invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

conformal blocks and generalized theta functions let yp be the moduli space of rank r vector bundles with trivial determinant on a riemann surface x this space carries a natural line bundle the determinant line bundle s we describe a canonical isomorphism of the space of global sections of sk with the space of conformal blocks defined in terms of representations of the lie algebra 5 c z it follows in particular that the dimension of h yk is given by the verlinde formula

an asymptotic and numerical description of self similar blow up in quasilinear parabolic equations we study the blow up behaviour of two reaction diffusion problems with a quasilinear degenerate diffusion and a superlinear reaction we show that in each case the blow up is self similar in contrast to the linear diffusion limit of each in which the diffusion is only approximately self similar we then investigate the limit of the self similar behaviour and describe the transition from a stable manifold blow up behaviour for quasilinear diffusion to a centre manifold one for the linear diffusion

infinite grassmannians and moduli spaces of g bundles let c be a smooth projective irreducible algebraic curve over c of any genus and g a connected simply connected simple affine algebraic group over c in this paper we elucidate the relationship between 1 the space of vacua conformal blocks defined in conformal field theory using an integrable highest weight representation of the affine kac moody algebra associated to g and 2 the space of regular sections generalised theta functions of a line bundle on the moduli space ta of semistable principal g bundles on c fix a point p in c and let p resp kp be the completion of the local ring p of c at p resp the quotient field of c p let g kp the kp rational points of the algebraic group g be the loop group of g and let g p be the standard maximal parahoric subgroup of then the generalised flag variety x n is an inductive limit of projective varieties in fact of generalised schubert varieties one has a natural r line bundle s on x cf sect 2 2 and the picard group pic x is isomorphic to z which is generated by s proposition 2 3 where f is the universal central extension of by the multiplicative group c cf sect 2 2 by an analogue of the borel weil theorem proved in the kac moody setting by kumar and also by mathieu the space h 162 s of the regular sections of the line bundle y dxo y xo for any d 0 is canonically isomorphic with the full vector space dual v dxo of the integrable highest weight irreducible module v dxo of the affine kacmoody group or of the associated affine kac moody algebra which is a certain one dimensional central extension of the loop algebra lie g with highest weight dxo cf sect 6 1

semistable bundles over an elliptic curve abstract we identify the moduli space m n d of semistable bundles of rank n and degree d on an elliptic curve c as a symmetric product of the curve its dimension being the greatest common divisor of n and d under this identification the determinant map det m n d corresponds to the abel jacobi map shc jh so that the fixed determinant moduli space is identified with a projective space we then describe the brill noether loci theta divisors and picard groups and compute the space of theta functions for both the full moduli space and the fixed determinant moduli space of any rank and any degree we extend to elliptic curves a theorem of drezet and narasimhan on the picard group of moduli spaces as well as two formuulas of beauville narasimhan and ramanan on theta functions of level one our results verify in particular for an elliptic curve a formula of e verlinde and an equality first noticed by bott and szenes relating theta functions over the full moduli space and over the fixed determinant moduli space

refined asymptotics for the blowup of ut δu up this work is concerned with positive blowing up solutions of the semilinear heat equation ut u up in rn our main contribution is a sort of center manifold analysis for the equation in similarity variables leading to refined asymptotics for u in a backward space time parabola near any blowup point we also explore a connection between the asymptotics of u and the local geometry of the blowup set

the boundary contour method for magneto electro elastic media with quadratic boundary elements abstract this paper presents a development of the boundary contour method bcm for magneto electro elastic media first the divergence free of the integrand of the magneto electro elastic boundary element is proved second the boundary contour method formulations are obtained by introducing quadratic shape functions and green s functions ding h j jiang a m 2004 a boundary integral formulation and solution for 2d problems in magneto electro elastic media computers and structures 82 20 21 1599 1607 for magneto electro elastic media and using the rigid body motion solution to regularize the bcm and avoid computation of the corner tensor the bcm is applied to the problem of magneto electro elastic media finally numerical solutions for illustrative examples are compared with exact ones the numerical results of the bcm coincide very well with the exact solution and the feasibility and efficiency of the method are verified

numerical simulation of singular solutions to the two dimensional cubic schrodinger equation numerical simulations of the cubic nonlinear schrodinger equation are presented in dimension n 2 the emphasis is on a detailed mechanism of blow up the numerical results indicate that the blow up is not restricted to the case where the problem is posed in the entire r2 space but also occurs with periodic boundary conditions the structure of the solution as the singular time t is approached has been investigated using examples with and without radial symmetry and or periodicity in most cases we observe that the amplitude develops something like the peaked self similar profile predicted by zakharov and synakh 38 which has a t t 2 3 maximum amplitude and a t t 2 3 width

the quantization conjecture revisited a strong version of the quantization conjecture of guillemin and sternberg is proved for a reductive group action on a smooth compact polarized variety x l the cohomologies of l over the git quotient x g equal the invariant part of the cohomologies over x this generalizes the theorem of gs on global sections and strengthens its subsequent extensions jk li to riemannroch numbers remarkable by products are the invariance of cohomology of vector bundles over x g under a small change in the defining polarization or under shift desingularization as well as a new proof of boutot s theorem also studied are equivariant holomorphic forms and the equivariant hodgeto de rham spectral sequences for x and its strata whose collapse is shown one application is a new proof of the borel weil bott theorem of ti for the moduli stack of c bundles over a curve and of analogous statements for the moduli stacks and spaces of bundles with parabolic structures collapse of the hodge to de rham sequences for these stacks is also shown

numerical analysis of the rescaling method for parabolic problems with blow up in finite time abstract in this work we study the numerical solution for parabolic equations whose solutions have a common property of blowing up in finite time and the equations are invariant under the following scaling transformation u u x t 2 p 1 u x 2 t for that purpose we apply the rescaling method proposed by berger and kohn 1988 to such problems the convergence of the method is proved under some regularity assumption some numerical experiments are given to derive the blow up profile verifying henceforth the theoretical results

a proposed stepwise regression method for model structure identification this paper proposes a new methodology for constructing groundwater models the proposed methodology which determines simultaneously both model structure and model parameters is based on the following ideas 1 when solving the inverse problem different model structures always produce different model parameters 2 since the number of possible model structures of an aquifer is infinite the number of possible representative parameters is also infinite 3 to obtain a set of appropriate representative model parameters we must have an appropriate model structure and 4 an appropriate model structure should be determined not only by observation data and prior information but also by the accuracy requirements of model applications in this proposed methodology we start with a homogeneous model structure and step by step gradually increase the complexity of the model structure at each level of complexity we calculate not only the fitting residual of parameter identification but also the error of model structure to determine if a more complex model structure is needed the model structure error of using one model structure to replace another model structure is defined by a maximum minimum max min problem that is based on the distance between the two models and is measured in parameter observation and prediction or decision spaces this proposed methodology is used to solve a hypothetical remediation design problem in which the true hydraulic conductivity is a random field with a certain trend we have found that for the example problem virtually identical pumping policy is obtained when a five zone model with an optimized zonation pattern is used to represent the nonstationary random field we have also found that observation errors have minimum impact on management solution in comparison with structure errors to calculate the model structure error for this example the inverse solution is coupled with a management problem we have also developed an effective iteration method to handle nonlinear water quality constraints

discrete maximal regularity and the finite element method for parabolic equations maximal regularity is a fundamental concept in the theory of partial differential equations in this paper we establish a fully discrete version of maximal regularity for parabolic equations on a polygonal or polyhedral domain varomega we derive various stability results in the discrete l p 0 t l q varomega norms for the finite element approximation with the mass lumping to the linear heat equation our method of analysis is an operator theoretical one using pure imaginary powers of operators and might be a discrete version of the result of dore and venni as an application optimal order error estimates in those norms are proved furthermore we study the finite element approximation for semilinear heat equations with locally lipschitz continuous nonlinear terms and offer a new method for deriving optimal order error estimates some interesting auxiliary results including discrete gagliardo nirenberg and sobolev inequalities are also presented

on the natural line bundle on the moduli space of stable parabolic bundles we construct the natural holomorphic line bundle on the moduli space of stable parabolic bundles on a compact marked riemann surface which is the prequantum line bundle for the chern simons gauge theory the fusion rule in the chern simons gauge theory can be viewed as the existence condition of this line bundle

a structure preserving scheme for the kolmogorov fokker planck equation in this paper we introduce a numerical scheme which preserves the behavior of solutions to the kolmogorov equation as time tends to infinity the method presented is based on a self similar change of variables technique to transform the kolmogorov equation into a new form such that the problem of designing structure preserving schemes for the original equation amounts to building a standard scheme for the transformed equation this transformation also has the added benefit of allowing for an exact operator splitting scheme whereas in the original form a standard operator splitting was only second order finally we verify the preservation of long time behavior through numerical simulations

on uniqueness of recovery of a discontinuous conductivity coefficient on considere le probleme de retrouver le coefficient a de l equation elliptique div a u 0 dans un domaine avec la condition aux limites u sur quand u n est donnee pour tout regulier on demontre l unicite pour a a 0 b discontinu ou est la fonction indicateur d un ensemble ouvert a o est une fonction c 2 donnee et b est une fonction c 2 inconnue

a rescaling algorithm for the numerical calculation of blowing up solutions on presente un algorithme a invariance d echelle pour le calcul numerique des solutions a singularites explosives des equations d evolution non lineaires

flat connections and geometric quantization using the space of holomorphic symmetric tensors on the moduli space of stable bundles over a riemann surface we construct a projectively flat connection on a vector bundle over teichmuller space the fibre of the vector bundle consists of the global sections of a power of the determinant bundle on the moduli space both dolbeault and cech techniques are used

blowup in a partial differential equation with conserved first integral a reaction diffusion equation with a nonlocal term is studied the nonlocal term acts to conserve the spatial integral of the unknown function as time evolves such equations give insight into biological and chemical problems where conservation properties predominate the aim of the paper is to understand how the conservation property affects the nature of blowup the equation studied has a trivial steady solution that is proved to be stable existence of nontrivial steady solutions is proved and their instability established numerically blowup is proved for sufficiently large initial data by using a comparison principle in fourier space the nature of the blowup is investigated by a combination of asymptotic and numerical calculations

groupe de picard des varietes de modules de fibres semi stables sur les courbes algebriques soient x une courbe algebrique projective lisse de genre g 2 sur r d des entiers avec r 2 on note u r d la variete de modules des fibres algebriques semi stables sur x de rang r et de degre d us r d l ouvert de u r d correspondant aux fibres stables on sait que u r d est une variete algebrique projective irreductible et normale si r et d ne sont pas premiers entre eux u r d n est pas lisse sauf dans une exception le cas ou g 2 r 2 et d est pair on supposera dans cet article qu on n est pas dans ce cas on a alors codimu r d u r d us r d 2 et u r d us r d est le lieu des points singuliers de u r d si l est un fibre en droites de degre d sur x on note u r l la sous variete fermee de u r d correspondant aux fibres vectoriels de determinant isomorphe a l on etudie dans cet article les groupes de picard pic u r d et pic u r l on montre d abord que les varietes u r d et u r l sont localement factorielles on calcule ensuite pic u r d et pic u r l qui sont decrits au moyen des diviseurs thetas generalises

on the adaptive numerical solution of nonlinear partial differential equations in wavelet bases this work develops fast and adaptive algorithms for numerically solving nonlinear partial differential equations of the formut lu nf u where l and n are linear differential operators andf u is a nonlinear function these equations are adaptively solved by projecting the solutionuand the operators l and n into a wavelet basis vanishing moments of the basis functions permit a sparse representation of the solution and operators using these sparse representations fast and adaptive algorithms that apply operators to functions and evaluate nonlinear functions are developed for solving evolution equations for a wavelet representation of the solutionuthat containsnssignificant coefficients the algorithms update the solution usingo ns operations the approach is applied to a number of examples and numerical results are given

symmetric singularity formation in lubrication type equations for interface motion fourth order degenerate diffusion equations arise in a lubrication approximation of a thin film or neck driven by surface tension numerical studies of the lubrication equation le h t h n h xxx x 0 with various boundary conditions indicate that singularity formation in which h x t t to 0 occurs for small enough n with anomalous or second type scaling inconsistent with usual dimensional analysis this paper considers locally symmetric or even singularities in the le and in the modified lubrication equation mle h t h n h xxxx 0 both equations have the property that entropy bounds forbid finite time singularities when n is sufficiently large power series expansions for local symmetric similarity solutions are proposed for equation le with n 1 and mle for all n in mathbb r in the latter case special boundary conditions that force singularity formation are required to produce singularities when n is large matching conditions at higher order terms in the

stability and spectra of blow up in problems with quasi linear gradient diffusivity we study a nonlinear diffusion equation which is invariant under a stretching group of transformations and which reduces to a linear diffusion equation in the limit of 0 we show that if 0 then the equation has solutions which form singularities which have a self similar profile by considering a nonlinear eigenvalue problem the existence and stability of the self similar profiles is discussed the existence of approximately self similar behaviour is also considered for evolution profiles with several maxima and minima this behaviour is compared to similar behaviour for the linear diffusion problem the paper uses a combination of techniques from analysis and the theory of dynamical systems

stability of interfaces with mesh refinement a study is conducted of the stability of mesh refinement in space and time for several different interface equations and finite difference approximations first a root condition which implies stability for the initial boundary value problem for this type of interface is derived from the root condition the stability of several interface equations is proved using the maximum principle in some cases the final verification steps can be done analytically in other cases a simple computer program has been written to check the condition for values of a parameter along the boundary of the unit circle using this method stability for lax wendroff with all the interface conditions considered and for leapfrog with interpolation interface conditions when the fine and coarse grids overlap is proved

a boundary integral formulation and 2d fundamental solutions for piezoelectric media abstract first based on the basic equations of piezoelectricity the boundary integral formulation is derived by utilizing the reciprocal work theorem second for the plane problem of piezoelectric media one general solution in terms of harmonic functions is derived and the fundamental solutions are therefore obtained numerical calculations by boundary element method bem are performed to give the stress concentration coefficient of an infinite piezoelectric plane with a circular hole as well as the stress and electric intensity factors of a central crack locating on an infinite plate results are found to agree well with the exact solutions

blow up profile for the complex ginzburg landau equation abstract we construct a solution to the complex ginzburg landau equation which blows up in finite time t only at one blow up point we also give a sharp description of its blow up profile the proof relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude two major difficulties arise in the proof the linearized operator around the profile is not self adjoint and it has a second neutral mode in the last section the interpretation of the parameters of the finite dimensional problem in terms of the blow up time and the blow up point gives the stability of the constructed solution with respect to perturbations in the initial data

on blow up and degeneracy for the semilinear heat equation with source the asymptotic behaviour of the solution of the semilinear parabolic equation u t u xx 1 u ln 2 l u for t 0 x u x t 0 for t 0 and u 0 x u 0 x 0 in which blows up at a finite time t 0 is investigated it is proved that for some two parametric set of initial functions u 0 the behaviour of u t x near t t 0 is described by the approximate self similar solution v a t x exp t 0 t 1 cos 2 x 2 1 satisfying the first order nonlinear hamilton jacobi equation v t v x 2 1 v 1 v ln 2 1 v some open problems of degeneracy near a finite blow up time for other semilinear or quasilinear parabolic equations with source u t u 1 u ln 1 u 1 u t u u l u t u e u u t ln 1 u u 1 u ln 1 u 0 1 are discussed

a symptotic behaviour for a numerical approximation of a parabolic problem with blowing up solutions abstract in this paper we study the asymptotic behaviour of a semidiscrete numerical approximation for u t u xx u p in a bounded interval 0 1 with dirichlet boundary conditions we focus in the behaviour of blowing up solutions we find that the blow up rate for the numerical scheme is the same as for the continuous problem also we find the blow up set for the numerical approximations and prove that it is contained in a neighbourhood of the blow up set of the continuous problem when the mesh parameter is small enough

movcol4 a moving mesh code for fourth order time dependent partial differential equations in this paper we develop and analyze a moving mesh code for the adaptive simulation of fourth order pdes based on collocation the scheme is shown to enforce discrete conservation for problems written in a generalized conservation form to demonstrate the breadth of applicability we present examples from both cahn hilliard and thin film type equations exhibiting metastable behavior finite time solution blow up finite time extinction and moving interfaces

numerical validation of blow up solutions with quasi homogeneous compactifications we provide a numerical validation method of blow up solutions for finite dimensional vector fields admitting asymptotic quasi homogeneity at infinity our methodology is based on quasi homogeneous compactifications containing quasi parabolic type and directional type compactifications divergent solutions including blow up solutions then correspond to global trajectories of associated vector fields with appropriate time variable transformation tending to equilibria on invariant manifolds representing infinity we combine standard methodology of rigorous numerical integration of differential equations with lyapunov function validations around equilibria corresponding to divergent directions which yields rigorous upper and lower bounds of blow up time as well as rigorous profile enclosures of blow up solutions

construction of the witten reshetikhin turaev tqft from conformal field theory in andersen and ueno j knot theory ramif 16 127 202 2007 we constructed the vacua modular functor based on the sheaf of vacua theory developed in tsuchiya et al adv stud pure math 19 459 566 1989 and the abelian analog in andersen and ueno int j math 18 919 993 2007 we here provide an explicit isomorphism from the modular functor underlying the skein theoretic model for the witten reshetikhin turaev tqft due to blanchet habbeger masbaum and vogel to the vacua modular functor this thus provides a geometric construction of the tqft first proposed by witten and constructed first by reshetikhin turaev from the quantum group u q text sl n

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

some comments on chern simons gauge theory following m f atiyah and r bott ab and e witten w we consider the space of flat connections on the trivialsu 2 bundle over a surfacem modulo the space of gauge transformations we describe on this quotient space a natural hermitian line bundle with connection and prove that if the surfacem is now endowed with a complex structure this line bundle is isomorphic to the determinant bundle we show heuristically how path integral quantisation of the chern simons action yields holomorphic sections of this bundle

deciphering singularities by discrete methods we consider the problem of estimating numerically the parameters of singularities of solutions of differential equations we propose a novel approach which is based on discretizing the governing equation and timestepping in the complex domain some applications to ordinary and partial differential equations are discussed

review of parameter identification procedures in groundwater hydrology the inverse problem the purpose of this survey is to review parameter identification procedures in groundwater hydrology and to examine computational techniques which have been developed to solve the inverse problem parameter identification methods are classified under the error criterion used in the formulation of the inverse problem the problem of ill posedness in connection with the inverse problem is addressed typical inverse solution techniques are highlighted the review also includes the evaluation of methods used for computing the sensitivity matrix statistics which can be used to estimate the parameter uncertainty are outlined attempts have been made to compare and contrast representative inverse procedures and direction for future research is suggested

an adaptive numerical method to handle blow up in a parabolic system we study numerical approximations to solutions of a system of two nonlinear diffusion equations in a bounded interval coupled at the boundary in a nonlinear way in certain cases the system develops a blow up singularity in finite time fixed mesh methods are not well suited to approximate the problem near the singularity as an alternative to reproduce the behaviour of the continuous solution we present an adaptive in space procedure the scheme recovers the conditions for blow up and non simultaneous blow up it also gives the correct non simultaneous blow up rate and set moreover the numerical simultaneous blow up rates coincide with the continuous ones in the cases when the latter are known finally we present numerical experiments that illustrate the behaviour of the adaptive method

adaptive numerical method for burgers type nonlinear equations abstract in this paper a moving mesh partial differential equation mmpde approach will be applied to some types of nonlinear partial differential equations the method is described in detail and a number of computational examples are presented using different monitor functions applied to the burgers type equations

asymptotic faithfulness of the quantum su n representations of the mapping class groups we prove that the sequence of projective quantum su n representations of the mapping class group of a closed oriented surface obtained from the projective flat su n verlinde bundles over teichm uller space is asymptotically faithful that is the intersection over all levels of the kernels of these representations is trivial whenever the genus is at least 3 for the genus 2 case this intersection is exactly the order 2 subgroup generated by the hyper elliptic involution in the case of even degree and n 2 otherwise the intersection is also trivial in the genus 2 case

the first chern form on moduli of parabolic bundles for moduli space of stable parabolic bundles on a compact riemann surface we derive an explicit formula for the curvature of its canonical line bundle with respect to quillen s metric and interprete it as a local index theorem for the family of overline partial operators in the associated parabolic endomorphism bundles the formula consists of two terms one standard proportional to the canonical kahler form on the moduli space and one nonstandard called a cuspidal defect that is defined by means of special values of the eisenstein maass series the cuspidal defect is explicitly expressed through the curvature forms of certain natural line bundles on the moduli space related to the parabolic structure we also compare our result with witten s volume computation

the witten reshetikhin turaev invariant for links in finite order mapping tori i abstract we state asymptotic expansion and growth rate conjectures for the witten reshetikhin turaev invariants of arbitrary framed links in 3 manifolds and we prove these conjectures for the natural links in mapping tori of finite order automorphisms of marked surfaces our approach is based upon geometric quantisation of the moduli space of parabolic bundles on the surface which we show coincides with the construction of the witten reshetikhin turaev invariants using conformal field theory as was recently completed by andersen and ueno

blowup in diffusion equations a survey this paper deals with quasilinear reaction diffusion equations for which a solution local in time exists if the solution ceases to exist for some finite time we say that it blows up in contrast to linear equations blowup can occur even if the data are smooth and well defined for all times depending on the equation either the solution or some of its derivatives become singular we shall concentrate on those cases where the solution becomes unbounded in finite time this can occur in quasilinear equations if the heat source is strong enough there exist many theoretical studies on the question on the occurrence of blowup in this paper we shall recount some of the most interesting criteria and most important methods for analyzing blowup the asymptotic behavior of solutions near their singularities is only completely understood in the special case where the source is a power a better knowledge would be useful also for their numerical treatment thus not surprisingly the numerical analysis of this type of problems is still at a rather early stage the goal of this paper is to collect some of the known results and algorithms and to direct the attention to some open problems

the boundary contour method for piezoelectric media with linear boundary elements this paper presents a development of the boundary contour method bcm for piezoelectric media first the divergence free property of the integrand of the piezoelectric boundary element is proved secondly the boundary contour method formulation is derived and potential functions are obtained by introducing linear shape functions and green s functions computer methods in applied mechanics and engineering 1998 158 65 for piezoelectric media the bcm is applied to the problem of piezoelectric media finally numerical solutions for illustrative examples are compared with exact ones and those of the conventional boundary element method bem the numerical results of the bcm coincide very well with the exact solution and the feasibility and efficiency of the method are verified copyright 2003 john wiley sons ltd

iterative approximation of solutions for semilinear parabolic equations system in this paper we present the convergence analysis of iterative schemes for solving semilinear systems resulting from multidimensional semilinear second order parabolic partial differential equations pdes defined in a domain n x and subject to cauchy boundary conditions on n using the method of integral successive iterative approximation and the principle of decay estimation we derive the bounded nonnegative local solutions and global solutions blowing up results of the solutions are also presented

the numerical study of blowup with application to a nonlinear schrodinger equation abstract we discuss the use of numerical methods in the study of the solutions of evolution problems which exhibit finite time unbounded growth we first examine a naive approach in which the growth rate of the numerical solution is accepted as an approximation of the true growth rate as we shall demonstrate for a radial nonlinear schrodinger equation this approach is inadequate since different discretizations exhibit different growth rates the spurious behaviour of discretizations in the neighbourhood of the singularity is discussed a reliable procedure for the estimation of the blowup parameters is considered which eliminates the discrepancies between different numerical methods

numerical approximation of blow up of radially symmetric solutions of the nonlinear schrodinger equation we consider the initial value problem for the radially symmetric nonlinear schrodin ger equation with cubic nonlinearity nls in d 2 and 3 space dimensions to approximate smooth solutions of this problem we construct and analyze a numerical method based on a standard galerkin finite element spatial discretization with piecewise linear continuous functions and on an implicit crank nicolson type time stepping procedure we then equip this scheme with an adaptive spatial and temporal mesh refinement mechanism that enables the numerical technique to approximate well singular solutions of the nls equation that blow up at the origin as the temporal variable t tends from below to a finite value t star for the blow up of the amplitude of the solution we recover numerically the well known rate t star t 1 2 for d 3 for d 2 our numerical evidence supports the validity of the log log law ln ln frac 1 t star t t star t 1 2 for t extremely close to t star the scheme also approximates well the details of the blow up of the phase of the solution at the origin as t to t star

identification of parameter structure in groundwater inverse problem this paper develops an optimization method that can be used to identify parameter structure as well as its values in connection with the inverse problem in groundwater modeling the complexity of the parameter structure is determined by the quantity and quality of observation data output least square error is used as the criterion of identification sensitivity coefficients at finite element nodes are evaluated by the variational method the problem of identifying the parameter structure is formulated as a problem of combinatorial optimization a new parameterization technique is presented for describing the parameter structure it can represent various distributions of parameters from piecewise constant to continuous variability based on this representation the combinatorial optimization problem for determining the parameter structure is then solved by a simple one at a time searching procedure numerical experimentations are used to demonstrate the efficiency and practicality of the proposed method

gauge theory on asymptotically periodic 4 manifolds il existe une famille non denombrable de classes de diffeomorphismes des 4 varietes orientees homeomorphes a r 4