brief paper adaptive hinging hyperplanes and its applications in dynamic system identification the model of adaptive hinging hyperplanes ahh is proposed in this paper it is based on multivariate adaptive regression splines mars and generalized hinging hyperplanes ghh and shares attractive properties of the two by making a modification to the basis function of mars ahh shows linear property in each subregion the ahh model is actually a special case of the ghh model which has a universal representation capability for continuous piecewise linear functions the approximation ability of the ahh model is proved the ahh algorithm is developed similar to the mars algorithm it is adaptive and can be executed efficiently hence has power and flexibility to model unknown relationships the ahh procedure is applied to identifying two dynamic systems and its potential is illustrated

brief paper identification of dynamic systems using piecewise affine basis function models piecewise affine pwa basis function autoregressive exogenous bpwarx models are proposed in this paper for nonlinear black box identification a bpwarx model is a weighted sum of pwa basis bpwa functions which are the minimum or maximum of n 1 affine functions in n dimensions since the bpwa functions have a universal representation capability for continuous pwa functions the bpwarx models provide better accuracy than the hinging hyperplane arx hharx models with the same number of parameters and the same order of computational complexity when using a modified gauss newton algorithm to build the models from input output data

error estimates for the implicit mac scheme for the compressible navier stokes equations we prove existence of a solution to the implicit mac scheme for the compressible navier stokes equations we derive error estimates for this scheme on two and three dimensional cartesian grids error estimates are obtained by using the discrete version of the relative energy method introduced on the continuous level in feireisl et al j math fluid mech 14 4 717 730 2012 a systematic use of the theoretical continuous analysis of the equations in combination with the numerical tools is crucial for the result this error estimate does not uses stability hypotheses on the solution of the numerical scheme

solution of the implicitly discretised reacting flow equations by operator splitting abstract a non iterative method for handling the coupling of the implicitly discretised time dependent fluid flow equations is described the method is based on the use of pressure and velocity as dependent variables and is hence applicable to both the compressible and incompressible versions of the transport equations the main feature of the technique is the splitting of the solution process into a series of steps whereby operations on pressure are decoupled from those on velocity at each step with the split sets of equations being amenable to solution by standard techniques at each time step the procedure yields solutions which approximate the exact solution of the difference equations the accuracy of this splitting procedure is assessed for a linearised form of the discretised equations and the analysis indicates that the solution yielded by it differs from the exact solution of the difference equations by terms proportional to the powers of the time step size by virtue of this it is possible to dispense with iteration thus resulting in an efficient implicit scheme while retaining simplicity of implementation relative to contemporary block simultaneous methods this is verified in a companion paper which presents results of computations carried out using the method

dimension of the torelli group for out fn let mathcal t n be the kernel of the natural map out fn gln we use combinatorial morse theory to prove that mathcal t n has an eilenberg maclane space which is 2n 4 dimensional and that h 2n 4 mathcal t n mathbb z is not finitely generated n 3 in particular this shows that the cohomological dimension of mathcal t n is equal to 2n 4 and recovers the result of krstic mccool that mathcal t 3 is not finitely presented we also give a new proof of the fact due to magnus that mathcal t n is finitely generated

on the completion of the mapping class group of genus two abstract in this paper we will study the lie algebra of the prounipotent radical of the relative completion of the mapping class group of genus two in particular we will partially determine a minimal presentation of the lie algebra by determining the generators and bounding the degree of the relations of it

nonlinear black box modeling in system identification a unified overview a nonlinear black box structure for a dynamical system is a model structure that is prepared to describe virtually any nonlinear dynamics there has been considerable recent interest in this area with structures based on neural networks radial basis networks wavelet networks and hinging hyperplanes as well as wavelet transform based methods and models based on fuzzy sets and fuzzy rules this paper describes all these approaches in a common framework from a user s perspective it focuses on what are the common features in the different approaches the choices that have to be made and what considerations are relevant for a successful system identification application of these techniques it is pointed out that the nonlinear structures can be seen as a concatenation of a mapping form observed data to a regression vector and a nonlinear mapping from the regressor space to the output space these mappings are discussed separately the latter mapping is usually formed as a basis function expansion the basis functions are typically formed from one simple scalar function which is modified in terms of scale and location the expansion from the scalar argument to the regressor space is achieved by a radial or a ridge type approach basic techniques for estimating the parameters in the structures are criterion minimization as well as two step procedures where first the relevant basis functions are determined using data and then a linear least squares step to determine the coordinates of the function approximation a particular problem is to deal with the large number of potentially necessary parameters this is handled by making the number of used parameters considerably less than the number of offered parameters by regularization shrinking pruning or regressor selection

generalization of hinging hyperplanes the model of hinging hyperplanes hh can approximate a large class of nonlinear functions to arbitrary precision but represent only a small part of continuous piecewise linear cpwl functions in two or more dimensions in this correspondence the influence of this drawback for black box modeling is first illustrated by a simple example then it is shown that the above shortcoming can be amended by adding a sufficient number of linear functions to current hinges it is proven that any cpwl function of n variables can be represented by a sum of hinges containing at most n 1 linear functions hence the model of a sum of such expanded hinges is a general representation for all cpwl functions the structure of the novel general representation is much simpler than the existing generalized canonical representation that consists of nested absolute value functions this characteristic is very useful for black box modeling based on the new general representation an upper bound on the number of nestings of nested absolute value functions of a generalized canonical representation is established which is much smaller than the known result

rank 1 phenomena for mapping class groups we prove that every element of the mapping class group g has linear growth confirming a conjecture of n ivanov and that g is not boundedly generated we also provide restrictions on linear representations of g and its finite index subgroups

convergence of the mac scheme for the compressible stokes equations we prove in this paper the convergence of the marker and cell mac scheme for the discretization of the steady state compressible stokes equations on two or three dimensional cartesian grids the existence of a solution to the scheme is proven followed by estimates on approximate solutions which yield the convergence of the approximate solutions up to a subsequence and in an appropriate sense we then prove that the limit of the approximate solutions satisfies the mass and momentum balance equations as well as the equation of state which is the main difficulty of this study

modelling thermal convection with large viscosity gradients in one block of the cubed sphere a numerical method solving thermal convection problems with variable viscosity in a spherical shell is presented several features of earlier programs solving the same problem in cartesian geometry are adopted because of their efficiency and robustness finite volume formulation multigrid flow solver parallel implementation a recent composite mesh gridding technique for a spherical surface termed the cubed sphere has proven to be successful in solving other partial differential equations in geophysical problems it is used here because of its various advantages absence of geometrical singularities same metric on each block simple coupling of adjacent blocks in addition it is a good tool to implement grid based methods proven efficient in the cartesian context since it provides a mesh reasonably close to uniform although as in the cartesian case convergence rates decrease with increasing viscosity gradients global contrasts up to 10 6 are obtained at a reasonable cost

overtraining regularization and searching for a minimum with application to neural networks in this paper we discuss the role of criterion minimization as a means for parameter estimation most traditional methods such as maximum likelihood and prediction error identification are based on these principles however somewhat surprisingly it turns out that it is not always optimal to try to find the absolute minimum point of the criterion the reason is that stopped minimization where the iterations have been terminated before the absolute minimum has been reached has more or less identical properties as using regularization adding a parametric penalty term regularization is known to have beneficial effects on the variance of the parameter estimates and it reduces the variance contribution of the misfit this also explains the concept of overtraining in neural nets how does one know when to terminate the iterations then a useful criterion would be to stop iterations when the criterion function applied to a validation data set no longer decreases however in this paper we show th

efficient agnostic learning of neural networks with bounded fan in we show that the class of two layer neural networks with bounded fan in is efficiently learnable in a realistic extension to the probably approximately correct pac learning model in this model a joint probability distribution is assumed to exist on the observations and the learner is required to approximate the neural network which minimizes the expected quadratic error as special cases the model allows learning real valued functions with bounded noise learning probabilistic concepts and learning the best approximation to a target function that cannot be well approximated by the neural network the networks we consider have real valued inputs and outputs an unlimited number of threshold hidden units with bounded fan in and a bound on the sum of the absolute values of the output weights the number of computation steps of the learning algorithm is bounded by a polynomial in 1 spl epsiv 1 spl delta n and b where spl epsiv is the desired accuracy spl delta is the probability that the algorithm fails n is the input dimension and b is the bound on both the absolute value of the target which may be a random variable and the sum of the absolute values of the output weights in obtaining the result we also extended some results on iterative approximation of functions in the closure of the convex hull of a function class and on the sample complexity of agnostic learning with the quadratic loss function

k plane regression in this paper we present a novel algorithm for piecewise linear regression which can learn continuous as well as discontinuous piecewise linear functions the main idea is to repeatedly partition the data and learn a linear model in each partition the proposed algorithm is similar in spirit to k means clustering algorithm we show that our algorithm can also be viewed as a special case of an em algorithm for maximum likelihood estimation under a reasonable probability model we empirically demonstrate the effectiveness of our approach by comparing its performance with that of the state of art algorithms on various datasets

harmonic volume symmetric products and the abel jacobi map the author generalizes b harris definition of harmonic volume to the algebraic cycle w k w k for k 1 in the jacobian of a nonsingular algebraic curve x we define harmonic volume determine its domain and show that it is related to the image of w k w k in the griffiths intermediate jacobian we derive a formula expressing harmonic volume as a sum of integrals over a nested sequence of submanifolds of the k fold symmetric product of x we show that when applied to a certain class of forms takes values in a discrete subgroup of r z and hence when suitably extended to complex valued forms is identically zero modulo periods on primitive forms if k 2

normal functions picard fuchs equations and elliptic fibrations onk3 surfaces using gauss manin derivatives of generalized normal functions we arrive at results on the non triviality of the transcendental regulator for km of a very general projective algebraic manifold our strongest results are for the transcendental regulator for k1 of a very general k3 surface and its self product we also construct an explicit family of k1 cycles on h e8 e8 polarized k3 surfaces and show they are indecomposable by a direct evaluation of the real regulator critical use is made of natural elliptic fibrations hypersurface normal forms and an explicit parametrization by modular functions

orbits of curves under the johnson kernel this paper has two main goals first we give a complete explicit and computable solution to the problem of when two simple closed curves on a surface are equivalent under the johnson kernel second we show that the johnson filtration and the johnson homomorphism can be defined intrinsically on subsurfaces and prove that both are functorial under inclusions of subsurfaces the key point is that the latter reduces the former to a finite computation which can be carried out by hand in particular this solves the conjugacy problem in the johnson kernel for separating twists using a theorem of putman we compute the first betti number of the torelli group of a subsurface

a simple lemma on greedy approximation in hilbert space and convergence rates for projection pursuit regression and neural network training a general convergence criterion for certain iterative sequences in hilbert space is presented for an important subclass of these sequences estimates of the rate of convergence are given under very mild assumptions these results establish an 0 1 f4n nonsampling convergence rate for projection pursuit regression and neural network training where n represents the number of ridge functions neurons or coefficients in a greedy basis expansion

hinging hyperplanes for time series segmentation division of a time series into segments is a common technique for time series processing and is known as segmentation segmentation is traditionally done by linear interpolation in order to guarantee the continuity of the reconstructed time series the interpolation based segmentation methods may perform poorly for data with a level of noise because interpolation is noise sensitive to handle the problem this paper establishes an explicit expression for segmentation from a compact representation for piecewise linear functions using hinging hyperplanes this expression enables the use of regression to obtain a continuous reconstructed signal and as a consequence application of advanced techniques in segmentation in this paper a least squares support vector machine with lasso using a hinging feature map is given and analyzed based on which a segmentation algorithm and its online version are established numerical experiments conducted on synthetic and real world datasets demonstrate the advantages of our methods compared to existing segmentation algorithms

an algebraic characterization of the point pushing subgroup abstract the point pushing subgroup p g of the mapping class group mod g 1 of a surface with marked point is an embedding of 1 g given by pushing the marked point around loops we prove that for g 3 the subgroup p g is the unique normal genus g surface subgroup of mod g 1 as a corollary to this uniqueness result we give a new proof that out mod g 1 1 where out denotes the outer automorphism group a proof which does not use automorphisms of complexes of curves ingredients in our proof of this characterization theorem include combinatorial group theory representation theory the johnson theory of the torelli group surface topology and the theory of lie algebras

complex multiplication cycles and a conjecture of beilinson and bloch a generalization of the conjecture of birch and swinnerton dyer is investigated using complex multiplication cycles on a particular kuga fiber variety a weak finiteness result consistent with the conjecture is proved the image of complex multiplication cycles under the abel jacobi map is computed explicitly the results provide numerical evidence supporting the conjecture they also give evidence for a relationship between complex multiplication cycles and a modular form of weight 5 2 and raise questions for further investigation

the second rational homology group of the moduli space of curves with level structures abstract let be a finite index subgroup of the mapping class group of a closed genus g surface that contains the torelli group for instance can be the level l subgroup or the spin mapping class group we show that h 2 q q for g 5 a corollary of this is that the rational picard groups of the associated finite covers of the moduli space of curves are equal to q we also prove analogous results for surface with punctures and boundary components

the computation of compressible and incompressible recirculating flows by a non iterative implicit scheme abstract the piso algorithm which is presented in a companion paper is a non iterative method for solving the implicity discretised time dependent fluid flow equations the algorithm is here applied in conjunction with a finite volume technique employing a backward temporal difference scheme to the computation of compressible and incompressible flow cases the results of calculations are compared with similar ones obtained with an existing iterative method it is shown that for time evolving flows the splitting error of piso is negligibly small at the level of time step required to eliminate the temporal truncation error while the avoidance of iteration results in a substantial reduction in computing effort over that required by iterative methods it is also demonstrated that piso is stable for fairly large time steps which renders it useful for steady state calculations as well

the johnson homomorphism and its kernel we give a new proof of a celebrated theorem of dennis johnson that asserts that the kernel of the johnson homomorphism on the torelli subgroup of the mapping class group is generated by separating twists in fact we prove a more general result that also applies to subsurface torelli groups using this we extend johnson s calculati on of the rational abelianization of the torelli group not only to the subsurface torelli groups but also to finite index subgroups of the torelli group that contain the kernel of the johnson homomorphism

a nontrivial algebraic cycle in the jacobian variety of the klein quartic we prove some value of the harmonic volume for the klein quartic c is nonzero modulo frac 1 2 mathbb z using special values of the generalized hypergeometric function 3 f 2 this result tells us the algebraic cycle c c is not algebraically equivalent to zero in the jacobian variety j c

on the abel jacobi maps of fermat jacobians we study the abel jacobi image of the ceresa cycle w k w k where wk is the image of the kth symmetric product of a curve x on its jacobian variety for the fermat curve of degree n we express it in terms of special values of generalized hypergeometric functions and give a criterion for the non vanishing of w k w k modulo algebraic equivalence which is verified numerically for some n and k

equivariant group presentations and the second homology group of the torelli group we develop a theory of equivariant group presentations and relate them to the second homology group of a group our main application says that the second homology group of the torelli subgroup of the mapping class group is finitely generated as a mathbb z mathrm sp 2g mathbb z module

estimation of a regression function by maxima of minima of linear functions in this paper estimation of a regression function from independent and identically distributed random variables is considered estimates are defined by minimization of the empirical l2 risk over a class of functions which are defined as maxima of minima of linear functions results concerning the rate of convergence of the estimates are derived in particular it is shown that for smooth regression functions satisfying the assumption of single index models the estimate is able to achieve up to some logarithmic factor the corresponding optimal one dimensional rate of convergence hence under these assumptions the estimate is able to circumvent the so called curse of dimensionality the small sample behavior of the estimates is illustrated by applying them to simulated data

a neural network of smooth hinge functions smooth hinging hyperplane shh has been proposed as an improvement over the well known hinging hyperplane hh by the fact that it retains the useful features of hh while overcoming hh s drawback of nondifferentiability this paper introduces a formal characterization of smooth hinge function shf which can be used to generate shh as a neural network a method for the general construction of shf is also given furthermore the work proves that shh is better than hh in functional approximation i e the optimal error of shh approximating a general function is always smaller or equal to that of hh particularly in the case that the shf is generated via the integration of a class of sigmoidal functions it is further proven that the corresponding shh of the 2m shfs would outperform a neural network with m of the sigmoidal function from which the shf is derived any upper bound established on the approximation error of a neural network of m sigmoidal activation functions can hence be translated to the shh of m shfs by replacing m with m 2 the work also includes an algorithm for the identification of shh making use of its differentiability property simulation experiments are presented to validate the theoretical conclusions to possible extent

unified formulation for compressible and incompressible flows by using multi integrated moments ii multi dimensional version for compressible and incompressible flows following our previous paper on a novel finite volume formulation for computing flows of any mach number we present the multi dimensional extension of the method in this paper treating two kinds of averages which are generically called moments and defined as the volume integrated average via and the surface integrated average sia in this paper as the prognostic variables makes the present method different from the conventional finite volume method where only via is the basic variable the resulting discretization approach which is a type of the cip multi moment finite volume method and called vsiam3 volume surface integrated average based multi moment method is combined with a pressure based projection formulation for the time integration to enable the simulations for both compressible and incompressible flows the numerical algorithm for multi dimensions will be reported in this paper numerical experiments with flows of a wide range mach numbers will be also presented

every mapping class group is generated by 6 involutions abstract let mod g b denote the mapping class group of a surface of genus g with b punctures luo asked in torsion elements in the mapping class group of a surface math gt 0004048 v1 8apr2000 if there is a universal upper bound independent of genus for the number of torsion elements needed to generate mod g b we answer luo s question by proving that 3 torsion elements suffice to generate mod g 0 we also prove the more delicate result that there is an upper bound independent of genus not only for the number of torsion elements needed to generate mod g b but also for the order of those elements in particular our main result is that 6 involutions i e orientation preserving diffeomorphisms of order two suffice to generate mod g b for every genus g 3 b 0 and g 4 b 1

a unified method for computing incompressible and compressible flows in boundary fitted coordinates a unified method for computing incompressible and compressible flows with mach uniform accuracy and efficiency is described the method is equally applicable to stationary and nonstationary flows a pressure based discretisation on a staggered grid in general boundary fitted coordinates is used for the euler equations extension to navier stokes is straightforward dimensionless variables that remain finite for all mach numbers are used mach number independent accuracy and efficiency is shown by numerical experiments

an oscillation free flow solver based on flux reconstruction abstract in this paper a segregated algorithm is proposed to suppress high frequency oscillations in the velocity field for incompressible flows in this context a new velocity formula based on a reconstruction of face fluxes is defined eliminating high frequency errors in analogy to the rhie chow interpolation this approach is equivalent to including a flux based pressure gradient with a velocity diffusion in the momentum equation in order to guarantee second order accuracy of the numerical solver a set of conditions are defined for the reconstruction operator to arrive at the final formulation an outlook over the state of the art regarding velocity reconstruction procedures is presented comparing them through an error analysis a new operator is then obtained by means of a flux difference minimization satisfying the required spatial accuracy the accuracy of the new algorithm is analyzed by performing mesh convergence studies for unsteady navier stokes problems with analytical solutions the stabilization properties of the solver are then tested in a problem where spurious numerical oscillations arise for the velocity field the results show a remarkable performance of the proposed technique eliminating high frequency errors without losing accuracy

the structure of the torelli group i a finite set of generators for j this is the first of three papers concerning the so called torelli group let m mg be a compact orientable surface of genus g having n boundary components and let 9 xg be its mapping class group that is the group of orientation preserving diffeomorphisms of m which are 1 on the boundary am modulo isotopies which fix 3m pointwise this group is also known to the complex analysts as the teichmuller group or modular group if n 0 or 1 let further 4 jg be the subgroup of d1 which acts trivially on h1 m z the topologists have no traditional name for a but the analysts tell me it was known classically and is called the torelli group several interesting problems and conjectures exist concerning f the principal one can be found in kirby s problem list k and asks if gg is finitely generated in this first paper we shall answer the question affirmatively for both gg 0 and fgg when g 3 and shall give a fairly simple set of generators two other conjectures were made by the author the first involves the subgroup j of f which is generated by twists on nulhomologous simple closed curves ji produces a surjective homomorphism t fgg 1 a3h1 m z which kills c and it is conjectured there that in fact 5y ker t the proof of this is the content of the second paper in the third paper we use the results of the second to compute the abelianization f f explicitly thereby verifying another conjecture in ji the first reasonably simple but infinite set of generators for fg0 was produced by powell in p his generators consist of two types a twists on bounding simple closed curves b opposite twists on a bounding pair of disjoint homologous simple closed curves each of which are nonbounding using his result the author showed in j2 that the maps of the second type which we call bp maps for bounding pair are actually sufficient to generate both kg 0 and fg 1 for g 3 and in fact that we need only those whose two curves bound a genus one subsurface of m note that for g 2 all bp maps are trivial and hence the result fails in this case in the finite set of generators produced in this paper only

canonical extensions of the johnson homomorphisms to the torelli groupoid abstract we prove that every trivalent marked bordered fatgraph comes equipped with a canonical generalized magnus expansion in the sense of kawazumi this magnus expansion is used to give canonical extensions of the higher johnson homomorphisms m for m 1 to the torelli groupoid and we provide a recursive combinatorial formula for tensor representatives of these extensions in particular we give an explicit 1 cocycle in the dual fatgraph complex which extends 2 and thus answer affirmatively a question of morita and penner to illustrate our techniques for calculating higher johnson homomorphisms in general we give explicit examples calculating m for m 3

identification of piecewise affine systems via mixed integer programming this paper addresses the problem of identification of hybrid dynamical systems by focusing the attention on hinging hyperplanes and wiener piecewise affine autoregressive exogenous models in which the regressor space is partitioned into polyhedra with affine submodels for each polyhedron in particular we provide algorithms based on mixed integer linear or quadratic programming which are guaranteed to converge to a global optimum for the special case where the estimation data only seldom switches between the different submodels we also suggest a way of trading off between optimality and complexity by using a change detection approach

p adic l functions and the coniveau filtration on chow groups we give evidence for the refined version of the bloch beilinson conjecture involving coniveau filtrations by studying several infinite families of cm motives indexed by the integers r 1 that are irreducible of hodge type 2r 1 0 0 2r 1 and whose l functions vanish at the center in each case we construct a corresponding algebraic cycle that is homologically trivial but nontrivial in the top graded piece for the coniveau filtration consequently we obtain new explicit examples of cycles that are nontorsion in the griffiths group the proof of the main result relies crucially on p adic hodge theoretic methods and in particular on the relation between images of homologically trivial cycles under the p adic abel jacobi map and the values of p adic l functions

hinging hyperplanes for regression classification and function approximation a hinge function y h x consists of two hyperplanes continuously joined together at a hinge in regression prediction classification pattern recognition and noiseless function approximation use of sums of hinge functions gives a powerful and efficient alternative to neural networks with computation times several orders of magnitude less than is obtained by fitting neural networks with a comparable number of parameters a simple and effective method for finding good hinges is presented

homological versus algebraic equivalence in a jacobian abstract let z be an algebraic p cycle homologous to zero in an algebraic complex manifold v associated to z is a linear function on holomorphic 2p 1 forms on v modulo periods that vanishes if z is algebraically equivalent to zero in v i give a formula for for the case of v the jacobian of an algebraic curve c and z c c c inverse of c in terms of iterated integrals of holomorphic 1 forms on c if c is the degree 4 fermat curve i use this formula to show that c c is not algebraically equivalent to zero

a general mass consistency algorithm for hybrid particle finite volume pdf methods an algorithm is devised to maintain a correct spatial distribution of computational particles in hybrid particle finite volume fv probability density function pdf methods for chemically reacting turbulent flows the approach is by design compatible with existing fv computational fluid dynamics cfd codes that are used to model practical engineering flows in complex geometric configurations the algorithm is suitable for general three dimensional incompressible or compressible steady or time dependent flows using structured or unstructured stationary or deforming computational meshes it is compatible with a variety of element shapes commonly used in research and engineering cfd codes including hexahedra prisms and tetrahedra robustness accuracy and efficiency of the approach are demonstrated via computations for several two and three dimensional steady and unsteady flow configurations using computational meshes that vary in element type and in mesh quality both composition pdf and velocity pdf methods are employed this work broadens the accessibility of pdf methods for practical turbulent combustion systems

influence of momentum interpolation methods on the accuracy and convergence of pressure velocity coupling algorithms in openfoam checkerboard pressure is one of the most significant problems arising from the use of collocated grids for fluid dynamic simulations using the finite volume method the original rhie chow momentum interpolation technique termed original momentum interpolation method omim was proposed to eliminate the non physical saw tooth pressure oscillations however it was soon proved that the steady state solutions obtained with this technique were under relaxation factor dependent nevertheless standard omim is still commonly used in several cfd codes like openfoam the software used for this work in this paper the omim and a possible correction for under relaxation dependency which has been implemented in openfoam are discussed in detail the proposed methodology is compared and contrasted with omim in terms of accuracy of the solution and speed of convergence for several classical pressure velocity segregated algorithms for steady state solvers namely simple simple c simple r and piso a classical laminar 2d cavity is used as the base test case the study is then extended to a more complex 2d airfoil profile naca0012 in the cases considered mesh uniformity and orthogonality are progressively reduced and turbulence starts playing an important role limiting therefore the convergence of the cases and the performance of the correction

stable and convergent approximation of two dimensional vector fields on unstructured meshes a new framework is proposed for analyzing staggered grid finite difference finite volume methods on unstructured meshes the new framework employs the concept of external approximation of function spaces and gauge convergence of numerical schemes through the quantities of vorticity and divergence instead of individual derivatives of the velocity components the construction of a stable and convergent external approximation of a simple but relevant vector valued function space is demonstrated and the new framework is applied to establish the convergence of the mac scheme for the incompressible stokes problem on unstructured meshes

finiteness properties of the johnson subgroups abstract the main goal of this note is to provide evidence that the first rational homology of the johnson subgroup k g 1 of the mapping class group of a genus g surface with one marked point is finite dimensional building on work of dimca papadima 4 we use symplectic representation theory to show that for all g 3 the completion of h 1 k g 1 q with respect to the augmentation ideal in the rational group algebra of z 2 g is finite dimensional we also show that the terms of the johnson filtration of the mapping class group have infinite dimensional rational homology in some degrees in almost all genera generalizing a result of akita

on the andreadakis johnson filtration of the automorphism group of a free group abstract families of non trivial cohomology classes are given for the discrete groups belonging to the johnson filtration of the automorphism group of a free group generated by n letters the methods are 1 to analyze analogous classes for filtrations of a subgroup of the pure symmetric automorphism group of a free group and 2 to analyze features of these classes which are preserved by the johnson homomorphism with values in the lie algebra of derivations of a free lie algebra one consequence is that the ranks of the cohomology groups in any fixed degree i for 1 i n 2 for the johnson filtrations of ia n increase without bound the actual classes constructed are fragile in the sense that they vanish after passage to successive filtrations furthermore these classes are all naturally in the image of the johnson homomorphism on the level of cohomology the methods are similar to those occurring within the theory of arrangements the authors congratulate corrado de concini on the occasion of his 60 th birthday one of the authors would like to thank corrado for a wonderful time arising from a discussion about cohomology of groups with friends in the basement of il palazzone

relative pro l completions of mapping class groups abstract fix a prime number l in this paper we develop the theory of relative pro l completion of discrete and profinite groups a natural generalization of the classical notion of pro l completion and show that the pro l completion of the torelli group does not inject into the relative pro l completion of the corresponding mapping class group when the genus is at least 2 see theorem 1 below as an application we prove that when g 2 the action of the pro l completion of the torelli group t g 1 on the pro l fundamental group of a pointed genus g surface is not faithful the choice of a first order deformation of a maximally degenerate stable curve of genus g determines an action of the absolute galois group g q on the relative pro l completion of the corresponding mapping class group we prove that for all g all such representations are unramified at all primes l when the first order deformation is suitably chosen this proof was communicated to us by mochizuki and tamagawa

the dimension of the torelli group we prove that the cohomological dimension of the torelli group for a closed connected orientable surface of genus g 2 is equal to 3g 5 this answers a question of mess who proved the lower bound and settled the case of g 2 we also find the cohomological dimension of the johnson kernel the subgroup of the torelli group generated by dehn twists about separating curves to be 2g 3 for g 2 we prove that the top dimensional homology of the torelli group is infinitely generated finally we give a new proof of the theorem of mess that gives a precise description of the torelli group in genus 2 the main tool is a new contractible complex called the complex of minimizing cycles on which the torelli group acts

the universal surface bundle over the torelli space has no sections for g 3 we give two proofs of the fact that the birman exact sequence for the torelli group begin aligned 1 rightarrow pi 1 s g rightarrow mathcal i g 1 rightarrow mathcal i g rightarrow 1 end aligned does not split this result was claimed by mess unit tangent bundle subgroups of the mapping class groups msri pre print 1990 but his proof has a critical and unrepairable error which will be discussed in the introduction let mathcal u mathcal i g n xrightarrow tu g n mathcal b mathcal i g n resp mathcal u mathcal p mathcal i g n xrightarrow tu g n mathcal b mathcal p mathcal i g n denote the universal surface bundle over the torelli space fixing n points as a set resp pointwise we also deduce that tu g n has no sections when n 1 and that tu g n has precisely n distinct sections for n ge 0 up to homotopy

arithmetic group symmetry and finiteness properties of torelli groups we examine groups whose resonance varieties characteristic varieties and sigma invariants have a natural arithmetic group symmetry and we explore implications on various niteness properties of subgroups we compute resonance varieties characteristic varieties and alexander polynomials of torelli groups and we show that all subgroups containing the johnson kernel have nite rst betti number when the genus is at least 4 we also prove that in this range the i adic completion of the alexander invariant is nite dimensional and the kahler property for the torelli group implies the nite generation of the johnson kernel

automorphisms of surface mapping class groups a recent theorem of n ivanov ivanov of the leningrad branch of the steklov mathematical institute has shown that the outer automorphism groups of surface mapping class groups are finite in this report we shall give explicit descriptions of the outer automorphism groups of both the mapping class groups and extended mapping class groups of closed connected orientable surfaces of negative euler characteristic the extended mapping class groups are the extensions of the mapping class groups by the orientation reversing isotopy classes for surfaces of genus greater than two these are respectively a cyclic group of order two and the trivial group for a surface of genus two these are both noncyclic groups of order four once again the hypergeometric involution in genus two plays a unique role

on the graded quotients of the rings of fricke characters of free groups abstract for a free group f n on n generators we consider the aut f n invariant ideal j generated by tr x 2 for all x f n in the ring x q f n of fricke characters of f n this ideal defines a descending filtration j j 2 j 3 of x q f n for each k 1 we introduce the normal subgroup e n k consisting of all automorphisms of f n which act on j j k 1 trivially these normal subgroups define a central filtration of aut f n this is a fricke character analogue of the andreadakis johnson filtration a n k of aut f n the main purpose of the paper is to show that e n 1 coincides with inn f n a n 2 where inn f n is the inner automorphism group of the free group f n and that a n 2 k e n k for any k 1

on the hinge finding algorithm for hingeing hyperplanes this correspondence concerns the estimation algorithm for hinging hyperplane hh models a piecewise linear model for approximating functions of several variables suggested in breiman 1993 the estimation algorithm is analyzed and it is shown that it is a special case of a newton algorithm applied to a sum of squared error criterion this insight is then used to suggest possible improvements of the algorithm so that convergence to a local minimum can be guaranteed in addition the way of updating the parameters in the hh model is discussed in breiman a stepwise updating procedure is proposed where only a subset of the parameters are changed in each step this connects closely to some previously suggested greedy algorithms and these greedy algorithms are discussed and compared to a simultaneous updating of all parameters

pressure based finite volume method for calculation of compressible viscous gas flows a pressure based iterative finite volume method is developed for calculation of compressible viscous heat conductive gas flows at all speeds the method does not need the use of under relaxation coefficient in order to ensure a convergence of the iterative process the method is derived from a general form of system of equations describing the motion of compressible viscous gas an emphasis is done on the calculation of gaseous microfluidic problems a fast transient process of gas wave propagation in a two dimensional microchannel is used as a benchmark problem the results obtained by using the new method are compared with the numerical solution obtained by using simple iterative and piso non iterative methods it is shown that the new iterative method is faster than simple for the considered problem the new method is slightly faster than piso as well calculated are also some typical microfluidic subsonic and supersonic flows and the rayleigh benard convection of a rarefied gas in continuum limit the numerical results are compared with other analytical and numerical solutions