asymptotic upper curvature bounds in coarse geometry we define a notion of an asymptotic upper curvature bound for gromov hyperbolic metric spaces that is invariant under rough isometries and examine the basic properties of this concept

logarithmic potentials and quasiconformal flows on the heisenberg group abstract let h be the sub riemannian heisenberg group that h supports a rich family of quasiconformal mappings was demonstrated by koranyi and reimann using the so called flow method here we supply further evidence of the flexible nature of this family constructing quasiconformal mappings with extreme behavior on small sets more precisely we establish criteria to determine when a given logarithmic potential on h is such that there exists a quasiconformal mapping of h with jacobian comparable to e 2 so that the jacobian is zero or infinity at the same points as e 2 when is continuous and meets the criteria we show the canonical sub riemannian metric g 0 and the weighted metric g e g 0 generate bi lipschitz equivalent distance functions these results rest on an extension to the theory of quasiconformal flows on h and constructions that adapt the iterative method of bonk heinonen and saksman

is gauss quadrature better than clenshaw curtis we compare the convergence behavior of gauss quadrature with that of its younger brother clenshaw curtis seven line matlab codes are presented that implement both methods and experiments show that the supposed factor of 2 advantage of gauss quadrature is rarely realized theorems are given to explain this effect first following o hara and smith in the 1960s the phenomenon is explained as a consequence of aliasing of coefficients in chebyshev expansions then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log z 1 z 1 in the complex plane gauss quadrature corresponds to pade approximation at z infty clenshaw curtis quadrature corresponds to an approximation whose order of accuracy at z infty is only half as high but which is nevertheless equally accurate near 1 1

compensated compactness and the heisenberg group jacobians of maps on the heisenberg group are shown to map suitable group sobolev spaces into the group hardy space h1 from this result and a weak convergence theorem for the hardy space h1 of the heisenberg group a compensated compactness property for these jacobians is obtained 0 introduction we investigate compensated compactness properties of jacobians of maps on the heisenberg group and we prove results analogous to those for the jacobians of maps on r let h c r be the lie group with multiplicative structure z t z t z z t t 2 im z z where z z1 zn z z 1 z n and z z n j 1 zj z j this is the n order heisenberg group it can be shown that the group operation is c on the manifold c r and hence h is a locally compact lie group let z1 zn t be coordinates on h write zj xj iyj and define the vector fields xj xj 2yj t yj yj 2xj t t t it turns out that the xj yj and t form a basis for the left invariant vector fields on h they satisfy the commutation relations xj yj 4t j 1 n and all other work partially supported by the national science foundation work partially supported by nsf grant dms 9007491 mathematics subject classification 1991 42 typeset by ams tex 1 commutators vanish the group h is equipped with a natural dilation structure r z t rz rt r 0 which is consistent with the group multiplication the associated norm z t z 4 t is homogeneous of degree 1 with respect to this group of dilations we denote by br u0 the heisenberg group ball u h u 1u0 0 f u where u 2n 2 1u if 2n 2 2n 3 the hardy space h h h is the set of all f such that f l h an alternative definition of h h can be given via the atomic decomposition see fos or cw for details given a c map f f1 fn from h into r define 0 3 jac f det l1f1 l1f2 l1fn l2f1 l2f2 l2fn lnf1 lnf2 lnfn where lj is either xj or yj we would like to show that jac f maps a product of suitable group sobolev spaces into the spaces h h for 1 2n 2 2n 3 the analogous result for the vector fields xj 1 j n on r n has been proved by p l lions and y meyer when 1 and extended by clms for 1 n n 1 note that the lower bound for in both cases is d d 1 where d is the homogeneous dimension of the group our first result is theorem 1 let n 2 and for 1 j n let 1 0 that depends only on n and on the pj s such that for every map f f1 fn on h we have 0 4 jac f h c n

a sinh transformation for evaluating two dimensional nearly singular boundary element integrals a new transformation technique is introduced for evaluating the two dimensional nearly singular integrals which arise in the solution of laplace s equation in three dimensions using the boundary element method when the source point is very close to the element of integration the integrals are evaluated using in a product fashion a transformation which has recently been used to evaluate one dimensional near singular integrals this sinh transformation method automatically takes into account the position of the projection of the source point onto the element and also the distance b between the source point and the element the method is straightforward to implement and when it is compared with a number of existing techniques for evaluating two dimensional near singular integrals it is found that the sinh method is superior to the existing methods considered both for potential integrals across the full range of b values considered 0 0 01 for smaller values of b the use of the lmethod is recommended for flux integrals copyright 2006 john wiley sons ltd

optimal mass transportation in the heisenberg group in this paper we consider the problem of optimal transportation of absolutely continuous masses in the heisenberg group hn in the case when the cost function is either the square of the carnot caratheodory distance or the square of the koranyi norm in both cases we show existence and uniqueness of an optimal transport map in the former case the proof requires a delicate analysis of minimizing geodesics of the group and of the differentiability properties of the squared distance function in the latter case the proof requires some fine properties of bv functions in the heisenberg group

groups quasi isometric to complex hyperbolic space we show that any finitely generated group quasi isometric to complex hyperbolic space is a finite extension of a properly discontinuous cocompact subgroup of the isometry group

area stationary surfaces in the heisenberg group h1 we consider area stationary surfaces perhaps with a volume constraint in the heisenberg group h1 endowed with its carnot caratheodory distance by analyzing the first variation of area we characterize c2 area stationary surfaces as those with mean curvature zero or constant if a volume preserving condition is assumed and such that the characteristic curves meet orthogonally the singular curves moreover a minkowski type formula relating the area the mean curvature and the volume is obtained for volume preserving area stationary surfaces enclosing a given region as a consequence of the characterization of area stationary surfaces we refine the bernstein type theorem given in jih hsin cheng jenn fang hwang andrea malchiodi paul yang minimal surfaces in pseudohermitian geometry ann sc norm super pisa cl sci 5 4 1 2005 129 177 mr 2165405 and nicola garofalo scott pauls the bernstein problem in the heisenberg group arxiv math dg 0209065 to describe all c2 entire area stationary graphs over the xy plane in a calibration argument shows that these graphs are globally area minimizing finally by using the description of the singular set in jih hsin cheng jenn fang hwang andrea malchiodi paul yang minimal surfaces in pseudohermitian geometry ann sc norm super pisa cl sci 5 4 1 2005 129 177 mr 2165405 the characterization of area stationary surfaces and the ruling property of constant mean curvature surfaces we prove our main results where we classify area stationary surfaces in with or without a volume constraint and non empty singular set in particular we deduce the following counterpart to alexandrov uniqueness theorem in euclidean space any compact connected c2 surface in area stationary under a volume constraint must be congruent to a rotationally symmetric sphere obtained as the union of all the geodesics of the same curvature joining two points as a consequence we solve the isoperimetric problem in assuming c2 smoothness of the solutions 13 completely reducible hypersurfaces in a pencil pages 672 688 j v pereira s yuzvinsky preview pdf 324 k related articles 14 modules over motivic cohomology pages 689 727 oliver rondigs paul arne ostvaer preview pdf 303 k related articles 15 the fundamental group of period domains over finite fields pages 728 741 sascha orlik preview pdf 154 k related articles articles 1 15

the length and other invariants of a real field the length of a field is the smallest integer m such that any totally positive quadratic form of dimension m represents all sums of squares we investigate this field invariant and compare it to others such as the u invariant the pythagoras number the hasse number and the mordell function related to sums of squares of linear forms

h type groups and iwasawa decompositions since their introduction by a kaplan kpl some ten years ago generalised heisenberg groups also known as groups of heisenberg type or h type groups have provided a framework in which to construct interesting examples in geometry and analysis see for instance c2 kp2 kp3 kpr k2 rl r2 tl tv the iwasawa n groups associated to all the real rank one simple lie groups are h type so one has a convenient vehicle for studying these in a unified way many problems on these simple lie groups can be reduced to a problem on h type groups via the so called noncompact picture and often problems on h type groups can be solved on all the groups of the family in one fell swoop as in for example ch ck dr out of this approach to studying simple lie groups several problems arise such as why only some h type groups correspond to simple lie groups of real rank one in this paper we discuss various features of iwasawa n groups which distinguish them in the class of all h type groups we shall show that all h type groups which possess certain geometric properties clearly possessed by iwasawa n groups satisfy a lie algebraic condition implicit in the work of b kostant kt2 that we shall call the j condition we shall also use elementary clifford algebra to classify the

weak differentiability of bv functions on stratified groups we prove the almost everywhere approximate differentiability of functions with bounded variation in stratified groups and we study their approximate discontinuity set we introduce functions of bounded higher order variation and obtain a weak version of alexandroff differentiability theorem in this context we present a nontrivial class of functions with second order bounded variation arising from inf convolution formula of a suitable cost function

the geometry of the eisenstein picard modular group the eisenstein picard modular group rm pu 2 1 mathbb z omega is defined to be the subgroup of rm pu 2 1 whose entries lie in the ring mathbb z omega where omega is a cube root of unity this group acts isometrically and properly discontinuously on bf h 2 mathbb c that is on the unit ball in mathbb c 2 with the bergman metric we construct a fundamental domain for the action of rm pu 2 1 mathbb z omega on bf h 2 mathbb c which is a 4 simplex with one ideal vertex as a consequence we elicit a presentation of the group see theorem 5 9 this seems to be the simplest fundamental domain for a finite covolume subgroup of rm pu 2 1

the kink phenomenon in fejer and clenshaw curtis quadrature the fejer and clenshawcurtis rules for numerical integration exhibit a curious phenomenon when applied to certain analytic functions when n the number of points in the integration rule increases the error does not decay to zero evenly but does so in two distinct stages for n less than a critical value the error behaves like o 2n where is a constant greater than 1 for these values of n the accuracy of both the fejer and clenshawcurtis rules is almost indistinguishable from that of the more celebrated gausslegendre quadrature rule for larger n however the error decreases at the rate o n i e only half as fast as before convergence curves typically display a kink where the convergence rate cuts in half in this paper we derive explicit as well as asymptotic error formulas that provide a complete description of this phenomenon

some examples of quasiisometries of nilpotent lie groups we construct quasiisometries of nilpotent lie groups in particular for any simply connected nilpotent lie group n we construct quasiisometries from n to itself that is not at finite distance from any map that is a composition of left translations and automorphisms we also construct bilipschitz maps of the heisenberg groups that send vertical lines to curves that are not vertical lines

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

highly accurate lagrangian flux calculation via algebraic quadratures on spline approximated donating regions abstract lagrangian flux through a fixed curve segment within a time interval t n t n k can be formulated as an integral at the initial time t n over a compact point set called donating region in which each particle will pass through the curve during the time interval and contribute to the flux based an explicit constructive and analytical solution of the donating region the author proposes algorithms of lagrangian flux calculation lfc as algebraic quadratures over the spline approximated donating regions this generic formulation yields high accuracies up to the 8th order both in time and space as another benefit lfc leads to a conservative semi lagrangian method whose time step size is free of the eulerian stability constraint c r 1 the high accuracy and robustness of the proposed lfc algorithms are demonstrated by various numerical tests with courant numbers ranging from 1 to 10 000 and with accuracies from the 2nd order to the 8th order lfc might also be useful in designing hybrid eulerian methods for complex geometries

clenshaw curtis and gauss legendre quadrature for certain boundary element integrals following a recent article by trefethen siam review 50 2008 pp 67 87 the use of clenshaw curtis quadrature rather than gauss legendre quadrature for nearly singular integrals which arise in the boundary element method has been investigated when these quadrature rules are used in association with the sinh transformation the authors have concluded after considering asymptotic estimates of the truncation errors for certain proto type functions arising in this context that gauss legendre quadrature should continue to be the preferred quadrature rule

quasiconformal mappings on the heisenberg group on etudie les applications conformes et quasi conformes sur le groupe de heisenberg la metrique invariante a gauche naturelle sur cet espace est non isotrope sous des dilatations et ne peut se decrire par une metrique de riemann ceci engendre de nouveaux phenomenes

eigenforms of the laplacian on real and complex hyperbolic spaces on decrit l espace total des solutions de l equation c ou est une p forme sur un espace x hyperbolique reel ou complexe a n dimensions et c une constante complexe

a sinh transformation for evaluating nearly singular boundary element integrals an implementation of the boundary element method requires the accurate evaluation of many integrals when the source point is far from the boundary element under consideration a straightforward application of gaussian quadrature suffices to evaluate such integrals when the source point is on the element the integrand becomes singular and accurate evaluation can be obtained using the same gaussian points transformed under a polynomial transformation which has zero jacobian at the singular point a class of integrals which lies between these two extremes is that of nearly singular integrals here the source point is close to but not on the element and the integrand remains finite at all points however instead of remaining flat the integrand develops a sharp peak as the source point moves closer to the element thus rendering accurate evaluation of the integral difficult this paper presents a transformation based on the sinh function which automatically takes into account the position of the projection of the source point onto the element which we call the nearly singular point and the distance from the source point to the element the transformation again clusters the points towards the nearly singular point but does not have a zero jacobian implementation of the transformation is straightforward and could easily be included in existing boundary element method software it is shown that for the two dimensional boundary element method several orders of magnitude improvement in relative error can be obtained using this transformation compared to a conventional implementation of gaussian quadrature asymptotic estimates for the truncation errors are also quoted copyright 2004 john wiley sons ltd

ideal triangle groups dented tori and numerical analysis we prove the goldman parker conjecture a complex hyperbolic ideal triangle group is discretely embedded in pu 2 1 if and only if the product of its three standard generators is not elliptic we also prove that such a group is indiscrete if the product of its three standard generators is elliptic a novel feature of this paper is that it uses a rigorous computer assisted proof to deal with difficult geometric estimates

on the convergence rate of clenshaw curtis quadrature for integrals with algebraic endpoint singularities abstract in this paper we are concerned with clenshaw curtis quadrature for integrals with algebraic endpoint singularities an asymptotic error expansion and convergence rate are derived by combining a delicate analysis of the chebyshev coefficients of functions with algebraic endpoint singularities and the aliasing formula of chebyshev polynomials numerical examples are provided to confirm our analysis

symmetry algebra for multi contact structures given by 2n vector fields on mathbb r 2n 1 the notion of multicontact structures has been introduced in 2 as the geometry of manifolds with several distinguished mutually transversal subbundles of the tangent bundle such structures appear naturally as secondary structures in parabolic geometry more precisely let g be a semi simple lie group and p a parabolic subgroup then the nilpotent factor n in the iwasawa decomposition g kan naturally gives rise to mutually transversal subbundles

singular solutions homogeneous norms and quasiconformal mappings in carnot groups in any carnot nilpotent stratified lie group g of homogeneous dimension q green s function u for the q laplace equation exists and is unique we prove that there exists a constant gamma gamma g so that n e gamma u is a homogeneous norm in g then the extremal lengths of spherical ring domains measured with respect to n can be computed and used to give estimates for the extremal lengths of ring domains measured with respect to the carnot caratheodory metric applications include regularity properties of quasiconformal mappings and a geometric characterization of bi lipschitz mappings

hardy space estimates of hausdorff operators on the heisenberg group abstract this is a continuation of a series of works about hausdorff operator on hardy spaces in the setting of the heisenberg group in this paper we obtain the boundedness from lebesgue spaces to hardy spaces for fractional hausdorff operators and their compositions with riesz transforms we also establish the boundedness for two kinds of special hausdorff operators the hausdorff poisson operator and the hausdorff gauss operator on hardy spaces

new quadrature formulas from conformal maps gauss and clenshaw curtis quadrature like legendre and chebyshev spectral methods make use of grids strongly clustered at boundaries from the viewpoint of polynomial approximation this seems necessary and indeed in certain respects optimal nevertheless such methods may waste a factor of pi 2 with respect to each space dimension we propose new nonpolynomial quadrature methods that avoid this effect by conformally mapping the usual ellipse of convergence to an infinite strip or another approximately straight sided domain the new methods are compared with related ideas of bakhvalov kosloff and tal ezer rokhlin and alpert and others an advantage of the conformal mapping approach is that it leads to theorems guaranteeing geometric rates of convergence for analytic integrands for example one of the formulas presented is proved to converge 50 faster than gauss quadrature for functions analytic in an varepsilon neighborhood of 1 1

quasiregular mappings of the heisenberg group the condition 1 1 means that for all d g the coordinate functions o f f belong to the sobolev space w d of functions in l d whose distributional first order partial derivatives are also l integrable in d in the above definition f x is the formal derivative of f a t x defined by means of the partial derivatives i f x l is the operator norm of f x and ji x is the jacobian determinant of f a t x these are uniquely defined almost everywhere by 1 1 it turns out that quasiregular mappings are differentiable almost everywhere and a nonconstant quasiregular map is discrete open and sense preserving for these and other basic properties of qr maps we refer to books re by reshetnyak vu by vuorinen and also to a forthcoming book r6 by the second author the definition extends immediately to the case f m n where m and n are connected oriented riemannian n manifolds we say that a continuous map f m n is quasiregular if it has a local representation at each point of m which is quasiregular in the above sense and if there exists k 1 k oe such that

error analysis for a sinh transformation used in evaluating nearly singular boundary element integrals in the two dimensional boundary element method one often needs to evaluate numerically integrals of the form 1 1g x j x f x a 2 b 2 dx where j 2 is a quadratic g is a polynomial and f is a rational logarithmic or algebraic function with a singularity at zero the constants a and b are such that 1

a notable family of entire intrinsic minimal graphs in the heisenberg group which are not perimeter minimizing one of the main objectives of this paper is to unravel a new interesting phenomenon of the sub riemannian bernstein problem with respect to its euclidean ancestor with the purpose of also indicating a possible line of attack toward its solution we show that the global intrinsic graphs 1 2 are unstable critical points of the horizontal perimeter as a consequence of this fact the study of the stability acquires a central position in the problem itself

sum of squares length of real forms for n d ge 1 let p n 2d denote the smallest number p such that every sum of squares of degree d forms in mathbb r x 1 ldots x n is a sum of p squares we establish lower bounds for p n 2d that are considerably stronger than the bounds known so far combined with known upper bounds they give p 3 2d in d 1 d 2 in the ternary case assuming a conjecture of iarrobino kanev on dimensions of tangent spaces to catalecticant varieties we show that p n 2d sim const cdot d n 1 2 for d rightarrow infty and all n ge 3 for ternary sextics and quaternary quartics we determine the exact value of the invariant showing p 3 6 4 and p 4 4 5

quasiconformal maps in metric spaces with controlled geometry this paper develops the foundations of the theory of quasiconformal maps in metric spaces that satisfy certain bounds on their mass and geometry the principal message is that such a theory is both relevant and viable the first main issue is the problem of definition which we next describe quasiconformal maps are commonly understood as homeomorphisms that distort the shape of infinitesimal balls by a uniformly bounded amount this requirement makes sense in every metric space given a homeomorphism f from a metric space x to a metric space y then for x x and r 0 set

on three dimensional cauchy riemann manifolds of the almost complex tensor of c2 multiplication by i the collection of these subspaces hp m forms a bundle hm c tm called the horizontal or contact bundle indeed strict pseudoconvexity of m implies that the plane field hpm is nondegenerate i e defines a contact structure the almost complex tensor of c2 then restricts to a bundle endomorphism j hm hm such that j2 id the bundle hm c tm together with this endomorphism j defines the cr structure of m slightly more generally let d d u m be a four dimensional compact manifold with boundary m interior d endowed with a smooth almost complex tensor which is integrable on d if d is strictly pseudoconvex i e in a neighborhood of any p e m there is a smooth strictly plurisubharmonic function u negative on d 0 on m but du 0 on m then m inherits a strictly pseudoconvex cr structure from d in this case we shall say that m bounds a strictly pseudoconvex surface this leads us to the abstract definition a strictly pseudoconvex threedimensional cr manifold is a compact manifold m without boundary dim m 3 endowed with a contact structure hm hpm p e m c tm and an endomorphism j of hm such that j2 id throughout this paper unless otherwise stated we shall be working with infinitely differentiable objects thus m hm j are assumed to be smooth in this sense furthermore a differentiable function f m c is a cr function if for any p e m the restriction df ihm hpm tf p c

sub riemannian calculus and monotonicity of the perimeter for graphical strips we consider the class of minimal surfaces given by the graphical strips mathcal s in the heisenberg group mathbb h 1 and we prove that for points p along the center of mathbb h 1 the quantity frac sigma h mathcal s cap b p r r q 1 is monotone increasing here q is the homogeneous dimension of mathbb h 1 we also prove that these minimal surfaces have maximum volume growth at infinity

conformality and q harmonicity in carnot groups we show that if f is a 1 quasiconformal map defined on an open subset of a carnot group g then composition with f preserves q harmonic functions we combine this with a regularity theorem for q harmonic functions and an algebraic regularity theorem for maps between carnot groups to show that f is smooth we give some applications to the study of rigidity

weighted sums of squares in local rings and their completions i let a be an excellent regular local ring of dimension two let t be a finitely generated preordering in a and let widehat t be the preordering generated by t in the completion widehat a of a we study the question when the property of being saturated descends from widehat t to t and establish conditions of geometric nature which allow to decide this question as an application we classify all principal preorderings in a of degree 3 which are saturated in the case where a has a real closed residue field these results have direct implications for nonnegativity certificates for real polynomials on two dimensional semi algebraic sets