on kawai theorem for orbifold riemann surfaces we prove a generalization of kawai theorem for the case of orbifold riemann surface the computation is based on a formula for the differential of a holomorphic map from the cotangent bundle of the teichmuller space to the mathrm psl 2 mathbb c character variety which allows to evaluate explicitly the pullback of goldman symplectic form in the spirit of riemann bilinear relations as a corollary we obtain a generalization of goldman s theorem that the pullback of goldman symplectic form on the mathrm psl 2 mathbb r character variety is a symplectic form of the weil petersson metric on the teichmuller space

stability of volterra diffusion equations with time delays this paper is concerned with asymptotic stability of a system of reaction diffusion equations which is expressed in terms of volterra integrals under homogeneous dirichlet boundary conditions the effect of diffusion and delays on the stability of the system are analyzed and sufficient conditions are given for the existence of positive solutions of the corresponding steady state problem their global attraction with respect to nonnegative solutions of the time dependent system is discussed the main tools are lyapunov functionals positive definite kernels and laplace transforms

dirichlet problem for the diffusive nicholson s blowflies equation abstract the object of this paper is to consider the asymptotic behavior of solutions of the diffusive nicholson s blowflies equation under dirichlet boundary condition a new approach is developed to study the global attractivity of the positive steady state for a reaction diffusion equation with time delay when monotone or quasi monotone conditions fail to apply this approach should be applicable to other dirichlet problems as well even though our analysis is tailored to the diffusive nicholson s blowflies equation

infimum of a pair of matrices abstract in this paper answers are provided to the following questions given a matrix e1 of order k m does there exist a matrix e2 of order m k m such that the matrix e e 1 e 2 is square idempotent and of the same rank as e1 further when is the choice of e2 unique given a subspace t virtually disjoint with the row span of e1 when does there exist a matrix g g 1 g 2 such that g is square idempotent of rank equal to rank g1 rank e1 and g1 is congruent to e1 mod t these results are then used to determine if a pair of matrices a and b has an infimum in the minus partial order further the existence of the sharp order infimum is studied

a pair of simultaneous linear matrix equations a1xb1 c1 a2xb2 c2 and a matrix programming problem necessary and sufficient conditions are obtained for a pair of matrix equations a1xb1 c1 a2xb2 c2 on a general field f to have a common solution along with the expression for a general common solution when certain conditions hold common solutions of minimum rank are described a matrix programming problem is solved en route

shorted matrices an extended concept and some applications the shorting of an operator hitherto considered by krein 11 and by anderson and trapp 3 only for positive operators is extended to rectangular matrices and square matrices not necessarily hermitian nonnegative definite some applications of the shorted matrix in mathematical statistics are discussed

symplectic geometry of the moduli space of projective structures in homological coordinates we study the symplectic geometry of the space of linear differential equations with holomorphic coefficients of the form varphi u varphi 0 on riemann surfaces of genus g this space coincides with the moduli space of projective connections which is an affine bundle modelled on the cotangent bundle t mathcal m g we show that for several choices of the origin or base holomorphic projective connection such as bergman wirtinger or schottky the canonical poisson structure on t mathcal m g induces the goldman bracket on the monodromy character variety these different choices give rise to equivalent symplectic structures on the space of projective connections but different symplectic polarizations we find the corresponding generating functions combined with a prior theorem of kawai our results show the symplectic equivalence between the embeddings of t mathcal m g induced by the bers and bergman projective connections into the space of projective structures the main technical tools are variational formulas with respect to homological darboux coordinates on the space of holomorphic quadratic differentials in particular we get a new system of differential equations for the prym matrix of the canonical two sheeted covering of a riemann surface defined by a quadratic differential

what are schur complements anyway abstract given a matrix m a b c d the schur complements of a in m are the matrices of the form s d cab where a is a generalized inverse of a we survey several recent characterizations of schur complements and discuss where they arose and how they are related

domain decomposition methods for a class of spatially heterogeneous delayed reaction diffusion equations abstract we derive and study a class of delayed reaction diffusion equations with spatial heterogeneity which models the population of a single species with different habitats for mature and immature individuals we introduce new solid cones obtain spectral bounds of several spatial heterogeneous operators and establish limiting non negativeness property for the whole space and the eventual comparison principle for bounded domains as a result we develop new domain decomposition methods so that one can compare solutions with those to associated equations from a suitable bounded spatial domain to the whole space then by employing domain decomposition methods and dynamical system approaches we obtain threshold results under the supremum norm these results are greatly different from the existing ones of other evolution equations in unbounded domains or the whole space the main results are applied to two examples with the ricker birth function and with the mackey glass birth function it reveals that the size of the immature habitat can affect the reproduction and spread of the population

algebraic and geometric isomonodromic deformations using the gauss manin connection picard fuchs differential equation and a result of malgrange a special class of algebraic solutions to isomonodromic deformation equations the geometric isomonodromic deformations i s defined from families of families of algebraic varieties geometric isomonodromic deformations arise naturally from combinatorial strata in the moduli spaces of elliptic surfaces over p1 the complete list of geometric

systems of parabolic equations with continuous and discrete delays abstract in this paper we investigate the global existence and the dynamics of a coupled system of nonlinear parabolic equations where the nonlinear reaction function may depend on both continuous infinite or finite and discrete delays it is shown that if the reaction function is locally lipschitz continuous and the system possesses a pair of coupled upper and lower solutions then there exists a unique global solution to the system without any quasimonotone condition on the reaction function for systems with mixed quasimonotone reaction functions we use the monotone method to establish more dynamic property of the parabolic system in terms of the quasisolutions of the corresponding elliptic system this approach yields a global attractor of the parabolic system and under some additional conditions this attractor leads to the existence and asymptotic stability of a solution of the elliptic system application to three model problems in population dynamics and chemical reactions is given

partial orders based on outer inverses classes of outer inverses are used to create new partial orders and unify the classification of some of the old partial orders these classes are then compared with those obtained by using g inverses inner inverses

on parallel summability of matrices abstract it is shown that the generalized inverses characterize the parallel sum the almost positive definite a p d matrices introduced by duffin and morley 2 are of two types whose intersection is the class of quasi positive definite matrices mitra and puri 7 the a p d matrices of any one type form a saturated subclass of pairwise parallel summable a p d matrices

propagation and heterogeneous steady states in a delayed nonlocal r d equation without spatial translation invariance abstract we consider a delayed reaction diffusion equation that models the population dynamics of a single species with the mature population living in the 1 d whole space r while the immature population only living in the half space r with homogeneous dirichlet condition for the immatures at the boundary point one of the important features of this model system is that it does have the translational invariance by linking the non translational invariant solution map for this equation to travelling wave maps for another related 1 d spatial homogeneous delay reaction diffusion equation we obtain some traveling like a priori estimates for nontrivial solutions we then establish the existence uniqueness and attractivity of heterogeneous steady states as a result we are able to describe the traveling like asymptotic behaviours of nontrivial solutions in space time region these enable us to develop a new method for exploring the spreading speeds and asymptotic propagation phenomena for a class of non translation invariant delay reaction diffusion equations on r as a corollary we also recover some results on the asymptotic spreading speeds and traveling waves for monostable and spatial homogeneous delay reaction diffusion equations in r

the symplectic nature of fundamental groups of surfaces soit 7b g la categorie des groupes de lie g avec une forme bilineaire symetrique non singuliere b sur l algebre de lie soit le groupe fondamental d une surface orientee close de genre 1 et soit 7b s la categorie des out espaces symplectiques alors la correspondance g b hom g g b definit un foncteur 7b g 7b s ou b est la structure symplectique sur hom g g definie par b

matrix partial orders through generalized inverses unified theory the unified theory presented here covers as special cases the star order of drazin the minus order of hartwig and nambooripad the sharp order and other partial orders introduced by the author

the minus partial order and the shorted matrix the minus partial order was defined by hartwig 6 weakening the conditions of the star partial order of drazin 5 several new properties of the minus partial order are established the minus partial order is used to redefine the shorted matrix 15 17 and to define the infimum a b and the supremum a b of a pair a b of matrices of the same order the definition of the shorted matrix given here is similar to the krein anderson trapp definition of the shorted positive operator

skinning maps are finite to one we show that thurston s skinning maps of teichmuller space have finite fibers the proof centers around a study of two subvarieties of the rm sl 2 mathbb c character variety of a surface one associated with complex projective structures and the other associated with a 3 manifold using the morgan shalen compactification of the character variety and author s results on holonomy limits of complex projective structures we show that these subvarieties have only a discrete set of intersections

the monodromy of meromorphic projective structures we study projective structures on a surface having poles of prescribed orders we obtain a monodromy map from a complex manifold parameterising such structures to the stack of framed mathrm pgl 2 mathbb c local systems on the associated marked bordered surface we prove that the image of this map is contained in the union of the domains of the cluster charts we discuss a number of open questions concerning this monodromy map

an area preserving action of the modular group on cubic surfaces and the painleve vi equation we construct an area preserving action of the modular group on a general 4 parameter family of affine cubic surfaces we present a geometrical background behind this construction that is a natural symplectic structure on a moduli space of rank two linear monodromy representations over the 2 dimensional sphere with four punctures and a natural symplectic action upon it of the braid group on three strings studying this action as a discrete dynamical system will be important in discussing the monodromy of the painleve vi equation

the matrix equations ax c xb d for the pair of matrix equations ax c xb d this paper gives common solutions of minimum possible rank and also other feasible specified ranks

simultaneous diagonalization of rectangular matrices abstract a matrix d is said to be diagonal if its i j th element is null whenever i and j are unequal for a set a of matrices a of the same order the paper gives necessary and sufficient conditions for nonsingular matrices s and t to exist such that sa t d is diagonal for each matrix a in the set

uniform bounds from bounded lp functionals in reaction diffusion equations to prove global existence of classical or mild solutions of reaction diffusion equations a priori bounds in the uniform norm are needed but for interesting examples often one can only derive bounds for some lp norms using the structure of the reaction term they can be used to obtain uniform bounds two propositions are stated which give conditions for this procedure the proofs use the smoothing properties of analytic semigroups and the multiplicative gagliardo nirenberg inequality to illustrate the method we prove global existence of solutions for the brusselator and a volterra lotka system with one diffusing and one sedentary species

on group inverses and the sharp order abstract in this work the group inverse of a matrix is used to define the order on square matrices of index 1 the order is similar to the order of drazin 2 and the minus order of hartwig 6 10 and nambooripad 17 the order and the order are compared and contrasted many conditions are given which assure the equivalence of the various partial orders studied

diagonally range dominant matrices abstract diagonally range dominant drd matrices were introduced and studied by mitra in the present paper which is partly expository mitra s results are reviewed and several additional results on drd matrices are obtained

roots of the canonical bundle of the universal teichmuller curve and certain subgroups of the mapping class group 1 1 throughout this paper let s be a fixed smooth c orientable closed surface of genus g 2 let s be equipped with a preferred complex structure we denote the riemann surface byx0 and the teichmtiller space ofx o by t 0 there is a fuchsian group fg operating in the unit disk so that x o u f o the universal teichmiiller curve v o is a fibre space over teichmtiller space with xt the fibre over te to conformally equivalent to the riemann surface represented by te t 0 each x t is diffeomorphic to s v o is a complex manifold of dimension 3 9 2 and we denote its canonical line bundle by k v0 we define l a holomorphic complex line bundle over vo to be an n th root of k v0 iffl k v0 the purpose of this paper is to study the n ttt roots of k v0 in particular we investigate an action of the teichmtiller modular group mod fg mapping class group on the finite set of n th roots denote the tangent and cotangent bundles of a manifold m by t m and t m respectively to m and t m denote the corresponding bundles with their zero sections removed david mumford observed informal communication that over a single riemann surface x the n th roots correspond to certain homomorphisms 2 hi to x z z he suggested trying to work out the action of mod f 0 in terms of its action on these homomorphisms 1 2 let f s s be a diffeomorphism and its equivalence class in mod fg mod fo acts on v 0 as a group of biholomorphic maps and that action induces by pullback an action on the set of holomorphic complex line bundles which a r e n th roots of k v0 the n tla roots are a finite set of order n 2g the action of mod fg on that set gives a homomorphism mod fg perm n2 where perm n 2g is the permutation group on a set of order n 2g let go denote the kernel of that homomorphism that is the subgroup of mod fo which acts trivially on all n tla roots these groups are of particular interest because they are normal subgroups of finite index in mod fg the study of the action of mod fg o n n th roots leads to the following characterization of these subgroups

monotone iterative methods for numerical solutions of nonlinear integro elliptic boundary problems abstract the aim of this paper is to obtain various monotone iterative schemes for numerical solutions of a class of nonlinear nonlocal reaction diffusion convection equations under linear boundary conditions the boundary value problem under consideration is discretized into a system of nonlinear algebraic equations by the finite difference method and the iterative schemes are given for the finite difference system using upper and lower solutions as the initial iterations the construction of the monotone sequences and the definition of upper and lower solutions depend on the quasimonotone property of the reaction function and the iterative schemes are presented for each of the three types of quasimonotone functions an application of the monotone iterations including some numerical results is given to a modified logistic diffusive equation where the kernel in the integral term may be positive negative or changing sign in its domain

left star and right star partial orderings abstract two partial orderings in the set of complex matrices are introduced by combining each of the conditions a a a b and aa ba which define the star partial ordering with one of the conditions m a m b and m a m b which define the space preordering several properties of these orderings are examined with main emphasis on comparing the new orderings with the star ordering the minus ordering and other related partial orderings moreover some further characterizations of partial orderings in terms of inclusions of appropriate classes of generalized inverses are derived with the main emphasis on characterizations involving reflexive generalized inverses