a splitting theorem for holomorphic banach bundles this paper is motivated by grothendieck s splitting theorem in the 1960s gohberg generalized this to a class of banach bundles we consider a compact complex manifold x and a holomorphic banach bundle e x that is a compact perturbation of a trivial bundle in a sense recently introduced by lempert we prove that e splits into the sum of a finite rank bundle and a trivial bundle provided h 1 x mathcal o 0

computing the roughening transition of ising and solid on solid models by bcsos model matching we study the roughening transition of the dual of the two dimensional 2d xy model of the discrete gaussian model of the absolute value solid on solid model and of the interface in an ising model on a three dimensional 3d simple cubic lattice the investigation relies on a renormalization group finite size scaling method that was proposed and successfully tested a few years ago the basic idea is to match the renormalization group flow of the interface observables with that of the exactly solvable body centred solid on solid bcsos model our estimates for the critical couplings are and for the xy model the discrete gaussian model and the absolute value solid on solid model respectively for the inverse roughening temperature of the ising interface we find to the best of our knowledge these are the most precise estimates for these parameters published so far

stability properties of autonomous homogeneous polynomial differential systems abstract a geometrical approach is used to derive a generalized characteristic value problem for dynamic systems described by homogeneous polynomials it is shown that a nonlinear homogeneous polynomial system possesses eigenvectors and eigenvalues quantities normally associated with a linear system these quantities are then employed in studying stability properties the necessary and sufficient conditions for all forms of stabilities characteristic of a two dimensional system are provided this result together with the classical theorem of frommer completes a stability analysis for a two dimensional homogeneous polynomial system

moving mesh generation using the parabolic monge ampere equation this article considers a new method for generating a moving mesh which is suitable for the numerical solution of partial differential equations pdes in several spatial dimensions the mesh is obtained by taking the gradient of a scalar mesh potential function which satisfies an appropriate nonlinear parabolic partial differential equation this method gives a new technique for performing r adaptivity based on ideas from optimal transportation combined with the equidistribution principle applied to a time varying scalar monitor function used successfully in moving mesh methods in one dimension detailed analysis of this new method is presented in which the convergence regularity and stability of the mesh is studied additionally this new method is shown to be straightforward to program and implement requiring the solution of only one simple scalar time dependent equation in arbitrary dimension with adaptivity along the boundaries handled automatically we discuss three preexisting methods in the context of this work examples are presented in which either the monitor function is prescribed in advance or it is given by the solution of a partial differential equation

a geometric interpretation of coherent structures in navier stokes flows the pressure in the incompressible three dimensional navier stokes and euler equations is governed by poisson s equation this equation is studied using the geometry of three forms in six dimensions by studying the linear algebra of the vector space of three forms l 3 wwhere w is a six dimensional real vector space we relate the characterization of non degenerate elements of l 3 wto the sign of the laplacian of the pressure and hence to the balance between the vorticity and the rate of strain when the laplacian of the pressure dp satisfies dpo0 the three form associated with poisson s equation is the real part of a decomposable complex form and an almost complex structure can be identified when dp 0 a real decomposable structure is identified these results are discussed in the context of coherent structures in turbulence

moduli of stability for germs of homogeneous vectorfields on r3 abstract in this paper we will show that there are an uncountable number of homogeneous quadratic vectorfields on r 3 so that no two of these vectorfields are topologically equivalent

extended integrability and bi hamiltonian systems the current notion of integrability of hamiltonian systems was fixed by liouville in a famous 1855 paper it describes systems in a 2k dimensional phase space whose trajectories are dense on tori q or wind on toroidal cylinders m q m within liouville s construction the dimension q cannot exceed k and is the main invariant of the system in this paper we generalize liouville integrability so that trajectories can be dense on tori k of arbitrary dimensions q 1 2k 1 2k and an additional invariant v 2 q k v 2 q 2 can be recovered the main theorem classifies all k k 1 2 canonical forms of hamiltonian systems that are integrable in a newly defined broad sense an integrable physical problem having engineering origin is presented the notion of extended compatibility of two poisson structures is introduced the corresponding bi hamiltonian systems are shown to be integrable in the broad sense

examples of quaternionic and kahler structures in hamiltonian models of nearly geostrophic flow we study 2 forms on phase spaces of hamiltonian models of nearly geostrophic flows a quaternionic structure is identified and the complex part of a symplectic representation of this structure corresponds to an elliptic monge ampere equation the real part is an invariant kahler structure

a pointwise finite dimensional reduction method for einstein lichnerowicz type systems the six dimensional case abstract we construct blowing up solutions for smooth perturbations of a focusing einstein lichnerowicz system on a closed 6 dimensional manifold our construction combines asymptotic analysis techniques with a finite dimensional reduction approach following the scheme of proof developed in premoselli 2016 the blow up examples constructed here highlight in particular the role of the coupling field in dimension 6

characterization of the hilbert ball by its automorphism group let be a bounded convex domain in a separable hilbert space the authors prove a version of the theorem of bun wong which asserts that if such a domain admits an automorphism orbit accumulating at a strongly pseudoconvex boundary point then it is biholomorphic to the ball key ingredients in the proof are a new localization argument using holomorphic peaking functions and the use of new normal families arguments in the construction of the limit biholomorphism

the role of the green s function in a non linear elliptic equation involving the critical sobolev exponent this paper is concerned with non linear elliptic problems of the type pe u u n 2 n 2 eu u 0 on u 0 on where is a smooth and bounded domain in rn n 4 and e 0 we show that if the ue are solutions of pe which concentrate around a point as e 0 then this point cannot be on the boundary of and is a critical point of the regular part of the green s function conversely we show that for n 5 and any non degenerate critical point x0 of the regular part of the green s function there exist solutions of pe concentrating around x0 as e 0

coherent states in quaternionic quantum mechanics we develop perelomov s coherent states formalism to include the case of a quaternionic hilbert space we find that because of the closure requirement an attempted quaternionic generalization of the special nilpotent or weyl group reduces to the normal complex case for the case of the compact group su 2 however coherent states can be formulated using the quaternionic half integer spin matrices of finkelstein jauch and speiser giving a nontrivial quaternionic analog of coherent states

blow up phenomena for the yamabe equation it is well known that the pde 1 has at least one positive solution for any choice of m g if n 6 and m g is not locally conformally flat this follows from results of t aubin 3 the remaining cases were solved by r schoen using the positive mass theorem 16 solutions to 1 are not usually unique as an example consider the product metric on 51 l x 5n 1 l if l is sufficiently small then the yamabe pde has a unique solution on the other hand there are many non minimizing solutions if l is large d pollack 14 has used gluing techniques to construct high energy solutions on more general background manifolds given any conformai class with positive yamabe constant and any positive integer iv there exists a new conformai class which is close to the original one in the c norm and contains at least n metrics of constant scalar curvature see 14 theorem 0 1 it is an interesting question whether the set of all solutions to the yamabe pde is compact in the c2 topology say a well known conjecture states that this should be true unless m p is conformally equivalent to the round sphere see 17 18 19 this conjecture has been verified in low dimensions and in the locally conformally flat case if m g is locally conformally flat compactness follows from work of r schoen 17 18 moreover schoen proposed a strategy for proving the conjecture in the non locally conformally flat case based on the pohozaev identity

inverses depending holomorphically on a parameter in a banach space abstract we show that the existence of a right inverse at each point for a holomorphic mapping from a pseudo convex domain in a banach space with an unconditional basis into a unital banach algebra implies the existence of a holomorphic right inverse variations of this result are given

non compactness and multiplicity results for the yamabe problem on sn the aim of this paper is to show the existence of metrics ge on sn where ge is a perturbation of the standard metric g0 for which the yamabe problem possesses a sequence of solutions unbounded in l sn the metrics ge that we find are of class ck on sn with k n 34 we also prove some new multiplicity results

families of lift and contact transformations a study of the contact transformation is carried out with special emphasis on the lift transformation the latter is the more basic concept because it applies to a single curve rather than to pairs of them in the first instance from that viewpoint the lift transformation can also transform an unfamiliar system of partial differential equations to a familiar one without reference to contact we illustrate this by showing how the semi geostrophic equations of meterology transform to hamiltonian form general and special lift transformations are distinguished new and very wide families of such transformations are derived these give a desirable perspective to particular cases such as the legendre transformation and the geostrophic transformation a brief study of generalized duality is described the distinction between regular and singular lifted curves is discussed in general terms and their participation in a contact transformation is illustrated by an accessible physical example

a dolbeault isomorphism theorem in infinite dimensions for a large class of separable banach spaces we prove the real analytic dolbeault isomorphism theorem for open subsets

mesh duality and legendre duality we describe a sense in which mesh duality is equivalent to legendre duality that is a general pair of meshes which satisfy a definition of duality for meshes are shown to be the projection of a pair of piecewise linear functions that are dual to each other in the sense of a legendre dual transformation in applications the latter functions can be a tangent plane approximation to a smoother function and a chordal plane approximation to its legendre dual convex examples include one from meteorology and also the relation between the delaunay mesh and the voronoi tessellation the latter are shown to be the projections of tangent plane and chordal approximations to the same paraboloid

cohomological characterization of pseudoconvexity in a banach space let x be a banach space with a countable unconditional basis e g x l2 x open we show that is pseudoconvex if and only if for each affine complex line l in x the sheaf cohomology group h 1 i vanishes where i is the ideal sheaf of all holomorphic functions on that vanish on l we also give an example that the condition h q 0 for all q 1 unlike in finite dimensions does not imply the pseudoconvexity of lastly we prove an interpolation result

dual variables in semigeostrophic theory this paper is a study of the duality structure present in the semigeostrophic equations of meterology we explore a new viewpoint represented by legendre transformations between alternative choices of independent variables including space coordinates geostrophic coordinates and isentropic coordinates detailed examples are given for both uniform and non uniform potential vorticity of the modelling of an atmospheric front at a given time in stable flow in each case the front is a half line of gradient discontinuity on a continuous convex surface obtained as the self intersection on a swallowtail surface after the latter is convexified so that non physical multivaluedness is removed

one dimensional quaternion homogeneous polynomial differential equations in this paper we study the dynamics of real four dimensional polynomial differential equations of the forms q aqkq m or q qkaqm where q a are quaternions q is the conjugate of q and k and m are non negative integers in many parts of the work the quaternion structure is used along the proofs

an embedding theorem for pseudoconvex domains in banach spaces we show that a pseudoconvex open subset of a banach space with an unconditional basis is biholomorphic to a closed direct submanifold of a banach space with an unconditional basis

holomorphic structures in hydrodynamical models of nearly geostrophic flow we study complex structures arising in hamiltonian models of nearly geostrophic flows in hydrodynamics in many of these models an elliptic monge ampere equation defines the relationship between a balanced velocity field defined by a constraint in the hamiltonian formalism and the materially conserved potential vorticity elliptic monge ampere operators define an almost complex structure and in this paper we show that a natural extension of the so called geostrophic momentum transformation of semi geostrophic theory which has a special importance in theoretical meteorology defines kahler and special kahler structures on phase space furthermore analogues of the geostrophic momentum coordinates are shown to be special lagrangian coordinates under conditions which depend upon the physical approximations under consideration certain duality properties of the operators are studied within the framework of the kahler geometry

holomorphic mappings on we describe the holomorphic mappings of bounded type and the arbitrary holomorphic mappings from the complex banach space 11 into a complex banach space x it is shown that these mappings have monomial expansions and the growth of the norms of the coefficients is characterized in each case this characterization is used to give new descriptions of the compact open topology and the nachbin ported topology on the space a 11 x of holomorphic mappings and to prove a lifting property for holomorphic mappings on 11 we also show that the monomials form an equicontinuous unconditional schauder basis for the space 11 so of holomorphic functions on 11 with the topology of uniform convergence on compact sets

multi symplectic formulation of near local hamiltonian balanced models we transform near local hamiltonian balanced models hbms describing nearly geostrophic fluid motion with constant coriolis parameter into multi symplectic ms systems this allows us to determine conservation of lagrangian momentum energy and potential vorticity for salmon s l1 dynamics a similar approach works for other near local balanced models such as the model the ms approach also enables us to determine a class of systems that have a contact structure similar to that of the semigeostrophic model the contact structure yields a contact transformation that makes the problem of front formation tractable the new class includes the first local model with a variable coriolis parameter that preserves all of the most useful geometric features of the semigeostrophic model

non compactness and infinite number of conformal initial data sets in high dimensions abstract on any closed riemannian manifold of dimension greater than 7 we construct examples of background physical coefficients for which the einstein lichnerowicz equation possesses a non compact set of positive solutions this yields in particular the existence of an infinite number of positive solutions in such cases

hyper kahler geometry and semi geostrophic theory we use the formalism of monge ampere operators to study the geometric properties of the monge ampere equations arising in semi geostrophic sg theory and related models of geophysical fluid dynamics we show how kahler and hyper kahler structures arise and the legendre duality arising in sg theory is generalized to other models of nearly geostrophic flows

a variational analysis of einstein scalar field lichnerowicz equations on compact riemannian manifolds we establish new existence and non existence results for positive solutions of the einstein scalar field lichnerowicz equation on compact manifolds this equation arises from the hamiltonian constraint equation for the einstein scalar field system in general relativity our analysis introduces variational techniques in the form of the mountain pass lemma to the analysis of the hamiltonian constraint equation which has been previously studied by other methods

a quaternionic structure in the three dimensional euler and ideal magneto hydrodynamics equations by considering the three dimensional incompressible euler equations a 4 vector is constructed out of a combination of scalar and vector products of the vorticity and the vortex stretching vector u s the evolution equation for can then be cast naturally into a quaternionic riccati equation this is easily transformed into a quaternionic zero eigenvalue schrodinger equation whose potential depends on the hessian matrix of the pressure with minor modifications this system can alternatively be written in complex notation an infinite set of solutions of scalar zero eigenvalue schrodinger equations has been found by adler and moser which are discussed in the context of the present problem similarly when the equations for ideal magneto hydrodynamics mhd are written in elsasser variables a pair of 4 vectors are governed by coupled

handbook of differential equations chapter headings definitions and concepts transformations exact analytical methods exact methods for odes exact methods for pdes approximate analytical methods numerical methods concepts numerical methods for odes numerical methods for pdes list of tables list of programs list of figures subject index

the dolbeault complex in infinite dimensions i infinite dimensional complex analysis was a popular subject in the sixties and seventies but in the last fifteen years enjoyed much less attention oddly enough this very same period began to see the emergence of examples of infinite dimensional complex manifolds in mathematical physics representation theory and geometry thus it appears to be a worthwhile undertaking to revisit infinite dimensional complex analysis and in particular to clarify fundamental properties of infinite dimensional complex manifolds the most fundamental questions at this point seem to be related to the solvability of the inhomogeneous cauchy riemann or 0 equations and more generally to the study of the dolbeault complex as an example in le3 we use a result on the a equation to describe the virasoro group as an infinite dimensional complex manifold up to now precious little has been known about the solvability of the infinite dimensional equation the available results pertain to solving the equation in domains in or over locally convex topological vector spaces and almost exclusively on the level of 0 1 forms postponing the precise definitions to sections 1 2 below we discuss those results that are of immediate relevance to this paper see d2 for more in li ligocka observes that h6rmander s proof of ehrenpreis theorem see e hor can be extended to infinite dimensions to solve the equation

on the cohomology groups of holomorphic banach bundles we consider a compact complex manifold m and introduce the notion of two holomorphic banach bundles e f over m being compact perturbations of one another given two such bundles we show that if the cohomology groups h q m e are finite dimensional then so are the cohomology groups h q m f as well as a more precise result in the same spirit

quaternionic valued ordinary differential equations the riccati equation abstract we give some sufficient conditions for the existence of at least two periodic solutions of the quaternionic riccati equation in some cases we are able to give a full description of dynamics and detect solutions heteroclinic to the periodic ones we also provide an example of the riccati equation without periodic solutions which appears in the euler vorticity dynamics

the effect of linear perturbations on the yamabe problem in conformal geometry the compactness conjecture asserts that the set of yamabe metrics on a smooth compact aspherical riemannian manifold left m g right is compact established in the locally conformally flat case by schoen lecture notes in mathematics vol 1365 pp 120 154 springer berlin 1989 surveys pure application and mathematics 52 longman science technology pp 311 320 harlow 1991 and for n le 24 by khuri marques schoen j differ geom 81 1 143 196 2009 it has revealed to be generally false for n ge 25 as shown by brendle j am math soc 21 4 951 979 2008 and brendle marques j differ geom 81 2 225 250 2009 a stronger version of it the compactness under perturbations of the yamabe equation is addressed here with respect to the linear geometric potential frac n 2 4 n 1 mathrm scal g mathrm scal g being the scalar curvature of left m g right we show that a priori l infty bounds fail for linear perturbations on all manifolds with n ge 4 as well as a priori gradient l 2 bounds fail for non locally conformally flat manifolds with n ge 6 and for locally conformally flat manifolds with n ge 7 in several situations the results are optimal our proof combines a finite dimensional reduction and the construction of a suitable ansatz for the solutions generated by a family of varying metrics in the conformal class of g

the parabolic umbilic and atmospheric fronts this paper describes how to calculate the legendre transformation of the beak to beak form of the parabolic umbilic the result is a multivalued surface in three dimensions with multiplicity of up to five which contains cusped edges of regression and lines of self intersection the calculation is self contained and is elementary in method but intricate in detail the ideas in it will be available in other such calculations of similar complexity and for any required context our underlying motivation is the study of atmospheric fronts in semigeostrophic theory we show how a front may be regarded as a selfintersection line terminating at a swallowtail point on the legendre transform surface a physical stability argument requires excision of certain parts of the surface to leave a jointly convex and single valued surface having a gradient discontinuity along the front a numerical illustration is presented this paper has two purposes the major part works out in detail the legendre transformation of the beak to beak form of the parabolic umbilic this calculation stands in its own right not oriented to any particular application and is addressed to a wide mathematical audience in a non committal notation the second purpose is to initiate and examine briefly a new way of studying atmospheric fronts which uses the result just obtained we explain how in suitably chosen variables a front may be regarded as terminating at a certain kind of swallowtail point qualitatively similar to one found on the legendre transform of the beak to beak potential this idea was the original motivation for the calculation mentioned above semigeostrophic theory in meteorology contains an analytical device which in fact is an example of a legendre transformation although previously unrecognized as such as remarked by sewell i987 p 157 the transformation is a certain kind of mapping from one single valued function to another function which may be multivalued in the first instance a single valued part of the latter is then retained by using arguments from atmospheric physics to delete unwanted branches for example those which represent unstable regimes the mapping which remains is from a smooth single valued convex function to another single valued convex function which is smooth everywhere except along a half line of gradient discontinuity and this half line represents the front the convexity thus retained by the physical stability argument is qualitatively similar