parameter uniformly convergent exponential spline difference scheme for singularly perturbed semilinear reaction diffusion problems abstract we consider a dirichlet boundary value problem for singularly perturbed semilinear reaction diffusion equation the problem is discretized using an exponential spline difference scheme derived on the basis of splines in tension on piecewise uniform shishkin type mesh the convergence analysis is given and the method is shown to be almost second order accurate in the discrete maximum norm uniformly in the perturbation parameter e numerical experiments are conducted to demonstrate the theoretical results

conformal deformation of a riemannian metric to a scalar flat metric with constant mean curvature on the boundary one of the most celebrated theorems in mathematics is the riemann mapping theorem it says that an open simply connected proper subset of the plane is conformally diffeomorphic to the disk in higher dimensions very few regions are conformally diffeomorphic to the ball however we can still ask whether a domain is conformally diffeomorphic to a manifold that resembles the ball in two ways namely it has zero scalar curvature and its boundary has constant mean curvature in this paper we generalize the riemann mapping theorem to higher dimensions in that sense

on the positivity of the effective action in a theory of random surfaces it is shown that the functional s eta frac 1 24 pi int left frac 1 2 left nabla eta right 2 2 eta right d mu 0 defined onc functions on the two dimensional sphere satisfies the inequalitys 0 if is subject to the constraint int e eta 1 d mu 0 0 the minimums 0 is attained at the solutions of the euler lagrange equations the proof is based on a sharper version of moser trudinger s inequality due to aubin which holds under the additional constraint int e eta xd mu 0 0 this condition can always be satisfied by exploiting the invariance ofs under the conformal transformations ofs2 the result is relevant for a recently proposed formulation of a theory of random surfaces

corner singularities and boundary layers in a simple convection diffusion problem abstract a singularly perturbed convection diffusion problem posed on the unit square is considered its solution may have exponential and parabolic boundary layers and corner singularities may also be present pointwise bounds on the solution and its derivatives are derived the dependence of these bounds on the small diffusion coefficient on the regularity of the data and on the compatibility of the data at the corners of the domain are all made explicit the bounds are derived by decomposing the solution into a sum of solutions of elliptic boundary value problems posed on half planes then analyzing these simpler problems

compactness of conformally compact einstein manifolds in dimension 4 abstract in this paper we establish some compactness results of conformally compact einstein metrics on 4 dimensional manifolds our results were proved under assumptions on the behavior of some local and non local conformal invariants on the compactness of the boundary metrics at the conformal infinity and on the topology of the manifolds

compact isospectral sets of surfaces on etudie des ensembles de surfaces qui sont isospectrales par rapport a l operateur de laplace beltrami

n widths and singularly perturbed boundary value problems using the theory of n widths the approximability of solutions of singularly perturbed linear reaction diffusion problems in two dimensions is quantified

singularly perturbed finite element methods this paper introduces a new piecewise linear finite element which is designed to handle singularly perturbed ordinary differential equations both pointwise and global estimates which are independent of the perturbation parameter are obtained

q curvature flow on 4 manifolds with boundary given a compact four dimensional smooth riemannian manifold m g with smooth boundary we consider the evolution equation by q curvature in the interior keeping the t curvature and the mean curvature to be zero using integral methods we prove global existence and convergence for the q curvature flow to a smooth metric conformal to g of prescribed q curvature zero t curvature and vanishing mean curvature under conformally invariant assumptions

a class of finite element methods for singularly perturbed second order differential equations abstract standard galerkin finite element methods give poor accuracy when applied to second order elliptic problems with a significant convective term an upwind finite element was introduced to overcome this difficulty for constant coefficient problems with zero source term this paper extends the use of this type of element to variable coefficient problems with nonzero source term by introducing a class of generalised upwind elements called comparison upwind finite elements two elements from this class are presented in detail in this paper we obtain nodal error estimates and global l1 and l2 error estimates for both methods finally some numerical results are presented which demonstrate the methods accuracy

the zeta functional determinants on manifolds with boundary abstract in this paper we derive some geometric formulas for the quotient of the zeta functional determinants for certain elliptic boundary value problems on riemannian 3 and 4 manifolds with smooth boundary

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part

on the decoupling of the homogeneous and inhomogeneous parts in inhomogeneous quantum groups we show that if there exists a realization of a hopf algebra h in a h module algebra then one can split their cross product into the tensor product algebra of itself with a subalgebra isomorphic to h and commuting with this result applies in particular to the algebra underlying inhomogeneous quantum groups like the euclidean groups which are obtained as cross products of the quantum euclidean spaces qn with the quantum groups of rotation uq so n of qn for which it has no classical analogue

a finite element analysis of a coupled system of singularly perturbed reaction diffusion equations we consider a system of two coupled reaction diffusion equations when the parameters multiplying the second order derivatives in the equations are small their solutions exhibit boundary layers moreover when the parameters are of different magnitudes two distinct but overlapping boundary layers are present we study a finite element discretization on general layer adapted meshes including the frequently studied shishkin mesh and the bakhvalov mesh supporting numerical results are presented

estimates and extremals for zeta function determinants on four manifolds leta be a positive integral power of a natural conformally covariant differential operator on tensor spinors in a riemannian manifold suppose thata is formally self adjoint and has positive definite leading symbol for example a could be the conformal laplacian yamabe operator l or the square of the dirac operator open image in new window within the conformal class left g e 2w g 0 w in c infty m right of an einstein locally symmetric background metricgo on a compact four manifoldm we use an exponential sobolev inequality of adams to show that bounds on the functional determinant ofa and the volume ofg imply bounds on thew2 2 norm of the conformal factorw provided that a certain conformally invariant geometric constantk k m g o a is strictly less than 32 2 we show for the operatorsl and open image in new window that indeedk 32 2 except when m g o is the standard sphere or a hyperbolic space form on the sphere a centering argument allows us to obtain a bound of the same type despite the fact thatk is exactly equal to 32 2 in this case finally we use an inequality of beckner to show that in the conformal class of the standard four sphere the determinant ofl or of open image in new window is extremized exactly at the standard metric and its images under the conformal transformation groupo 5 1

on the chern gauss bonnet integral for conformal metrics on r4 where m is the euler number ofm later huber hu extended this inequality to metrics with much weaker regularity more importantly he proved that such a surface m is conformally equivalent to a closed surface with finitely many punctures the deficit in formula 1 1 has an interpretation as an isoperimetric constant on a complete and open surface with gaussian curvature absolutely integrable one may represent each end conformally as r2 k for some compact set k we consider the isoperimetric ratio

prescribing q curvature problem on sn let pn be the n th order paneitz operator on sn n 3 we consider the following prescribing q curvature problem on sn pnu n 1 q x enuon sn where q is a smooth positive function on sn satisfying the following non degeneracy condition q 2 q 2 0 let g sn rn 1 be defined by g x q x q x we show that if q 0 is non degenerate and deg g g sn 0 then the above equation has a solution when n is even this has been established in our earlier work j wei x xu on conformal deformation of metrics on sn j funct anal 157 1998 292 325 when n is odd pn becomes a pseudo differential operator here we develop a unified approach to treat both even and odd cases the key idea is to write it as an integral equation and use liapunov schmidt reduction method

a shishkin mesh for a singularly perturbed riccati equation in this paper a singularly perturbed riccati initial value problem is examined parameter explicit bounds on the solution and its derivatives are given a numerical method composed of an implicit difference operator and a piecewise uniform shishkin mesh is constructed a theoretical parameter independently bound on the errors in the numerical approximations is established numerical results are presented which are in agreement with the theoretical error bound

moser trudinger and beckner onofri s inequalities on the cr sphere we derive sharp moser trudinger inequalities on the cr sphere the rst type is in the adams form for powers of the sublaplacian and for general spectrally dened operators on the space of cr pluriharmonic functions we will then obtain the sharp beckner onofri inequality for crpluriharmonic functions on the sphere and as a consequence a sharp logarithmic hardy littlewood sobolev inequality in the form given by carlen and loss

geometrical tools for quantum euclidean spaces we apply one of the formalisms of noncommutative geometry to n q the quantum space covariant under the quantum group so q n over n q there are two so q n covariant differential calculi for each we find a frame a metric and two torsion free covariant derivatives which are metric compatible up to a conformal factor and which have a vanishing linear curvature this generalizes results found in a previous article for the case of 3 q as in the case n 3 one has to slightly enlarge the algebra n q for n odd one needs only one new generator whereas for n even one needs two as in the particular case n 3 there is a conformal ambiguity in the natural metrics on the differential calculi over n q while in our previous article the frame was found by hand here we disclose the crucial role of the quantum group covariance and exploit it in the construction as an intermediate step we find a homomorphism from the cross product of n q with u q so n into n q an interesting result in itself

q analogues of clifford and weyl algebras spinor and oscillator representations of quantum enveloping algebras we introduceq analogues of clifford and weyl algebras using these we construct spinor and oscillator representations of quantum enveloping algebras of typean 1 bn cn dn andan 1 1 also we discuss the irreducibility and the unitarity of these representations

extremals of determinants of laplacians on etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

a sharp inequality of j moser for higher order derivatives in fact go nwl 1 where wnis the area of the surface of the unit n ball also he notes that if 3 exceeds go then 1 can no longer hold with some co independent of the u satisfying 2 the fact that this result holds for some 3 0 independent of u has been observed now by several authors with a variety of proofs see 24 20 23 22 10 4 1 2 13 14 for example the earliest of these arguments expands the exponential function in a power series then observing that the norm in the imbedding

regularity of a fourth order nonlinear pde with critical exponent in this paper we demonstrate the regularity of minimizers for a variational problem a special case of which arises in spectral theory and conformal geometry the associated euler lagrange equation is fourth order semilinear the leading term is the bilaplacian and lower order terms appear at critical powers

group representations arising from lorentz conformal geometry abstract it is shown that there exist conformally covariant differential operators d 2 l k of all even orders 2 l on differential forms of all orders k in the double cover n of the n dimensional compactified minkowski space n these act as intertwining differential operators for natural representations of o 2 n the conformal group of n for even n the resulting decompositions of differential form representations of o 2 n the orthochronous conformal group produce infinite families of unitary representations the most interesting of which are carried by positive mass squared positive frequency quotients for 2l n 2k physically these generalize unitary representations of the conformal group associated with the modified wave operator d 2 0 n 2 2 2 and the maxwell operator on vector potentials d 2 n 2 2 d all the representation spaces produced unitary and nonunitary may be viewed as infinite systems of harmonic oscillators as a by product of the spectral resolution of the d 2 l k one gets some striking wave propagative properties for all of the equations d 2 l k 0 including huygens principle in the curved spacetime n the operators d 2 l k have not been seen before except in the special cases k 0 or n and k n 2 2 l 1 the maxwell operator thus much new information is obtained even in the physical case n 4

unbraiding the braided tensor product we show that the braided tensor product algebra a1 a2 of two module algebras a1 a2 of a quasitriangular hopf algebra h is isomorphic to the ordinary tensor product a1 a2 provided there exists a realization of h within a1 in other words under this assumption we construct a transformation of generators which decouples a1 a2 i e makes them commuting we apply the theorem to the braided tensor product algebras of two or more quantum group covariant quantum spaces deformed heisenberg algebras and q deformed fuzzy spheres

extremal metrics of zeta function determinants on 4 manifolds in conformal geometry the sobolev inequality at a critical exponent has received much attention in particular the determination of the best constants has played a crucial role in the yamabe problem in dimension two the analogous problem deals with the moser trudinger inequality on a compact riemann surface m2 there exists a constant c c m so that f e47w2 c m if f ivw12 1 and f w 0 the connection of this inequality with geometry comes through the zeta functional determinant of the laplacian as defined by ray singer for a riemannian metric g let 0 a1 a2 be the spectrum

extremal metrics for spectral functions of dirac operators in even and odd dimensions abstract let m n g be a closed smooth riemannian spin manifold and denote by its atiyah singer dirac operator we study the variation of riemannian metrics for the zeta function and functional determinant of 2 and prove finiteness of the morse index at stationary metrics and local extremality at such metrics under general i e not only conformal change of metrics in even dimensions which is also a new case for the conformal laplacian the relevant stability operator is of log polyhomogeneous pseudodifferential type and we prove new results of independent interest on the spectrum for such operators we use this to prove local extremality under variation of the riemannian metric which in the important example when m n g is the round n sphere gives a partial verification of branson s conjecture on the pattern of extremals thus det 2 has a local max max min min when the dimension is 4 k 4 k 1 4 k 2 4 k 3 respectively

sobolev trace inequalities of order four we establish sharp trace sobolev inequalities of order four on euclidean d balls for d 4 when d 4 our inequality generalizes the classical second order lebedev milin inequality on euclidean 2 balls our method relies on the use of scattering theory on hyperbolic d balls as an application we characterize the extremal metric of the main term in the log determinant formula corresponding to the conformal laplacian coupled with the boundary robin operator on euclidean 4 balls which surprisingly is not the flat metric on the ball

a new stabilized finite element formulation for scalar convection diffusion problems the streamline and approximate upwind petrov galerkin method abstract a new stabilized and accurate finite element formulation for convection dominated problems is herein developed the basis of the new formulation is the choice of a new upwind function the upwind function chosen for the new method provokes its degeneration into the supg or cau methods depending on the approximate solution s regularity the accuracy and stability of the new formulation for the linear and scalar advection diffusion equation is demonstrated in several numerical examples

a survey of numerical techniques for solving singularly perturbed ordinary differential equations this survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984 2000 the work done in this area during the period 1905 1984 has already been surveyed by the first author of this paper see appl math comput 30 1989 223 for details due to the space constraints we have covered only singularly perturbed one dimensional problems

the euclidean hopf algebra u q e n and its fundamental hilbert space representations the euclidean hopf algebra uq en dual of fun rnq soq 1 n is constructed by realizing it as a subalgebra of the differential algebra diff rnq on the quantum euclidean space rnq in fact the previous realization g fiore commun math phys 169 475 500 1995 of uq 1 so n is extended within diff rnq through the introduction of q derivatives as generators of q translations the fundamental hilbert space representations of uq en turn out to be of highest weight type and rather simple lattice regularized versions of the classical ones the vectors of a basis of the singlet i e zero spin irrep can be realized as normalizable functions on rnq going to distributions in the limit q 1

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

realization of u q so n within the differential algebra on r q n we realize the hopf algebrauq 1 so n as an algebra of differential operators on the quantum euclidean spacer q n the generators are suitableq deformed analogs of the angular momentum components on ordinaryr n the algebra fun r q n of functions onr q n splits into a direct sum of irreducible vector representations ofuq 1 so n the latter are explicitly constructed as highest weight representations

moduli space heights and isospectral sets of plane domains let q be a plane domain of finite connectivity n with smooth boundary and choose a fixed domain 2 of the same type then there exists a flat metric g on 2 such that q is isometric with 29g in what follows we do not distinguish between isometric domains by the spectrum of 29 we mean the spectrum of the laplace beltrami operator ag on 29 with dirichlet boundary conditions the height h eg log det ag is a spectral invariant and plays a central role in this paper among all suitably normalized flat metrics on 2 conformal to a given metric g there is a unique flat metric for which the height is a minimum this metric is characterized by the fact that d 2 has constant geodesic curvature we call such a metric uniform and denote it by u the set of all such metrics is denoted by u 2 we can therefore identify with the moduli space 2 of conformal structures on e for n 3 we introduce a special parametrization for mu 2 by means of which we show that h u oo as u approaches the boundary of fu 2 using this along with the heat invariants for the laplacian we then show that any isospectral set of plane domains is compact in the c topology similar results hold for n 1 and 2

petrov galerkin finite element methods with a hinged test space for singularly perturbed problems on introduit des methodes d elements finis du type de petrov galerkin pour la solution approchee des problemes a deux points limites relatifs a des equations differentielles lineaires du second ordre a perturbations singulieres

q deformed poincare algebra theq differential calculus for theq minkowski space is developed the algebra of theq derivatives with theq lorentz generators is found giving theq deformation of the poincare algebra the reality structure of theq poincare algebra is given the reality structure of theq differentials is also found the real laplacian is constructed finally the comultiplication counit and antipode for theq poincare algebra are obtained making it a hopf algebra

hilbert space representations of cross product algebras in this paper we study and classify hilbert space representations of cross product ast algebras of the quantized enveloping algebra user1 mathcal u q left text e text 2 right with the coordinate algebras user1 mathcal o left text e q left 2 right right of the quantum motion group and user1 mathcal o left mathbb c q right of the complex plane and of the quantized enveloping algebra user1 mathcal u q left text su 1 1 right with the coordinate algebras user1 mathcal o left text su q left 1 1 right right of the quantum group mathrm su q 1 1 and user1 mathcal o left text u p right of the quantum disc invariant positive functionals and the corresponding heisenberg representations are explicitly described

algebraic topological methods for the supercritical q curvature problem abstract we study the problem of existence of conformal metrics with prescribed q curvature on closed four dimensional riemannian manifolds this problem has a variational structure and in the case of interest here it is noncompact in the sense that accumulations points of some noncompact flow lines of a pseudogradient of the associated euler lagrange functional the so called true critical points at infinity of the associated variational problem occur using the characterization of the critical points at infinity of the associated variational problem which is established in 42 combined with some arguments from morse theory some algebraic topological methods and some tools from dynamical system originating from conley s isolated invariant sets and isolated blocks theory we derive a new kind of existence results under an algebraic topological hypothesis involving the topology of the underling manifold stable and unstable manifolds of some of the critical points at infinity of the associated euler lagrange functional

maximum norm error analysis of a 2d singularly perturbed semilinear reaction diffusion problem a semilinear reaction diffusion equation with multiple solutions is considered in a smooth two dimensional domain its diffusion parameter e 2 is arbitrarily small which induces boundary layers constructing discrete sub and super solutions we prove existence and investigate the accuracy of multiple discrete solutions on layer adapted meshes of bakhvalov and shishkin types it is shown that one gets second order convergence with in the case of the shishkin mesh a logarithmic factor in the discrete maximum norm uniformly in e for 0 is the maximum side length of mesh elements while the number of mesh nodes does not exceed ch 2 numerical experiments are performed to support the theoretical results

compactness of conformally compact einstein 4 manifolds ii abstract in this paper we establish a number of compactness results for some class of conformally compact einstein 4 manifolds in the first part of the paper we improve the earlier compactness results obtained by chang ge 15 in the second part of the paper as applications we derive some further compactness results under the perturbation condition when the l 2 norm of the weyl curvature is small we also derive the global uniqueness of conformally compact einstein metrics on the 4 ball constructed in the earlier work of graham lee 31

a priori estimates and analysis of a numerical method for a turning point problem bounds are obtained for the derivatives of the solution of a turning point problem these results suggest a modification of the el mistikawy werle finite difference scheme at the turning point a uniform error estimate is obtained for the resulting method and illustrative numerical results are given i introduction we will examine the following two point boundary value problem with dirichlet data at the endpoints l la ly eyx x p x yx x q x y x f x for i x 1