similarity reduction nonlocal and master symmetries of sixth order korteweg devries equation a systematic investigation to derive the lie point symmetries nonlocal and master symmetries of sixth order korteweg de vries equation kdv6 is presented using the obtained point symmetries similarity reductions are derived and constructed their particular solutions wherever possible it is shown that kdv6 admits infinitely many nonlocal and master symmetries the existence of infinitely many master symmetries ensures that kdv6 is completely integrable

classification of reduction operators and exact solutions of variable coefficient newell whitehead segel equations abstract a class of the newell whitehead segel equations also known as generalized fisher equations and newell whitehead equations is studied with lie and nonclassical symmetry points of view the classifications of lie reduction operators and of regular nonclassical reduction operators are performed the set of admissible transformations the equivalence groupoid of the class is described exhaustively the criterion of reducibility of variable coefficient newell whitehead segel equations to their constant coefficient counterparts is derived wide families of exact solutions for such variable coefficient equations are constructed

efficient empirical bayes variable selection and estimation in linear models we propose an empirical bayes method for variable selection and coefficient estimation in linear regression models the method is based on a particular hierarchical bayes formulation and the empirical bayes estimator is shown to be closely related to the lasso estimator such a connection allows us to take advantage of the recently developed quick lasso algorithm to compute the empirical bayes estimate and provides a new way to select the tuning parameter in the lasso method unlike previous empirical bayes variable selection methods which in most practical situations can be implemented only through a greedy stepwise algorithm our method gives a global solution efficiently simulations and real examples show that the proposed method is very competitive in terms of variable selection estimation accuracy and computation speed compared with other variable selection and estimation methods

analytical and numerical studying of the perturbed korteweg de vries equation the perturbed korteweg de vries equation is considered this equation is used for the description of one dimensional viscous gas dynamics nonlinear waves in a liquid with gas bubbles and nonlinear acoustic waves the integrability of this equation is investigated using the painleve approach the condition on parameters for the integrability of the perturbed korteweg de vries equation equation is established new classical and nonclassical symmetries admitted by this equation are found all corresponding symmetry reductions are obtained new exact solutions of these reductions are constructed they are expressed via trigonometric and airy functions stability of the exact solutions of the perturbed korteweg de vries equation is investigated numerically

model averaged double robust estimation researchers estimating causal effects are increasingly challenged with decisions on how to best control for a potentially high dimensional set of confounders typically a single propensity score model is chosen and used to adjust for confounding while the uncertainty surrounding which covariates to include into the propensity score model is often ignored and failure to include even one important confounder will results in bias we propose a practical and generalizable approach that overcomes the limitations described above through the use of model averaging we develop and evaluate this approach in the context of double robust estimation more specifically we introduce the model averaged double robust ma dr estimators which account for model uncertainty in both the propensity score and outcome model through the use of model averaging the ma dr estimators are defined as weighted averages of double robust estimators where each double robust estimator corresponds to a specific choice of the outcome model and the propensity score model the ma dr estimators extend the desirable double robustness property by achieving consistency under the much weaker assumption that either the true propensity score model or the true outcome model be within a specified possibly large class of models using simulation studies we also assessed small sample properties and found that ma dr estimators can reduce mean squared error substantially particularly when the set of potential confounders is large relative to the sample size we apply the methodology to estimate the average causal effect of temozolomide plus radiotherapy versus radiotherapy alone on one year survival in a cohort of 1887 medicare enrollees who were diagnosed with glioblastoma between june 2005 and december 2009

accounting for uncertainty in confounder and effect modifier selection when estimating average causal effects in generalized linear models summary confounder selection and adjustment are essential elements of assessing the causal effect of an exposure or treatment in observational studies building upon work by wang et al 2012 biometrics 68 661 671 and lefebvre et al 2014 statistics in medicine 33 2797 2813 we propose and evaluate a bayesian method to estimate average causal effects in studies with a large number of potential confounders relatively few observations likely interactions between confounders and the exposure of interest and uncertainty on which confounders and interaction terms should be included our method is applicable across all exposures and outcomes that can be handled through generalized linear models in this general setting estimation of the average causal effect is different from estimation of the exposure coefficient in the outcome model due to noncollapsibility we implement a bayesian bootstrap procedure to integrate over the distribution of potential confounders and to estimate the causal effect our method permits estimation of both the overall population causal effect and effects in specified subpopulations providing clear characterization of heterogeneous exposure effects that may vary considerably across different covariate profiles simulation studies demonstrate that the proposed method performs well in small sample size situations with 100 150 observations and 50 covariates the method is applied to data on 15 060 us medicare beneficiaries diagnosed with a malignant brain tumor between 2000 and 2009 to evaluate whether surgery reduces hospital readmissions within 30 days of diagnosis

new classes of solutions for the schwarzian korteweg de vries equation in 2 1 dimensions we have obtained new classes of solutions for the 2 1 dimensional schwarzian korteweg de vries equation by considering several types of reductions of a system equivalent to this equation the first analysis is done by studying the nonclassical reductions of the system further reductions are attained by means of other types of symmetry reductions or by ansatz based reductions most of the new classes of solutions depend on jacobian elliptic functions and solutions of a riemann wave equation including the cnoidal waves solutions the new classes of solutions can display several types of coherent structures and can exhibit the overturning or intertwining phenomena according to the suitable selection of the functions these solutions depend on

reduction of the n dimensional static vacuum einstein equation and generalized schwarzschild solutions abstract we consider the static vacuum einstein spacetime when the spatial factor is conformal to a n dimensional pseudo euclidean space the most general ansatz that reduces the resulting system of partial differential equations to a system of ordinary differential equations is completely described we obtain the entire set of solutions of the reduced system where the classical schwarzschild solution arises as a particular solution in addition we show that the riemannian spatial factors associated to these solutions are foliated by parallel hypersurfaces of constant mean curvature

iterations of the non classical symmetries method and conditional lie backlund symmetries iterations of the non classical symmetries method give rise to new nonlinear equations which inherit the lie point symmetry algebra of the original equation invariant solutions of these heir equations supply new solutions of the original equation we show that particular cases of such invariant solutions correspond to zhdanov s conditional lie backlund symmetries

model averaged confounder adjustment for estimating multivariate exposure effects with linear regression in environmental and nutritional epidemiology and in many other fields there is increasing interest in estimating the effect of simultaneous exposure to several agents e g multiple nutrients pesticides or air pollutants on a health outcome we consider estimating the effect of a multivariate exposure that includes several continuous agents and their interactions on an outcome when the true confounding variables are an unknown subset of a potentially large relative to sample size set of measured covariates our approach is rooted in the ideas of bayesian model averaging the exposure effect is estimated as a weighted average of the estimated exposure effects obtained under several linear regression models that include different sets of the potential confounders we introduce a data driven prior that assigns to the likely confounders a higher probability of being included into the regression model we show that our approach can also be formulated as a penalized likelihood formulation with an interpretable tuning parameter through a simulation study we demonstrate that the proposed approach identifies parsimonious models that are fully adjusted for observed confounding and estimates the multivariate exposure effect with smaller mean squared error compared to several alternatives we apply the method to an environmental wide association study using national heath and nutrition examination survey to estimate the effect of mixtures of nutrients and pesticides on lipid levels

nonclassical analysis and painleve property for the kupershmidt equations by means of the nonclassical lie approach and the direct method all the similarity reductions of the kupershmidt equations are found two reduction equations which can also be obtained by the classical lie approach are the painleve type ii and iv respectively the other three reduction equations which cannot be obtained by the classical lie approach are only linear or riccati type

non classical symmetry reduction example of the boussinesq equation a symmetry of an equation will leave the set of all solutions invariant a conditional symmetry will leave only a subset of solutions defined by some differential condition invariant the authors show how a specific class of conditional symmetries can be used to reduce a partial differential equation to an ordinary one in particular for the boussinesq equation these conditional symmetries together with the ordinary ones provide all possible reductions to ordinary differential equations a group theoretical explanation of the recently obtained new reductions is provided

partial symmetry of initial value problems abstract it is generally believed that the symmetry of initial value problems ivp must leave both the partial differential equations pdes and initial conditions invariant in this paper we propose partial symmetry of ivp which needs less restrictive conditions on the pdes and initial conditions and only leaves the ivp invariant on some non empty subset of the whole solution set of the governing pdes thus some symmetries which either only leave the pdes invariant or only leave initial conditions invariant are partial symmetries of ivp then considering from two different starting points we define two types of partial symmetry and give the corresponding algorithms where one starts with the pdes and the other is from the initial conditions applications to boussinesq equation and a class of nonlinear heat equation are performed

prediction via orthogonalized model mixing abstract we introduce an approach and algorithms for model mixing in large prediction problems with correlated predictors we focus on the choice of predictors in linear models and mix over possible subsets of candidate predictors our approach is based on expressing the space of models in terms of an orthogonalization of the design matrix advantages are both statistical and computational statistically orthogonalization often leads to a reduction in the number of competing models by eliminating correlations computationally large model spaces cannot be enumerated recent approaches are based on sampling models with high posterior probability via markov chains based on orthogonalization of the space of candidate predictors we can approximate the posterior probabilities of models by products of predictor specific terms this leads to an importance sampling function for sampling directly from the joint distribution over the model space without resorting to markov chains compared to the latter ortho

analytic investigation of the 2 1 dimensional schwarzian korteweg de vries equation for traveling wave solutions abstract by means of the two distinct methods the exp function method and the extended g g expansion method we successfully performed an analytic study on the 2 1 dimensional schwarzian korteweg de vries equation we exhibited its further closed form traveling wave solutions which reduce to solitary and periodic waves new rational solutions are also revealed

non classical reductions of initial value problems for a class of nonlinear evolution equations we classify initial value problems for a class of one dimensional evolution equations which can be reduced to cauchy problems for some systems of first order ordinary differential equations the technique applied relies heavily on higher conditional symmetries of the equations under study which means in particular that the obtained reductions cannot be derived within the framework of the standard lie approach

on new ways of group methods for reduction of evolution type equations new exact solutions of the evolution type equations are constructed by means of a non point contact symmetries also we analyzed the discrete symmetries of maxwell equations in vacuum and decoupled ones to the four independent equations that can be solved independently

direct reduction and differential constraints direct reductions of partial differential equations to systems of ordinary differential equations are in one to one correspondence with compatible differential constraints the differential constraint method is applied to prove that a parabolic evolution equation admits infinitely many characteristic second order reductions but admits a non characteristic second order reduction if and only if it is linearizable

partial lie point symmetries of differential equations when we consider a differential equation 0 whose set of solutions is a lie point exact symmetry of this is a lie point invertible transformation t such that t i e such that any solution to 0 is transformed into a generally different solution to the same equation here we define partial symmetries of 0 as lie point invertible transformations t such that there is a non empty subset such that t i e such that there is a subset of solutions to 0 which are transformed into one another we discuss how to determine both partial symmetries and the invariant set and show that our procedure is effective by means of concrete examples we also discuss relations with conditional symmetries and how our discussion applies to the special case of dynamical systems our discussion will focus on continuous lie point partial symmetries but our approach would also be suitable for more general classes of transformations the discussion is indeed extended to partial generalized or lie backlund symmetries along the same lines and in the appendix we will discuss the case of discrete partial symmetries

stratification and weighting via the propensity score in estimation of causal treatment effects a comparative study estimation of treatment effects with causal interpretation from observational data is complicated because exposure to treatment may be confounded with subject characteristics the propensity score the probability of treatment exposure conditional on covariates is the basis for two approaches to adjusting for confounding methods based on stratification of observations by quantiles of estimated propensity scores and methods based on weighting observations by the inverse of estimated propensity scores we review popular versions of these approaches and related methods offering improved precision describe theoretical properties and highlight their implications for practice and present extensive comparisons of performance that provide guidance for practical use

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

conditional symmetry and spectrum of the one dimensional schrodinger equation we develop an algebraic approach to studying the spectral properties of the stationary schrodinger equation in one dimension based on its high order conditional symmetries this approach makes it possible to obtain in explicit form representations of the schrodinger operator by n n matrices for any n n and thus to reduce a spectral problem to a purely algebraic one of finding eigenvalues of constant n n matrices the connection to so called quasiexactly solvable models is discussed it is established in particular that the case when conditional symmetries reduce to high order lie symmetries corresponds to exactly solvable schrodinger equations a symmetry classification of schrodinger equation admitting nontrivial high order lie symmetries is carried out which yields a hierarchy of exactly solvable schrodinger equations exact solutions of these are constructed in explicit form possible applications of the technique developed to multidimensional linear and one dimensional nonlinear schrodinger equations are briefly discussed

conditional lie backlund symmetry and reduction of evolution equations we suggest a generalization of the notion of invariance of a given partial differential equation with respect to a lie backlund vector field such a generalization proves to be effective and enables us to construct principally new ansatz reducing evolution type equations to several ordinary differential equations in the framework of the said generalization we obtain principally new reductions of a number of nonlinear heat conductivity equations ut uxx f u ux with poor lie symmetry and obtain their exact solutions it is shown that these solutions cannot be constructed by means of the symmetry reduction procedure

bayesian model averaging for linear regression models abstract we consider the problem of accounting for model uncertainty in linear regression models conditioning on a single selected model ignores model uncertainty and thus leads to the underestimation of uncertainty when making inferences about quantities of interest a bayesian solution to this problem involves averaging over all possible models i e combinations of predictors when making inferences about quantities of interest this approach is often not practical in this article we offer two alternative approaches first we describe an ad hoc procedure occam s window which indicates a small set of models over which a model average can be computed second we describe a markov chain monte carlo approach that directly approximates the exact solution in the presence of model uncertainty both of these model averaging procedures provide better predictive performance than any single model that might reasonably have been selected in the extreme case where there are many candidate predictors but

the role of the propensity score in estimating dose response functions summary estimation of average treatment effects in observational studies often requires adjustment for differences in pre treatment variables if the number of pre treatment variables is large standard covariance adjustment methods are often inadequate rosenbaum rubin 1983 propose an alternative method for adjusting for pre treatment variables for the binary treatment case based on the so called propensity score here an extension of the propensity score methodology is proposed that allows for estimation of average casual effects with multi valued treatments estimation of average treatment effects in observational studies often requires adjustment for differences in pre treatment variables if the number of pre treatment variables is large and their distribution varies substantially with treatment status standard adjustment methods such as covari ance adjustment are often inadequate rosenbaum rubin 1983 1984 propose an alternative method for adjusting for pre treatment variables based on the propensity score the conditional probability of receiving the treatment given pre treatment variables they demonstrate that adjusting solely for the propensity score removes all bias associated with differences in the pre treatment variables the rosenbaum rubin proposals deal exclusively with binary valued treat ments in many cases of interest however treatments take on more than two values here an extension of the propensity score methodology is proposed that allows for estimation of average causal effects with multi valued treatments the key insight is that for estimation of average causal effects it is not necessary to divide the population into subpopulations where causal comparisons are valid as the propensity score does it is sufficient to divide the population into subpopulations where average potential outcomes can be estimated

correlation clustering in general weighted graphs we consider the following general correlation clustering problem n bansal a blum s chawla correlation clustering in proc 43rd annu ieee symp on foundations of computer science vancouver canada november 2002 pp 238 250 given a graph with real nonnegative edge weights and a edge labelling partition the vertices into clusters to minimize the total weight of cut edges and uncut edges thus edges with large weights representing strong correlations between endpoints encourage those endpoints to belong to a common cluster while edges with large weights encourage the endpoints to belong to different clusters in contrast to most clustering problems correlation clustering specifies neither the desired number of clusters nor a distance threshold for clustering both of these parameters are effectively chosen to be best possible by the problem definition correlation clustering was introduced by bansal et al correlation clustering in proc 43rd annu ieee syrup on foundations of computer science vancouver canada november 2002 pp 238 250 motivated by both document clustering and agnostic learning they proved np hardness and gave constant factor approximation algorithms for the special case in which the graph is complete full information and every edge has the same weight we give an o log n approximation algorithm for the general case based on a linear programming rounding and the region growing technique we also prove that this linear program has a gap of log n and therefore our approximation is tight under this approach we also give an o r3 approximation algorithm for kr r minor free graphs on the other hand we show that the problem is equivalent to minimum multicut and therefore apx hard and difficult to approximate better than log n

complexity of the cluster deletion problem on subclasses of chordal graphs we consider the following vertex partition problem on graphs given a graph with real nonnegative edge weights partition the vertices into clusters in this case cliques to minimize the total weight of edges out of the clusters this optimization problem is known to be an np complete problem even for unweighted graphs and has been studied extensively in the scope of fixed parameter tractability fpt where it is commonly known as the cluster deletion problem many of the recently developed fpt algorithms rely on being able to solve cluster deletion in polynomial time on restricted graph structures in this paper the complexity of the cluster deletion problem is investigated for the family of chordal graphs it is shown that this problem is np complete for edge weighted split graphs edge weighted interval graphs and edge unweighted chordal graphs we also prove that the cluster deletion is an np complete problem for edge weighted cographs some polynomial time solvable cases are also identified in particular cluster deletion for unweighted split graphs unweighted proper interval graphs and weighted block graphs

group foliation and non invariant solutions of the heavenly equation the main physical results of this paper are new exact analytical solutions of the heavenly equation of importance in the general theory of relativity these solutions are not invariant under any subgroup of the symmetry group of the equation the main mathematical result is a new method of obtaining non invariant solutions of partial differential equations with infinite dimensional symmetry groups the method involves the compatibility of the given equations with a differential constraint which is automorphic under a specific symmetry subgroup the latter acting transitively on the submanifold of the common solutions by studying the integrability of the resulting conditions one can provide an explicit foliation of the entire solution manifold of the considered equations

strong triadic closure in cographs and graphs of low maximum degree abstract the maxstc problem is an assignment of the edges with two types of labels namely strong and weak that maximizes the number of strong edges such that any two vertices that have a common neighbor with a strong edge are adjacent the cluster deletion problem seeks for the minimum number of edge removals of a given graph such that the remaining graph is a disjoint union of cliques both problems are known to be np hard and an optimal solution for the cluster deletion problem provides a feasible solution for the maxstc problem however not necessarily an optimal one in this work we conduct the first systematic study that reveals graph families for which the optimal solutions for maxstc and cluster deletion coincide we first show that maxstc coincides with cluster deletion on cographs and thus maxstc is solvable in polynomial time on cographs as a side result we give an interesting computational characterization of the maximum independent set on the cartesian product of two cographs furthermore we address the influence of the low degree bounds to the complexity of the maxstc problem we show that this problem is polynomial time solvable on graphs of maximum degree three whereas maxstc becomes np complete on graphs of maximum degree four the proof of the latter result implies that there is no subexponential time algorithm for maxstc unless the exponential time hypothesis fails

cluster analysis and mathematical programming given a set of entities cluster analysis aims at finding subsets called clusters which are homogeneous and or well separated as many types of clustering and criteria for homogeneity or separation are of interest this is a vast field a survey is given from a mathematical programming viewpoint steps of a clustering study types of clustering and criteria are discussed then algorithms for hierarchical partitioning sequential and additive clustering are studied emphasis is on solution methods i e dynamic programming graph theoretical algorithms branch and bound cutting planes column generation and heuristics

differential constraints and exact solutions of nonlinear diffusion equations the differential constraints are applied to obtain explicit solutions of nonlinear diffusion equations certain linear determining equations with parameters are used to find such differential constraints they generalize the determining equations used in the search for classical lie symmetries

rational solutions for schwarzian integrable hierarchies we give an approach to finding rational solutions of completely intagrable hierarchies which makes use of the relationship between modifications and the schwarzian equations obtained via the singular manifold method this extends the recent work of kudryashov which allowed a simple derivation of the iteration used to construct sequences of such solutions we also give a closed form for the index polynomial of the schwarzian korteweg de vries hierarchy in addition we consider the representation of rational solutions using lower families of the hierarchy we give a simple representation under which the rational solutions remain solutions of every flow of the hierarchy this representation also allows the inclusion of arbitrary data corresponding to negative indices we use our method to derive an alternative form of the backlund transformation for the hierarchy of the second painleve equation as well as new solutions of a hierarchy of breaking soliton equations we also present here for the first time a schwarzian version of this breaking soliton hierarchy

on higher order riccati equations as backlund transformations in this short paper we illustrate by few examples the special role played by higher order riccati equations in the construction of backlund transformations for integrable systems

similarity reductions of the kp equation by a direct method basing on a direct method developed by clarkson and kruskal 1989 the kadomtsev petviashvili kp equation has been reduced to three types of 1 1 dimensional partial differential equations which are equivalent to the three types of the similarity reduction equations obtained by the classical lie approach but with different independent variables more arbitrary functions which have been missed by the classical lie approach have been included in the solutions of the kp equation for instance the third type of reduction obtained by the direct method can be divided into three subcases and the third type of solution of the kp equation obtained by the classical lie approach is only a special case of one subcase of the authors results

similarity reductions of the 1 3 dimensional burgers equation we consider the 1 3 dimensional burgers equation u t u x x u y y u z z uu x which has considerable interest in mathematical physics we complete the list of similarity reductions that are obtained from the lie symmetries admitted by this equation to achieve this goal we employ two and three dimensional subalgebras of the lie symmetry algebra in addition to the one dimensional subalgebras that appears in the literature we solve analytically where it is possible the reduced ordinary differential equations and hence we obtain closed form solutions for the original equation finally we derive certain non lie solutions

conditional symmetries and riemann invariants for hyperbolic systems of pdes this paper contains an analysis of rank k solutions in terms of riemann invariants obtained from interrelations between two concepts that of the symmetry reduction method and of the generalized method of characteristics for first order quasilinear hyperbolic systems of pdes in many dimensions a variant of the conditional symmetry method for obtaining this type of solutions is proposed a lie module of vector fields which are symmetries of an overdetermined system defined by the initial system of equations and certain first order differential constraints is constructed it is shown that this overdetermined system admits rank k solutions expressible in terms of riemann invariants finally examples of applications of the proposed approach to the fluid dynamics equations in k 1 dimensions are discussed in detail several new soliton like solutions among them kinks bumps and multiple wave solutions have been obtained these solutions remain bounded even when the riemann invariants admit a gradient catastrophe some physical interpretation of these results is discussed

new similarity reductions of the steady state boundary layer equations new similarity reductions and exact solutions of the steady state boundary layer equations which are of considerable importance in physics are presented besides similarity reductions obtained by applying the non classical group method due to bluman and cole some similarity solutions which figure neither among classical nor among non classical reductions are determined using an extension of the non classical method

the painleve property for partial differential equations ii backlund transformation lax pairs and the schwarzian derivative in this paper we investigate the painleve property for partial differential equations by application to several well known partial differential equations burgers kdv mkdv bousinesq higher order kdv and kp equations it is shown that consideration of the singular manifold leads to a formulation of these equations in terms of the schwarzian derivative this formulation is invariant under the moebius group acting on dependent variables and is shown to obtain the appropriate lax pair linearization for the underlying nonlinear pde

a reduction procedure for generalized riemann problems with application to nonlinear transmission lines generalized simple wave solutions to quasilinear hyperbolic nonhomogeneous systems of pdes are obtained through the differential constraint method these solutions prove to be flexible enough to solve generalized riemann problems where discontinuous initial data are involved within such a theoretical framework the governing model of nonlinear transmission lines is investigated throughout

approximate similarity reduction for singularly perturbed boussinesq equation via symmetry perturbation and direct method we investigate the singularly perturbed boussinesq equation in terms of the approximate symmetry perturbation method and the approximate direct method the similarity reduction solutions and similarity reduction equations of different orders display formal coincidence for both methods series reduction solutions are consequently derived for the approximate symmetry perturbation method similarity reduction equations of the zero order are equivalent to the painleve iv painleve i and weierstrass elliptic equations for the approximate direct method similarity reduction equations of the zero order are equivalent to the painleve iv painleve ii painleve i or weierstrass elliptic equations the approximate direct method yields more general approximate similarity reductions than the approximate symmetry perturbation method

weak transversality and partially invariant solutions new exact solutions are obtained for several nonlinear physical equations namely the navier stokes and euler systems an isentropic compressible fluid system and a vector nonlinear schrodinger equation the solution method makes use of the symmetry group of the system in situations when the standard lie method of symmetry reduction is not applicable

on the definition of a confounder the causal inference literature has provided a clear formal definition of confounding expressed in terms of counterfactual independence the causal inference literature has not however produced a clear formal definition of a confounder as it has given priority to the concept of confounding over that of a confounder we consider a number of candidate definitions arising from various more informal statements made in the literature we consider the properties satisfied by each candidate definition principally focusing on i whether under the candidate definition control for all confounders suffices to control for confounding and ii whether each confounder in some context helps eliminate or reduce confounding bias several of the candidate definitions do not have these two properties only one candidate definition of those considered satisfies both properties we propose that a confounder be defined as a pre exposure covariate c for which there exists a set of other covariates x such that effect of the exposure on the outcome is unconfounded conditional on x c but such that for no proper subset of x c is the effect of the exposure on the outcome unconfounded given the subset a variable that helps reduce bias but not eliminate bias we propose referring to as a surrogate confounder

symmetry reductions of partial differential equations related to singular manifold expansions it is shown that all the information yielded by the reduction methods of bluman and cole 1974 and clarkson and kruskal 1989 can be obtained using the singular manifold expansion the burgers equation modified korteweg de vries equation caudrey dodd gibbon equation and the fitzhugh nagumo equation are used as illustrative examples several new exact solutions are presented

a precise definition of reduction of partial differential equations we give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry classical and non classical reductions of partial differential equations using the introduced definition of reduction of differential equations we establish equivalence of the non classical conditional symmetry and direct ansatz approaches to reduction of partial differential equations as an illustration we give an example of non classical reduction of the nonlinear wave equation in 1 3 dimensions the conditional symmetry approach when applied to the equation in question yields a number of non lie reductions which are far reaching generalizations of the well known symmetry reductions of the nonlinear wave equations

new conditional symmetries and exact solutions of reaction diffusion convection equations with exponential nonlinearities abstract a complete description of q conditional symmetries for a class of reaction diffusion convection equations with exponential diffusivities is derived it is shown that all the known results for reaction diffusion equations with exponential diffusivities follow as particular cases from those obtained here but not vice versa the symmetries obtained for constructing exact solutions of the relevant equations are successfully applied the exact solutions are compared with those found earlier by means of different approaches an application of the solutions for solving a boundary value problem arising in population dynamics is presented

solutions of 2 1 dimensional spin systems we use the methods of group theory to reduce the equations of motion of two spin systems in 2 1 dimensions to sets of coupled ordinary differential equations we present solutions of some classes of these sets and discuss their physical significance

approximate conditional symmetries and approximate solutions of the perturbed fitzhugh nagumo equation we consider a perturbation of the fitzhugh nagumo equation the perturbation is proportional to the electric potential across the cell membrane the purpose of this investigation is to determine the effects of a change in electric potential across the cell membrane exact solutions of the perturbed equation are easily obtained from the well known solutions of the unperturbed fitzhugh nagumo equation the method of approximate conditional symmetries is used to obtain first order approximate solutions of the perturbed fitzhugh nagumo equation the approximate solutions are compared with the exact solutions of the perturbed equation the exact solutions of the perturbed equation do not indicate a change in the wave front connecting one constant state to another there is only a proportional increase or decrease in the constant nonzero state the approximate solutions do show a change in the shape of the wave front connecting two constant states as well as a proportional increase or decrease in the constant nonzero state

algorithms for the nonclassical method of symmetry reductions in this article the authors first present an algorithm for calculating the determining equations associated with so called nonclassical method of symmetry reductions a la bluman and cole for systems of partial differential equations this algorithm requires significantly less computation time than that standardly used and avoids many of the difficulties commonly encountered the proof of correctness of the algorithm is a simple application of the theory of grobner bases in the second part they demonstrate some algorithms which may be used to analyse and often to solve the resulting systems of overdetermined nonlinear pdes the authors take as their principal example a generalised boussinesq equation which arises in shallow water theory although the equation appears to be nonintegrable the authors obtain an exact two soliton solution from a nonclassical reduction

the schwarzian korteweg de vries equation in 2 1 dimensions in this paper the complete lie group classification of a 2 1 dimensional integrable schwarzian korteweg de vries equation is obtained the reduction to systems of partial differential equations in 1 1 dimension is derived from the optimal system of subalgebras the invariance study of these systems leads to second order odes these odes provide several classes of solutions all of them are expressible in terms of known functions some of them are expressible in terms of the second and third painleve transcendents the corresponding solutions of the 2 1 dimensional equation involve up to three arbitrary smooth functions they even appear in the form z f x t consequently the solutions exhibit a rich variety of qualitative behaviour indeed by making appropriate choices for the arbitrary functions we are able to exhibit large families of solitary waves coherent structures and different types of bound states

adjustment uncertainty in effect estimation summary often there is substantial uncertainty in the selection of confounders when estimating the association between an exposure and health we define this type of uncertainty as adjustment uncertainty we propose a general statistical framework for handling adjustment uncertainty in exposure effect estimation for a large number of confounders we describe a specific implemen tation and we develop associated visualization tools theoretical results and simulation studies show that the proposed method provides consistent estimators of the exposure effect and its variance we also show that when the goal is to estimate an exposure effect accounting for ad justment uncertainty bayesian model averaging with posterior model probabilities approximated using information criteria can fail to estimate the exposure effect and can over or underestimate its variance we compare our approach to bayesian model averaging using time series data on levels of fine particulate matter and mortality

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

all solutions of standard symmetric linear partial differential equations have classical lie symmetry abstract it is proven that every solution of any linear partial differential equation with an independent variable deforming classical lie point symmetry is invariant under some classical lie point symmetry this is true for any number of independent variables and for equations of any order higher than one although this result makes use of the infinite dimensional component of the lie symmetry algebra due to linear superposition it is shown that new similarity solutions previously thought not to be classical can be recovered prospectively by allowing symmetries to include superposition of similarity solutions already known from the finite part of the symmetry algebra this result applies to all constant coefficient equations and to many variable coefficient equations such as fokker planck equations

nonclassical symmetry reduction and riemann wave solutions abstract in this paper we employ the nonclassical symmetry method in order to obtain riemann multiple wave solutions of a system of first order quasilinear differential equations we show how to construct a lie module of vector fields which are symmetries of the system supplemented by certain first order differential constraints we demonstrate the usefulness of our approach on several examples

iterating the nonclassical symmetries method we show that iterations of the nonclassical symmetries method for any evolution equation give raise to new nonlinear equations which inherit the symmetry algebra of the given equation invariant solutions of these heir equations supply new solutions of the original equation the same method generates heir equations for hyperbolic and elliptic equations some examples are given

a combinatorial strongly polynomial algorithm for minimizing submodular functions this paper presents a combinatorial polynomial time algorithm for minimizing submodular functions answering an open question posed in 1981 by grotschel lovasz and schrijver the algorithm employs a scaling scheme that uses a flow in the complete directed graph on the underlying set with each arc capacity equal to the scaled parameter the resulting algorithm runs in time bounded by a polynomial in the size of the underlying set and the length of the largest absolute function value the paper also presents a strongly polynomial version in which the number of steps is bounded by a polynomial in the size of the underlying set independent of the function values

on non linear partial differential equations with an infinite dimensional conditional symmetry abstract the invariance of non linear partial differential equations under a certain infinite dimensional lie algebra a n z in n spatial dimensions is studied the special case a 1 2 was introduced in j stat phys 75 1994 1023 and contains the schrodinger lie algebra sch 1 as a lie subalgebra it is shown that there is no second order equation which is invariant under the massless realizations of a n z however a large class of strongly non linear partial differential equations is found which are conditionally invariant with respect to the massless realization of a n z such that the well known monge ampere equation is the required additional condition new exact solutions are found for some representatives of this class

vertex deletion on split graphs beyond 4 hitting set abstract vertex deletion problems on graphs involve finding a set of minimum number of vertices whose deletion results in a graph with some specific property in a recent study on vertex deletion problems on split graphs it was shown that transforming a split graph to a block graph or a threshold graph using minimum number of vertex deletions is np hard we call the decision version of these problems as split to block vertex deletion sbvd and split to threshold vertex deletion stvd respectively in this paper we study the parameterized complexity of these problems with the number of vertex deletions k as a parameter these problems are implicit 4 hitting set and thus admit an algorithm with running time o 3 0755 k a kernel with o k 3 vertices and a 4 approximation algorithm in this paper we exploit the structural properties of the concerned graph classes and obtain a kernel for sbvd with o k 2 vertices and fpt algorithms for sbvd and stvd with running times o 2 3028 k and o 2 7913 k respectively we further give improved approximation algorithms for sbvd and stvd

nonclassical symmetry reductions for the kadomtsev petviashvili equation abstract in this paper we discuss dimensional reductions of the physically and mathematically significant kadomtsev petviashvili equation in particular we exhibit some new reductions of the kadomtsev petviashvili equation which are obtained by a generalization of the direct method for determining similarity reductions recently developed by clarkson and kruskal j math phys 30 1989 2201 new solutions of the kadomtsev petviashvili equation are obtained in terms of weierstrass elliptic functions solutions of the lame equations and the first second and fourth painleve transcendents a group theoretical explanation of the new solutions is provided in terms of conditional symmetries leaving only specific subsets of solutions of the kadomtsev petviashvili equation invariant

on symmetries of radially symmetric nonlinear diffusion equations in this paper symmetries are sought for radially symmetric nonlinear diffusion equations of the type ut r1 n rn 1f u ur r the nonlinear forms of this equation which admit lie classical symmetries are completely classified and are shown to exist when f un eu also a trivial symmetry exists for arbitrary f lie backlund symmetries are also considered when f un and shown to exist when n 0 linear case and when n 2 and n 1

the investigation into the schwarz korteweg de vries equation and the schwarz derivative in 2 1 dimensions in this note we shall introduce a new integrable equation and the schwarz derivative in 2 1 dimensions first we show the existence of the lax pair for an equation which has the relation to the schwarz korteweg de vries skdv equation next we derive a new equation in 2 1 dimensions by using a well known higher dimensional manner to the lax pair for the skdv equation the 2 1 dimensional schwarz derivative is defined here finally we briefly discuss various results which we have obtained about the new equation

nonclassical symmetries as special solutions of heir equations in phys d 78 1994 124 we have found that iterations of the nonclassical symmetries method give rise to new nonlinear equations which inherit the lie point symmetry algebra of the given equation in the present paper we show that special solutions of the right order heir equation correspond to classical and nonclassical symmetries of the original equations an infinite number of nonlinear equations which possess nonclassical symmetries are derived

partially invariant solutions of nonlinear klein gordon and laplace equations the concept of partially invariant solutions is discussed in the framework of the group analysis of systems of partial differential equations using a general and systematic approach based on subgroup classification methods nontrivial partially invariant solutions for specific equations are explicitly obtained belonging to the nlkge and nlle classes in two dimensions it is proven that the obtained partially invariant solutions are distinct from the invariant ones and from solutions obtained by other methods of reducing pdes to odes

bayesian effect estimation accounting for adjustment uncertainty model based estimation of the effect of an exposure on an outcome is generally sensitive to the choice of which confounding factors are included in the model we propose a new approach which we call bayesian adjustment for confounding bac to estimate the effect of an exposure of interest on the outcome while accounting for the uncertainty in the choice of confounders our approach is based on specifying two models 1 the outcome as a function of the exposure and the potential confounders the outcome model and 2 the exposure as a function of the potential confounders the exposure model we consider bayesian variable selection on both models and link the two by introducing a dependence parameter denoting the prior odds of including a predictor in the outcome model given that the same predictor is in the exposure model in the absence of dependence 1 bac reduces to traditional bayesian model averaging bma in simulation studies we show that bac with 1 estimates the exposure effect with smaller bias than traditional bma and improved coverage we then compare bac a recent approach of crainiceanu dominici and parmigiani 2008 biometrika 95 635 651 and traditional bma in a time series data set of hospital admissions air pollution levels and weather variables in nassau ny for the period 1999 2005 using each approach we estimate the short term effects of on emergency admissions for cardiovascular diseases accounting for confounding this application illustrates the potentially significant pitfalls of misusing variable selection methods in the context of adjustment uncertainty

new family of overturning soliton solutions for a typical breaking soliton equation the breaking soliton equations are of current interest while the application of computer algebra to sciences has a bright future in this paper a new family of overturning soliton solutions for a typical breaking soliton equation is obtained via a computer algebra based method an example of explicit solutions from the family is given solitary waves are also shown to be merely a simple case belonging to the family

constructing conditionally integrable evolution systems in 1 1 dimensions a generalization of invariant modules approach given a generalized lie backlund vector field satisfying certain nondegeneracy assumptions we explicitly describe all 1 1 dimensional evolution systems that admit this vector field as a generalized conditional symmetry the connection with the theory of symmetries of systems of odes and with the theory of invariant modules is discussed

on a new method for constructing exact solutions of the nonlinear differential equations of mathematical physics a simple new method for constructing solutions of multidimensional nonlinear wave equations is proposed

symmetry reductions of a generalized cylindrical nonlinear schrodinger equation in this work symmetry reductions for a generalized cylindrical nonlinear schrodinger equation are presented these are obtained using an extension of the direct method originally developed by clarkson and kruskal 1989 which involves no group theoretic techniques

partially invariant solutions of a class of nonlinear schrodinger equations partially invariant solutions of a general class of nonlinear schrodinger equations involving four arbitrary functions of the modulus rho of the solution and its derivative rho x are obtained the modulus rho xi is assumed to depend on a symmetry variable xi whereas the phase omega x t depends on both independent variables both rho and omega are obtained explicitly as are the conditions on the coefficients in the equation necessary for such solutions to exist

group invariant solutions of differential equations we introduce the concept of a weak symmetry group of a system of partial differential equations that generalizes the nonclassical method introduced by bluman and cole for finding group invariant