

global homeomorphisms and covering projections on metric spaces for a large class of metric spaces with nice local structure which includes banach finsler manifolds and geodesic spaces of curvature bounded above we give sufficient conditions for a local homeomorphism to be a covering projection we first obtain a general condition in terms of a path continuation property as a consequence we deduce several conditions in terms of path liftings involving a generalized derivative and in particular we obtain an extension of hadamard global inversion theorem in this context next we prove that in the case of quasi isometric mappings some of these sufficient conditions are also necessary finally we give an application to the existence of global implicit functions

derivative computations for the log likelihood function this note improves on a method for computing derivatives of the log likelihood function used in the identification of discrete linear time invariant gaussian systems a formula for the derivative is obtained in terms of the derivatives of the matrices used to define the system under identification

analytic first and second derivatives for the recursive prediction error algorithm s log likelihood function this note improves on a method for computing analytic derivatives of the likelihood function of a discrete linear time invariant gaussian state space system and extends it to handle correlation between the state transition and measurement noise terms and to compute the analytic hessian matrix

totally reflexive modules and poincare series abstract we study cohen macaulay non gorenstein local rings r m k admitting certain totally reflexive modules more precisely we give a description of the poincare series of k by using the poincare series of a non zero totally reflexive module with minimal multiplicity our results generalize a result of yoshino to higher dimensional cohen macaulay local rings moreover from a quasi gorenstein ideal satisfying some conditions we construct a family of non isomorphic indecomposable totally reflexive modules having an arbitrarily large minimal number of generators

on the existence of pareto solutions for polynomial vector optimization problems we are interested in the existence of pareto solutions to the vector optimization problem begin aligned mathrm min mathbb r m f x x in mathbb r n end aligned where f mathbb r n rightarrow mathbb r m is a polynomial map by using the tangency variety of f we first construct a semi algebraic set of dimension at most m 1 containing the set of pareto values of the problem then we establish connections between the palais smale conditions m tameness and properness for the map f based on these results we provide some sufficient conditions for the existence of pareto solutions of the problem we also introduce a generic class of polynomial vector optimization problems having at least one pareto solution

on globally diffeomorphic polynomial maps via newton polytopes and circuit numbers in this article we analyze the global diffeomorphism property of polynomial maps f mathbb r n rightarrow mathbb r n by studying the properties of the newton polytopes at infinity corresponding to the sum of squares polynomials vert f vert 2 2 this allows us to identify a class of polynomial maps f for which their global diffeomorphism property on mathbb r n is equivalent to their jacobian determinant det jf vanishing nowhere on mathbb r n in other words we identify a class of polynomial maps for which the real jacobian conjecture which was proven to be false in general still holds

on the volumorphism group the first conjugate point is always the hardest we find a simple local criterion for the existence of conjugate points on the group of volume preserving diffeomorphisms of a 3 manifold with the riemannian metric of ideal fluid mechanics in terms of an ordinary differential equation along each lagrangian path using this criterion we prove that the first conjugate point along a geodesic in this group is always pathological the differential of the exponential map always fails to be fredholm

global inversion and covering maps on length spaces in order to obtain global inversion theorems for mappings between length metric spaces we investigate sufficient conditions for a local homeomorphism to be a covering map in this context we also provide an estimate of the domain of invertibility of a local homeomorphism around a point in terms of a kind of lower scalar derivative as a consequence we obtain an invertibility result using an analog of the hadamard integral condition in the frame of length spaces some applications are given to the case of local diffeomorphisms between banach finsler manifolds finally we derive a global inversion theorem for mappings between stratified groups

nonlinear fredholm operators with noncompact fibers and applications to elliptic problems on rn under suitable conditions an equation f x y between banach spaces involving a nonlinear fredholm mapping f of nonnegative index is shown to have a noncompact and hence infinite set of solutions for almost every y for which the equation is solvable the proof of this nonuniqueness but not existence result relies on an entirely new line of arguments in which the concept of generalized critical value plays a central role when f w2 p rn lp rn is associated with a quasilinear elliptic pde on rn with constant coefficient it often happens that a crucial denseness hypothesis in the abstract theorem is equivalent to the existence of a nontrivial solution to the equation f x f 0 and hence can be verified in practice generalizations exist for some classes of problems with nonconstant coefficients and for problems on exterior domains

free resolutions over short gorenstein local rings let r be a local ring with maximal ideal mathfrak m admitting a non zero element a in mathfrak m for which the ideal 0 a is isomorphic to r ar we study minimal free resolutions of finitely generated r modules m with particular attention to the case when mathfrak m 4 0 let e denote the minimal number of generators of mathfrak m if r is gorenstein with mathfrak m 4 0 and e 3 we show that rm p m r t is rational with denominator h r t 1 et et 2 t 3 for each finitely generated r module m in particular this conclusion applies to generic gorenstein algebras of socle degree 3

superlinear convolution equations in rn abstract we give existence results for convolution equations u g u v on r n based on degree theory for fredholm mappings of index 0 existence is obtained in orlicz sobolev spaces w 1 l a adapted to the nonlinearity g and by a limiting process and under weaker conditions in l loc 1 the case of r n volterra equations is also treated and applications to pdes of the form p d u q d g u f are discussed

global inversion of nonsmooth mappings using pseudo jacobian matrices we study the global inversion of a continuous nonsmooth mapping f r n r n which may be non locally lipschitz to this end we use the notion of pseudo jacobian map associated to f introduced by jeyakumar and luc and we consider a related index of regularity for f we obtain a characterization of global inversion in terms of its index of regularity furthermore we prove that the hadamard integral condition has a natural counterpart in this setting providing a sufficient condition for global invertibility

ascoli s theorem for functions vanishing at infinity and selected applications abstract we give a new form of the ascoli theorem for functions on r n tending to some given closed subset z of a complete metric space e at infinity for instance when e is a normed space and z 0 the usual uniform decay requirement is replaced by the assumption that the 0 function is the only continuous function produced by some limiting process this formulation which has significant practical value in concrete applications is described in its general form but with emphasis on the case when z is totally disconnected variants in sobolev spaces and the properness of nonlinear ordinary differential operators are discussed

on the limit set at infinity of a gradient trajectory of a semialgebraic function given any c2 semialgebraic function f defined on a non bounded open set of rn we prove that the limit of the secants at infinity of a non bounded trajectory of the gradient of f does exist as a corollary we find a new sufficient condition to ensure the trivialisation by the gradient flow of f nearby a regular asymptotic critical value at infinity

ideal magnetohydrodynamics and passive scalar motion as geodesics on semidirect product groups three dimensional ideal magnetohydrodynamic 3d imhd equations are shown to be a geodesic equation on an infinite dimensional lie group the group is the semidirect product of a group of volume preserving diffeomorphisms particle motion and a linear space of divergenceless vector fields current density the present theory includes the work of zeitlin and kambe 1993 for two dimensional imhd flow as a special case passive scalar motion in n dimensional ideal hydrodynamic flow is also generally shown to be described by a geodesic equation

the diameter of the symplectomorphism group is infinite shnirelman has proved that the group of volume preversing diffeomorphisms of the cube in r 3 has finite diameter and has announced the result that this is false for the square despite the fact that shnirelman formulated his theorem only for the 3 dimensional cube his proof can be modified for the case of the group of volume preserving diffeomorphisms of any compact simply connected riemannian manifold of dimension 2 it turns out that the situation with the group of symplectic diffeomorphisms is completely different we prove that the diameter of the symplectomorphism group of any compact exact symplectic manifold necessarily with boundary is infinite

detecting bifurcation values at infinity of real polynomials we present a new approach for estimating the set of bifurcation values at infinity this yields a significant shrinking of the number of coefficients in the recent algorithm introduced by jelonek and kurdyka for reaching critical values at infinity by rational arcs

the discriminant of a system of equations abstract we introduce and study the discriminant of a system of polynomial equations with indeterminate coefficients generalizing a number of known special cases such as the sparse resultant and the a determinant and inheriting their nice properties one such property is the fact that the discriminant is always a hypersurface in the space of systems of equations which seems unexpected when compared to the existence of dual defect toric varieties this fact reflects the combinatorics of a certain partial order relation on the set of tuples of faces of a tuple of polytopes the newton polytopes of the equations in our case generalizing the poset of faces of one polytope this combinatorics is the main challenge in our work and can be reduced to a certain fact of tropical geometry which resembles the geometry of the discriminants that we study as a sample application of our version of the multivariate discriminant we prove that atypical fibers of a generic polynomial map are distinguished by their euler characteristics and the bifurcation set of such a map is a hypersurface whose degree we explicitly compute

finite gorenstein representation type implies simple singularity let r be a commutative noetherian local ring and consider the set of isomorphism classes of indecomposable totally reflexive r modules we prove that if this set is finite then either it has exactly one element represented by the rank 1 free module or r is gorenstein and an isolated singularity if r is complete then it is even a simple hypersurface singularity the crux of our proof is to argue that if the residue field has a totally reflexive cover then r is gorenstein or every totally reflexive r module is free

exact pairs of homogeneous zero divisors abstract let s be a standard graded artinian algebra over a field k we identify constraints on the hilbert function of s which are imposed by the hypothesis that s contains an exact pair of homogeneous zero divisors as a consequence we prove that if s is a compressed level algebra then s does not contain any homogeneous zero divisors

an improved log likelihood gradient for continuous time stochastic systems with deterministic input using a covariance operator approach we derive an explicit expression for the log likelihood ratio gradient for system parameter estimation for continuous time stochastic systems with deterministic inputs the gradient formula includes the smoother estimates and derivatives of system matrices with no derivatives of estimates or covariance matrices a deterministic input is also permitted and the state noise covariance is not required to be nonsingular stable numerical techniques to calculate the gradient are also discussed

a sard theorem for tame set valued mappings abstract if f is a set valued mapping from r n into r m with closed graph then y r m is a critical value of f if for some x with y f x f is not metrically regular at x y we prove that the set of critical values of a set valued mapping whose graph is a definable tame set in an o minimal structure containing additions and multiplications is a set of dimension not greater than m 1 respectively a porous set as a corollary of this result we get that the collection of asymptotically critical values of a set valued mapping with a semialgebraic graph has dimension not greater than m 1 we also give an independent proof of the fact that a definable continuous real valued function is constant on components of the set of its subdifferentiably critical points

on complementary models and fixed interval smoothing a new algorithm is derived for the standard fixed interval linear smoothing problem in which the signal is generated by a state model the structure of this new algorithm allows the smoothed estimate to be easily updated in response to a change in the initial state covariance matrix pi 0 since unlike in existing algorifiuns the relevant riccati equation is entirely independent of pi 0 the derivation of the algorithm is based on properties of complementary models

on the singularities of the exponential map in infinite dimensional riemannian manifolds using symplectic techniques and spectral analysis of smooth paths of self adjoint operators we characterize the set of conjugate instants along a geodesic in an infinite dimensional riemannian hilbert manifold

a new method for evaluating the log likelihood gradient score of linear dynamic systems a novel method is presented that is based on the optimal smoothing equations the result can be used for efficient calculations and approximations of gradient search algorithms for maximum likelihood estimation of the unknown system parameters the method is applied to the continuous discrete case

global łojasiewicz type inequality for non degenerate polynomial maps abstract let f f 1 f p r n r p be a polynomial map this paper studies the existence of the following global ojasiewicz type inequality f x f x c d x f 1 0 for all x r n for some constants c 0 0 and 0 we show that the above inequality holds if one of the following conditions is satisfied i f is convenient and khovanskii non degenerate at infinity ii f is convenient and non degenerate at infinity iii f is mikhailov gindikin non degenerate further in cases ii and iii the exponents and can be determined explicitly

semialgebraic sard theorem for generalized critical values we prove that a semialgebraic differentiable mapping has a generalized critical values set of measure zero moreover if the mapping is c 2 we obtain bya generalisation of ehresmann s fibration theorem due to p j rabier 20 a locallytrivial fibration over the complement of this set in the complex case we prove that the set of generalized critical values of a polynomial mapping is a proper algebraic set

pseudo jacobian and global inversion of nonsmooth mappings on riemannian manifolds abstract we introduce a notion of pseudo jacobian for continuous nonsmooth mappings between riemannian manifolds then we discuss the global inversion of continuous mappings between riemannian manifolds which may be non locally lipschitz for this purpose we obtain a version of the hadamard integral condition for invertibility by using pseudo jacobian

boundedness versus periodicity over commutative local rings over commutative graded local artinian rings examples are constructed of periodic modules of arbitrary minimal period and modules with bounded betti numbers which are not eventually periodic they provide counterexamples to a conjecture of d eisenbud that every module with bounded betti numbers over a commutative local ring is eventually periodic of period 2 it is proved however that the conjecture holds over rings of small length

a new formula for the log likelihood gradient for continuous time stochastic systems using a finitely additive white noise approach we obtain an explicit expression for the gradient of the log likelihood ratio for system parameter estimation for continuous time linear stochastic systems with noisy observations our gradient formula includes the smoother estimates of the state vector and derivatives of only the system matrices and not the estimates or error covariances a scheme to calculate the log likelihood gradient without solving a riccati equation is described when only one matrix and the initial covariance depend on the unknown parameter

submanifolds that are level sets of solutions to a second order elliptic pde abstract we consider the problem of characterizing which noncompact hypersurfaces in r n can be regular level sets of a harmonic function modulo a c diffeomorphism as well as certain generalizations to other pdes we prove a versatile sufficient condition that shows in particular that any nonsingular algebraic hypersurface whose connected components are all noncompact can be transformed onto a union of components of the zero set of a harmonic function via a diffeomorphism of r n the technique we use combines robust but not explicit local constructions with appropriate global approximation theorems in view of applications to a problem posed by berry and dennis intersections of level sets are also studied

two dimensional ideal magnetohydrodynamics and differential geometry it is shown that equations of two dimensional ideal magnetohydrodynamics may be regarded as geodesic equations on appropriate infinite dimensional lie group the physical interpretation of such a geometric picture is given using an analogy with two and a half dimensional ideal hydrodynamics the sectional curvature responsible for the separation of neighbouring geodesics and hence for stability is calculated

brauer thrall for totally reflexive modules let r be a commutative noetherian local ring that is not gorenstein it is known that the category of totally reflexive modules over r is representation infinite provided that it contains a non free module the main goal of this paper is to understand how complex the category of totally reflexive modules can be in this situation local rings r m with m3 0 are commonly regarded as the structurally simplest rings to admit diverse categorical and homological characteristics for such rings we obtain conclusive results about the category of totally reflexive modules modeled on the brauer thrall conjectures starting from a non free cyclic totally reflexive module we construct a family of indecomposable totally reflexive r modules that contains for every n n a module that is minimally generated by n elements moreover if the residue field r m is algebraically closed then we construct for every n n an infinite family of indecomposable and pairwise non isomorphic totally reflexive r modules each of which is minimally generated by n elements the modules in both families have periodic minimal free resolutions of period at most 2

a nonsmooth morse sard theorem for subanalytic functions abstract according to the morse sard theorem any sufficiently smooth function on a euclidean space remains constant along any arc of critical points we prove here a theorem of morse sard type suitable as a tool in variational analysis we broaden the definition of a critical point to the standard notion in nonsmooth optimization while we restrict the functions under consideration to be semialgebraic or subanalytic we make no assumption of subdifferential regularity ojasiewicz type inequalities for nonsmooth functions follow quickly from tools of the kind we develop leading to convergence theory for subgradient dynamical systems

testing sets for properness of polynomial mappings in this paper we characterize testing sets for properness of polynomial mappings f mathbb c n rightarrow mathbb c m we also study the set of points at which such mappings are not proper as the first application we give a proof of the complementary conjecture of mckay and wang conjecture 12 in 16 the second application is an answer to the problem of kraft russell about a characterization of mathbb c n the third application is given in 13 a solution of the problem of van den essen and shpilrain about endomorphism of the ring mathbb c x 1 x n the fourth application is a theorem about extensions of affine varieties

a frank wolfe type theorem for nondegenerate polynomial programs in this paper we study the existence of optimal solutions to a constrained polynomial optimization problem more precisely let f 0 f 0 and f 1 ldots f p mathbb r n rightarrow mathbb r f 1 f p r n r be convenient polynomial functions and let s x in mathbb r n f i x le 0 i 1 ldots p ne emptyset s x r n f i x 0 i 1 p under the assumption that the map f 0 f 1 ldots f p mathbb r n rightarrow mathbb r p 1 f 0 f 1 f p r n r p 1 is non degenerate at infinity we show that if f 0 f 0 is bounded from below on s s then f 0 f 0 attains its infimum on s s

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

on some problems on smooth approximation and smooth extension of lipschitz functions on banach finsler manifolds abstract let us consider a riemannian manifold m either separable or non separable we prove that for every e 0 every lipschitz function f m r can be uniformly approximated by a lipschitz c 1 smooth function g with lip g lip f e as a consequence every riemannian manifold is uniformly bumpable these results extend to the non separable setting those given in 1 for separable riemannian manifolds the results are presented in the context of c l finsler manifolds modeled on banach spaces sufficient conditions are given on the finsler manifold m and the banach space x where m is modeled so that every lipschitz function f m r can be uniformly approximated by a lipschitz c k smooth function g with lip g c lip f for some c depending only on x some applications of these results are also given as well as a characterization on the separable case of the class of c l finsler manifolds satisfying the above property of approximation finally we give sufficient conditions on the c 1 finsler manifold m and x to ensure the existence of lipschitz and c 1 smooth extensions of every real valued function f defined on a submanifold n of m provided f is c 1 smooth on n and lipschitz with the metric induced by m

on gorenstein projective injective and flat dimensions a functorial description with applications gorenstein homological dimensions are refinements of the classical homological dimensions and finiteness singles out modules with amenable properties reflecting those of modules over gorenstein rings as opposed to their classical counterparts these dimensions do not immediately come with practical and robust criteria for finiteness not even over commutative noetherian local rings in this paper we enlarge the class of rings known to admit good criteria for finiteness of gorenstein dimensions

a feasibility test for linear interference alignment in mimo channels with constant coefficients in this paper we consider the feasibility of linear interference alignment ia for multiple input multiple output mimo channels with constant coefficients for any number of users antennas and streams per user and propose a polynomial time test for this problem combining algebraic geometry techniques with differential topology ones we first prove a result that generalizes those previously published on this topic in particular we consider the input set complex projective space of mimo interference channels the output set precoder and decoder grassmannians and the solution set channels decoders and precoders satisfying the ia polynomial equations not only as algebraic sets but also as smooth compact manifolds using this mathematical framework we prove that the linear alignment problem is feasible when the algebraic dimension of the solution variety is larger than or equal to the dimension of the input space and the linear mapping between the tangent spaces of both smooth manifolds given by the first projection is generically surjective if that mapping is not surjective then the solution variety projects into the input space in a singular way and the projection is a zero measure set this result naturally yields a simple feasibility test which amounts to checking the rank of a matrix we also provide an exact arithmetic version of the test which proves that testing the feasibility of ia for generic mimo channels belongs to the bounded error probabilistic polynomial complexity class

gradient of the log likelihood ratio for infinite dimensional stochastic systems using a covariance operator approach we derive an expression for the log likelihood ratio gradient for system parameter estimation for continuous time infinite dimensional stochastic systems the gradient formula includes the smoother estimates and derivatives of system operators with no derivatives of estimates or covariance operators the unbounded operators typically found in partial differential equations limit how much the gradient formula can be simplified a random heat equation is considered

surjection and inversion for locally lipschitz maps between banach spaces abstract we study the global invertibility of non smooth locally lipschitz maps between infinite dimensional banach spaces using a kind of palais smale condition to this end we consider the chang version of the weighted palais smale condition for locally lipschitz functionals in terms of the clarke subdifferential as well as the notion of pseudo jacobians in the infinite dimensional setting which are the analog of the pseudo jacobian matrices defined by jeyakumar and luc using these tools we obtain the existence and uniqueness of solution for certain nonlinear equations defined by locally lipschitz mappings along the way we also obtain a global surjection theorem for locally lipschitz maps in terms of pseudo jacobians

fredholm properties of riemannian exponential maps on diffeomorphism groups we prove that exponential maps of right invariant sobolev h r metrics on a variety of diffeomorphism groups of compact manifolds are nonlinear fredholm maps of index zero as long as r is sufficiently large this generalizes the result of ebin et al geom funct anal 16 2006 for the l 2 metric on the group of volume preserving diffeomorphisms important in hydrodynamics in particular our results apply to many other equations of interest in mathematical physics we also prove an infinite dimensional morse index theorem settling a question raised by arnold and khesin topological methods in hydrodynamics springer new york 1998 on stable perturbations of flows in hydrodynamics finally we include some applications to the global geometry of diffeomorphism groups

tame nonsmooth inverse mapping theorems we give several versions of local and global inverse mapping theorems for tame nonnecessarily smooth mappings here tame mapping means a continuous mapping which is locally definable in some o minimal structure our sufficient conditions are formulated in terms of various properties convexity positivity of some principal minors of the space of jacobian matrices at smooth points