convergence of vortex methods for three dimensional euler equations in bounded domains an approach is introduced where continuous norms and high order estimates are used to study the convergence of vortex methods for the euler equations it is applied to initial boundary value problems in three dimensional bounded domains convergence results are proved

on the weak monotonicity of gini means and other mixture functions weak monotonicity was recently proposed as a relaxation of the monotonicity condition for averaging aggregation and weakly monotone functions were shown to have desirable properties when averaging data corrupted with outliers or noise we extended the study of weakly monotone averages by analyzing their transforms and we established weak monotonicity of several classes of averaging functions in particular gini means and mixture operators mixture operators with gaussian weighting functions were shown to be weakly monotone for a broad range of their parameters this study assists in identifying averaging functions suitable for data analysis and image processing tasks in the presence of outliers

on model complete differential fields we develop a geometric approach to definable sets in differentially closed fields with emphasis on the question of orthogonality to a given strongly minimal set equivalently within a family of ordinary differential equations we consider those equations that can be transformed by differential algebraic transformations so as to yield solutions of a given fixed first order ode x we show that this sub family is usually definable in particular if x lives on a curve of positive genus as a corollary we show the existence of many model complete superstable theories of differential fields

local andre oort conjecture for the universal abelian variety we prove a p adic version of the andre oort conjecture for subvarieties of the universal abelian varieties let g and n be integers with n 3 and p a prime number not dividing n let r be a finite extension of w mathbb f p alg the ring of witt vectors of the algebraic closure of the field of p elements the moduli space mathcal a mathcal a g 1 n of g dimensional principally polarized abelian varieties with full level n structure as well as the universal abelian variety pi mathcal x to mathcal a over mathcal a may be defined over r we call a point xi in mathcal x r r special if mathcal x pi xi is a canonical lift and is a torsion point of its fibre employing the model theory of difference fields and work of moonen on special subvarieties of mathcal a we show that an irreducible subvariety of mathcal x r containing a dense set of r special points must be a special subvariety in the sense of mixed shimura varieties

periodic subvarieties of semiabelian varieties and annihilators of irreducible representations abstract let g be a semiabelian variety defined over a field of characteristic 0 endowed with an endomorphism we prove there is no proper subvariety y g which intersects the orbit of each periodic point of g under the action of as an application we are able to give a topological characterization of the annihilator ideals of irreducible representations in certain skew polynomial algebras

invariant varieties for polynomial dynamical systems we study algebraic dynamical systems and more generally s varieties f a n c a n c given by coordinatewise univariate polynomials by refining an old theorem of ritt on compositional identities amongst polynomials more precisely we find a nearly canonical way to write a polynomial as a composition of clusters from which one may easily read off possible compositional identities our main result is an explicit description of the weakly skew invariant varieties that is for a fixed field automorphism s c c those algebraic varieties x a n c for which f x x s as a special case we show that if f x c x is a polynomial of degree at least two that is not conjugate to a monomial chebyshev polynomial or a negative chebyshev polynomial and x a 2 c is an irreducible curve that is invariant under the action of x y f x f y and projects dominantly in both directions then x must be the graph of a polynomial that commutes with f under composition as consequences we deduce a variant of a conjecture of zhang on the existence of rational points with zariski dense forward orbits and a strong form of the dynamical manin mumford conjecture for liftings of the frobenius we also show that in models of acfa 0 a disintegrated set defined by s x f x for a polynomial f has morley rank one and is usually strongly minimal that model theoretic algebraic closure is a locally finite closure operator on the nonalgebraic points of this set unless the skew conjugacy class of f is defined over a fixed field of a power of s and that nonorthogonality between two such sets is definable in families if the skew conjugacy class of f is defined over a fixed field of a power of s

finite index theorems for iterated galois groups of cubic polynomials let k be a number field or a function field let f in k x be a rational function of degree d ge 2 and let beta in mathbb p 1 overline k for all n in mathbb n cup infty the galois groups g n beta mathrm gal k f n beta k beta embed into mathrm aut t n the automorphism group of the d ary rooted tree of level n a major problem in arithmetic dynamics is the arboreal finite index problem determining when mathrm aut t infty g infty beta infty when f is a cubic polynomial and k is a function field of transcendence degree 1 over an algebraic extension of mathbb q we resolve this problem by proving a list of necessary and sufficient conditions for finite index this is the first result that gives necessary and sufficient conditions for finite index and can be seen as a dynamical analog of the serre open image theorem when k is a number field our proof is conditional on both the abc conjecture for k and vojta s conjecture for blowups of mathbb p 1 times mathbb p 1 we also use our approach to solve some natural variants of the finite index problem for modified trees

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

fully discrete convergence estimates for vortex methods in bounded domains in this paper the authors study vortex method for 2 dimensionsal euler equations of incompressible flow in bounded domains to approximate the initial vortex field by a sum of vortex blobs with arbitrary high accuracy this field is extended smoothly to a small neighborhood of the boundary and the computation is carried out in the extended domain to construct the velocity field from vorticity a second order isoparametric finite element method is applied and to solve the characteristic equations the explicit euler s scheme is considered optimal error bounds for this fully discrete scheme are obtained

the dual decomposition of aggregation functions and its application in welfare economics in this paper we review the role of self duality in the theory of aggregation functions the dual decomposition of aggregation functions into a self dual core and an anti self dual remainder and some applications to welfare inequality and poverty measures

combined effect of explicit time stepping and quadrature for curvature driven flows the flow of a closed surface of codimension 1 in bbb r n driven by curvature is first approximated by a singularly perturbed parabolic double obstacle problem with small parameter varepsilon 0 conforming piecewise linear finite elements with mass lumping over a quasi uniform and weakly acute mesh of size h are further used for space discretization and combined with forward differences for time discretization with uniform time step tau the resulting explicit schemes are the basis for an efficient algorithm the so called dynamic mesh algorithm and exhibit finite speed of propagation and discrete ondegeneracy no iteration is required not even to handle the obstacle constraints the zero level set of the fully discrete solution is shown to converge past singularities to the true interface provided tau h 2 approx o varepsilon 4 and no fattening occurs if the more stringent relations tau h 2 approx o varepsilon 6 are enforced then an interface rate of convergence o varepsilon is derived in the vicinity of regular points along with a companion o varepsilon 1 2 for type i singularities for smooth flows an interface rate of convergence of o varepsilonq is proven provided tau h 2 approx o varepsilon 5 and exact integration is used for the potential term the analysis is based on constructing fully discrete barriers via an explicit parabolic projection with quadrature which bears some intrinsic interest lipschitz properties of viscosity solutions of the level set approach and discrete nondegeneracy these basic ingredients are also discussed

interior maximum norm estimates for finite element methods part ii we consider bilinear forms a connected with second order elliptic problems and assume that for u h in a finite element space s h we have a u u h x f x for x in s h with local compact support we give local estimates for u u h in l and w 1 of the type local best approximation plus weak outside influences plus the local size of f

weak monotonicity of lehmer and gini means we analyze directional monotonicity of several mixture functions in the direction 1 1 1 called weak monotonicity our particular focus is on power weighting functions and the special cases of lehmer and gini means we establish limits on the number of arguments of these means for which they are weakly monotone these bounds significantly improve the earlier results and hence increase the range of applicability of gini and lehmer means we also discuss the case of affine weighting functions and find the smallest constant which ensures weak monotonicity of such mixture functions

strengthened ordered directionally monotone functions links between the different notions of monotonicity abstract in this work we propose a new notion of monotonicity strengthened ordered directional monotonicity this generalization of monotonicity is based on directional monotonicity and ordered directional monotonicity two recent weaker forms of monotonicity we discuss the relation between those different notions of monotonicity from a theoretical point of view additionally along with the introduction of two families of functions and a study of their connection to the considered monotonicity notions we define an operation between functions that generalizes the choquet integral and the ukasiewicz implication

a new numerical method for backward parabolic problems in the maximum norm setting a new method for solving numerically backward parabolic problems is proposed as usual for this kind of ill posed problems it is assumed that an a priori bound for the solution is available the algorithm consists of two basic steps first a standard forward integration is performed in order to approximate the solution at suitable future time levels second a holomorphic recovery procedure is carried out providing the required approximations for the preceding times the analysis is valid in the maximum norm setting and rigorous estimates are derived among other advantages the method can also be applied to nonlinear problems and it produces a continuous output some numerical illustrations are presented

model theory of difference fields ii periodic ideals and the trichotomy in all characteristics we classify all possible combinatorial geometries associated with one dimensional difference equations in any characteristic the theory of difference fields admits a proper interpretation of itself namely the reduct replacing the automorphism by its nth power we show that these reducts admit a successively smoother theory as n becomes large and we succeed in defining a limit structure to these reducts or rather to the structure they induce on one dimensional sets this limit structure is shown to be a zariski geometry in roughly the sense of hrushovski and zil ber the trichotomy is thus obtained for the limit structure as a consequence of a general theorem and then shown to be inherited by the original theory 2000 mathematical subject classification 03c60 primary 03c45 03c98 08a35 12h10 secondary

vortex method for two dimensional euler equations in bounded domains with boundary correction the vortex method for the initial boundary value problems of the euler equations for incompressible flow is studied a boundary correction technique is introduced to generate second order accuracy convergence and error estimates are proved

pointwise directional increasingness and geometric interpretation of directionally monotone functions abstract the relaxation of monotonicity requirements is a trend in the theory of aggregation functions in the recent literature we can find several relaxed forms of monotonicity such as weak directional cone ordered directional and strengthened directional monotonicity all these forms of monotonicity are global properties in the sense that they are imposed for all the points in the domain of a function in this work we introduce a local notion of monotonicity called pointwise directional monotonicity or directional monotonicity at a point based on this concept we characterize all the previously defined notions of monotonicity and in the final part of the paper we present some geometric aspects of the global weaker forms of monotonicity stressing their relations and singularities

density of orbits of endomorphisms of abelian varieties let a be an abelian variety defined over q and let be a dominant endomorphism of a as an algebraic variety we prove that either there exists a non constant rational fibration preserved by or there exists a point x a q whose orbit is zariski dense in a this provides a positive answer for abelian varieties of a question raised by medvedev and the second author in ms14 we prove also a stronger statement of this result in which is replaced by any commutative finitely generated monoid of dominant endomorphisms of a

dynamical anomalous subvarieties structure and bounded height theorems abstract according to medvedev and scanlon 14 a polynomial f x q x of degree d 2 is called disintegrated if it is not conjugate to x d or to c d x where c d is the chebyshev polynomial of degree d let n n let f 1 f n q x be disintegrated polynomials of degrees at least 2 and let f 1 f n be the corresponding coordinate wise self map of p 1 n let x be an irreducible subvariety of p 1 n of dimension r defined over q we define the anomalous locus of x which is related to the periodic subvarieties of p 1 n we prove that the anomalous locus of x is zariski closed this is a dynamical analogue of a theorem of bombieri masser and zannier 4 we also prove that the points in the intersection of x with the union of all irreducible periodic subvarieties of p 1 n of codimension r have bounded height outside the anomalous locus of x this is a dynamical analogue of habegger s theorem 8 which was previously conjectured in 4 the slightly more general self maps f 1 f n where each f i q x is a disintegrated rational function are also treated at the end of the paper

from finite differences to finite elements a short history of numerical analysis of partial differential equations abstract this is an account of the history of numerical analysis of partial differential equations starting with the 1928 paper of courant friedrichs and lewy and proceeding with the development of first finite difference and then finite element methods the emphasis is on mathematical aspects such as stability and convergence analysis