the em algorithm an old folk song sung to a fast new tune celebrating the 20th anniversary of the presentation of the paper by dempster laird and rubin which popularized the em algorithm we investigate after a brief historical account strategies that aim to make the em algorithm converge faster while maintaining its simplicity and stability e g automatic monotone convergence in likelihood first we introduce the idea of a working parameter to facilitate the search for efficient data augmentation schemes and thus fast em implementations second summarizing various recent extensions of the em algorithm we formulate a general alternating expectation conditional maximization algorithm aecm that couples flexible data augmentation schemes with model reduction schemes to achieve efficient computations we illustrate these methods using multivariate t models with known or unknown degrees of freedom and poisson models for image reconstruction we show through both empirical and theoretical evidence the potential for a dramatic reduction in computational time with little increase in human effort we also discuss the intrinsic connection between em type algorithms and the gibbs sampler and the possibility of using the techniques presented here to speed up the latter the main conclusion of the paper is that with the help of statistical considerations it is possible to construct algorithms that are simple stable and fast

missing data dial m for the question mark is common notation for the missing data that occur in most applied statistical analyses over the past century statisticians and other scientists not only have invented numerous methods for handling missing incomplete data but also have invented many forms of missing data including data augmentation hidden states latent variables potential outcome and auxiliary variables purposely constructing unobserved unobservable variables offers an extraordinarily flexible and powerful framework for both scientific modeling and computation and is one of the central statistical contributions to natural engineering and social sciences in parallel much research has been devoted to better understanding and modeling of real life missing data mechanisms that is the unintended data selection process that prevents us from observing our intended data this article is a very brief and personal tour of these developments and thus necessarily has much missing history and citations the tour consists of a number of m s starting with a historic story of the mysterious method of mckendrick for analyzing an epidemic study and its link to the em algorithm the most popular and powerful method of the twentieth century for fitting models involving missing data and latent variables the remaining m s touch on theoretical methodological and practical aspects of missingdata problems highlighted with some common applications in social computational biological medical and physical sciences

on the capacity of free space optical intensity channels upper and lower bounds are derived on the capacity of the free space optical intensity channel this channel has a nonnegative input representing the transmitted optical intensity which is corrupted by additive white gaussian noise to preserve the battery and for safety reasons the input is constrained in both its average and its peak power for a fixed ratio of the allowed average power to the allowed peak power the difference between the upper and the lower bound tends to zero as the average power tends to infinity and their ratio tends to one as the average power tends to zero when only an average power constraint is imposed on the input the difference between the bounds tends to zero as the allowed average power tends to infinity and their ratio tends to a constant as the allowed average power tends to zero

galerkin finite element methods with symmetric pressure stabilization for the transient stokes equations stability and convergence analysis we consider the stability and convergence analysis of pressure stabilized finite element approximations of the transient stokes equation the analysis is valid for a class of symmetric pressure stabilization operators but also for standard inf sup stable velocity pressure spaces without stabilization provided the initial data are chosen as a specific method dependent ritz projection we get unconditional stability and optimal convergence for both pressure and velocity approximations in natural norms for arbitrary interpolations of the initial data a condition between the space and time discretization parameters has to be verified in order to guarantee pressure stability

on the correct convergence of the em algorithm for gaussian mixtures it is well known that the em algorithm generally converges to a local maximum likelihood estimate however there have been many evidences to show that the em algorithm can converge correctly to the true parameters as long as the overlap of gaussians in the sample data is small enough this paper studies this correct convergence problem asymptotically on the em algorithm for gaussian mixtures it has been proved that the em algorithm becomes a contraction mapping of the parameters within a neighborhood of the consistent solution of the maximum likelihood when the measure of average overlap among gaussians in the original mixture is small enough and the number of samples is large enough that is if the initial parameters are set within the neighborhood the em algorithm will always converge to the consistent solution i e the expected result moreover the simulation results further demonstrate that this correct convergence neighborhood becomes larger as the average overlap becomes smaller

on really locking free mixed finite element methods for the transient incompressible stokes equations inf sup stable mixed methods for the steady incompressible stokes equations that relax the divergence constraint are often claimed to deliver locking free discretizations however this relaxation leads to a pressure dependent contribution in the velocity error which is proportional to the inverse of the viscosity thus giving rise to a different locking phenomenon however a recently proposed modification of the right hand side alone leads to a discretization that is really locking free i e its velocity error converges with optimal order and is independent of the pressure and the smallness of the viscosity in this contribution we extend this approach to the transient incompressible stokes equations where besides the right hand side also the velocity time derivative requires an improved space discretization semidiscrete and fully discrete a priori velocity and pressure error estimates are derived which show remarkable robustness properties two numerical examples illustrate the superior accura

on calculating the capacity of an infinite input finite infinite output channel a version of the arimoto blahut algorithm for continuous channels involves evaluating integrals over an entire input space and thus is not tractable two generalized discrete versions of the arimoto blahut algorithm are presented for this purpose instead of calculating integrals both algorithms require only the computation of a sequence of finite sums this significantly reduces numerical computational complexity

maximum likelihood estimation via the ecm algorithm a general framework two major reasons for the popularity of the em algorithm are that its maximum step involves only complete data maximum likelihood estimation which is often computationally simple and that its convergence is stable with each iteration increasing the likelihood when the associated complete data maximum likelihood estimation itself is complicated em is less attractive because the m step is computationally unattractive in many cases however complete data maximum likelihood estimation is relatively simple when conditional on some function of the parameters being estimated

capacity of the l p norm constrained poisson channel the capacity of the poisson channel is derived with a specified constraint imposed on the lp norm t p of the encoder intensity formulae are given for the capacity which establish previously known results for p 1 p 2 and p to be part of a system of capacity problems with common form and solution these formulae apply with or without causal feedback and whether or not the encoder intensity is ook on off keying constrained

large sample joint posterior approximations when full conditionals are approximately normal modern bayesian statistical methods such as gibbs and metropolis hastings sampling were developed to liberate statisticians from the necessity of making large sample assumptions and to facilitate the numerical approximation of problems that had previously been analytically intractable counter to this trend we develop a method for constructing asymptotic joint posterior approximations based on models with k blocks of parameters and where the corresponding properly normalized full conditionals are themselves asymptotically normal we illustrate these techniques by applying them to particular linear and generalized linear mixed models glmms we also consider the relevance of different parameterizations with regard to our asymptotics recent work has indicated that gibbs samplers based on so called centering parameterizations result in better convergence properties for the resulting markov chains our results for the one way random effects model shed some light on this issue for this example we also

successive refinement of abstract sources in successive refinement of information the decoder refines its representation of the source progressively as it receives more encoded bits the rate distortion region of successive refinement describes the minimum rates required to attain the target distortions at each decoding stage in this paper we derive a parametric characterization of the rate distortion region for successive refinement of abstract sources our characterization extends csiszar s result to successive refinement and generalizes a result by tuncel and rose applicable for finite alphabet sources to abstract sources this characterization spawns a family of outer bounds to the rate distortion region it also enables an iterative algorithm for computing the rate distortion region which generalizes blahut s algorithm to successive refinement finally it leads a new nonasymptotic converse bound in all the scenarios where the dispersion is known this bound is second order optimal in our proof technique we avoid karush kuhn tucker conditions of optimality and we use basic tools of probability theory we leverage the donsker varadhan lemma for the minimization of relative entropy on abstract probability spaces

estimating the parameters of general frequency modulated signals a general frequency modulated gfm signal characterizes the vibrations produced by compressors turbines propellers gears and other rotating machines in a dynamic environment a gfm signal is defined as the composition of a real or complex periodic or almost periodic carrier function with a real differentiable modulation function a gfm signal therefore contains sinusoids whose frequencies are possibly nonintegral multiples of a fundamental to distinguish a gfm signal from a set of unrelated sinusoids it is necessary to track them as a group this paper develops the general frequency modulation tracker gfmt for one or more gfm signals in noise using the expectation conditional maximization ecm algorithm that is an extension of the expectation maximization em algorithm three advantages of this approach are that the ratios harmonic numbers of the carrier functions do not need to be known a priori that the parameters of multiple signals are estimated simultaneously and that the gfmt algorithm exploits knowledge of the noise spectrum so that a separate normalization procedure is not required several simulated examples are presented to illustrate the algorithm s performance

convergence of a stochastic approximation version of the em algorithm the expectation maximization em algorithm is a powerful computational technique for locating maxima of functions it is widely used in statistics for maximum likelihood or maximum a posteriori estimation in incomplete data models in certain situations however this method is not applicable because the expectation step cannot be performed in closed form to deal with these problems a novel method is introduced the stochastic approximation em saem which replaces the expectation step of the em algorithm by one iteration of a stochastic approximation procedure the convergence of the saem algorithm is established under conditions that are applicable to many practical situations moreover it is proved that under mild additional conditions the attractive stationary points of the saem algorithm correspond to the local maxima of the function presented to support our findings

squeezing the arimoto blahut algorithm for faster convergence the arimoto blahut algorithm for computing the capacity of a discrete memoryless channel is revisited a so called squeezing strategy is used to design algorithms that preserve its simplicity and monotonic convergence properties but have provably better rates of convergence

a gradient robust well balanced scheme for the compressible isothermal stokes problem abstract a novel notion for constructing a well balanced scheme a gradient robust scheme is introduced and a showcase application for the steady compressible isothermal stokes equations in a nearly hydrostatic situation is presented gradient robustness means that gradient fields in the momentum balance are well balanced by the discrete pressure gradient which is possible on arbitrary unstructured grids the scheme is asymptotic preserving in the sense that it reduces for low mach numbers to a recent inf sup stable and pressure robust discretization for the incompressible stokes equations the convergence of the coupled fem fvm scheme for the nonlinear isothermal stokes equations is proved by compactness arguments numerical examples illustrate the numerical analysis and show that the novel approach can lead to a dramatically increased accuracy in nearly hydrostatic low mach number flows numerical examples also suggest that a straightforward extension to barotropic situations with nonlinear equations of state is feasible

grad div stablilization for stokes equations in this paper a stabilizing augmented lagrangian technique for the stokes equations is studied the method is consistent and hence does not change the continuous solution we show that this stabilization improves the well posedness of the continuous problem for small values of the viscosity coefficient we analyze the influence of this stabilization on the accuracy of the finite element solution and on the convergence properties of the inexact uzawa method

information rates of photon limited overlapping pulse position modulation channels a direct detection photon limited optical communication channel that uses pulse position modulation ppm under a pulsewidth constraint is considered overlapping ppm oppm allows multiple positions per pulsewidth as well as fractional modulation indices number of pulsewidths per frame requiring more refined timing than that needed for conventional disjoint ppm djppm it is shown that even at moderate values of the expected photon count per pulse q such as needed for high data rates oppm outperforms on off keying ook in both capacity and cutoff rate even though ook is uniformly superior to djppm moreover efficient use of oppm is possible with equiprobable input symbols whereas ook requires inconvenient asymmetrical inputs to achieve capacity and high cutoff rate efficiencies nats photon at lower data rates where capacity efficiency is the prime criterion a significant advantage 20 percent over djppm can be achieved up to efficiencies of about 0 7 nats photon the m ary photon limited oppm channel can be viewed as an ambiguity and erasure channel in the sense that some channel outputs are ambiguous in only some input symbols and only if no photons are counted is there ambiguity in all input symbols for large m ambiguities cause bursts of erasures of data symbols massey s interlaced encoding as well as conventional encoding followed by interleaving are adaptable to this bursty channel and effect an increase in its cutoff rate comparable to the increase obtainable with djppm by the same techniques

on the divergence constraint in mixed finite element methods for incompressible flows the divergence constraint of the incompressible navier stokes equations is revisited in the mixed finite element framework while many stable and convergent mixed elements have been developed throughout the past four decades most classical methods relax the divergence constraint and only enforce the condition discretely as a result these methods introduce a pressure dependent consistency error which can potentially pollute the computed velocity these methods are not robust in the sense that a contribution from the right hand side which influences only the pressure in the continuous equations impacts both velocity and pressure in the discrete equations this article reviews the theory and practical implications of relaxing the divergence constraint several approaches for improving the discrete mass balance or even for computing divergence free solutions will be discussed grad div stabilization higher order mixed methods derived on the basis of an exact de rham complex h div conforming finite

on the capacity of the discrete time poisson channel the large inputs asymptotic capacity of a peak power and average power limited discrete time poisson channel is derived using a new firm nonasymptotic lower bound and an asymptotic upper bound the upper bound is based on the dual expression for channel capacity and the notion of capacity achieving input distributions that escape to infinity the lower bound is based on a lower bound on the entropy of a conditionally poisson random variable in terms of the differential entropy of its conditional mean

on the role of the helmholtz decomposition in mixed methods for incompressible flows and a new variational crime abstract according to the helmholtz decomposition the irrotational parts of the momentum balance equations of the incompressible navier stokes equations are balanced by the pressure gradient unfortunately nearly all mixed methods for incompressible flows violate this fundamental property resulting in the well known numerical instability of poor mass conservation the origin of this problem is the lack of l 2 orthogonality between discretely divergence free velocities and irrotational vector fields therefore a new variational crime for the nonconforming crouzeix raviart element is proposed where divergence free lowest order raviart thomas velocity reconstructions reestablish l 2 orthogonality this approach allows to construct a cheap flow discretization for general 2d and 3d simplex meshes that possesses the same advantageous robustness properties like divergence free flow solvers in the stokes case optimal a priori error estimates for the velocity gradients and the pressure are derived moreover the discrete velocity is independent of the continuous pressure several detailed linear and nonlinear numerical examples illustrate the theoretical findings

capacity achieving probability measure for conditionally gaussian channels with bounded inputs a conditionally gaussian channel is a vector channel in which the channel output given the channel input has a gaussian distribution with well behaved input dependent mean and covariance we study the capacity achieving probability measure for conditionally gaussian channels subject to bounded input constraints and average cost constraints many practical communication systems including additive gaussian noise channels certain optical channels fading channels and interference channels fall within this framework subject to bounded input constraint and average cost constraints we show that the channel capacity is achievable and we derive a necessary and sufficient condition for a probability measure to be capacity achieving under certain conditions the capacity achieving measure is proved to be discrete

on locking and robustness in the finite element method a numerical scheme for the approximation of a parameter dependent problem is said to exhibit locking if the accuracy of the approximations deteriorates as the parameter tends to a limiting value a robust numerical scheme for the problem is one that is essentially uniformly convergent for all values of the parameter precise mathematical definitions for these terms are developed their quantitative characterization is given and some general theorems involving locking and robustness are proven a model problem involving heat transfer is analyzed in detail using this mathematical framework and various related computational results are described applications to some different problems involving locking are presented

pressure robustness and discrete helmholtz projectors in mixed finite element methods for the incompressible navier stokes equations abstract recently it was understood how to repair a certain l 2 orthogonality of discretely divergence free vector fields and gradient fields such that the velocity error of inf sup stable discretizations for the incompressible stokes equations becomes pressure independent these new pressure robust stokes discretizations deliver a small velocity error whenever the continuous velocity field can be well approximated on a given grid on the contrary classical inf sup stable stokes discretizations can guarantee a small velocity error only when both the velocity and the pressure field can be approximated well simultaneously in this contribution pressure robustness is extended to the time dependent navier stokes equations in particular steady and time dependent potential flows are shown to build an entire class of benchmarks where pressure robust discretizations can outperform classical approaches significantly speedups will be explained by a new theoretical concept the discrete helmholtz projector of an inf sup stable discretization moreover different discrete nonlinear convection terms are discussed and skew symmetric pressure robust discretizations are proposed

conditions for a monotonic channel capacity motivated by results in optical communications where the performance can degrade dramatically if the transmit power is sufficiently increased the channel capacity is characterized for various kinds of memoryless vector channels it is proved that for all static point to point channels the channel capacity under an equal power constraint is a nondecreasing function of power as a consequence maximizing the mutual information over all input distributions with a certain power is for such channels equivalent to maximizing it over the larger set of input distributions with upperbounded power the channel coding theorem is formally proved for an equal power constraint for interference channels such as optical wavelength division multiplexing systems the primary channel capacity is always nondecreasing with power if all interferers transmit with identical distributions as the primary user also if all input distributions in an interference channel are optimized jointly then the achievable sum rate capacity is again nondecreasing the results generalize to the channel capacity as a function of a wide class of costs not only power

a new family of stable mixed finite elements for the 3d stokes equations a natural mixed element approach for the stokes equations in the velocity pressure formulation would approximate the velocity by continuous piecewise polynomials and would approximate the pressure by discontinuous piecewise polynomials of one degree lower however many such elements are unstable in 2d and 3d this paper is devoted to proving that the mixed finite elements of this p k p k 1 type when k 3 satisfy the stability condition the babuska brezzi inequality on macro tetrahedra meshes where each big tetrahedron is subdivided into four subtetrahedra this type of mesh simplifies the implementation since it has no restrictions on the initial mesh the new element also suits the multigrid method

on the global and componentwise rates of convergence of the em algorithm abstract the em algorithm is a very general and popular iterative algorithm in statistics for finding maximum likelihood estimates in the presence of incomplete data in the paper that defined and popularized em dempster laird and rubin 1977 showed that its global rate of convergence is governed by the largest eigenvalue of the matrix of fractions of missing information due to incomplete data it was also mentioned that componentwise rates of convergence can differ from each other when the fractions of information loss vary across different components of a parameter vector in this article using the well known diagonability theorem we present a general description on how and when the componentwise rates differ as well as their relationships with the global rate we also provide an example a standard contaminated normal model to show that such phenomena are not necessarily pathological but can occur in useful statistical models

on velocity errors due to irrotational forces in the navier stokes momentum balance this contribution studies the influence of the pressure on the velocity error in finite element discretisations of the navier stokes equations four simple benchmark problems that are all close to real world applications convey that the pressure can be comparably large and is not to be underestimated in fact the velocity error can be arbitrarily large in such situations only pressure robust mixed finite element methods whose velocity error is pressure independent can avoid this influence indeed the presented examples show that the pressure dependent component in velocity error estimates for classical mixed finite element methods is sharp in consequence classical mixed finite element methods are not able to simulate some classes of real world flows even in cases where dominant convection and turbulence do not play a role

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

a scaling law from discrete to continuous solutions of channel capacity problems in the low noise limit an analog communication channel typically achieves its full capacity when the distribution of inputs is discrete composed of just k symbols such as voltage levels or wavelengths as the effective noise level goes to zero for example by sending the same message multiple times it is known that optimal codes become continuous here we derive a scaling law for the optimal number of symbols in this limit finding a novel rational scaling exponent the number of symbols in the optimal code grows as log k sim i 4 3 where the channel capacity i increases with decreasing noise the same scaling applies to other problems equivalent to maximizing channel capacity over a continuous distribution

a discontinuous skeletal method for the viscosity dependent stokes problem we devise and analyze arbitrary order nonconforming methods for the discretization of the viscosity dependent stokes equations on simplicial meshes we keep track explicitly of the viscosity and aim at pressure robust schemes that can deal with the practically relevant case of body forces with large curl free part in a way that the discrete velocity error is not spoiled by large pressures the method is inspired from the recent hybrid high order hho methods for linear elasticity after elimination of the auxiliary variables by static condensation the linear system to be solved involves only discrete face based velocities which are polynomials of degree k ge 0 and cell wise constant pressures our main result is a pressure independent energy error estimate on the velocity of order k 1 the main ingredient to achieve pressure independence is the use of a divergence preserving velocity reconstruction operator in the discretization of the body forces we also prove an l 2 pressure estimate of order k 1 and an l 2 velocity estimate of order k 2 the latter under elliptic regularity the local mass and momentum conservation properties of the discretization are also established finally two and three dimensional numerical results are presented to support the analysis

iterative and recursive estimators for hidden markov errors in variables models in this paper we propose maximum likelihood ml estimation of errors in variables models with finite state markovian disturbances such models have applications in econometrics speech processing communication systems and neurobiological signal processing we derive the maximum likelihood ml model estimates using the expectation maximization em algorithm then two recursive or on line estimation schemes are derived for estimating such models the first on line algorithm is based on the em algorithm and uses stochastic approximations to maximize the kullback leibler kl information measure the second on line algorithm we propose is a gradient based scheme and uses stochastic approximations to maximize the log likelihood

asymptotic convergence properties of the em algorithm for mixture of experts mixture of experts me is a modular neural network architecture for supervised classification the double loop expectation maximization em algorithm has been developed for learning the parameters of the me architecture and the iteratively reweighted least squares irls algorithm and the newton raphson algorithm are two popular schemes for learning the parameters in the inner loop or gating network in this letter we investigate asymptotic convergence properties of the em algorithm for me using either the irls or newton raphson approach with the help of an overlap measure for the me model we obtain an upper bound of the asymptotic convergence rate of the em algorithm in each case moreover we find that for the newton approach as a specific newton raphson approach to learning the parameters in the inner loop the upper bound of asymptotic convergence rate of the em algorithm locally around the true solution i is where i 0 is an arbitrarily small number o x means that it is a higher order infinitesimal as x 0 and e i is a measure of the average overlap of the me model that is as the average overlap of the true me model with large sample tends to zero the em algorithm with the newton approach to learning the parameters in the inner loop tends to be asymptotically superlinear finally we substantiate our theoretical results by simulation experiments

the ecme algorithm a simple extension of em and ecm with faster monotone convergence a generalisation of the ecm algorithm meng rubin 1993 which is itselfan extension of the em algorithm dempster laird rubin 1977 can be obtained by replacing some cm steps of ecm which maximise the constrained expected complete data loglikelihood function with steps that maximise the correspondingly constrained actual likelihood function this algorithm which we call ecme algorithm for expectation conditional maximisation either shares with both em and ecm their stable monotone convergence and basic simplicity of implementation relative to competing faster converging methods moreover ecme can have a substantially faster convergence rate than either em or ecm measured using either the number of iterations or actual computer time

capacity and error exponent for the direct detection photon channel ii for pt i see ibid vol 34 no 6 p 1449 61 1980 the discussion of the capacity and error exponent of the direct detection optical channel is continued the channel input in a t second interval is a waveform satisfying certain peak and average power constraints for the optical signals the channel output is a poisson process with an intensity parameter that accounts for the dark current component an upper bound is obtained on the error exponent which coincides with the lower bound thus this channel is one for which the error exponent can be known exactly

a rigorous approach to high resolution entropy constrained vector quantization the nonnegativity of relative entropy implies that the differential entropy of a random vector x with probability density function pdf f is upper bounded by mathsf e left log g text x right for any arbitrary pdf g using this inequality with a cleverly chosen g we derive a lower bound on the asymptotic excess rate of entropy constrained vector quantization for d dimensional sources and r th power distortion where the asymptotic excess rate is defined as the difference between the smallest output entropy of a vector quantizer satisfying the distortion constraint and the rate distortion function in the limit as the distortion tends to zero specialized to the one dimensional case this lower bound coincides with the asymptotic excess rate achieved by a uniform quantizer thereby recovering the result by gish and pierce that uniform quantizers are asymptotically optimal as the allowed distortion tends to zero furthermore in the one dimensional case the derivation of the lower bound reveals a necessary condition for a sequence of quantizers to be asymptotically optimal this condition implies that any sequence of asymptotically optimal almost regular quantizers must converge to a uniform quantizer as the distortion tends to zero while the obtained lower bound itself is not novel to the best of our knowledge we present the first rigorous derivation that follows the direct approach by gish and pierce without resorting to heuristic high resolution approximations commonly found in the quantization literature furthermore our derivation holds for all d dimensional sources having finite differential entropy and whose integer part has finite entropy in contrast to gish and pierce we do not require additional constraints on the continuity or decay of the source pdf

divergence free reconstruction operators for pressure robust stokes discretizations with continuous pressure finite elements classical inf sup stable mixed finite elements for the incompressible navier stokes equations are not pressure robust i e their velocity errors depend on the continuous pressure however a modification only in the right hand side of a stokes discretization is able to reestablish pressure robustness as shown recently for several inf sup stable stokes elements with discontinuous discrete pressures in this contribution this idea is extended to low and high order taylor hood and mini elements which have continuous discrete pressures for the modification of the right hand side a velocity reconstruction operator is constructed that maps discretely divergence free test functions to exactly divergence free ones the reconstruction is based on local h mathrm div conforming flux equilibration on vertex patches and fulfills certain orthogonality properties to provide consistency and optimal a priori error estimates numerical examples for the incompressible stokes and navier stokes equations conf

the fading number of single input multiple output fading channels with memory we derive the fading number of stationary and ergodic not necessarily gaussian single input multiple output simo fading channels with memory this is the second term after the double logarithmic term of the high signal to noise ratio snr expansion of channel capacity the transmitter and receiver are assumed to be cognizant of the probability law governing the fading but not of its realization it is demonstrated that the fading number is achieved by independent and identically distributed i i d circularly symmetric inputs of squared magnitude whose logarithm is uniformly distributed over an snr dependent interval the upper limit of the interval is the logarithm of the allowed transmit power and the lower limit tends to infinity sublogarithmically in the snr the converse relies inter alia on a new observation regarding input distributions that escape to infinity lower and upper bounds on the fading number for gaussian fading are also presented these are related to the mean squared errors of the one step predictor and the one gap interpolator of the fading process respectively the bounds are computed explicitly for stationary mth order autoregressive ar m gaussian fading processes

higher order discontinuous galerkin time stepping and local projection stabilization techniques for the transient stokes problem abstract we introduce and analyze discontinuous galerkin time discretizations coupled with continuous finite element methods based on equal order interpolation in space for velocity and pressure in transient stokes problems spatial stability of the pressure is ensured by adding a stabilization term based on local projection we present error estimates for the semi discrete problem after discretization in space only and for the fully discrete problem the fully discrete pressure shows an instability in the limit of small time step length numerical tests are presented which confirm our theoretical results including the pressure instability

pressure stabilization of finite element approximations of time dependent incompressible flow problems in this paper we develop and analyze pressure stabilized finite element methods for the solution of the transient stokes problem which is a linear model problem of transient incompressible flow a model for the bubble enrichment is proposed to stabilize the numerical solution and an implicit backward euler method is considered for the time approximation different methods are obtained depending on whether bubbles are allowed to evolve with time or they are considered quasi static stability estimates are provided for the two methods considered both for the velocity and the pressure solutions numerical results obtained on test cases are also given which confirm the theoretical stability results

bounds on the capacity of a spectrally constrained poisson channel an upper bound is derived on the capacity of a poisson channel that has a stationary input process of a given spectrum and is subjected to peak and average power constraints the bound is shown to be asymptotically tight with the relaxation of the spectral constraints its maximization over a given set of admissible spectra is closely related to an analogous problem in the awgn regime the results are used for bounding the capacity of a poisson channel under a second moment bandwidth constraint as well as the capacity under a strict bandwidth constraint asymptotically tight lower bounds on the channel capacity for the above two cases are also presented the approach for lower bounding the capacity for the latter case yields as a by product improved bounds on the bit error probability in uncoded amplitude shift keying and on off modulation as a special case operating over a poisson channel impaired by intersymbol interference

stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection abstract we consider the problem of poor mass conservation in mixed finite element algorithms for flow problems with large rotation free forcing in the momentum equation we provide analysis that suggests for such problems obtaining accurate solutions necessitates either the use of pointwise divergence free finite elements such as scott vogelius or heavy grad div stabilization of weakly divergence free elements the theory is demonstrated in numerical experiments for a benchmark natural convection problem where large irrotational forcing occurs with high rayleigh numbers

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

variable selection using mm algorithms variable selection is fundamental to high dimensional statistical modeling many variable selection techniques may be implemented by maximum penalized likelihood using various penalty functions optimizing the penalized likelihood function is often challenging because it may be nondifferentiable and or nonconcave this article proposes a new class of algorithms for finding a maximizer of the penalized likelihood for a broad class of penalty functions these algorithms operate by perturbing the penalty function slightly to render it differentiable then optimizing this differentiable function using a minorize maximize mm algorithm mm algorithms are useful extensions of the well known class of em algorithms a fact that allows us to analyze the local and global convergence of the proposed algorithm using some of the techniques employed for em algorithms in particular we prove that when our mm algorithms converge they must converge to a desirable point we also discuss conditions under which this convergence may be guaranteed we exploit the newton raphson like aspect of these algorithms to propose a sandwich estimator for the standard errors of the estimators our method performs well in numerical tests

high order exactly divergence free hybrid discontinuous galerkin methods for unsteady incompressible flows abstract in this paper we present an efficient discretization method for the solution of the unsteady incompressible navier stokes equations based on a high order hybrid discontinuous galerkin formulation the crucial component for the efficiency of the discretization method is the distinction between stiff linear parts and less stiff non linear parts with respect to their temporal and spatial treatment exploiting the flexibility of operator splitting time integration schemes we combine two spatial discretizations which are tailored for two simpler sub problems a corresponding hyperbolic transport problem and an unsteady stokes problem for the hyperbolic transport problem a spatial discretization with an upwind discontinuous galerkin method and an explicit treatment in the time integration scheme is rather natural and allows for an efficient implementation the treatment of the stokes part involves the solution of linear systems in this case a discretization with hybrid discontinuous galerkin methods is better suited we consider such a discretization for the stokes part with two important features h div conforming finite elements to guarantee exactly divergence free velocity solutions and a projection operator which reduces the number of globally coupled unknowns we present the method discuss implementational aspects and demonstrate the performance on two and three dimensional benchmark problems

analysis of the pspg method for the transient stokes problem abstract we propose a new analysis for the pspg method applied to the transient stokes problem stability and convergence are obtained under different conditions on the discretization parameters depending on the approximation used in space for the pressure we prove optimal stability and convergence only in the case of piecewise affine approximation under the standard condition on the time step finally we show that the stability problems of the pspg method can be circumvented using an appropriate discrete reconstruction of the laplace operator

grad div stabilization and subgrid pressure models for the incompressible navier stokes equations in this paper the grad div stabilization for the incompressible navier stokes finite element approximations is considered from two different viewpoints i as a variational multiscale approach for the pressure subgrid modeling and ii as a stabilization procedure of least square type some new error estimates for the linearized problem with the grad div stabilization are proved with the help of norms induced by the pressure schur complement operator we discuss the stabilization parameter choice arising in the frameworks of least square and multiscale methods and consider assumptions which allow to relate both approaches

two iterative algorithms for finding minimax solutions two iterative minimax algorithms are presented with associated convergence theorems both algorithms consist of iterative procedures based on a sequence of finite parameter sets in general these finite parameter sets are subsets of an infinite parameter space to show their applicabilities several commonly used examples are presented it is also shown that minimax problems with or without finite parameter sets can be solved by these two algorithms numerically to any assigned degree of accuracy

discussion one step sparse estimates in nonconcave penalized likelihood models who cares if it is a white cat or a black cat 1 an insider s minor comments section 2 3 seems to be the reason that i am a discussant there it was first stated that the proposed lla algorithm is an instance of the mm algorithm as termed by lange hunter and yang 8 then it was shown under certain conditions that it is also an em algorithm my initial re action was hmmm the authors reading of lange hunter and yang 8 must have ceased before reaching its discussions because a more general mm em re sult using the same laplace transform technique was the base for a key inquiry of meng 10 a discussion of lange hunter and yang 8 upon a more careful reading i realized that the authors construction though mathematically equiva lent to mine gives a different interpretation to the constructed missing data latent variable this is rather interesting especially if my initial reaction was correct for the current paper this mm em result appears to be of minor interest especially because its potential benefit is not explored in the paper instead the only punch line seems to be a logically unsubstantiated one thus theorem 3 also indicates that mm algorithms are more flexible than em algorithms to clar ify these issues which seems to be what i have been asked to do i ll focus my discussion first on this algorithmic connection and qualify the statements that i just made i will then pose some questions going beyond those algorithmic and computational considerations the two parts are connected via the cat title for those who are patient enough to read both to do so most effectively let me invoke an author s privilege to reproduce sec tion 2 of meng 10 in its entirety to map out its mathematical connections with section 2 3 of the current paper the acronym sm below was my suggestion because the general recipe lange hunter and yang 8 put forward consists of a surrogate step and a maximization minimization step in their rejoinder lange hunter and yang 8 coined the term mm partially in fear of the as sociation of sm with sm in deed for readers who enjoy collecting acronyms the gaecm below is another treasure to hunt

stokes complexes and the construction of stable finite elements with pointwise mass conservation two families of conforming finite elements for the two dimensional stokes problem are developed guided by two discrete smoothed de rham complexes which we coin stokes complexes we show that the finite element pairs are inf sup stable and also provide pointwise mass conservation on very general triangular meshes

fitting full information item factor models and an empirical investigation of bridge sampling abstract based on item response theory bock and aitken introduced a method of item factor analysis termed full information item factor fiif analysis by bartholomew because it uses all distinct item response vectors as data but a limitation of their fitting algorithm is its reliance on fixed point gauss hermite quadrature which can produce appreciable numerical errors especially in high dimension problems the first purpose of this article is to offer more reliable methods by using recent advances in statistical computation specifically we illustrate two ways of implementing monte carlo expectation maximization em algorithm to fit a fiif model using the gibbs sampler to carry out the computation for the e steps we also show how to use bridge sampling to simulate the likelihood ratios for monitoring the convergence of a monte carlo em a strategy that is useful in general simulations demonstrate substantial improvement over bock and aitken s algorithm in recovering known factor loadings in hig

information capacity of the poisson channel the information capacity of the poisson channel with random or time varying noise intensity is obtained for time varying peak and average constraints on the encoder intensity the channel model is specified and some definitions from information theory are given causal feedback is shown not to increase the channel capacity for the case of nonrandom noise intensity for random noise intensity use of causal feedback does increase capacity the extent of that increase is quantified jamming is considered and the optimal jamming signal is given poisson channels with thinning are introduced

on the convergence properties of the em algorithm two convergence aspects of the em algorithm are studied i does the em algorithm find a local maximum or a stationary value of the incomplete data likelihood function ii does the sequence of parameter estimates generated by em converge several convergence results are obtained under conditions that are applicable to many practical situations two useful special cases are a if the unobserved complete data specification can be described by a curved exponential family with compact parameter space all the limit points of any em sequence are stationary points of the likelihood function b if the likelihood function is unimodal and a certain differentiability condition is satisfied then any em sequence converges to the unique maximum likelihood estimate a list of key properties of the algorithm is included

efficient universal noiseless source codes although the existence of universal noiseless variable rate codes for the class of discrete stationary ergodic sources has previously been established very few practical universal encoding methods are available efficient implementable universal source coding techniques are discussed in this paper results are presented on source codes for which a small value of the maximum redundancy is achieved with a relatively short block length a constructive proof of the existence of universal noiseless codes for discrete stationary sources is first presented the proof is shown to provide a method for obtaining efficient universal noiseless variable rate codes for various classes of sources for memoryless sources upper and lower bounds are obtained for the minimax redundancy as a function of the block length of the code several techniques for constructing universal noiseless source codes for memoryless sources are presented and their redundancies are compared with the bounds consideration is given to possible applications to data compression for certain nonstationary sources

hybridized globally divergence free ldg methods part i the stokes problem we devise and analyze a new local discontinuous galerkin ldg method for the stokes equations of incompressible fluid flow this optimally convergent method is obtained by using an ldg method to discretize a vorticity velocity formulation of the stokes equations and by applying a new hybridization to the resulting discretization one of the main features of the hybridized method is that it provides a globally divergence free approximate velocity without having to construct globally divergence free finite dimensional spaces only elementwise divergence free basis functions are used another important feature is that it has significantly less degrees of freedom than all other ldg methods in the current literature in particular the approximation to the pressure is only defined on the faces of the elements on the other hand we show that as expected the condition number of the schur complement matrix for this approximate pressure is of order h 2 in the mesh size h finally we present numerical experiments that confirm the sharpness of our theoretical a priori error estimates

efficient approximation of channel capacities we propose an iterative method for approximately computing the capacity of discrete memoryless channels possibly under additional constraints on the input distribution based on duality of convex programming we derive explicit upper and lower bounds for the capacity the presented method requires o m 2 n sqrt log n varepsilon to provide an estimate of the capacity to within varepsilon where n and m denote the input and output alphabet size a single iteration has a complexity o m n we also show how to approximately compute the capacity of memoryless channels having a bounded continuous input alphabet and a countable output alphabet under some mild assumptions on the decay rate of the channel s tail it is shown that discrete time poisson channels fall into this problem class as an example we compute sharp upper and lower bounds for the capacity of a discrete time poisson channel with a peak power input constraint

capacity bounds via duality with applications to multiple antenna systems on flat fading channels a technique is proposed for the derivation of upper bounds on channel capacity it is based on a dual expression for channel capacity where the maximization of mutual information over distributions on the channel input alphabet is replaced with a minimization of average relative entropy over distributions on the channel output alphabet we also propose a technique for the analysis of the asymptotic capacity of cost constrained channels the technique is based on the observation that under fairly mild conditions capacity achieving input distributions escape to infinity the above techniques are applied to multiple antenna flat fading channels with memory where the realization of the fading process is unknown at the transmitter and unknown or only partially known at the receiver it is demonstrated that for high signal to noise ratio snr the capacity of such channels typically grows only double logarithmically in the snr to better understand this phenomenon and the rates at which it occurs we introduce the fading number as the second order term in the high snr asymptotic expansion of capacity and derive estimates on its value for various systems it is suggested that at rates that are significantly higher than the fading number communication becomes extremely power inefficient thus posing a practical limit on practically achievable rates upper and lower bounds on the fading number are also presented for single input single output siso systems the bounds coincide thus yielding a complete characterization of the fading number for general stationary and ergodic fading processes we also demonstrate that for memoryless multiple input single output miso channels the fading number is achievable using beam forming and we derive an expression for the optimal beam direction this direction depends on the fading law and is in general not the direction that maximizes the snr on the induced siso channel using a new closed form expression for the expectation of the logarithm of a noncentral chi square distributed random variable we provide some closed form expressions for the fading number of some systems with gaussian fading including siso systems with circularly symmetric stationary and ergodic gaussian fading the fading number of the latter is determined by the fading mean fading variance and the mean squared error in predicting the present fading from its past it is not directly related to the doppler spread for the rayleigh ricean and multiple antenna rayleigh fading channels we also present firm nonasymptotic upper and lower bounds on channel capacity these bounds are asymptotically tight in the sense that their difference from capacity approaches zero at high snr and their ratio to capacity approaches one at low snr

general formulas for obtaining the mles and the asymptotic variance covariance matrix in mapping quantitative trait loci when using the em algorithm we present in this paper general formulas for deriving the maximum likelihood estimates and the asymptotic variance covariance matrix of the positions and effects of quantitative trait loci qtls in a finite normal mixture model when the em algorithm is used for mapping qtls the general formulas are based on two matrices d and q where d is the genetic design matrix characterizing the genetic effects of the qtls and q is the conditional probability matrix of qtl genotypes given flanking marker genotypes containing the information on qtl positions with the general formulas it is relatively easy to extend qtl mapping analysis to using multiple marker intervals simultaneously for mapping multiple qtls for analyzing qtl epistasis and for estimating the heritability of quantitative traits simulations were performed to evaluate the performance of the estimates of the asymptotic variances of qtl positions and effects

statistical guarantees for the em algorithm from population to sample based analysis we develop a general framework for proving rigorous guarantees on the performance of the em algorithm and a variant known as gradient em our analysis is divided into two parts a treatment of these algorithms at the population level in the limit of infinite data followed by results that apply to updates based on a finite set of samples first we characterize the domain of attraction of any global maximizer of the population likelihood this characterization is based on a novel view of the em updates as a perturbed form of likelihood ascent or in parallel of the gradient em updates as a perturbed form of standard gradient ascent leveraging this characterization we then provide non asymptotic guarantees on the em and gradient em algorithms when applied to a finite set of samples we develop consequences of our general theory for three canonical examples of incompletedata problems mixture of gaussians mixture of regressions and linear regression with covariates missing completely at random in each case our theory guarantees that with a suitable initialization a relatively small number of em or gradient em steps will yield with high probability an estimate that is within statistical error of the mle we provide simulations to confirm this theoretically predicted behavior

explicit versus implicit representations of subsets of the herbrand universe in lassez and marriott j automat reson 3 3 1987 301 317 explicit and implicit generalizations were studied as representations of subsets of some fixed herbrand universe h an explicit generalization e r1v v rl represents all ground terms that are instances of at least one of the terms ti whereas an implicit generalization i t t1 v v tm represents all h ground instances of t that are not instances of any term ti more generally a disjunction i i1 v v in of implicit generalizations contains all ground terms that are contained in at least one of the implicit generalizations ij implicit generalizations have applications to many areas of computer science like machine learning unification specification of abstract data types logic programming functional programming etc in these areas the so called finite explicit representability problem plays an important role i e given a disjunction of implicit generalizations i i1 v v in does there exist an explicit generalization e s t i and e are equivalent we shall prove the conp completeness of this decision problem implicit generalizations can be represented as equational formulae i e first order formulae whose only predicate symbol is syntactic equality closely related to the finite explicit representability problem is the so called negation elimination problem of equational formulae i e given an arbitrary equational formula p is p semantically equivalent to an equational formula without universal quantifiers and negation in this work we study the negation elimination problem of equational formulae with purely existential quantifier prefix we prove the conp completeness for such formulae in dnf and the 2p hardness in case of cnf

a connection between scott vogelius and grad div stabilized taylor hood fe approximations of the navier stokes equations this article studies two methods for obtaining excellent mass conservation in finite element computations of the navier stokes equations using continuous velocity fields with a particular mesh construction the scott vogelius element pair has recently been shown to be inf sup stable and have optimal approximation properties while also providing pointwise mass conservation we present herein the first numerical tests of this element pair for the time dependent navier stokes equations we also prove that the limit of the grad div stabilized taylor hood solutions to the navier stokes problem converges to the scott vogelius solution as the stabilization parameter tends to infinity that is we provide theoretical justification that choosing the grad div parameter large does not destroy the solution numerical tests are provided which verify the theory and show how both scott vogelius and grad div stabilized taylor hood with large stabilization parameter elements can provide accurate results with excellent mass conservation for navier stokes approximations

an em based semi parametric mixture model approach to the regression analysis of competing risks data we consider a mixture model approach to the regression analysis of competing risks data attention is focused on inference concerning the effects of factors on both the probability of occurrence and the hazard rate conditional on each of the failure types these two quantities are specified in the mixture model using the logistic model and the proportional hazards model respectively we propose a semi parametric mixture method to estimate the logistic and regression coefficients jointly whereby the component baseline hazard functions are completely unspecified estimation is based on maximum likelihood on the basis of the full likelihood implemented via an expectation conditional maximization ecm algorithm simulation studies are performed to compare the performance of the proposed semi parametric method with a fully parametric mixture approach the results show that when the component baseline hazard is monotonic increasing the semi parametric and fully parametric mixture approaches are comparable for mildly and moderately censored samples when the component baseline hazard is not monotonic increasing the semi parametric method consistently provides less biased estimates than a fully parametric approach and is comparable in efficiency in the estimation of the parameters for all levels of censoring the methods are illustrated using a real data set of prostate cancer patients treated with different dosages of the drug diethylstilbestrol

refined a posteriori error estimation for classical and pressure robust stokes finite element methods recent works showed that pressure robust modifications of mixed finite element methods for the stokes equations outperform their standard versions in many cases this is achieved by divergence free reconstruction operators and results in pressure independent velocity error estimates which are robust with respect to small viscosities in this paper we develop a posteriori error control which reflects this robustness the main difficulty lies in the volume contribution of the standard residual based approach that includes the l 2 norm of the right hand side however the velocity is only steered by the divergence free part of this source term an efficient error estimator must approximate this divergence free part in a proper manner otherwise it can be dominated by the pressure error to overcome this difficulty a novel approach is suggested that uses arguments from the stream function and vorticity formulation of the navier stokes equations the novel error estimators only take the hbox curl of the right hand side into account and so lead to provably reliable efficient and pressure independent upper bounds in case of a pressure robust method in particular in pressure dominant situations this is also confirmed by some numerical examples with the novel pressure robust modifications of the taylor hood and mini finite element methods

working with arms an atomic representation of a herbrand model arm is a finite set of not necessarily ground atoms over a given herbrand universe each arm represents a possibly infinite herbrand interpretation this concept has emerged independently in different branches of computer science as a natural and useful generalization of the concept of finite herbrand interpretation it was shown that several recursively decidable problems on finite herbrand models or interpretations remain decidable on arms the following problems are essential when working with arms deciding the equivalence of two arms deciding subsumption between arms and evaluating clauses over arms these problems were shown to be decidable but their computational complexity has remained obscure so far the previously published decision algorithms require exponential space in this paper we prove that all mentioned problems are conp complete

asymptotic convergence rate of the em algorithm for gaussian mixtures it is well known that the convergence rate of the expectation maximization em algorithm can be faster than those of convention first order iterative algorithms when the overlap in the given mixture is small but this argument has not been mathematically proved yet this article studies this problem asymptotically in the setting of gaussian mixtures under the theoretical framework of xu and jordan 1996 it has been proved that the asymptotic convergence rate of the em algorithm for gaussian mixtures locally around the true solution is o e0 5 e where e 0 is an arbitrarily small number o x means that it is a higher order infinitesimal as x 0 and e is a measure of the average overlap of gaussians in the mixture in other words the large sample local convergence rate for the em algorithm tends to be asymptotically superlinear when e tends to zero

divergence free finite elements on tetrahedral grids for k 6 it was shown two decades ago that the p k p k 1 mixed element on triangular grids approximating the velocity by the continuous p k piecewise polynomials and the pressure by the discontinuous p k 1 piecewise polynomials is stable for all k 4 provided the grids are free of a nearly singular vertex the problem with the method in 3d was posted then and remains open the problem is solved partially in this work it is shown that the p k p k 1 element is stable and of optimal order in approximation on a family of uniform tetrahedral grids for all k 6 the analysis is to be generalized to non uniform grids when we can deal with the complicity of 3d geometry for the divergence free elements the finite element spaces for the pressure can be avoided in computation if a classic iterated penalty method is applied the finite element solutions for the pressure are computed as byproducts from the iterate solutions for the velocity numerical tests are provided

an analysis of thep version of the finite element method for nearly incompressible materials in this paper we analyze the behavior of the so calledp version of the finite element method when applied to the equations of plane strain linear elasticity we establish optimal rate error estimates that are uniformly valid independent of the value of the poisson ratio v in the interval 0 1 2 this shows that thep versiondoes not exhibit the degeneracy phenomenon which has led to the use of various only partially justified techniques of reduced integration or mixed formulations for more standard finite element schemes and the case of a nearly incompressible material

spatiotemporal multi resolution approximation of the amari type neural field model neural fields are spatially continuous state variables described by integro differential equations which are well suited to describe the spatiotemporal evolution of cortical activations on multiple scales here we develop a multi resolution approximation mra framework for the integro difference equation ide neural field model based on semi orthogonal cardinal b spline wavelets in this way a flexible framework is created whereby both macroscopic and microscopic behavior of the system can be represented simultaneously state and parameter estimation is performed using the expectation maximization em algorithm a synthetic example is provided to demonstrate the framework

analysis of the pressure stabilized petrov galerkin method for the evolutionary stokes equations avoiding time step restrictions optimal error estimates for the pressure stabilized petrov galerkin method for the evolutionary stokes equations are proved in the case of regular solutions without restriction on the length of the time step these results clarify the instability of the discrete pressure for small time steps reported in the literature first the limit situation of the continuous in time discretization is considered and second the numerical analysis for the backward euler scheme is carried out the main results are applicable to higher order finite elements the analytical results are strongly based on an appropriate approach for computing the initial velocity which is suggested in this paper numerical studies confirm the theoretical results showing in particular that this instability does not occur for the proposed initial condition

equational formulas with membership constraints we propose a set of transformation rules for first order formulae whose atoms are either equations between terms or membership constraints t can be interpreted as a regular tree language is called a sort in the algebraic specification community or as a tree language in any class of languages which satisfies some adequate closure and decidability properties this set of rules is proved to be correct terminating and complete this provides a quantifier elimination procedure for every regular tree language l the first order theory of some structure defining l is decidable this extends the results of several previously published results we also show how to apply our results to automatic inductive proofs in equational theories