linear time nearest point algorithms for coxeter lattices the coxeter lattices are a family of lattices containing many of the important lattices in low dimensions this includes an e 7 e 8 and their duals an e 7 and e 8 we consider the problem of finding a nearest point in a coxeter lattice we describe two new algorithms one with worst case arithmetic complexity o nlogn and the other with worst case complexity o n where n is the dimension of the lattice we show that for the particular lattices an and an the algorithms are equivalent to nearest point algorithms that already exist in the literature

approximate maximum likelihood period estimation from sparse noisy timing data the problem of estimating the period of a series of periodic events is considered under the condition where the measurements of the times of occurrence are noisy and sparse the problem is common to bit synchronisation in telecommunications and pulse train parameter estimation in electronic support among other applications two new algorithms are presented which represent different compromises between computational and statistical efficiency the first extends the separable least squares line search sls2 algorithms of sidiropoulos et al having very low computational complexity while attaining good statistical accuracy the second is an approximate maximum likelihood algorithm based on a low complexity lattice search and is found to achieve excellent accuracy

a general mass conservative numerical solution for the unsaturated flow equation numerical approximations based on different forms of the governing partial differential equation can lead to significantly different results for unsaturated flow problems numerical solution based on the standard h based form of richards equation generally yields poor results characterized by large mass balance errors and erroneous estimates of infiltration depth conversely numerical solutions based on the mixed form of richards equation can be shown to possess the conservative property so that mass is perfectly conserved this leads to significant improvement in numerical solution performance while requiring no additional computational effort however use of the mass conservative method does not guarantee good solutions accurate solution of the unsaturated flow equation also requires use of a diagonal time or mass matrix only when diagonal time matrices are used can the solution be shown to obey a maximum principle which guarantees smooth nonoscillatory infiltration profiles this highlights the fact that proper treatment of the time derivative is critical in the numerical solution of unsaturated flow

selecting wavelengths for least squares range estimation we consider the problem of estimating the distance or range between two locations by measuring the phase of multiple sinusoidal signals transmitted between the locations traditional estimators developed for optical interferometry include the beat wavelength and excess fractions methods more recently estimators based on the chinese remainder theorem crt and least squares have appeared recent research suggests the least squares estimator to be the most accurate in many cases the accuracy of all of these range estimators depends upon the wavelengths chosen this leads to the problem of selecting wavelengths that maximize accuracy procedures for selecting wavelengths for the beat wavelength and excess fractions methods have previously been described but procedures for the crt and least squares estimators are yet to be developed in this paper we develop an algorithm to automatically select wavelengths for use with the least square range estimator the algorithm minimizes an optimization criterion connected with the mean square error interesting properties of a particular class of lattices simplify the criterion allowing minimization by depth first search monte carlo simulations indicate that wavelengths that minimize the criterion can result in considerably more accurate range estimates than wavelengths selected by ad hoc means

a new class of bayesian cyclic bounds for periodic parameter estimation many practical signal processing applications involve estimation of parameters with periodic nature such as phase frequency and direction of arrival estimation the commonly used mean squared error mse risk does not take periodicity into account and thus is inappropriate for periodic parameter estimation in which one is interested in the modulo t error rather than the plain error value as a result mse lower bounds are not valid for periodic estimation in addition conventional bayesian mse lower bounds such as the bayesian cramer rao bound bcrb and the bobrovsky zakai bound bzb require restrictive regularity conditions and usually do not exist in periodic settings an alternative risk which is commonly used in periodic parameter estimation problems is the mean cyclic error mce in this paper we establish a new class of bayesian lower bounds on the mce of any estimator the new class includes cyclic versions of the bcrb and the bzb that have less restrictive regularity conditions than those of the conventional bcrb and bzb respectively the tightest bound in the proposed class is derived and its tightness is discussed in addition the proposed class is extended to mixed vector parameter estimation with both periodic and nonperiodic parameters the new cyclic lower bounds are compared with the mce performance of the minimum mce and maximum a posteriori probability estimators for von mises parameter estimation and for frequency estimation

monotonicity considerations for saturated unsaturated subsurface flow it is demonstrated that monotonicity is a sufficient condition to ensure that no new nonphysical local maxima and minima can be produced in the discrete nonlinear unsaturated flow equation monotonicity conditions are derived for various types of weighting for the mobility term the basic discretization is of finite element type but the results can be extended to finite volume discretizations central weightings are only conditionally monotone while upstream weightings are unconditionally monotone sample computations are given for highly heterogeneous problems

large sample theory of intrinsic and extrinsic sample means on manifolds ii this article develops nonparametric inference procedures for estimation and testing problems for means on manifolds a central limit theorem for frechet sample means is derived leading to an asymptotic distribution theory of intrinsic sample means on riemannian manifolds central limit theorems are also obtained for extrinsic sample means w r t an arbitrary embedding of a differentiable manifold in a euclidean space bootstrap methods particularly suitable for these problems are presented applications are given to distributions on the sphere s d directional spaces real projective space rp n 1 axial spaces complex projective space cp k 2 planar shape spaces w r t veronese whitney embeddings and a three dimensional shape space 4 3

estimating the common mean direction of several circular or spherical distributions with differing dispersions summary this paper considers the problem of forming a pooled estimate of the common mean direction of several circular or spherical populations with possibly differing dispersions based on samples from each population methods for obtaining approximate confidence cones for the mean direction of a single circular or spherical population are developed then applied to the case of several samples

polynomial phase estimation by least squares phase unwrapping estimating the coefficients of a noisy polynomial phase signal is important in fields including radar biology and radio communications one approach attempts to perform polynomial regression on the phase of the signal this is complicated by the fact that the phase is wrapped modulo 2 and must be unwrapped before regression can be performed in this paper we consider an estimator that performs phase unwrapping in a least squares manner we call this the least squares unwrapping lsu estimator the lsu estimator can be computed in a reasonable amount of time for data sets of moderate size using existing general purpose algorithms from algebraic number theory under mild conditions on the distribution of the noise we describe the asymptotic properties of this estimator showing that it is strongly consistent and asymptotically normally distributed a key feature is that the lsu estimator is accurate over a far wider range of parameters than many popular existing estimators monte carlo simulations support our theoretical results and demonstrate the excellent statistical performance of the lsu estimator when compared with existing state of the art estimators

estimation of finite difference interblock conductivities for simulation of infiltration into initially dry soils simulations of transient unsaturated vertical flow in different initially dry homogeneous soils using richards equation suggest that for a relatively coarse numerical grid there is no universal weighting scheme to estimate interblock conductivities capable of giving an accurate resolution of the resulting steep wetting fronts except for the scheme associated with the kirchhoff transformation kt the principal drawback of kt is that it can hardly be applied to heterogeneous soils a new asymptotic weighting aw scheme based on the asymptotic behavior of the hydraulic conductivity function in the vicinity of residual saturation is proposed numerical tests demonstrate that the aw scheme describes wetting fronts in homogeneous soils nearly as accurately as a scheme based on kt applicability and robustness of the aw scheme for simulating water infiltration into heterogeneous soils having substantial vertical variation of hydraulic properties are also demonstrated extension of the proposed scheme to higher dimensions is straightforward

three dimensional simulation of unstable gravity driven infiltration of water into a porous medium infiltration of water in dry porous media is subject to a powerful gravity driven instability although the phenomenon of unstable infiltration is well known its description using continuum mathematical models has posed a significant challenge for several decades the classical model of water flow in the unsaturated flow the richards equation is unable to reproduce the instability here we present a computational study of a model of unsaturated flow in porous media that extends the richards equation and is capable of predicting the instability and captures the key features of gravity fingering quantitatively the extended model is based on a phase field formulation and is fourth order in space the new model poses a set of challenges for numerical discretizations such as resolution of evolving interfaces stiffness in space and time treatment of singularly perturbed equations and discretization of higher order spatial partial differential operators we develop a numerical algorithm based on isogeometric analysis a generalization of the finite element method that permits the use of globally smooth basis functions leading to a simple and efficient discretization of higher order spatial operators in variational form we illustrate the accuracy efficiency and robustness of our method with several examples in two and three dimensions in both homogeneous and strongly heterogeneous media we simulate for the first time unstable gravity driven infiltration in three dimensions and confirm that the new theory reproduces the fundamental features of water infiltration into a porous medium our results are consistent with classical experimental observations that demonstrate a transition from stable to unstable fronts depending on the infiltration flux

cyclic barankin type bounds for non bayesian periodic parameter estimation in many practical periodic parameter estimation problems the appropriate performance criteria are periodic in the parameter space the existing mean square error mse lower bounds such as cramer rao bound crb and barankin type bounds do not provide valid lower bounds in such problems in this paper cyclic versions of the crb and the barankin type bounds hammersley chapman robbins and mcaulay seidman are derived for non bayesian parameter estimation the proposed bounds are lower bounds on the mean cyclic error mce of any cyclic unbiased estimator where the cyclic unbiasedness is defined by using lehmann unbiasedness these mce lower bounds can be readily obtained from existing mse lower bounds and thus can be easily calculated the cyclic barankin type bounds and the performance of the maximum likelihood ml estimator are compared in terms of mce in von mises distributed measurements problem and for frequency and amplitude estimation with gaussian noise in these problems the ml estimator is found to be cyclic unbiased

non bayesian periodic cramer rao bound the cramer rao bound crb is one of the most important tools for performance analysis in parameter estimation in many practical periodic parameter estimation problems the appropriate criterion is periodic in the parameter space however the crb does not provide a valid lower bound in such problems in this paper the periodic crb for non bayesian periodic parameter estimation is derived the proposed periodic crb is a lower bound on the mean square periodic error mspe of any periodic unbiased estimator where the periodic unbiasedness is defined by using lehmann unbiasedness it is shown that if there exists a periodic unbiased estimator which achieves the bound then the maximum likelihood produces it the periodic crb and the performance of some periodic unbiased estimators are compared in terms of mspe in a linear gaussian problem with modulo measurements and for phase estimation problems

on the incompatibility of richards equation and finger like infiltration in unsaturated homogeneous porous media 1 it is demonstrated by means of a mathematical proof that richards equation in principle cannot admit finger like solutions for three dimensional homogeneous unsaturated porous media flow subject to monotone boundary conditions this is demonstrated for any reasonable type of homogeneous porous material the result is not dependent on any particular form of the hydraulic conductivity or the retention curve moreover it is explained why hysteresis of the retention curve does not play any role in the proof consequently the proof is true for any type of hysteretic behavior of the retention curve an alternative approach to finger flow modeling is discussed which uses the ideas of cellular automata

biased search trees we consider the problem of storing items from a totally ordered set in a search tree so that the access time for a given item depends on a known estimate of the access frequency of the item we describe two related classes of biased search trees whose average access time is within a constant factor of the minimum and that are easy to update under insertions deletions and more radical update operations we present and analyze efficient update algorithms for biased search trees we list several applications of such trees

development and evaluation of closed form expressions for hysteretic soil hydraulic properties a concise representation of hysteretic soil hydraulic properties is given based on a combination of m t van genuchten s 1980 parametric k h model and p s scott et al s 1983 empirical hysteresis model modified to account for air entrapment the resulting model yields compact closed form expressions for the hysteretic water retention curve h and soil water capacity c h as well as for the hydraulic conductivity function k h depending on the degree of simplification involved the model entails a total of 6 9 parameters which can be calibrated from direct measurements of h and saturated conductivity or by an inverse solution approach from transient flow experiments comparison of model predicted and measured k h relations for eight soils revealed one case in which model predictions were very poor model accuracy was judged to be acceptably good in the other cases mualern s modified y mualem 1984 dependent domain model was found to be more accurate for soils with very narrow pore size distributions use of a simplified version of the proposed model with two parameters eliminated provided overall accuracy very similar to that of the more complex model numerical simulations of flow during transient infiltration and drainage using the proposed model and a variant of y mualem s 1984 modified dependent domain model did not differ greatly and agreed reasonably well with experimental water content distributions even when scanning curves were not described very accurately by contrast simulations without consideration of hysteresis produced highly unacceptable results it is concluded that the proposed model provides a convenient and simple means of incorporating hysteretic effects into numerical flow models to provide significant improvement in prediction accuracy with little additional effort and with minimal data requirements

direction estimation by minimum squared arc length circular statistics has found substantial application in science and engineering one of the fundamental problems in circular statistics is that of estimating the mean direction of a circular random variable from a number of observations the standard approach in the literature is called the sample circular mean and its asymptotic properties are well known it can also be computed efficiently in a number of arithmetic operations that is linear in the number of observations in this paper we consider an alternative estimator called the sample intrinsic mean that is based on minimizing squared arc length we show how this estimator can be computed efficiently in a linear number of operations using an algorithm from algebraic number theory and we derive its asymptotic properties in some scenarios the sample circular mean and the sample intrinsic mean are estimators of the same quantity and can therefore be compared we show both theoretically and by simulation that in some of these scenarios the sample intrinsic mean is statistically more accurate than the sample circular mean as such the results in this paper potentially have implications for the wide variety of fields in science engineering and statistics that currently use the sample circular mean

an algorithm to compute the nearest point in the lattice a_ n the lattice a n is an important lattice because of its covering properties in low dimensions clarkson described an algorithm to compute the nearest lattice point in a n that requires o n log n arithmetic operations in this correspondence we describe a new algorithm while the complexity is still o n log n it is significantly simpler to describe and verify in practice we find that the new algorithm also runs faster

closest point search in lattices in this semitutorial paper a comprehensive survey of closest point search methods for lattices without a regular structure is presented the existing search strategies are described in a unified framework and differences between them are elucidated an efficient closest point search algorithm based on the schnorr euchner 1995 variation of the pohst 1981 method is implemented given an arbitrary point x spl isin spl ropf sup m and a generator matrix for a lattice spl lambda the algorithm computes the point of spl lambda that is closest to x the algorithm is shown to be substantially faster than other known methods by means of a theoretical comparison with the kannan 1983 1987 algorithm and an experimental comparison with the pohst 1981 algorithm and its variants such as the viterbo boutros see ibid vol 45 p 1639 42 1999 decoder modifications of the algorithm are developed to solve a number of related search problems for lattices such as finding a shortest vector determining the kissing number computing the voronoi 1908 relevant vectors and finding a korkine zolotareff 1873 reduced basis

frequency estimation by phase unwrapping single frequency estimation is a long studied problem with application domains including radar sonar telecommunications astronomy and medicine one method of estimation called phase unwrapping attempts to estimate the frequency by performing linear regression on the phase of the received signal this procedure is complicated by the fact that the received phase is wrapped modulo and therefore must be unwrapped before the regression can be performed in this paper we propose an estimator that performs phase unwrapping in the least squares sense the estimator is shown to be strongly consistent and its asymptotic distribution is derived we then show that the problem of computing the least squares phase unwrapping is related to a problem in algorithmic number theory known as the nearest lattice point problem we derive a polynomial time algorithm that computes the least squares estimator the results of various simulations are described for different values of sample size and snr

on the continuum scale modeling of gravity driven fingers in unsaturated porous media the inadequacy of the richards equation with standard monotonic constitutive relations and hysteretic equations of state we consider the ability of the continuum scale richards equation re with standard monotonic constitutive relations and hysteretic equations of state to model gravity driven fingers in unsaturated porous media and in particular the underlying nonmonotonicity observed along the vertical extent of fingers as published in a number of recent papers finger like solutions with nonmonotonic profiles can be simulated numerically using a downwind averaging method however we find these fingers to be artifacts generated by the combined effects of a truncation error induced oscillation or oversaturation at the wetting front and capillary hysteresis as the oscillation is removed either through grid refinement or the use of inherently monotone schemes the numerically generated fingers disappear and the solution becomes monotonic thus the re along with standard monotonic hydraulic properties does not contain the critical physics required to model gravity driven fingers and must be considered inadequate for unsaturated flow in initially dry highly nonlinear and hysteretic media where these fingers occur

some new procedures for numerical solution of variably saturated flow problems persistent difficulties that arise in forming numerical solutions of variably saturated flow problems include controlling the stability of the nonlinear equation solvers and devising a reliable yet efficient method for determining the positions of seepage surfaces new techniques for addressing these problems are applied to a subdomain finite element discretization of the governing flow equations a series of test problems demonstrates that the techniques are reliable and efficient for a wide variety of problems

fast quantizing and decoding and algorithms for lattice quantizers and codes for each of the lattices a n n geq 1 d n n geq 2 e 6 e 7 e 8 and their duals a very fast algorithm is given for finding the closest lattice point to an arbitrary point if these lattices are used for vector quantizing of uniformly distributed data the algorithm finds the minimum distortion lattice point if the lattices are used as codes for a gaussian channel the algorithm performs maximum likelihood decoding