conics in the grothendieck ring abstract the subring of the grothendieck ring of k varieties generated by smooth conics is described giving many zero divisors the proof uses only elementary projective geometry

uniqueness of weak solutions of 1 1 dimensional wave maps in this paper we prove that finite energy distributional solutions to the cauchy problem of wave maps from 1 1 dimensional minkowsky space to any complete riemannian manifold are unique

global strong solution to the thirring model in critical space abstract we obtain a strong solution in charge critical space l 2 r of the thirring system and federbusch equations in one space dimension by using solution representation of the models the uniqueness is obtained for the solution l 0 t l 2 r l 4 r a decay of local charge and asymptotic behavior of the field can be shown directly

optimal by accuracy and optimal by order cubature formulae in interpolational classes in this paper the problem of optimal integration for fast oscillatory functions of two variables is solved constructively in the case where a priori information is limited the connection of this problem with the problem of optimal recovery of a function from interpolational classes is analysed using properties of majorants and minorants for these functional classes

lyapunov functionals and l 1 stability for discrete velocity boltzmann equations we devise lyapunov functionals and prove uniform l 1 stability for one dimensional semilinear hyperbolic systems with quadratic nonlinear source terms these systems encompass a class of discrete velocity models for the boltzmann equation the lyapunov functional is equivalent to the l 1 distance between two weak solutions and non increasing in time they result from computations of two point interactions in the phase space for certain models with only transversal collisional terms there exist generalizations for three and multi point interactions

numerical evaluation of one dimensional diffraction integrals an efficient technique is described for performing one dimensional 1 d integrals of oscillatory complex functions such as might arise in the numerical solution of diffraction problems the upper integration limit may be infinite provided that the phase of the integrand varies rapidly beyond some point the integration is done in segments chosen so that both the phase and amplitude may be approximated as cubic polynomials on each segment with a small cubic term for phase and continued to infinity from the last segment three methods are given for integrating at each step 1 an asymptotic series method 2 an almost quadratic phase method and 3 an almost linear phase method criteria are given for choosing among the methods an example calculation verifies the method in the special case of wave diffraction over two knife edges

the grothendieck ring of varieties and piecewise isomorphisms let k0 var k be the grothendieck ring of algebraic varieties over a field k l et x y be two algebraic varieties over k which are piecewise isomorphic i e x and y admit finite partitions x1 xn y1 yn into locally closed subvarieties such that xi is iso morphic to yi for all i n then x y in k0 var k larsen and lunts ask whether the converse is true for characteristic zero and algebraically closedfield k we answer positively this question when dim x 1o rx is a smooth connected projective surface or if x contains only finitely many rational curves

variational methods in relativistic quantum mechanics this review is devoted to the study of stationary solutions of lin ear and nonlinear equations from relativistic quantum mechanics involving the dirac operator the solutions are found as critical points of an energy func tional contrary to the laplacian appearing in the equations of nonrelativistic quantum mechanics the dirac operator has a negative continuous spectrum which is not bounded from below this has two main consequences first the energy functional is strongly indefinite second the euler lagrange equations are linear or nonlinear eigenvalue problems with eigenvalues lying in a spectral gap between the negative and positive continuous spectra moreover since we work in the space domain r 3 the palais smale condition is not satisfied for these reasons the problems discussed in this review pose a challenge in the calculus of variations the existence proofs involve sophisticated tools from nonlinear analysis and have required new variational methods which are now applied to other problems in the first part we consider the fixed eigenvalue problem for models of a free self interacting relativistic particle they allow to describe the localized state of a spin 1 2 particle a fermion which propagates without changing its shape this includes the soler models and the maxwell dirac or klein gordon dirac equations the second part is devoted to the presentation of min max principles al lowing to characterize and compute the eigenvalues of linear dirac operators with an external potential in the gap of their essential spectrum many con sequences of these min max characterizations are presented among them a new kind of hardy like inequalities and a stable algorithm to compute the eigenvalues in the third part we look for normalized solutions of nonlinear eigenvalue problems the eigenvalues are lagrange multipliers lying in a spectral gap we review the results that have been obtained on the dirac fock model which is a nonlinear theory describing the behavior of n interacting electrons in an external electrostatic field in particular we focus on the problematic definition of the ground state and its nonrelativistic limit in the last part we present a more involved relativistic model from quan tum electrodynamics in which the behavior of the vacuum is taken into ac count it being coupled to the real particles the main interesting feature of this model is that the energy functional is now bounded from below providing us with a good definition of a ground state

a numerical criterion for uniruledness an n dimensional variety x over an algebraically closed field k is said to be uniruled if there exist an n 1 dimensional k variety w and a dominant rational map f p x w x x is called separably uniruled if f is a separable map i e k p x w is a finite separable extension of k x via f if x is uniniled then there exists a rational curve passing through a general k valued point of x and the converse is true by the countability of the components of the hilbert scheme provided that the ground field k is uncountable uniruled varieties are of importance in the classification of varieties it is conjectured that in characteristic 0 uniruledness is equivalent to k xc this article gives a numerical criterion for a projective variety to be uniruled

a trace formula for varieties over a discretely valued field we study the motivic serre invariant of a smoothly bounded alge braic or rigid variety x over a complete discretely valued field k with perfect residue field k if k has characteristic zero we extend the definition to arbi trary k varieties using bittner s presentation of the grothendieck ring and a process of neron smoothening of pairs of varieties the motivic serre invariant can be considered as a measure for the set of unramified points on x under certain tameness conditions it admits a cohomological interpretation by means of a trace formula in the curve case we use t saito s geometric criterion for cohomological tameness to obtain more detailed results we discuss some applications to weil chatelet groups chow motives and the structure of the grothendieck ring of varieties msc 2000 11s15 14g05 14g22

approximation of certain functions given by integrals with highly oscillatory integrands there is often a need to approximate integrals of highly oscillatory functions when studying scattering and diffraction of electromagnetic waves this paper presents a method of estimating certain types of these integrals by evaluating one interpolating function and performing one or two relatively easy numerical integrations the method is demonstrated for the case of a fresnel integral

global strong solution to a nonlinear dirac type equation in one dimension abstract this paper studies a class of nonlinear massless dirac equations in one dimension which include the equations for the massless thirring model and the massless gross neveu model under the assumptions of the initial data having small charge and being bounded the global existence of the strong solution is established the decay of the local charge is also proved

global solution to nonlinear dirac equation for gross neveu model in 1 1 dimensions abstract this paper studies a class of nonlinear dirac equations with cubic terms in r 1 1 which include the equations for the massive thirring model and the massive gross neveu model under the assumption that the initial data has bounded l 2 norm the global existence and the uniqueness of the strong solution in c 0 l 2 r 1 are proved

global small solutions to the critical radial dirac equation with potential abstract we solve globally a radial cubic dirac equation perturbed with a small potential with data of small critical norm h 1 the main tool is a new endpoint estimate of the perturbed dirac flow for a class of radial type initial data

algebraic independence in the grothendieck ring of varieties we give sufficient cohomological criteria for the classes of given varieties over a field k to be algebraically independent in the grothendieck ring of varieties over k and construct some examples

families of rationally connected varieties recall that a proper variety x is said to be rationally connected if two general points p q x are contained in the image of a map g p x this is clearly a birationally invariant property when x is smooth this turns out to be equivalent to the a priori weaker condition that two general points can be joined by a chain of rational curves and also to the a priori stronger condition that for any finite subset x there is a map g p x whose image contains and such that gtx is an ample bundle

categorical resolution of singularities abstract building on the concept of a smooth dg algebra we define the notion of a smooth derived category we then propose the definition of a categorical resolution of singularities our main example is the derived category d x of quasi coherent sheaves on a scheme x we prove that d x has a canonical categorical resolution if the base field is perfect and x is a separated scheme of finite type with a dualizing complex

initial boundary value problem for nonlinear dirac equation of gross neveu type in 1 1 dimensions abstract this paper studies an initial boundary value problem for a class of nonlinear dirac equations with cubic terms which include the equations for the massive thirring model and the massive gross neveu model under the assumptions that the initial data has bounded l 2 norm and the boundary satisfies suitable conditions the global existence and the uniqueness of the strong solution are proved

optimal by order quadrature formulae for fast oscillatory functions with inaccurately given a priori information abstract in this article the authors construct optimal by order quadrature formulae for integration of fast oscillatory functions in interpolational classes c 1 l n 1 and c 1 l n e 1 the construction of efficient formulae for numerical integration of fast oscillatory functions is based on the application of the residual method and the method of quasi solutions both cases weak and strong oscillations are considered results of numerical examples are presented