a note on convergence under dynamical thresholds with delays we complement the study of the asymptotic behaviour of the dynamical threshold neuron model with delay introduced by gopalsamy and leung 1997 by providing a description of the dynamics of the system in the remaining parameters range we characterize the regions of harmless delays and those in which delay induced oscillations appear

exact likelihood evaluation in a markov mixture model for time series of seizure counts this paper provides an alternative to albert s 1991 biometrics 47 1371 1381 approximation to the e step when using the em algorithm for parameter estimation in markov mixture models use of a recursive algorithm of baum et al 1970 annals of mathematical statistics 41 164 171 results in exact evaluation of the likelihood optimal parameter estimates and very efficient computation applications to time series of seizure counts and fetal movements clearly show the advantages of this exact approach

discrete and continuous time probabilistic models and algorithms for inferring neuronal up and down states up and down states the periodic fluctuations between increased and decreased spiking activity of a neuronal population are a fundamental feature of cortical circuits understanding up down state dynamics is important for understanding how these circuits represent and transmit information in the brain to date limited work has been done on characterizing the stochastic properties of up down state dynamics we present a set of markov and semi markov discrete and continuous time probability models for estimating up and down states from multiunit neural spiking activity we model multiunit neural spiking activity as a stochastic point process modulated by the hidden up and down states and the ensemble spiking history we estimate jointly the hidden states and the model parameters by maximum likelihood using an expectation maximization em algorithm and a monte carlo em algorithm that uses reversible jump markov chain monte carlo sampling in the e step we apply our models and algorithms in the analysis of both simulated multiunit spiking activity and actual multi unit spiking activity recorded from primary somatosensory cortex in a behaving rat during slow wave sleep our approach provides a statistical characterization of up down state dynamics that can serve as a basis for verifying and refining mechanistic descriptions of this process

asymptotic normality of the maximum likelihood estimator for general hidden markov models hidden markov models hmms have during the last decade become a widespread tool for modeling sequences of dependent random variables inference for such models is usually based on the maximum likelihood estimator mle and consistency of the mle for general hmms was recently proved by leroux in this paper we show that under mild conditions the mle is also asymptotically normal and prove that the observed information matrix is a consistent estimator of the fisher information

finite element exterior calculus from hodge theory to numerical stability this article reports on the confluence of two streams of research one emanating from the fields of numerical analysis and scientific computation the other from topology and geometry in it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de rham cohomology and hodge theory are key tools for exploring the well posedness of the continuous problem the discretization methods we consider are finite element methods in which a variational or weak formulation of the pde problem is approximated by restricting the trial subspace to an appropriately constructed piecewise polynomial subspace after a brief introduction to finite element methods we develop an abstract hilbert space framework for analyzing the stability and convergence of such discretizations in this framework the differential complex is represented by a complex of hilbert spaces and stability is obtained by transferring hodgetheoretic structures that ensure well posedness of the continuous problem from the continuous level to the discrete we show stable discretization is achieved if the finite element spaces satisfy two hypotheses they can be arranged into a subcomplex of this hilbert complex and there exists a bounded cochain projection from that complex to the subcomplex in the next part of the paper we consider the most canonical example of the abstract theory in which the hilbert complex is the de rham complex of a domain in euclidean space we use the koszul complex to construct two families of finite element differential forms show that these can be arranged in subcomplexes of the de rham complex in numerous ways and for each construct a bounded cochain projection the abstract theory therefore applies to give the stability and convergence of finite element approximations of the hodge laplacian other applications are considered as well especially the elasticity complex and its application to the equations of elasticity background material is included to make the presentation self contained for a variety of readers

the poincare bendixson theorem for monotone cyclic feedback systems with delay abstract we consider cyclic nearest neighbor systems of differential delay equations in which the coupling between neighbors possesses a monotonicity property using a discrete integer valued lyapunov function we prove that the poincare bendixson theorem holds for such systems we also obtain results on piecewise monotonicity and stability of periodic solutions of such systems

oscillations in i o monotone systems under negative feedback oscillatory behavior is a key property of many biological systems the small gain theorem sgt for input output monotone systems provides a sufficient condition for global asymptotic stability of an equilibrium and hence its violation is a necessary condition for the existence of periodic solutions one advantage of the use of the monotone sgt technique is its robustness with respect to all perturbations that preserve monotonicity and stability properties of a very low dimensional in many interesting examples just one dimensional model reduction this robustness makes the technique useful in the analysis of molecular biological models in which there is large uncertainty regarding the values of kinetic and other parameters however verifying the conditions needed in order to apply the sgt is not always easy this paper provides an approach to the verification of the needed properties and illustrates the approach through an application to a classical model of circadian oscillations as a nontrivial case study and provides a theorem in the converse direction of predicting oscillations when the sgt conditions fail

stability analysis of a single neuron model with delay in this paper we study the asymptotic behavior and numerical approximation of the single neuron model equation x t dx t af x t bf x t i t 0 1 where d 0 and f x 0 5 x 1 x 1 we obtain new sufficient conditions for global asymptotic stability of constant equilibriums of 1 give several numerical examples to illustrate our results and formulate conjectures on the asymptotic behavior of the solutions based on our numerical experiments

a staggered dg method of minimal dimension for the stokes equations on general meshes abstract in this paper a locally conservative lowest order staggered discontinuous galerkin method is developed for the stokes equations the proposed method allows rough grids and is based on the partition of the domain into arbitrary shapes of quadrilaterals or polygons which makes the method highly desirable for practical applications a priori error analysis covering low regularity is demonstrated a new postprocessing scheme for the velocity earning faster convergence is constructed next adaptive mesh refinement is highly appreciated on quadrilateral and polygonal meshes since hanging nodes are allowed therefore we propose two guaranteed type error estimators in l 2 error of stress and energy error of the postprocessed velocity respectively numerical experiments confirm our theoretical findings and illustrate the flexibility of the proposed method and accuracy of the guaranteed upper bounds

h div finite elements based on nonaffine meshes for 3d mixed formulations of flow problems with arbitrary high order accuracy of the divergence of the flux

pullback attractors of a multi valued process generated by parabolic differential equations with unbounded delays abstract the theory of nonautonomous multi valued dynamical systems and a method of asymptotic compactness based on the concept of the kuratowski measure of the noncompactness of a bounded set are used to prove the existence of pullback attractors in the weighted space c v for the multi valued process associated with the nonautonomous parabolic differential equation with infinite delay for which the uniqueness of solutions need not hold

two dimensional mixed finite element approximations for elliptic problems with enhanced accuracy for the potential and flux divergence abstract the purpose of the present paper is to analyse two new different possibilities of choosing balanced pairs of approximation spaces for dual flux and primal potential variables one for triangles and the other one for quadrilateral elements to be used in discrete versions of the mixed finite element method for elliptic problems they can be interpreted as enriched versions of b d f m k 1 spaces based on triangles and of r t k spaces for quadrilateral elements the new flux approximations are incremented with properly chosen internal shape functions with vanishing normal components on the edges of degree k 2 and matching primal functions of degree k 1 higher than the border fluxes which are kept of degree k in all these cases the divergence of the flux space coincide with the primal approximation space on the master element producing stable simulations using static condensation the global condensed system to be solved in the enriched cases has same dimension and structure of the original ones which is proportional to the space dimension of the border fluxes for each element geometry measuring the errors with l 2 norms the enriched space configurations give higher convergence rate of order k 2 for the primal variable while keeping the order k 1 for the flux for affine meshes the divergence errors have the same improved accuracy rate as for the error in the primal variable for quadrilateral non affine meshes for instance trapezoidal elements the divergence error has order k 1 one unit more than the order k occurring for r t k spaces on this kind of deformed meshes this fact also holds for a b f k elements but for them the potential order of accuracy does not improve keeping order k 1

nontrivial periodic solutions for delay differential systems via morse theory abstract the existence of the nontrivial periodic solutions to the system of delay differential equations 1 1 x t f x t is investigated where x r n f c r n r n we prove several new existence results via morse theory and the galerkin methods

slowly oscillating wave solutions of a single species reaction diffusion equation with delay we study positive bounded wave solutions u t x x ct 0 of equation ut t x u t x u t x g u t h x x rm this equation is assumed to have two non negative equilibria u1 0 and u2 0 the birth function g c r r is unimodal and differentiable at 0 and some results also require the feedback condition g s s c our results are based on a detailed geometric description of the wave profile in particular the monotonicity of its leading edge is established we also discuss the uniqueness problem indicating a subclass g of asymmetric tent maps such that given g g there exists exactly one positive travelling front for each fixed admissible speed

a sufficient condition for additive d stability and application to reaction diffusion models abstract the matrix a is said to be additively d stable if a d remains hurwitz for all non negative diagonal matrices d in reaction diffusion models additive d stability of the matrix describing the reaction dynamics guarantees the stability of the homogeneous steady state thus ruling out the possibility of diffusion driven instabilities we present a new criterion for additive d stability using the concept of compound matrices we first give conditions under which the second additive compound matrix has non negative off diagonal entries we then use this metzler property of the compound matrix to prove additive d stability with the help of an additional determinant condition this result is then applied to investigate the stability of cyclic reaction networks in the presence of diffusion finally a reaction network structure that fails to achieve additive d stability is exhibited

traveling waves in the nonlocal kpp fisher equation different roles of the right and the left interactions abstract we consider the nonlocal kpp fisher equation u t t x u x x t x u t x 1 k u t x which describes the evolution of population density u t x with respect to time t and location x the non locality is expressed in terms of the convolution of u t with kernel k 0 r k s d s 1 the restrictions k s s 0 and k s s 0 are responsible for interactions of an individual with his left and right neighbors respectively we show that these two parts of k play quite different roles as for the existence and uniqueness of traveling fronts to the kpp fisher equation in particular if the left interaction is dominant the uniqueness of fronts can be proved while the dominance of the right interaction can induce the co existence of monotone and oscillating fronts we also present a short proof of the existence of traveling waves without assuming various technical restrictions usually imposed on k

global estimates for mixed methods for second order elliptic equations global error estimates in l2 92 l 9 and h s q 2 in r2 or a re derived for a mixed finite element method for the dirichlet problem for the elliptic operator lp div a grad p bp cp based on the raviart thomas nedelec space vh x wh c h div 92 x l2 g2 optimal order estimates are obtained for the approximation of p and the associated velocity field u a grad p bp in l2 g2 and h s 92 0 6 s k 1 and if q c r2 forp in lv2

nonconforming polynomial mixed finite element for the brinkman problem over quadrilateral meshes abstract this work provides a new mixed finite element method for the brinkman problem over arbitrary convex quadrilateral meshes the velocity is approximated by piecewise polynomial element space which is h div nonconforming and the pressure is approximated by piecewise constant we give the convergence analysis of our element and especially show the robustness with respect to the darcy limit moreover via a discrete de rham complex a higher order approximation error term is obtained for incompressible flow numerical examples verify our theoretical findings

hidden markov processes an overview of statistical and information theoretic aspects of hidden markov processes hmps is presented an hmp is a discrete time finite state homogeneous markov chain observed through a discrete time memoryless invariant channel in recent years the work of baum and petrie 1966 on finite state finite alphabet hmps was expanded to hmps with finite as well as continuous state spaces and a general alphabet in particular statistical properties and ergodic theorems for relative entropy densities of hmps were developed consistency and asymptotic normality of the maximum likelihood ml parameter estimator were proved under some mild conditions similar results were established for switching autoregressive processes these processes generalize hmps new algorithms were developed for estimating the state parameter and order of an hmp for universal coding and classification of hmps and for universal decoding of hidden markov channels these and other related topics are reviewed

saddle node bifurcation of periodic orbits for a delay differential equation abstract we consider the scalar delay differential equation x t x t f k x t 1 with a nondecreasing feedback function f k depending on a parameter k and we verify that a saddle node bifurcation of periodic orbits takes place as k varies the nonlinearity f k is chosen so that it has two unstable fixed points hence the dynamical system has two unstable equilibria and these fixed points remain bounded away from each other as k changes the generated periodic orbits are of large amplitude in the sense that they oscillate about both unstable fixed points of f k

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

quadratic serendipity finite elements on polygons using generalized barycentric coordinates we introduce a nite element construction for use on the class of convex planar polygons and show it obtains a quadratic error convergence estimate on a convex n gon satisfying simple geometric criteria our construction produces 2n basis functions associated in a lagrange like fashion to each vertex and each edge midpoint by transforming and combining a set of n n 1 2 basis functions known to obtain quadratic convergence the technique broadens the scope of the so called serendipity elements previously studied only for quadrilateral and regular hexahedral meshes by employing the theory of generalized barycentric coordinates uniform a priori error estimates are established over the class of convex quadrilaterals with bounded aspect ratio as well as over the class of generic convex planar polygons satisfying additional shape regularity conditions to exclude large interior angles and short edges numerical evidence is provided on a trapezoidal quadrilateral mesh previously not amenable to serendipity constructions and applications to adaptive meshing are discussed

mixed methods for two phase darcy stokes mixtures of partially melted materials with regions of zero porosity the earth s mantle or e g a glacier involves a deformable solid matrix phase within which a second phase a fluid may form due to melting processes the system is modeled as a dual continuum mixture with at each point of space the solid matrix being governed by a stokes flow and the fluid melt if it exists being governed by darcy s law this system is mathematically degenerate when the porosity volume fraction of fluid vanishes assuming the porosity is given we develop a mixed variational framework for the mechanics of the system by carefully scaling the darcy variables by powers of the porosity we prove that the variational problem is well posed even when there are regions of one and two phases we then develop an accurate mixed finite element method for solving this darcy stokes system and prove a convergence result numerical results are presented that illustrate and verify the convergence of the method

the poincare bendixson theorem for a class of delay equations with state dependent delay and monotonic feedback abstract we consider the real valued differential equation x t f x t x t d x t with state dependent delay where f is strictly monotonic in its second argument we describe a class of such equations for which a version of the poincare bendixson theorem holds

machine learning and transport simulations for groundwater anomaly detection abstract this paper presents studies on modeling and algorithms for groundwater anomaly detection specifically conductivity along with four other surrogates are used for identifying anomaly in groundwater the one class support vector machine 1 svm technique is utilized for model training and real data from colorado water watch is used for testing the model and algorithms design of code modules in python is briefly discussed since groundwater contamination rarely happens in reality we also use synthetic data from numerical simulations of flow and transport in porous media to test this model for anomaly detection

full h div approximation of linear elasticity on quadrilateral meshes based on abf finite elements abstract for meshes of nondegenerate convex quadrilaterals we present a family of stable mixed finite element spaces for the mixed formulation of planar linear elasticity the problem is posed in terms of the stress tensor the displacement vector and the rotation scalar fields with the symmetry of the stress tensor weakly imposed the proposed spaces are based on the arnold boffi falk abf k k 0 elements for the stress and piecewise polynomials for the displacement and the rotation we prove that these finite elements provide full h div approximation of the stress field in the sense that it is approximated to order h k 1 where h is the mesh diameter in the h div norm we show that displacement and rotation are also approximated to order h k 1 in the l 2 norm the convergence is optimal order for k 1 while the lowest order case index k 0 requires special treatment the spaces also apply to both compressible and incompressible isotropic problems i e the poisson ratio may be one half the implementation as a hybrid method is discussed and numerical results are given to illustrate the effectiveness of these finite elements

construction of h mathrm div conforming mixed finite elements on cuboidal hexahedra we generalize the two dimensional mixed finite elements of arbogast and correa siam j numer anal 54 3332 3356 2016 defined on quadrilaterals to three dimensional cuboidal hexahedra the construction is similar in that polynomials are used directly on the element and supplemented with functions defined on a reference element and mapped to the hexahedron using the piola transform the main contribution is providing a systematic procedure for defining supplemental functions that are divergence free and have any prescribed polynomial normal flux general procedures are also presented for determining which supplemental normal fluxes are required to define the finite element space both full and reduced h mathrm div approximation spaces may be defined so the scalar variable vector variable and vector divergence are approximated optimally the spaces can be constructed to be of minimal local dimension if desired

systems of differential delay equations floquet multipliers and discrete lyapunov functions abstract we define a discrete integer valued lyapunov function v for cyclic nearest neighbor systems of differential delay equations possessing a feedback condition this extends analogous definitions for cyclic systems of ode s and for scalar differential delay equations we relate the values of v to the real parts of the floquet multipliers for such linear periodic systems and thereby prove all floquet subspaces are at most two dimensional

asymptotic constancy and periodicity for a single neuron model with delay abstract we study the single neuron model equation x t d x t a f x t b f x t 1 i t 0 where d 0 b 0 and f x 1 2 x 1 x 1 we examine the asymptotic behaviour of the solutions and confirm that although most of them are convergent there exists a periodic solution if d a b i

the lowest order weak galerkin finite element method for the darcy equation on quadrilateral and hybrid meshes abstract this paper investigates the lowest order weak galerkin finite element method for solving the darcy equation on quadrilateral and hybrid meshes consisting of quadrilaterals and triangles in this approach the pressure is approximated by constants in element interiors and on edges the discrete weak gradients of these constant basis functions are specified in local raviart thomas spaces specifically r t 0 for triangles and unmapped r t 0 for quadrilaterals these discrete weak gradients are used to approximate the classical gradient when solving the darcy equation the method produces continuous normal fluxes and is locally mass conservative regardless of mesh quality and has optimal order convergence in pressure velocity and normal flux when the quadrilaterals are asymptotically parallelograms implementation is straightforward and results in symmetric positive definite discrete linear systems we present numerical experiments and comparisons with other existing methods

existence of periodic solutions for a system of delay differential equations abstract in this paper we mainly study the existence of periodic solutions for a system of delay differential equations representing a simple two neuron network model of hopfield type with time delayed connections between the neurons we first examine the local stability of the trivial solution propose some sufficient conditions for the uniqueness of equilibria and then apply the poincare bendixson theorem for monotone cyclic feedback delayed systems to establish the existence of periodic solutions in addition a sufficient condition that ensures the trivial solution to be globally exponentially stable is also given numerical examples are provided to support the theoretical analysis

a passivity based approach to stability of spatially distributed systems with a cyclic interconnection structure a class of distributed systems with a cyclic interconnection structure is considered these systems arise in several biochemical applications and they can undergo diffusion driven instability which leads to a formation of spatially heterogeneous patterns in this paper a class of cyclic systems in which addition of diffusion does not have a destabilizing effect is identified for these systems global stability results hold if the ldquosecantrdquo criterion is satisfied in the linear case it is shown that the secant condition is necessary and sufficient for the existence of a decoupled quadratic lyapunov function which extends a recent diagonal stability result to partial differential equations for reaction diffusion equations with nondecreasing coupling nonlinearities global asymptotic stability of the origin is established all of the derived results remain true for both linear and nonlinear positive diffusion terms similar results are shown for compartmental systems

modeling flat stretches bursts and outliers in time series using mixture transition distribution models abstract the class of mixture transition distribution mtd time series models is extended to general non gaussian time series in these models the conditional distribution of the current observation given the past is a mixture of conditional distributions given each one of the last p observations they can capture non gaussian and nonlinear features such as flat stretches bursts of activity outliers changepoints in a single unified model class they can also represent time series defined on arbitrary state spaces univariate or multivariate continuous discrete or mixed which need not even be euclidean they perform well in the usual case of gaussian time series without obvious nonstandard behaviors the models are simple analytically tractable easy to simulate readily estimated the stationarity and autocorrelation properties of the models are derived a simple em algorithm is given and shown to work well for estimation the models are applied to several real and simulated datasets with satisfacto

a proof of jones conjecture abstract in this paper we prove that wright s equation y t y t 1 1 y t has a unique slowly oscillating periodic solution for parameter values 2 1 9 up to time translation this result proves jones conjecture formulated in 1962 that there is a unique slowly oscillating periodic orbit for all 2 furthermore there are no isolas of periodic solutions to wright s equation all periodic orbits arise from hopf bifurcations

multi valued characteristics and morse decompositions abstract a rapid growth of molecular and systems biology in recent years challenges mathematicians to develop robust modeling and analytical tools for this area we combine a theory of monotone input output systems with a classical theory of morse decompositions in the context of ordinary differential equations models of biochemical reactions we show that a multi valued input output characteristic can be used to define non trivial morse decompositions which provide information about a global structure of the attractor the previous work on input output characteristics is shown to apply locally to individual morse sets and is seamlessly incorporated into our global theory we apply our tools to a model of cell cycle maintenance we show that changing the strength of the negative feedback loop can lead to cessation of cell cycle in two different ways it can either lead to globally attracting equilibrium or to a pair of equilibria that attract almost all solutions

high order polygonal discontinuous petrov galerkin polydpg methods using ultraweak formulations abstract this work represents the first endeavor in using ultraweak formulations to implement high order polygonal finite element methods via the discontinuous petrov galerkin dpg methodology ultraweak variational formulations are nonstandard in that all the weight of the derivatives lies in the test space while most of the trial space can be chosen as copies of l 2 discretizations that have no need to be continuous across adjacent elements additionally the test spaces are broken along the mesh interfaces this allows one to construct conforming polygonal finite element methods termed here as polydpg methods by defining most spaces by restriction of a bounding triangle or box to the polygonal element the only variables that require nontrivial compatibility across elements are the so called interface or skeleton variables which can be defined directly on the element boundaries unlike other high order polygonal methods polydpg methods do not require ad hoc stabilization terms thanks to the crafted stability of the dpg methodology a proof of convergence of the form h p is provided and corroborated through several illustrative numerical examples these include polygonal meshes with n sided convex elements and with highly distorted concave elements as well as the modeling of discontinuous material properties along an arbitrary interface that cuts a uniform grid since polydpg methods have a natural a posteriori error estimator a polygonal adaptive strategy is developed and compared to standard adaptivity schemes based on constrained hanging nodes this work is also accompanied by an open source polydpg software supporting polygonal and conventional elements

mixed poisson regression models with covariate dependent rates this paper studies a class of poisson mixture models that includes covariates in rates this model contains poisson regression and independent poisson mixtures as special cases estimation methods based on the em and quasi newton algorithms properties of these estimates a model selection procedure residual analysis and goodness of fit test are discussed a monte carlo study investigates implementation and model choice issues this methodology is used to analyze seizure frequency and ames salmonella assay data

biochemical oscillations in delayed negative cyclic feedback existence and profiles in this paper we propose a theoretical framework to systematically analyze the existence and the profiles of chemical oscillations in gene regulatory networks with negative cyclic feedback in particular we analytically derive the existence conditions and the profiles of oscillations in terms of reaction kinetic parameters and reveal dimensionless quantities that essentially characterize the oscillatory dynamics these discoveries then allow us to provide general biological insights that are useful for the design of synthetic gene circuits and wet lab experiments we point out that time delays due to splicing and transport play an important role for both of the existence and the profiles of oscillations to this end we first show that local instability leads to oscillations in cyclic gene regulatory networks and we derive the existence conditions based on local instability analysis then we analyze the period phase and amplitude of oscillations using multivariable harmonic balance analysis these results are demonstrated with two existing biochemical networks the pentilator and a self repression network of a hes protein

two families of h div mixed finite elements on quadrilaterals of minimal dimension we develop two families of mixed finite elements on quadrilateral meshes for approximating mathbf u p solving a second order elliptic equation in mixed form standard raviart thomas rt and brezzi douglas marini bdm elements are defined on rectangles and extended to quadrilaterals using the piola transform which are well known to lose optimal approximation of nabla cdot mathbf u arnold boffi falk spaces rectify the problem by increasing the dimension of rt so that approximation is maintained after piola mapping our two families of finite elements are uniformly inf sup stable achieve optimal rates of convergence and have minimal dimension the elements for mathbf u are constructed from vector polynomials defined directly on the quadrilaterals rather than being transformed from a reference rectangle by the piola mapping and then supplemented by two one for the lowest order basis functions that are piola mapped one family has full h div approximation mathbf u p

hierarchical high order finite element spaces in h div ω h1 ω for a stabilized mixed formulation of darcy problem abstract the classical dual mixed finite element method for flow simulations is based on h div conforming approximation spaces for the flux which guarantees continuous normal components on element interfaces and discontinuous approximations in l 2 for the pressure however stability and convergence can only be obtained for compatible approximation spaces stabilized finite element methods may provide an alternative stable procedure to avoid this kind of delicate balance the main purpose of this paper is to present a high order finite element methodology to solve the darcy problem based on the combination of an unconditionally stable mixed finite element method with a hierarchical methodology for the construction of finite dimensional subspaces of h div and h 1 the chosen stabilized method is free of mesh dependent stabilization parameters and allows for the use of different high order finite element approximations for the flux and the pressure variables without requiring any compatibility constraint as required in mixed methods for these problems convergence studies are presented comparing the numerical solutions obtained for different approximation orders on quadrilateral elements with the ones given by classical mixed formulation with raviart thomas elements

rehabilitation of the lowest order raviart thomas element on quadrilateral grids a recent study d n arnold d boffi and r s falk siam j numer anal 42 2005 pp 2429 2451 reveals that convergence of finite element methods using h mathrm div omega compatible finite element spaces deteriorates on nonaffine quadrilateral grids this phenomena is particularly troublesome for the lowest order raviart thomas elements because it implies loss of convergence in some norms for finite element solutions of mixed and least squares methods in this paper we propose reformulation of finite element methods based on the natural mimetic divergence operator m shashkov conservative finite difference methods on general grids crc press boca raton fl 1996 which restores the order of convergence reformulations of mixed galerkin and least squares methods for the darcy equation illustrate our approach we prove that reformulated methods converge optimally with respect to a norm involving the mimetic divergence operator furthermore we prove that standard and reformulated versions of the mixed galerkin method lead to identical linear systems but the two versions of the least squares method are veritably different the surprising conclusion is that the degradation of convergence in the mixed method on nonaffine quadrilateral grids is superficial and that the lowest order raviart thomas elements are safe to use in this method however the breakdown in the least squares method is real and there one should use our proposed reformulation

maximum penalized likelihood estimation for independent and markov dependent mixture models summary this paper concerns the use and implementation of maximum penalized likelihood procedures for choosing the number of mixing components and estimating the parameters in independent and markov dependent mixture models computation of the estimates is achieved via algorithms for the automatic generation of starting values for the em algorithm computation of the information matrix is also discussed poisson mixture models are applied to a sequence of counts of movements by a fetal lamb in utero obtained by ultrasound the resulting estimates are seen to provide plausible mechanisms for the physiological process the analysis of count data that are overdispersed relative to the poisson distribution i e variance mean has received considerable recent attention such data might arise in a clinical study in which overdispersion is caused by unexplained or random subject effects alternatively we might observe a time series of counts in which temporal patterns in the data suggest that a poisson model and its implied randomness are inappropriate this paper is motivated by analysis of a time series of overdispersed count data generated in a study of central nervous system development in fetal lambs our data set consists of observed movement counts in 240 consecutive 5 second intervals obtained from a single animal in analysing these data we focus on the use of poisson mixture models assuming independent observations and also markov dependent mixture models or hidden markov models these models assume that the counts follow independent poisson distributions conditional on the rates which are generated from a mixing distribution either independently or with markov dependence we believe finite mixture models are particularly attractive because they provide plausible explanations for variation in the data this paper will emphasize the following issues concerning estimation inference and application of mixture models i choosing the number of model components ii applying the em algorithm to obtain parameter estimates iii generating sufficiently many starting values to identify a global maximum of the likelihood iv avoiding numerical instability

cones of rank 2 and the poincare bendixson property for a new class of monotone systems abstract we study systems that are monotone in a generalized sense with respect to cones of rank 2 the main result of the paper is the existence of a poincare bendixson property for some solutions of those systems