enhancing random heterogeneity representation by mixing the kriging method with the zonation structure 1 this research develops a zonation kriging zk method and extends the geostatistical theory to investigate the spatially distributed random field and obtain an optimal nonsmooth conditional estimate in the zk method the zonal structure honors the parameter measurements such that each sample location represents a distinct zone while the kriging method also honors the same data set and results in a smooth distribution the zk method integrates the conditional estimates of the zonal structure and the kriged field over a set of weighting coefficients on the sample data the extended geostatistical theory using the zk method is applied to the intrinsic field stationary field and nonstationary field the study shows that the conditional variance using the zk method is bounded between the kriging variance and variogram when estimating hydraulic conductivity by inversion of a groundwater model the optimal values of weighting coefficients are determined via predictive probability of the bayesian decision theory the numerical examples show that the zk method has less dependence on the semivariogram model the zk method outperforms the kriging method in both smooth and nonsmooth fields the zk method shows high efficiency to better the conditional estimate of hydraulic conductivity distribution with low conditional uncertainty and good agreement to the groundwater head observations

on model selection criteria in multimodel analysis 1 hydrologic systems are open and complex rendering them prone to multiple conceptualizations and mathematical descriptions there has been a growing tendency to postulate several alternative hydrologic models for a site and use model selection criteria to 1 rank these models 2 eliminate some of them and or 3 weigh and average predictions and statistics generated by multiple models this has led to some debate among hydrogeologists about the merits and demerits of common model selection also known as model discrimination or information criteria such as aic aicc bic and kic and some lack of clarity about the proper interpretation and mathematical representation of each criterion we examine the model selection literature to find that 1 all published rigorous derivations of aic and aicc require that the true model having generated the observational data be in the set of candidate models 2 though bic and kic were originally derived by assuming that such a model is in the set bic has been rederived by cavanaugh and neath 1999 without the need for such an assumption and 3 kic reduces to bic as the number of observations becomes large relative to the number of adjustable model parameters implying that it likewise does not require the existence of a true model in the set of alternatives we explain why kic is the only criterion accounting validly for the likelihood of prior parameter estimates elucidate the unique role that the fisher information matrix plays in kic and demonstrate through an example that it imbues kic with desirable model selection properties not shared by aic aicc or bic our example appears to provide the first comprehensive test of how aic aicc bic and kic weigh and rank alternative models in light of the models predictive performance under cross validation with real hydrologic data

assessment of parametric uncertainty for groundwater reactive transport modeling the validity of using gaussian assumptions for model residuals in uncertainty quantification of a groundwater reactive transport model was evaluated in this study least squares regression methods explicitly assume gaussian residuals and the assumption leads to gaussian likelihood functions model parameters and model predictions while the bayesian methods do not explicitly require the gaussian assumption gaussian residuals are widely used this paper shows that the residuals of the reactive transport model are non gaussian heteroscedastic and correlated in time characterizing them requires using a generalized likelihood function such as the formal generalized likelihood function developed by schoups and vrugt 2010 for the surface complexation model considered in this study for simulating uranium reactive transport in groundwater parametric uncertainty is quantified using the least squares regression methods and bayesian methods with both gaussian and formal generalized likelihood functions while the least squares methods and bayesian methods with gaussian likelihood function produce similar gaussian parameter distributions the parameter distributions of bayesian uncertainty quantification using the formal generalized likelihood function are non gaussian in addition predictive performance of formal generalized likelihood function is superior to that of least squares regression and bayesian methods with gaussian likelihood function the bayesian uncertainty quantification is conducted using the differential evolution adaptive metropolis dream zs algorithm as a markov chain monte carlo mcmc method it is a robust tool for quantifying uncertainty in groundwater reactive transport models for the surface complexation model the regression based local sensitivity analysis and morris and dream zs based global sensitivity analysis yield almost identical ranking of parameter importance the uncertainty analysis may help select appropriate likelihood functions improve model calibration and reduce predictive uncertainty in other groundwater reactive transport and environmental modeling

effects of error covariance structure on estimation of model averaging weights and predictive performance 1 when conducting model averaging for assessing groundwater conceptual model uncertainty the averaging weights are often evaluated using model selection criteria such as aic aicc bic and kic akaike information criterion corrected akaike information criterion bayesian information criterion and kashyap information criterion respectively however this method often leads to an unrealistic situation in which the best model receives overwhelmingly large averaging weight close to 100 which cannot be justified by available data and knowledge it was found in this study that this problem was caused by using the covariance matrix c of measurement errors for estimating the negative log likelihood function common to all the model selection criteria this problem can be resolved by using the covariance matrix cek of total errors including model errors and measurement errors to account for the correlation between the total errors an iterative two stage method was developed in the context of maximum likelihood inverse modeling to iteratively infer the unknown cek from the residuals during model calibration the inferred cek was then used in the evaluation of model selection criteria and model averaging weights while this method was limited to serial data using time series techniques in this study it can be extended to spatial data using geostatistical techniques the method was first evaluated in a synthetic study and then applied to an experimental study in which alternative surface complexation models were developed to simulate column experiments of uranium reactive transport it was found that the total errors of the alternative models were temporally correlated due to the model errors the iterative two stage method using cek resolved the problem that the best model receives 100 model averaging weight and the resulting model averaging weights were supported by the calibration results and physical understanding of the alternative models using cek obtained from the iterative two stage method also improved predictive performance of the individual models and model averaging in both synthetic and experimental studies

oscillation criteria for half linear elliptic inequalities with p x laplacians via riccati method abstract riccati inequality for half linear elliptic inequalities with p x laplacians is established and oscillation criteria are derived by using the riccati inequality our method is to reduce oscillation problems for half linear elliptic inequalities with p x laplacians to one dimensional riccati inequality with variable exponents generalizations to more general elliptic inequalities with p x laplacians are also discussed

inverses of weighted graphs abstract we consider only connected bipartite graphs g with a unique perfect matching m let g w be the weighted graph obtained from g by giving weights to its edges using the positive weight function w e g 0 such that w e 1 for each e m an unweighted graph g may be viewed as a weighted graph with the weight function w 1 a weighted graph g w is nonsingular if its adjacency matrix a g w is nonsingular the inverse of a nonsingular weighted graph g w is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix a g w via a diagonal matrix of 1s by g m we denote the graph obtained from g by contracting each edge of m to a single vertex it is known that if g m is bipartite then g w is invertible for each weight function w in this article we consider the converse of this result and show that if g w is invertible for each w then the graph g m is bipartite we also supply exact conditions under which g w is invertible for one w will force that the graph g m is bipartite

variable exponent linear growth functionals in image restoration we study a functional with variable exponent 1 leq p x leq 2 which provides a model for image denoising enhancement and restoration the diffusion resulting from the proposed model is a combination of total variation tv based regularization and gaussian smoothing the existence uniqueness and long time behavior of the proposed model are established experimental results illustrate the effectiveness of the model in image restoration

an adaptive sparse grid high order stochastic collocation method for bayesian inference in groundwater reactive transport modeling 1 bayesian analysis has become vital to uncertainty quantification in groundwater modeling but its application has been hindered by the computational cost associated with numerous model executions required by exploring the posterior probability density function ppdf of model parameters this is particularly the case when the ppdf is estimated using markov chain monte carlo mcmc sampling in this study a new approach is developed to improve the computational efficiency of bayesian inference by constructing a surrogate of the ppdf using an adaptive sparse grid high order stochastic collocation asg hsc method unlike previous works using first order hierarchical basis this paper utilizes a compactly supported higher order hierarchical basis to construct the surrogate system resulting in a significant reduction in the number of required model executions in addition using the hierarchical surplus as an error indicator allows locally adaptive refinement of sparse grids in the parameter space which further improves computational efficiency to efficiently build the surrogate system for the ppdf with multiple significant modes optimization techniques are used to identify the modes for which high probability regions are defined and components of the asg hsc approximation are constructed after the surrogate is determined the ppdf can be evaluated by sampling the surrogate system directly without model execution resulting in improved efficiency of the surrogate based mcmc compared with conventional mcmc the developed method is evaluated using two synthetic groundwater reactive transport models the first example involves coupled linear reactions and demonstrates the accuracy of our high order hierarchical basis approach in approximating high dimensional posteriori distribution the second example is highly nonlinear because of the reactions of uranium surface complexation and demonstrates how the iterative asg hsc method is able to capture multimodal and non gaussian features of ppdf caused by model nonlinearity both experiments show that asg hsc is an effective and efficient tool for bayesian inference

inverse groundwater modeling for hydraulic conductivity estimation using bayesian model averaging and variance window 1 this study proposes a bayesian model averaging bma method to address parameter estimation uncertainty arising from nonuniqueness in parameterization methods bma is able to incorporate multiple parameterization methods for prediction through the law of total probability and to obtain an ensemble average of hydraulic conductivity estimates two major issues in applying bma to hydraulic conductivity estimation are discussed the first problem is using occam s window in usual bma applications to measure approximated posterior model probabilities occam s window only accepts models in a very narrow range tending to single out the best method and discard other good methods we propose a variance window to replace occam s window to cope with this problem the second problem is the kashyap information criterion kic in the approximated posterior model probabilities which tends to prefer highly uncertain parameterization methods by considering the fisher information matrix with sufficient amounts of observation data the bayesian information criterion bic is a good approximation and is able to avoid controversial results from using kic this study adopts multiple generalized parameterization gp methods such as the bma models to estimate spatially correlated hydraulic conductivity numerical examples illustrate the issues of using kic and occam s window and show the advantages of using bic and the variance window in bma application finally we apply bma to the hydraulic conductivity estimation of the 1500 foot sand in east baton rouge parish louisiana

a picone identity for half linear elliptic equations and its applications to oscillation theory abstract a picone identity is established for a class of half linear elliptic operators and sturmian comparison and oscillation results are derived on the basis of the picone identity wirtinger inequalities and riccati inequalities are also obtained as corollaries of the picone identity and a new class of half linear elliptic equations is introduced

on reciprocal eigenvalue property of weighted trees an undirected unweighted graph g is called nonsingular if its adjacency matrix a g is nonsingular a graph is said to have property r if with every eigenvalue of a g 1 is also an eigenvalue if further the multiplicity of and 1 as eigenvalues of a g are the same then g is said to have property sr in a previous paper the classes of unweighted trees with property r and sr where shown to be the same and that this common class coincides with the class of corona trees that is trees which have been obtained from smaller trees by adding a pendent edge to each vertex in this paper we continue the study of nonsingular trees in two ways first we characterize the set of all graphs whose inverses are nonsingular trees second we investigate the extent of property r for weighted trees and characterize the property for all trees with 8 vertices or more under some conditions on the weights

on some graphs which satisfy reciprocal eigenvalue properties abstract we consider only simple graphs consider connected bipartite graphs with unique perfect matchings such that the graph obtained by contracting all matching edges is also bipartite on the class h g of such graphs g the equivalence of the following statements are known i the reciprocal of the spectral radius of the adjacency matrix a g is the least positive eigenvalue of the adjacency matrix ii the graph is isomorphic to its inverse where the inverse of a graph g is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix a g via a diagonal matrix of 1s iii the graph has the reciprocal eigenvalue property that is the reciprocal of each eigenvalue of the adjacency matrix a g is also an eigenvalue of a g iv the graph has the strong reciprocal eigenvalue property that is the reciprocal of each eigenvalue of the adjacency matrix a g is also an eigenvalue of a g and they both have the same multiplicities v the graph is a corona graph that is it is obtained by taking a bipartite graph and then by inserting a new adjacent vertex of degree one at each vertex it is known that the statements iii and iv are not equivalent if we allow nonbipartite graphs however the equivalence of these two is not yet known for the class of bipartite graphs with unique perfect matchings where we relax the condition of bipartiteness of the contracted graph in this article we establish the equivalence of the first four statements for a class larger than h g we then supply a constructive structural description of those graphs

principal eigenvalues and sturm comparison via picone s identity we characterize the principal eigenvalues and eigenfunctions in rn and present comparison results for higher dimensional p laplacian our main tool is picone s identity in this way we extend several recent results on spectral and comparison properties for differential equations

regression and time series model selection in small samples summary a bias correction to the akaike information criterion aic is derived for regression and autoregressive time series models the correction is of particular use when the sample size is small or when the number of fitted parameters is a moderate to large fraction of the sample size the corrected method called aicc is asymptotically efficient if the true model is infinite dimensional furthermore when the true model is of finite dimension aicc is found to provide better model order choices than any other asymptotically efficient method applications to nonstationary autoregressive and mixed autoregressive moving average time series models are also discussed

characterization and identification of aquifer heterogeneity with generalized parameterization and bayesian estimation 1 this research proposes a generalized parameterization method referred to as gp and bayesian estimation for parameter heterogeneity characterization and identification in groundwater modeling gp unifies zonation and interpolation through a set of weighting coefficients and is capable of creating a zonation structure a continuous distribution or a mixed structure gp shows greater flexibility not only in manipulating the highly complex spatial distribution but also in identifying the parameter structure with gp parameter structure identification seeks to identify the parameter dimension parameter pattern parameter values as well as the values of the weighting coefficients simultaneously through a set of basis points additionally this study develops an embedded genetic algorithm ga for solving the structure identification problem a bayesian estimator that estimates the basis point values as well as the values of the weighing coefficients is embedded in the ga which searches for the best basis point locations we demonstrate the inverse methodology by a numerical example in which the distributed transmissivity in a two dimensional confined aquifer is identified we calculate the jacobian matrix by the adjoint state method with gp we have successfully identified the transmissivity structure with four basis points that results in a good fitting in groundwater heads and captures the nonsmooth characteristic as well as the trend of the true transmissivity field we compare gp with voronoi tessellation zonation and natural neighbor interpolation results show that gp outperforms the other two parameterization methods in that gp identified the transmissivity field with a smaller parameter uncertainty along with a sufficiently small fitting residual and without over parameterization

a reciprocal eigenvalue property for unicyclic weighted directed graphs with weights from 1 i abstract a weighted directed graph is a directed graph g whose underlying undirected graph is simple and whose edges have nonzero directional complex weights that is the presence of an edge u v of weight w is as good as the presence of the edge v u with weight w the complex conjugate of w let g be a weighted directed graph on vertices 1 2 n denote by w u v the weight of an edge u v e g the adjacency matrix a g of g is an n n matrix with entries a i j w i j or w j i or 0 depending on whether i j e g or j i e g or otherwise respectively we supply a characterization of those unicyclic weighted directed graphs g whose edges have weights from the set 1 i and whose adjacency matrix a g satisfies the following property is an eigenvalue of a g with multiplicity k if and only if 1 is an eigenvalue of a g with the same multiplicity

model selection on solid ground rigorous comparison of nine ways to evaluate bayesian model evidence bayesian model selection or averaging objectively ranks a number of plausible competing conceptual models based on bayes theorem it implicitly performs an optimal trade off between performance in fitting available data and minimum model complexity the procedure requires determining bayesian model evidence bme which is the likelihood of the observed data integrated over each model s parameter space the computation of this integral is highly challenging because it is as high dimensional as the number of model parameters three classes of techniques to compute bme are available each with its own challenges and limitations 1 exact and fast analytical solutions are limited by strong assumptions 2 numerical evaluation quickly becomes unfeasible for expensive models 3 approximations known as information criteria ics such as the aic bic or kic akaike bayesian or kashyap information criterion respectively yield contradicting results with regard to model ranking our study features a theory based intercomparison of these techniques we further assess their accuracy in a simplistic synthetic example where for some scenarios an exact analytical solution exists in more challenging scenarios we use a brute force monte carlo integration method as reference we continue this analysis with a real world application of hydrological model selection this is a first time benchmarking of the various methods for bme evaluation against true solutions results show that bme values from ics are often heavily biased and that the choice of approximation method substantially influences the accuracy of model ranking for reliable model selection bias free numerical methods should be preferred over ics whenever computationally feasible

hydrologic consistency as a basis for assessing complexity of monthly water balance models for the continental united states methods to select parsimonious and hydrologically consistent model structures are useful for evaluating dominance of hydrologic processes and representativeness of data while information criteria appropriately constrained to obey underlying statistical assumptions can provide a basis for evaluating appropriate model complexity it is not sufficient to rely upon the principle of maximum likelihood ml alone we suggest that one must also call upon a principle of hydrologic consistency meaning that selected ml structures and parameter estimates must be constrained as well as possible to reproduce desired hydrological characteristics of the processes under investigation this argument is demonstrated in the context of evaluating the suitability of candidate model structures for lumped water balance modeling across the continental united states using data from 307 snow free catchments the models are constrained to satisfy several tests of hydrologic consistency a flow space transformation is used to ensure better consistency with underlying statistical assumptions and information criteria are used to evaluate model complexity relative to the data the results clearly demonstrate that the principle of consistency provides a sensible basis for guiding selection of model structures and indicate strong spatial persistence of certain model structures across the continental united states further work to untangle reasons for model structure predominance can help to relate conceptual model structures to physical characteristics of the catchments facilitating the task of prediction in ungaged basins

a reference bayesian test for nested hypotheses and its relationship to the schwarz criterion abstract to compute a bayes factor for testing h 0 0 in the presence of a nuisance parameter priors under the null and alternative hypotheses must be chosen as in bayesian estimation an important problem has been to define automatic or reference methods for determining priors based only on the structure of the model in this article we apply the heuristic device of taking the amount of information in the prior on equal to the amount of information in a single observation then after transforming to be null orthogonal to we take the marginal priors on to be equal under the null and alternative hypotheses doing so and taking the prior on to be normal we find that the log of the bayes factor may be approximated by the schwarz criterion with an error of order o p n rather than the usual error of order o p 1 this result suggests the schwarz criterion should provide sensible approximate solutions to bayesian testing problems at least when the hypotheses are nested when

strong reciprocal eigenvalue property of a class of weighted graphs let h be the class of connected bipartite graphs g with a unique perfect matching m for g h let wg be the set of weight functions w on the edge set e g such that w e 1 for each matching edge and w e 0 for each nonmatching edge let gw denote the weighted graph with g h and w wg the graph gw is said to satisfy the reciprocal eigenvalue property property r if 1 is an eigenvalue of the adjacency matrix a gw whenever is an eigenvalue of a gw moreover if the multiplicities of the reciprocal eigenvalues are the same we say gw has the strong reciprocal eigenvalue property property sr let hg g h g m is bipartite where g m is the graph obtained from g by contracting each edge in m to a vertex recently in 12 it was shown that if g hg then gw has property sr for some w wg if and only if gw has property sr for each w wg if and only if g is a corona graph obtained from another graph h by adding a new pendant vertex to each vertex of h now we have the following questions is there a graph g h hg such that gw has property sr for each w wg are there graphs g h hg such that gw never has property sr not even for one w wg are there graphs g h such that gw has property sr for some w wg but not for all w wg in this article we supply answers to these three questions we also supply a graph class larger than hg where for any graph g if gw has property sr for one w wg then g is a corona graph

evaluating marginal likelihood with thermodynamic integration method and comparison with several other numerical methods evaluating marginal likelihood is the most critical and computationally expensive task when conducting bayesian model averaging to quantify parametric and model uncertainties the evaluation is commonly done by using laplace approximations to evaluate semianalytical expressions of the marginal likelihood or by using monte carlo mc methods to evaluate arithmetic or harmonic mean of a joint likelihood function this study introduces a new mc method i e thermodynamic integration which has not been attempted in environmental modeling instead of using samples only from prior parameter space as in arithmetic mean evaluation or posterior parameter space as in harmonic mean evaluation the thermodynamic integration method uses samples generated gradually from the prior to posterior parameter space this is done through a path sampling that conducts markov chain monte carlo simulation with different power coefficient values applied to the joint likelihood function the thermodynamic integration method is evaluated using three analytical functions by comparing the method with two variants of the laplace approximation method and three mc methods including the nested sampling method that is recently introduced into environmental modeling the thermodynamic integration method outperforms the other methods in terms of their accuracy convergence and consistency the thermodynamic integration method is also applied to a synthetic case of groundwater modeling with four alternative models the application shows that model probabilities obtained using the thermodynamic integration method improves predictive performance of bayesian model averaging the thermodynamic integration method is mathematically rigorous and its mc implementation is computationally general for a wide range of environmental problems

some comments on cp i study the typical configuration of a cp plot when the number of variables in the regression problem is large and there are many weak effects i show that a particular configuration that is very commonly seen can arise in a simple way i give a formula by means of which the risk incurred by the minimum cp rule can be estimated

the risk inflation criterion for multiple regression a new criterion is proposed for the evaluation of variable selection procedures in multiple regression this criterion which we call the risk inflation is based on an adjustment to the risk essentially the risk inflation is the maximum increase in risk due to selecting rather than knowing the correct predictors a new variable selection procedure is obtained which in the case of orthogonal predictors substantially improves on aic c p and bic and is close to optimal in contrast to aic c p and bic which use dimensionality penalties of 2 2 and log n respectively this new procedure uses a penalty 2 log p where p is the number of available predictors for the case of nonorthogonal predictors bounds for the optimal penalty are obtained

global local optimization for parameter structure identification in three dimensional groundwater modeling 1 this paper develops two global local optimization methods for three dimensional parameter structure identification in groundwater modeling parameter structure identification is formulated in terms of solving a generalized inverse problem which allows for a determination of an appropriate level of parameter structure complexity and the identification of its pattern and the associated parameter values the structure complexity is determined by calculating the parameter structure error measured in the prediction space or management space while the structure pattern and the associated parameter values are identified simultaneously by minimizing the fitting residual measured in the observation space the former requires solving a continuous maximum minimum max min optimization problem and the latter requires solving a combinatorial optimization problem in this paper voronoi zonation is used to parameterize the unknown distributed parameter for each given level of structure complexity a sequential global local optimization method which consists of a genetic algorithm a quasi newton method and local search is developed to solve the combinatorial problem in addition the continuous max min problem is transformed to a hierarchical optimization problem and solved by a hybrid global local method sensitivities of state variables to the unknown parameters are calculated by the sensitivity equations the validity and applicability of the proposed methodology are demonstrated by numerical experiments we have shown that the choice of an objective function in model application impacts the determination of the parameter structure complexity

singular solutions of nonlinear differential equations with p t laplacian abstract this paper deals with the singular solutions of the nonlinear differential equation a t x p t 2 x b t x q t 2 x involving p t laplacian sufficient conditions are given for the existence and the nonexistence of singular solutions of the first second kind in addition we study the existence of proper solutions and the coexistence of proper solutions and singular solutions of the second kind finally some examples are given to illustrate our results

assessment and propagation of model uncertainty in most examples of inference and prediction the expression of uncertainty about unknown quantities y on the basis of known quantities x is based on a model m that formalizes assumptions about how x and y are related m will typically have two parts structural assumptions s such as the form of the link function and the choice of error distribution in a generalized linear model and parameters whose meaning is specific to a given choice of s it is common in statistical theory and practice to acknowledge parametric uncertainty about given a particular assumed structure s it is less common to acknowledge structural uncertainty about s itself a widely used approach involves enlisting the aid of x to specify a plausible single best choice s for s and then proceeding as if s were known to be correct in general this approach fails to assess and propagate structural uncertainty fully and may lead to miscalibrated uncertainty assessments about y given x when miscalibration occurs it will often result in understatement of inferential or predictive uncertainty about y leading to inaccurate scientific summaries and overconfident decisions that do not incorporate sufficient hedging against uncertainty in this paper i discuss a bayesian approach to solving this problem that has long been available in principle but is only now becoming routinely feasible by virtue of recent computational advances and examine its implementation in examples that involve forecasting the price of oil and estimating the chance of catastrophic failure of the us space shuttle

inverses of triangular matrices and bipartite graphs abstract to a given nonsingular triangular matrix a with entries from a ring we associate a weighted bipartite graph g a and give a combinatorial description of the inverse of a by employing paths in g a under a certain condition nonsingular triangular matrices a such that a and a 1 have the same zero nonzero pattern are characterized a combinatorial construction is given to construct outer inverses of the adjacency matrix of a weighted tree

picone identities for half linear elliptic operators with p x laplacians and applications to sturmian comparison theory picone identities are established for a class of half linear elliptic operators with p x laplacians and sturmian comparison theorems are obtained on the basis of the picone identities specializations to half linear partial or ordinary differential inequalities with p x laplacians are shown

overview of differential equations with non standard growth abstract differential equations with non standard growth have been a very active field of investigation in recent years in this survey we present an overview of the field as well as several of the most important results we consider both existence and regularity questions finally we provide a comprehensive list of papers published to date

picone identity for quasilinear elliptic equations with p x laplacians and sturmian comparison theory picone identities for quasilinear elliptic equations with p x laplacians are established and sturmian comparison theory is developed on the basis of picone identities as specializations various elliptic equations such as linear elliptic equations half linear elliptic equations with p laplacians half linear elliptic equations with p x laplacians pseudo p laplacian with variable power are discussed generalizations are also investigated

sturmian comparison and oscillation theorems for a class of half linear elliptic equations abstract sturmian comparison theorems are established for a class of half linear elliptic equations via the picone identity and oscillation results are derived on the basis of comparison results obtained