structure of some noetherian si rings abstract we describe the structure of rings over which every cyclic or finitely generated right module is a direct sum of a projective module and an injective module

boundary behavior of solutions to some singular elliptic boundary value problems abstract let be a bounded domain with smooth boundary in r n n 1 for the more general weight b some nonlinearities f and singularities g by two kinds of nonlinear transformations a new perturbation method which was introduced by garcia melian in j garcia melian boundary behavior of large solutions to elliptic equations with singular weights nonlinear anal 67 2007 818 826 and comparison principles we show that the boundary behavior of solutions to a boundary blow up elliptic problem w b x f w w 0 x w and a singular dirichlet problem u b x g u u 0 x u 0 has the same form under the nonlinear transformations which can be determined in terms of the inverses of the transformations

the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

boundary regularity of conformally compact einstein metrics we show that c conformally compact riemannian einstein metrics have conformal compactifications that are smooth up to the boundary in dimension 3 and all even dimensions and polyhomogeneous in odd dimensions greater than 3

existence of solutions and boundary asymptotic behavior of p r laplacian equation multi point boundary value problems abstract this paper investigates the problem p r u r f r u 0 r 0 r u 0 i 1 m i u i e 0 u r as r r where p r u u p r 2 u is called the p r laplacian r is a singular coefficient the existence and nonexistence of solutions are discussed and the exact boundary blow up rate of solutions is given

asymptotic integration of some nonlinear differential equations with fractional time derivative we establish that under some simple integral conditions regarding the nonlinearity the 1 order fractional differential equation 0d t x f t x 0 t 0 has a solution with which can be expanded asymptotically as a bt o t 1 when t for given real numbers a b our arguments are based on fixed point theory here 0d t designates the riemann liouville derivative of order 0 1

boundary behavior in the loewner nirenberg problem abstract let r n be a bounded domain of class c 2 0 1 we show that if n 3 and u is the maximal solution of equation u n n 2 u n 2 n 2 in then the hyperbolic radius v u 2 n 2 is of class c 2 up to the boundary the argument rests on a reduction to a nonlinear fuchsian elliptic pde

painleve analysis of the non linear schrodinger family of equations the authors apply the painleve test to the generalised derivative non linear schrodinger equation iut uxx iauu ux ibu2ux cu3u 2 where u denotes the complex conjugate of u and a b and c are real constants to determine under what conditions the equation might be completely integrable it is shown that apart from a trivial multiplicative factor this equation possesses the painleve property for partial differential equations as formulated by weiss tabor and carnevale 1983 only if c 1 4b 2b a when then this relation holds this is equivalent under a gauge transformation to the derivative non linear schrodinger equation dnls of kaup and newell which is known to be completely integrable or else to a linear equation

existence nonexistence and asymptotic behavior of boundary blow up solutions to p x laplacian problems with singular coefficient abstract this paper investigates the problem p x u x f x u 0 in u x as d x 0 where p x u d i v u p x 2 u is called the p x laplacian and x is a singular coefficient the existence and nonexistence of boundary blow up solutions is discussed and the asymptotic behavior of boundary blow up solutions is given in particular we do not assume radial symmetric conditions and the pointwise different exact blow up rate of solutions has been discussed

the structure of wtc expansions and applications we construct generalized painleve expansions with logarithmic terms for a general class of non integrable scalar equations and describe their structure in detail these expansions were introduced without logarithms by weiss tabor carnevale wtc 1983 the construction of the formal solutions is shown to involve semi invariants of binary forms and tools from invariant theory are applied to the determination of the type of logarithmic terms that are required for the most general singular series the structure of the series depends strongly on whether 1 is or is not a resonance the convergence of these series is obtained as a consequence of the general results of littman and kichenassamy 1993 the results are illustrated on a family of fifth order models for water waves and other examples we also give necessary and sufficient conditions for 1 to be a resonance

mechanics with fractional derivatives lagrangian and hamiltonian mechanics can be formulated to include derivatives of fractional order f riewe phys rev 53 1890 1996 lagrangians with fractional derivatives lead directly to equations of motion with nonconservative classical forces such as friction the present work continues the development of fractional derivative mechanics by deriving a modified hamilton s principle introducing two types of canonical transformations and deriving the hamilton jacobi equation using generalized mechanics with fractional and higher order derivatives the method is illustrated with a frictional force proportional to velocity in contrast to conventional mechanics with integer order derivatives quantization of a fractional derivative hamiltonian cannot generally be achieved by the traditional replacement of momenta with coordinate derivatives instead a quantum mechanical wave equation is proposed that follows from the hamilton jacobi equation by application of the correspondence principle

the fractional finite hankel transform and its applications in fractal space in the present work a generalized finite hankel transform is derived which is useful in solving equations in fractal dimension df and involving a fractal diffusion coefficient d0r the corresponding inversion formula is established and some properties are given then the transform is successfully used to solve a class of time fractional diffusion equations in fractional spatial dimension with an absorbent term and schrodinger equation in fractional dimensional space green s functions and exact wave function of the above problems are found

boundary behaviors for liouville s equation in planar singular domains abstract we study asymptotic behaviors near the boundary of complete metrics of constant curvature in planar singular domains and establish an optimal estimate of these metrics by the corresponding metrics in tangent cones near isolated singular points on boundary the conformal structure plays an essential role

constructing bach flat manifolds of signature 2 2 using the modified riemannian extension we use the modified riemannian extension of an affine surface to construct bach flat manifolds as all these examples have vanishing scalar invariants we shall construct scalar invariants which are not of weyl type to distinguish them we illustrate these phenomena in the context of homogeneous affine surfaces we use the modified riemannian extension of an affine surface to construct bach flat manifolds as all these examples have vanishing scalar invariants we shall construct scalar invariants which are not of weyl type to distinguish them we illustrate these phenomena in the context of homogeneous affine surfaces

necessary and sufficient conditions for finite time singularities in ordinary differential equations abstract this paper gives a theorem which provides necessary and sufficient conditions for the existence of a real finite value of the independent variable at which the general solution of a certain class of ordinary differential equations diverges to infinity this class is a large subset of the set of all autonomous non linear polynomial ordinary differential equation these conditions involve the asymptotic form of the local series representation for the general solutions around the singularities and can be checked algorithmically

a local existence result for poincare einstein metrics abstract given a closed riemannian manifold m g m of dimension n 3 we prove the existence of a conformally compact einstein metric g defined on a collar neighborhood m 0 1 whose conformal infinity is g m

invariant modules and the reduction of nonlinear partial differential equations to dynamical systems abstract we completely characterize all nonlinear partial differential equations leaving a given finite dimensional vector space of analytic functions invariant existence of an invariant subspace leads to a reduction of the associated dynamical partial differential equations to a system of ordinary differential equations and provides a nonlinear counterpart to quasi exactly solvable quantum hamiltonians these results rely on a useful extension of the classical wronskian determinant condition for linear independence of functions in addition new approaches to the characterization of the annihilating differential operators for spaces of analytic functions are presented

harmonic radius and concentration of energy hyperbolic radius and liouville s equations d u e u and d u u n 2 n 2 the conformal radius of a simply connected planar domain omega is defined as r x 1 f x where f denotes a riemann map from omega to the unit disk mapping the given point x to the origin beyond geometric function theory the conformal radius has various applications in partial differential equations therefore it is desirable to extend it to multiply connected and higher dimensional domains we discuss two natural extensions 1 the harmonic radius is defined in terms of the green s function its maximum points the harmonic centers play an important role in variational problems involving concentration of energy they are of the following form sup left int limits omega f u u in h 0 1 omega int omega nabla u 2 leqslant varepsilon 2 right as varepsilon to 0 the solutions concentrate at a harmonic center our examples include bernoulli s free boundary problem the plasma problem and the elastic membrane with a movable obstacle an efficient method

controllability for a class of fractional neutral integro differential equations with unbounded delay abstract in this paper we consider a class of fractional neutral integro differential equations with unbounded delay in banach spaces this paper deals with the exact controllability for fractional integro differential neutral control systems first we establish a new set of sufficient conditions for the exact controllability of nonlinear fractional systems by using a fixed point analysis approach further we extend the result to study the exact controllability concept with nonlocal conditions examples are provided to illustrate our main results

boundary blow up and degenerate equations let r2 be a bounded domain of class c2 0 1 we show that if u is the solution of u 4exp 2u which tends to as x y then the hyperbolic radius v exp u is also of class c2 up to the boundary the proof relies on new schauder estimates for degenerate elliptic equations of fuchsian type

on a conjecture of fefferman and graham abstract in 1985 fefferman and graham have constructed a local embedding of an arbitrary real analytic manifold of odd dimension n into a ricci flat manifold of dimension n 2 admitting a homothety they conjectured that their result remains valid in even dimensions if logarithms are allowed in the expansion of the metric in this paper we i prove that such expansions exist and converge and ii establish the degree of non uniqueness of the solution in terms of the coefficients in the expansion

mathscr h space with a cosmological constant it is demonstrated that a real analytic 3 manifold with riemannian conformal metric is naturally the conformal infinity of a germ unique real analytic 4 manifold with real analytic riemannian metric satisfying the self dual einstein equations with cosmological constant 1 moreover this result holds if riemannian is replaced in the first case by lorentzian i e signature and in the second case by pseudo riemannian with signature or if real analytic is replaced by complex analytic and 9riemannian is replaced by holomorphic this provides a cosmological constant analogue of newman s space construction newman 1976 1977

the analytic structure of dynamical systems and self similar natural boundaries abstract in this paper we investigate the analytic complex time structure of the movable singularities for several dynamical systems in general it is found that there exists a direct connection between the occurencce of a certain type of multiple valuedness of the singularities and the existence of a class of remarkable self similar natural boundaries for these systems an asymptotic description of the distribution of singularities in the natural boundary is developed this provides a description of the fine scale structure of these natural boundaries that agrees closely with the numerical calculations

integrating the nonintegrable analytic structure of the lorenz system revisited abstract a study of the complex time analytic structure of the lorenz system in nonintegrable parameter regimes reveals the special sets of parameter values for which one time dependent integral of motion exists furthermore the analysis yields the exact form of the part of the integral with the highest homogeneous weight and a method to construct the rest of the integral recursive clustering of singularities in the chaotic regimes of the system is observed in computer studies and explained by a simple analytic argument the analytic techniques used in these studies a systematic resummation of a logarithmic psi series appears to be quite general and can provide explicit representations of a solution even in the chaotic regimes in the neighborhood of a given movable singularity furthermore we suggest that this technique provides a type of renormalization program to study a wide class of nonintegrable systems

analytic structure of the henon heiles hamiltonian in integrable and nonintegrable regimes the solutions of the henon heiles hamiltonian are investigated in the complex time plane the use of the painleve property i e the property that the only movable singularities exhibited by the solution are poles enables successful prediction of the values of the nonlinear coupling parameter for which the system is integrable special attention is paid to the structure of the natural boundaries that are found in some of the nonintegrable regimes these boundaries have a remarkable self similar structure whose form changes as a function of the nonlinear coupling

hodograph transformations of linearizable partial differential equations in this paper an algorithmic method is developed for transforming quasilinear partial differential equations of the form u t g u u nx f u u x cdots u n 1 x u mx equiv partial m u partial x m where dg du not equiv 0 into semilinear equations i e equations of the above form with g u 1 this crucially involves the use of hodograph transformations i e transformations involving the interchange of dependent and independent variables furthermore the most general quasilinear equation of the above form is found that can be mapped via a hodograph transformation to a semilinear form this algorithm provides a method for establishing whether a given quasilinear equation is linearizable i e is solvable in terms of either a linear partial differential equation or of a linear integral equation in particular this method is used to show how the painleve tests may be applied to quasilinear equations this appears to resolve the problem that solutions of lineariz

optimal boundary regularity for nonlinear singular elliptic equations abstract in this paper we prove the optimal boundary regularity under natural structural conditions for a large class of nonlinear elliptic equations with singular terms near the boundary by a careful construction of super and sub solutions we obtain precise growth estimates for solutions at the boundary and reduce the boundary regularity to the interior one by a rescaling argument

convexity of solutions of semilinear elliptic equations on considere le probleme elliptique suivant u f u dans u m sur ou est un domaine convexe de r 2 et m est soit une constante reelle soit on etablit que pour une bonne fonction monotone g t pout toute solution u on a g u est strictement convexe dans

aspects of the painleve property for partial differential equations abstract aspects of the painleve property are reviewed special attention is paid to the connection with hirota s method the formulation of a painleve type property suitable for differential difference equations and certain results concerning the self dual yang mills equations

ambient metrics for n dimensional pp waves we provide an explicit formula for the fefferman graham ambient metric of an n dimensional conformal pp wave in those cases where it exists in even dimensions we calculate the obstruction explicitly furthermore we describe all 4 dimensional pp waves that are bach flat and give a large class of bach flat examples which are conformally cotton flat but not conformally einstein finally as an application we use the obtained ambient metric to show that even dimensional pp waves have vanishing critical q curvature

the thomas fermi von weizsacker theory of atoms and molecules we place the thomas fermi von weizsacker model of atoms on a firm mathematical footing we prove existence and uniqueness of solutions of the thomas fermi von weizsacker equation as well as the fact that they minimize the thomas fermi von weizsacker energy functional moreover we prove the existence of binding for two very dissimilar atoms in the frame of this model

painleve analysis and normal forms theory nonlinear vector fields have two important types of singularities the fixed points in phase space and the time singularities in the complex plane the first singularities are locally analyzed via normal form theory whereas the second ones are studied by the painleve analysis in this paper normal form theory is used to describe the solutions around their complex time singularities to do so a transformation mapping the local series around the singularities to the local series around a fixed point of a new system is introduced regular normal form theory is then used in this new system it is shown that a vector field has the painleve property only if the associated system is locally linearizable around its fixed points a problem analogous to the classical problem of the center moreover the connection between partial and complete integrability and the structure of local series around both types of singularities are established a new proof of the convergence of the local psi series is given and an explicit method to prove the existence of finite time blow up manifold in phase space is presented

gradient bounds for elliptic problems singular at the boundary let be a bounded smooth domain in r n n geqq 2 and let us denote by d x the distance function d x we study a class of singular hamilton jacobi equations arising from stochastic control problems whose simplest model is alpha delta u u frac nabla u cdot b x d x c x nabla u 2 f x quad rm in omega where f belongs to w 1 infty rm loc omega and is possibly singular at partial omega c in w 1 infty omega with no sign condition and the field b in w 1 infty omega n has an outward direction and satisfies b cdot nu geqq alpha at is the outward normal despite the singularity in the equation we prove gradient bounds up to the boundary and the existence of a globally lipschitz solution we show that in some cases this is the unique bounded solution we also discuss the stability of such estimates with respect to as vanishes obtaining lipschitz solutions for first order problems with similar features the main tool is a refined weighted version of the classical bernstein method to get gradient bounds the key role is played here by the orthogonal transport component of the hamiltonian

instability of pole singularities for the chazy equation we prove that the negative resonances of the chazy equation in the sense of painleve analysis can be related directly to its group invariance properties these resonances indicate in this case the instability of pole singularities depending on the value of a parameter in the equation an unstable isolated pole may turn into the familiar natural boundary or split into several isolated singularities in the first case a convergent series representation involving exponentially small corrections can be given this reconciles several earlier approaches to the interpretation of negative resonances on the other hand we also prove that pole singularities with the maximum number of positive resonances are stable the proofs rely on general properties of nonlinear fuchsian equations

a boundary blow up elliptic problem with an inhomogeneous term abstract by a perturbation method and constructing comparison functions we reveal how the inhomogeneous term h affects the exact asymptotic behaviour of solutions near the boundary to the problem u b x g u h x u 0 in u where is a bounded domain with smooth boundary in r n 0 g c 1 0 is increasing on 0 g 0 0 g is regularly varying at infinity with positive index the weight b which is non trivial and non negative in may be vanishing on the boundary and the inhomogeneous term h is non negative in and may be singular on the boundary

rings whose modules have maximal or minimal injectivity domains in a recent paper alahmadi alkan and lopez permouth defined a module m to be poor if m is injective relative only to semisimple modules and a ring to have no right middle class if every right module is poor or injective we prove that every ring has a poor module and characterize rings with semisimple poor modules next a ring with no right middle class is proved to be the ring direct sum of a semisimple artinian ring and a ring t which is either zero or of one of the following types i morita equivalent to a right pci domain ii an indecomposable right si ring which is either right artinian or a right v ring and such that soc tt is homogeneous and essential in tt and t has a unique simple singular right module or iii an indecomposable right artinian ring with homogeneous right socle coinciding with the jacobson radical and the right singular ideal and with unique non injective simple right module in case iii either tt is poor or t is a qf ring with j t 2 0 converses of these cases are discussed it is shown in particular that a qf ring r with j r 2 0 and homogeneous right socle has no middle class

on the asymptotic integration of nonlinear differential equations the aim of the present paper is twofold firstly the paper surveys the literature concerning a specific topic in asymptotic integration theory of ordinary differential equations the class of second order equations with bihari like nonlinearity secondly some general existence results are established with regard to a condition that has been found recently to be of significant use in the theory of elliptic partial differential equations

nonsingular semiperfect cs rings in this paper the structure of right nonsingular semiperfect right cs rings has been considered under certain conditions these rings are shown to be direct sums of triangular blocked matrix rings over ore domains as a by product it follows that then nmatrix ring over a local domaind n 1 is a right cs ring if and only ifdis a right and left valuation domain

explicit ambient metrics and holonomy we present three large classes of examples of conformal structures whose fefferman graham ambient metrics can be found explicitly our method for constructing these examples rests upon a set of sufficiency conditions under which the fefferman graham equations are assured to reduce to a system of inhomogeneous linear partial differential equations our examples include conformal pp waves and more importantly conformal structures that are defined by generic co rank 3 distributions in dimensions 5 and 6 our examples illustrate various aspects of the ambient metric construction the holonomy algebras of our ambient metrics are studied in detail in particular we exhibit a large class of metrics with holonomy equal to the exceptional non compact lie group operatorname g 2 as well as ambient metrics with holonomy contained in operatorname spin 4 3

a simple global characterization for normal forms of singular vector fields we derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems using an appropriate scalar product in the space of homogeneous vector polynomials we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator this leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form explicit examples and results obtained when there are additional symmetries are also presented

holographic formula for q curvature ii we extend the holographic formula for the critical q curvature in graham and juhl 2007 9 to all q curvatures moreover we confirm a conjecture of juhl 2009 11

global existence of solutions with prescribed asymptotic behavior for second order nonlinear differential equations where the nonlinearity f satis9es certain monotonicity conditions speci9ed in the sequel we always assume without mentioning that the function f t u v is continuous in the domain d t u v t t0 u v r where t0 1 it is supposed also that all equations and inequalities involving t hold for t t0 unless otherwise speci9ed the symbols o and o have their standard meanings as t and we use the notation r def 0 eq 1 is often used for mathematical modelling of various physical chemical and biological systems and attracts constant interest of researchers a great deal of papers published during the last three decades are concerned with local and global existence of solutions of eq 1 and its numerous particular cases uniqueness of solutions

on the solution set for a class of sequential fractional differential equations we establish here that under some simple restrictions on the functional coefficient a t the solution set of the fractional differential equation 0d tx a t x 0 splits between eventually small and eventually large solutions as t where 0d t designates the riemann liouville derivative of the order 0 1

on the boundary blow up solutions of p x laplacian equations with singular coefficient abstract this paper investigates the problem p x u x f x u 0 in u x as d x 0 where p x u div u p x 2 u is called the p x laplacian x is a singular coefficient the existence and nonexistence of boundary blow up solutions are discussed and the singularity of the boundary blow up solution is given

on the asymptotic integration of a class of sublinear fractional differential equations we estimate the growth in time of the solutions to a class of nonlinear fractional differential equations d0 x x0 f t x which includes d0 x x0 h t x with 0 1 for the case of slowly decaying coefficients h the proof is based on the triple interpolation inequality on the real line and the growth estimate reads as x t o ta when t for 1 1 a 0 our result can be thought of as a noninteger counterpart of the classical bihari asymptotic integration result for nonlinear ordinary differential equations by a carefully designed example we show that in some circumstances such an estimate is optimal

a perturbative painleve approach to nonlinear differential equations abstract we further improve the painleve test so that negative indices resonances can be treated we demand single valuedness not only for any pole like expansion as in the usual painleve test but also for every solution close to it represented as a perturbation series in a small parameter e order zero is the usual test order one already treated in a preliminary paper reduces to a linear fuchs analysis near a regular singularity and allows the introduction of all missing arbitrary coefficients higher orders lead to the analysis of a linear fuchsian type inhomogeneous equation we obtain an infinite sequence of necessary conditions for the absence of movable logarithmic branch points arising at every integer index whether positive or negative and at every order those arising at negative indices including 1 are new while some conditions may not arise before some high perturbation order we present several illustrative examples we discuss the understanding of negative indices and conclude that they are indistinguishable from positive indices just as in the fuchs theory in particular negative indices give rise to doubly infinite laurent series