asymptotic dimension and small cancellation for hierarchically hyperbolic spaces and groups we prove that all hierarchically hyperbolic groups have finite asymptotic dimension one application of this result is to obtain the sharpest known bound on the asymptotic dimension of the mapping class group of a finite type surface improving the bound from exponential to at most quadratic in the complexity of the surface we also apply the main result to various other hierarchically hyperbolic groups and spaces we also prove a small cancellation result namely if g is a hierarchically hyperbolic group h g is a suitable hyperbolically embedded subgroup and n h is sufficiently deep in h then g n is a relatively hierarchically hyperbolic group this new class provides many new examples to which our asymptotic dimension bounds apply along the way we prove new results about the structure of hhss for example the associated hyperbolic spaces are always obtained up to quasi isometry by coning off canonical coarse product regions in the original space generalizing a relation established by masur and minsky between the complex of curves of a surface and teichmuller space

estimation of aquifer parameters under transient and steady state conditions 2 uniqueness stability and solution algorithms paper 2 of this three part series starts with a discussion of the question under what conditions is the aquifer inverse problem well posed after defining the terms uniqueness identifiability and stability theoretical considerations and synthetic examples are used to demonstrate that ill posedness can be mitigated by including prior information about the parameters in the estimation criterion to be minimized at the same time the inclusion of such information is shown to be insufficient to guarantee uniqueness and stability in all cases several test problems in the recent literature which have resulted in pessimistic conclusions about the solvability of the aquifer inverse problem are shown to be ill posed a question is thus raised about the validity of these conclusions in the general case various conjugate gradient algorithms coupled with the adjoint state finite element method for computing the gradient of the estimation criterion and with newton s method for determining the optimum step size downgradient are compared a marked improvement in the rate at which these algorithms converge is shown to be achieved by switching from one method to another when the former slows down or fails to converge

deviation inequalities for random walks we study random walks on groups with the feature that roughly speaking successive positions of the walk tend to be aligned we formalize and quantify this property by means of the notion of deviation inequalities we show that deviation inequalities have several consequences including central limit theorems the local lipschitz continuity of the rate of escape and entropy as well as linear upper and lower bounds on the variance of the distance of the position of the walk from its initial point in the second part of this article we show that the exponential deviation inequality holds for measures with exponential tail on acylindrically hyperbolic groups these include nonelementary relatively hyperbolic groups mapping class groups many groups acting on cat 0 spaces and small cancellation groups

small cancellation theory over burnside groups abstract we develop a version of small cancellation theory in the variety of burnside groups more precisely we show that there exists a critical exponent n 0 such that for every odd integer n n 0 the well known classical c 1 6 small cancellation theory as well as its graphical generalization and its version for free products produce examples of infinite n periodic groups our result gives a powerful tool for producing uncountable collections of examples of n periodic groups with prescribed properties it can be applied without any prior knowledge in the subject of n periodic groups as applications we show the undecidability of markov properties in classes of n periodic groups we produce n periodic groups whose cayley graph contains an embedded expander graphs and we give an n periodic version of the rips construction we also obtain simpler proofs of some known results like the existence of uncountably many finitely generated n periodic groups and the sq universality in the class of n periodic groups of free burnside groups

median structures on asymptotic cones and homomorphisms into mapping class groups the main goal of this paper is a detailed study of asymptotic cones of the mapping class groups in particular we prove that every asymptotic cone of a mapping class group has a bi lipschitz equivariant embedding into a product of real trees sending limits of hierarchy paths onto geodesics and with image a median subspace one of the applications is that a group with kazhdan s property t can have only finitely many pairwise non conjugate homomorphisms into a mapping class group we also give a new proof of the rank conjecture of brock and farb previously proved by behrstock and minsky and independently by hamenstaedt

the direct inverse problem in aquifers solution of the inverse problem for aquifers requires identification of a distributed parameter system in addition to ill posedness the inverse problem may be underdetermined if recharge is treated as a spatially and temporally variable distribution parameter galerkin s method is applied to a weak formulation of the direct inverse problem leading to a possibly nonlinear weighted least squares problem consideration of generic errors suggests that if the estimate for some derivative of head contains a significant error on one element for all time then significant estimation errors are possible for all nodes and for all reservoir parameters in contrast if some derivative of head is small in some spatial region for all time causing the matrix to be inverted in the least squares algorithm to be nearly singular then significant estimation errors can occur in the corresponding reservoir parameters in that spatial region although it is still possible that other reservoir parameters in that region as well as all reservoir parameters elsewhere may be satisfactorily estimated

geometry of the mapping class groups i boundary amenability we construct a geometric model for the mapping class group mathcal m mathcal c mathcal g of a non exceptional oriented surface s of genus g with k punctures and use it to show that the action of mathcal m mathcal c mathcal g on the compact metrizable hausdorff space of complete geodesic laminations for s is topologically amenable as a consequence the novikov higher signature conjecture holds for every subgroup of mathcal m mathcal c mathcal g

tight geodesics in the curve complex the curve graph mathcal g associated to a compact surface is the 1 skeleton of the curve complex defined by harvey masur and minsky showed that this graph is hyperbolic and defined the notion of a tight geodesic therein we prove some finiteness results for such geodesics for example we show that a slice of the union of tight geodesics between any pair of points has cardinality bounded purely in terms of the topological type of we deduce some consequences for the action of the mapping class group on mathcal g in particular we show that it satisfies an acylindricity condition and that the stable lengths of pseudoanosov elements are rational with bounded denominator

estimation of aquifer parameters under transient and steady state conditions 1 maximum likelihood method incorporating prior information in this series of three papers a method is presented to estimate the parameters of groundwater flow models under steady and nonsteady state conditions the parameters include values and directions of principal hydraulic conductivities or transmissivities in anisotropic media specific storage or storativity interior and boundary recharge or leakage rates coefficients of head dependent interior and boundary sources and boundary heads in transient situations the initial head distribution can also be estimated if the system is originally at a steady state paper 1 of the series discusses some of the advantage in treating the inverse problem statistically and in regularizing its solution by means of penalty criteria based on prior estimates of the parameters the inverse problem is posed in the framework of maximum likelihood theory cast in a manner that accounts for prior information about the parameters since not all the factors which contribute to the prior errors can be quantified statistically at the outset the covariance matrices of these errors are expressed in terms of several parameters which if unknown can be estimated jointly with the hydraulic parameters by a stagewise optimization process when transient head data are separated by a fixed time interval the temporal structure of these data is approximated by a lag one autoregressive model with a correlation coefficient that can be treated as another unknown parameter estimation errors are analyzed by examining the lower bound of their covariance matrix in the eigenspace paper 1 concludes by suggesting that certain model identification criteria developed in the time series literature all of which are based on the maximum likelihood concept might be useful for selecting the best groundwater model or the best method of parameterizing a particular model among a number of given alternatives

the virtual cohomological dimension of the mapping class group of an orientable surface let f f r be the mapping class group of a surface f of genus g with s punctures and r boundary components the purpose of this paper is to establish cohomology properties of f parallel to those of the arithmetic groups if g is a linear algebraic group defined over q x is the symmetric space associated to g and a is an arithmetic subgroup of g then a is virtually torsion free and acts properly discontinuously on x the rational homology of a is the same as that of x a furthermore there is a bordification of x bs to a manifold with corners 3 and an extension of the action of a to a properly discontinuous action on jf so that the quotient x a is compact the boundary of jf is homotopy equivalent to a wedge of spheres say of dimension d and the virtual cohomological dimension of a is n d 1 where n is the dimension of x in the case of the mapping class group there is no analog for g ivanov unpublished has proven that f is not arithmetic nevertheless f acts properly discontinuously on teichmiiller space r which is homeomorphic to euclidean space of dimension 6 g 6 2 s 3 will play the role of the symmetric space the quotient of 9by f is the moduli space of curves whose rational homology is then identified with that of e harvey har has constructed a borel serre bordification j of 3 by analytic methods in the case where f has punctures we will build g by a different combinatorial method in addition we will explicitly describe inside y a cell complex y of dimension 4 g 4 s onto which j may be f equivariantly retracted thus establishing an analog of the constructions for sl n by serre soul6 sol and ash a this complex will be of the lowest possible dimension because we will use j to prove our main result

acylindrical accessibility for groups we define the notion of acylindrical graph of groups of a group we bound the combinatorics of these graphs of groups for f g freely indecomposable groups our arguments imply the finiteness of acylindrical surfaces in closed 3 manifolds ha finiteness of isomorphism classes of small splittings of torsion free freely indecomposable hyperbolic groups as well as finiteness results for small splittings of f g kleinian and semisimple discrete groups acting on non positively curved simply connected manifolds in order to get our accessibility for f g groups we generalize parts of rips analysis of stable actions of f p groups on real trees to f g groups the concepts we present play an essential role in constructing the canonical jsj decomposition se1 ri se2 in obtaining the hopf property for hyperbolic groups se2 and in our study of sets of solutions to equations in a free group se3

simultaneously continuous retraction and bishop phelps bollobas type theorem abstract the dual space x of a banach space x is said to admit a uniformly simultaneously continuous retraction if there is a retraction r from x onto its unit ball b x which is uniformly continuous in norm topology and continuous in weak topology we prove that if a banach space resp complex banach space x has a normalized unconditional schauder basis with unconditional basis constant 1 and if x is uniformly monotone resp uniformly complex convex then x admits a uniformly simultaneously continuous retraction it is also shown that x admits such a retraction if x x i c 0 or x x i l 1 where x i is a family of separable banach spaces whose duals are uniformly convex with moduli of convexity i e with inf i i e 0 for all 0 e 1 let k be a locally compact hausdorff space and let c 0 k be the real banach space consisting of all real valued continuous functions vanishing at infinity as an application of simultaneously continuous retractions we show that a pair x c 0 k has the bishop phelps bollobas property for operators if x admits a uniformly simultaneously continuous retraction as a corollary c 0 s c 0 k has the bishop phelps bollobas property for operators for every locally compact metric space s

error estimates for the numerical identification of a variable coefficient error estimates are derived for the approximate identification of an unknown transmissivity coefficient in a partial differential equation describing a model problem in groundwater now the approximation scheme considered determines the coefficient by least squares fitting of the observed pressure data 1 introduction in this paper we wish to present an error analysis of a common numerical scheme used in the identification of parameters in distributed systems specifically we shall concern ourselves here with a model problem in groundwater flow the problem is to identify a spatially varying transmissivity coefficient a x from observations of the piezometric head u x in a two dimensional static aquifer s where a and u are related by the equations 1 div avw ins2

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

recovery of a variable coefficient in a coastal evolution equation a method is presented for reconstructing an unknown coefficient in a linear diffusion equation from measured data this equation arises in the description of coastline evolution and preliminary results are presented here the unknown term may vary with both space and time although time variation is assumed to be slow inversion is carried out by first expressing the solution of the direct problem formally in terms of the governing operators and making explicit approximations to these expressions using data at two time steps this then allows equations to be derived and solved to give explicit expressions for the required function

cauchy data requirement of the inverse problem of the mean age equation a classical theoretical inverse problem in groundwater hydraulics is the calibration of a heterogeneous isotropic hydraulic conductivity field using known hydraulic head data at each point in the domain it has been long known that a necessary condition for the uniqueness of the solution to this theoretical inverse problem is the cauchy data i e the hydraulic conductivity values or equivalently flux along a line crossing all streamtubes in the entire domain this result has been cited in identifying the value of boundary flux data in practical inverse problems although in general such data are generally not available in this work we show that groundwater mean age data replace this cauchy data requirement