applications of fixed point methods to discrete variational and quasi variational inequalities in this paper discrete analogues of variational inequalities v i and quasi variational inequalities q v i encountered in stochastic control and mathematical physics are discussed it is shown that those discrete v i s and q v i s can be written in the fixed point formx tx such that eithert or some power oft is a contraction this leads to globally convergent iterative methods for the solution of discrete v i s and q v i s which are very suitable for implementation on parallel computers with single instruction multiple data architecture particularly on massively parallel processors m p p s

adaptive sparse polynomial chaos expansion based on least angle regression polynomial chaos pc expansions are used in stochastic finite element analysis to represent the random model response by a set of coefficients in a suitable so called polynomial chaos basis the number of terms to be computed grows dramatically with the size of the input random vector which makes the computational cost of classical solution schemes may it be intrusive i e of galerkin type or non intrusive unaffordable when the deterministic finite element model is expensive to evaluate to address such problems the paper describes a non intrusive method that builds a sparse pc expansion first an original strategy for truncating the pc expansions based on hyperbolic index sets is proposed then an adaptive algorithm based on least angle regression lar is devised for automatically detecting the significant coefficients of the pc expansion beside the sparsity of the basis the experimental design used at each step of the algorithm is systematically complemented in order to avoid the overfitting phenomenon the accuracy of the pc metamodel is checked using an estimate inspired by statistical learning theory namely the corrected leave one out error as a consequence a rather small number of pc terms are eventually retained sparse representation which may be obtained at a reduced computational cost compared to the classical full pc approximation the convergence of the algorithm is shown on an analytical function then the method is illustrated on three stochastic finite element problems the first model features 10 input random variables whereas the two others involve an input random field which is discretized into 38 and 30 500 random variables respectively

contractive type non self mappings on metric spaces of hyperbolic type abstract let x d be a metric space of hyperbolic type and k a nonempty closed subset of x in this paper we study a class of mappings from k into x not necessarily self mappings on k which are defined by the contractive condition 2 1 below and a class of pairs of mappings from k into x which satisfy the condition 2 28 below we present fixed point and common fixed point theorems which are generalizations of the corresponding fixed point theorems of ciric l b ciric quasi contraction non self mappings on banach spaces bull acad serbe sci arts 23 1998 25 31 l b ciric j s ume m s khan h k t pathak on some non self mappings math nachr 251 2003 28 33 rhoades b e rhoades a fixed point theorem for some non self mappings math japon 23 1978 457 459 and many other authors some examples are presented to show that our results are genuine generalizations of known results from this area

multivalued f weakly picard mappings recently m berinde and v berinde m berinde v berinde on a general class of multi valued weakly picard mappings j math anal appl in press extended the notion of weak contraction from single valued mappings to multivalued mappings and obtained some convergence theorems for the picard iteration associated with multivalued weak contractions we further extended the notion of weak contraction and introduced the notions of multivalued f weak contraction and generalized multivalued f weak contraction we obtain some coincidence and fixed point theorems our results extend m berinde v berinde on a general class of multi valued weakly picard mappings j math anal appl in press and those contained therein

approximating common fixed points of asymptotically nonexpansive maps in uniformly convex banach spaces we introduce three step iterative schemes with errors for two and three nonexpansive maps and establish weak and strong convergence theorems for these schemes mann type and ishikawa type convergence results are included in the analysis of these new iteration schemes the results presented in this paper substantially improve and extend the results due to s h khan h fukhar ud din weak and strong convergence of a scheme with errors for two nonexpansive mappings nonlinear anal 8 2005 1295 1301 n shahzad approximating fixed points of non self nonexpansive mappings in banach spaces nonlinear anal 61 2005 1031 1039 w takahashi t tamura convergence theorems for a pair of nonexpansive mappings j convex anal 5 1995 45 58 k k tan h k xu approximating fixed points of nonexpansive mappings by the ishikawa iteration process j math anal appl 178 1993 301 308 and h f senter w g dotson approximating fixed points of nonexpansive mappings proc amer math soc 44 1974 375 380

compressive sampling of polynomial chaos expansions sampling orthogonal polynomial bases via monte carlo is of interest for uncertainty quantification of models with random inputs using polynomial chaos pc expansions it is known that bounding a probabilistic parameter referred to as coherence yields a bound on the number of samples necessary to identify coefficients in a sparse pc expansion via solution to an 1 minimization problem utilizing results for orthogonal polynomials we bound the coherence parameter for polynomials of hermite and legendre type under their respective natural sampling distribution in both polynomial bases we identify an importance sampling distribution which yields a bound with weaker dependence on the order of the approximation for more general orthonormal bases we propose the coherence optimal sampling a markov chain monte carlo sampling which directly uses the basis functions under consideration to achieve a statistical optimality among all sampling schemes with identical support we demonstrate these different sampling strategies numerically in both high order and high dimensional manufactured pc expansions in addition the quality of each sampling method is compared in the identification of solutions to two differential equations one with a high dimensional random input and the other with a high order pc expansion in both cases the coherence optimal sampling scheme leads to similar or considerably improved accuracy

some common fixed point theorems for a family of mappings in metrically convex spaces abstract in the present paper some common fixed point theorems for a sequence and a pair of nonself mappings in complete metrically convex metric spaces are proved which generalize such results due to khan et al m s khan h k pathak m d khan some fixed point theorems in metrically convex spaces georgian math j 7 3 2000 523 530 assad n a assad on a fixed point theorem of kannan in banach spaces tamkang j math 7 1976 91 94 chatterjea s k chatterjea fixed point theorems c r acad bulgare sci 25 1972 727 730 and several others some related results are also discussed

common fixed point theorems for a mixed family of fuzzy and crisp mappings we prove a common fixed point theorem for the nonfuzzy mappings and a sequence of fuzzy mappings in a linear metric space under hybrid d compatible condition our result is more general than those of lee et al fuzzy sets and systems 2000a submitted and many others biologically relevant involvements are raised with respect to brain functioning

common fixed point theorems for contractive maps two common fixed point theorems have been proved by using minimal type commutativity and contractive conditions the last theorem extends known results on compatible maps to a wider class of mappings

one local retracts and banach operator pairs in metric spaces abstract in this paper we first introduce the concept of nr map and then use this concept to establish the existence of common fixed points for banach operator pairs in the context of uniformly convex geodesic metric spaces new proofs of main results theorems 2 1 and 3 5 of chen and li j chen z li banach operator pair and common fixed points for nonexpansive maps nonlinear anal 74 2011 3086 3090 are presented further we prove de marr s theorem for the family of symmetric banach operator pairs in metric spaces and r trees for single and multivalued mappings satisfying conditions that generalize the concept of nonexpansivity

common fixed point and invariant approximation results for noncommuting generalized f g nonexpansive maps we present common fixed point results for noncommuting generalized f g nonexpansive maps as application invariant approximation results are obtained our results unify and generalize various known results existing in the literature

coincidence points generalized i nonexpansive multimaps and applications we establish coincidence point results for multimaps satisfying generalized i nonexpansive type conditions several invariant approximation results are obtained as applications a random coincidence point result is also proved and as applications random invariant approximation results are obtained our results unify extend and complement several well known results

multi resolution analysis of wiener type uncertainty propagation schemes a multi resolution analysis mra is applied to an uncertainty propagation scheme based on a generalized polynomial chaos pc representation the mra relies on an orthogonal projection of uncertain data and solution variables onto a multi wavelet basis consisting of compact piecewise smooth polynomial functions the coefficients of the expansion are computed through a galerkin procedure the mra scheme is applied to the simulation of the lorenz system having a single random parameter the convergence of the solution with respect to the resolution level and expansion order is investigated in particular results are compared to two monte carlo sampling strategies demonstrating the superiority of the mra for more complex problems however the mra approach may require excessive cpu times adaptive methods are consequently developed in order to overcome this drawback two approaches are explored the first is based on adaptive refinement of the multi wavelet basis while the second is based on adaptive block partitioning of the space of random variables computational tests indicate that the latter approach is better suited for large problems leading to a more efficient flexible and parallelizable scheme

some common fixed point theorems for generalized nonlinear contractive mappings sufficient conditions for the existence of a common fixed point for generalized nonlinear contractive mappings are derived as applications some results on the set of best approximation for this class of mappings are also obtained the proved results generalize and extend various known results in the literature

polynomial chaos expansions for dependent random variables abstract polynomial chaos expansions pce are well suited to quantifying uncertainty in models parameterized by independent random variables the assumption of independence leads to simple strategies for building multivariate orthonormal bases and for sampling strategies to evaluate pce coefficients in contrast the application of pce to models of dependent variables is much more challenging three approaches can be used to construct pce of models of dependent variables the first approach uses mapping methods where measure transformations such as the nataf and rosenblatt transformation can be used to map dependent random variables to independent ones however we show that this can significantly degrade performance since the jacobian of the map must be approximated a second strategy is the class of dominating support methods in these approaches a pce is built using independent random variables whose distributional support dominates the support of the true dependent joint density we provide evidence that this approach appears to produce approximations with suboptimal accuracy a third approach the novel method proposed here uses gram schmidt orthogonalization gso to numerically compute orthonormal polynomials for the dependent random variables this approach has been used successfully when solving differential equations using the intrusive stochastic galerkin method and in this paper we use gso to build pce using a non intrusive stochastic collocation method the stochastic collocation method treats the model as a black box and builds approximations of the input output map from a set of samples building pce from samples can introduce ill conditioning which does not plague stochastic galerkin methods to mitigate this ill conditioning we generate weighted leja sequences which are nested sample sets to build accurate polynomial interpolants we show that our proposed approach gso with weighted leja sequences produces pce which are orders of magnitude more accurate than pce constructed using mapping or dominating support methods

a robust and efficient stepwise regression method for building sparse polynomial chaos expansions abstract polynomial chaos pc expansions are widely used in various engineering fields for quantifying uncertainties arising from uncertain parameters the computational cost of classical pc solution schemes is unaffordable as the number of deterministic simulations to be calculated grows dramatically with the number of stochastic dimension this considerably restricts the practical use of pc at the industrial level a common approach to address such problems is to make use of sparse pc expansions this paper presents a non intrusive regression based method for building sparse pc expansions the most important pc contributions are detected sequentially through an automatic search procedure the variable selection criterion is based on efficient tools relevant to probabilistic method two benchmark analytical functions are used to validate the proposed algorithm the computational efficiency of the method is then illustrated by a more realistic cfd application consisting of the non deterministic flow around a transonic airfoil subject to geometrical uncertainties to assess the performance of the developed methodology a detailed comparison is made with the well established lar based selection technique the results show that the developed sparse regression technique is able to identify the most significant pc contributions describing the problem moreover the most important stochastic features are captured at a reduced computational cost compared to the lar method the results also demonstrate the superior robustness of the method by repeating the analyses using random experimental designs

rhoades type fixed point theorems for a pair of nonself mappings abstract using suitable conditions of weak commutativity we prove theorems for a pair of nonself mappings which generalize earlier results due to rhoades and assad et al some illustrative examples and applications are also discussed

representation type of the blocks of category os abstract in this paper the authors investigate the representation type of the blocks of the relative parabolic category o s for complex semisimple lie algebras a complete classification of the blocks corresponding to regular weights is given the main results of the paper provide a classification of the blocks in the mixed case when the simple roots corresponding to the singular set and s do not meet

common fixed points of almost generalized contractive mappings in ordered metric spaces the existence theorems of common fixed points for two weakly increasing mappings satisfying an almost generalized contractive condition in ordered metric spaces are proved some comparative example are constructed which illustrate the values of the obtained results in comparison to some of the existing ones in literature

common fixed point theorems of gregus type for weakly compatible mappings satisfying generalized contractive conditions abstract we prove a common fixed point theorem of gregus type for four mappings satisfying a generalized contractive condition in metric spaces using the concept of weak compatibility which generalizes theorems of i altun d turkoglu b e rhoades fixed points of weakly compatible mappings satisfying a general contractive condition of integral type fixed point theory appl 2007 2007 article id 17301 a djoudi l nisse gregus type fixed points for weakly compatible mappings bull belg math soc 10 2003 369 378 a djoudi a aliouche common fixed point theorems of gregus type for weakly compatible mappings satisfying contractive conditions of integral type j math anal appl 329 1 2007 31 45 p vijayaraju b e rhoades r mohanraj a fixed point theorem for a pair of maps satisfying a general contractive condition of integral type int j math math sci 15 2005 2359 2364 x zhang common fixed point theorems for some new generalized contractive type mappings j math anal appl 333 2 2007 780 786 we prove also a common fixed point theorem which generalizes theorem 3 5 of h k pathak m s khan t rakesh a common fixed point theorem and its application to nonlinear integral equations comput math appl 53 2007 961 971 and common fixed point theorems of gregus type using a strict generalized contractive condition a property e a and a common property e a

invariant approximations and r subweakly commuting maps the existence of invariant approximations for r subweakly commuting mappings is proved our results extend most of the known results to a new class of noncommuting mappings

a near optimal sampling strategy for sparse recovery of polynomial chaos expansions abstract compressive sampling has become a widely used approach to construct polynomial chaos surrogates when the number of available simulation samples is limited originally these expensive simulation samples would be obtained at random locations in the parameter space it was later shown that the choice of sample locations could significantly impact the accuracy of resulting surrogates this motivated new sampling strategies or design of experiment approaches such as coherence optimal sampling which aim at improving the coherence property in this paper we propose a sampling strategy that can identify near optimal sample locations that lead to improvement in local coherence property and also enhancement of cross correlation properties of measurement matrices we provide theoretical motivations for the proposed sampling strategy along with several numerical examples that show that our near optimal sampling strategy produces substantially more accurate results compared to other sampling strategies

some results on existence and approximation in metric fixed point theory some extensions and applications of the synthetic approaches to fixed point theorems introduced recently by angrisani and clavelli are discussed

a common fixed point theorem for r weakly commuting mappings in probabilistic spaces with nonlinear contractive conditions abstract this paper presents a common fixed point theorem for a family of r weakly commuting mappings satisfying nonlinear type contractive conditions this result improves and unifies previous results due to o regan and saadati d o regan r saadati nonlinear contraction theorems in probabilistic spaces appl math comput 195 2008 86 93 also the following results will be probabilistic versions and improvements of the results given in d w boyd j s w wong on nonlinear contractions proc am math soc 20 1969 458 464 g jungck commuting maps and fixed points am math month 83 1976 261 263 r p pant common fixed points of noncommuting mappings j math anal appl 188 1994 436 440 and many other

common fixed point and approximation results for h operator pair with applications abstract the existence of common fixed point results for new classes of noncommuting selfmaps satisfying generalized i contraction or i nonexpansive type conditions is established as applications several invariant best approximation results are proved which unify extend and complement the well known results

banach operator pairs and common fixed points in hyperconvex metric spaces the purpose of this paper is to establish demarr s well known theorem for an arbitrary family of symmetric banach operator pairs in hyperconvex metric spaces without the compactness assumption we also give necessary and sufficient criteria for the existence of a common fixed point of a semigroup of isometric mappings as an application several results on the invariant best approximation are proved

on common fixed points in g metric spaces using e a property in this paper we introduce some new types of pairs of mappings f g on g metric spaces called g weakly commuting of type g f and g r weakly commuting of type g f we obtain also several common fixed point results by using the e a property

modeling uncertainty in flow simulations via generalized polynomial chaos we present a new algorithm to model the input uncertainty and its propagation in incompressible flow simulations the stochastic input is represented spectrally by employing orthogonal polynomial functionals from the askey scheme as trial basis to represent the random space a standard galerkin projection is applied in the random dimension to obtain the equations in the weak form the resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode this approach can be considered as a generalization of the original polynomial chaos expansion first introduced by wiener am j math 60 1938 897 the original method employs the hermite polynomials one of the 13 members of the askey scheme as the basis in random space the algorithm is applied to micro channel flows with random wall boundary conditions and to external flows with random freestream efficiency and convergence are studied by comparing with exact solutions as well as numerical solutions obtained by monte carlo simulations it is shown that the generalized polynomial chaos method promises a substantial speed up compared with the monte carlo method the utilization of different type orthogonal polynomials from the askey scheme also provides a more efficient way to represent general non gaussian processes compared with the original wiener hermite expansions

gaussian process regression and conditional polynomial chaos for parameter estimation abstract we present a new approach for constructing a data driven surrogate model and using it for parameter estimation in partial differential equation pde models we first use parameter observations and gaussian process regression to condition the karhunen loeve kl expansion of the unknown space dependent parameters and then build the conditional generalized polynomial chaos gpc surrogate model of the pde states next we estimate the unknown parameters by computing coefficients in the kl expansion by minimizing the square difference between the gpc predictions and measurements of the states our approach addresses two major challenges in the parameter estimation first it reduces dimensionality of the parameter space and replaces expensive direct solutions of pdes with the conditional gpc surrogates second the estimated parameter field exactly matches the parameter measurements in addition we show that the conditional gpc surrogate can be used to estimate the state variance that in turn can be used to guide data acquisition we demonstrate that our approach improves the accuracy of parameter estimation with application to one and two dimensional darcy equations that have unknown space dependent hydraulic conductivity we also discuss the effect of hydraulic conductivity and head locations on the accuracy of the hydraulic conductivity estimations

convergence theorems for nonself asymptotically nonexpansive mappings in this paper we prove some strong and weak convergence theorems using a modified iterative process for nonself asymptotically nonexpansive mappings in a uniformly convex banach space this will improve and generalize the corresponding results in the existing literature finally we will state that our theorems can be generalized to the case of finitely many mappings

banach operator pairs common fixed points invariant approximations and nonexpansive multimaps abstract common fixed point results are proved for i nonexpansive maps ciric type maps and nonexpansive multimaps invariant approximation results are obtained for these types of maps as applications our results extend or generalize several known results including the recent results of chen and li j chen z li common fixed points for banach operators pairs in best approximation j math anal appl 336 2007 1466 1475 in particular we show that the results of chen and li on banach operator pairs are particular cases of the results of al thagafi and shahzad m a al thagafi n shahzad noncommuting selfmaps and invariant approximations nonlinear anal 64 2006 2778 2786

common fixed points under strict contractive conditions abstract the aim of the present paper is to obtain common fixed point theorems under strict contractive conditions by using a minimal commutativity condition our results extend the recent results due to pant 1998 j math anal appl 226 251 258 1999 soochow j math 25 37 42

common fixed points for banach operator pairs in best approximation the notion of banach operator pairs is introduced as a new class of noncommuting maps some common fixed point theorems for banach operator pairs and the existence of the common fixed points of best approximation are presented these results are proved without the assumption of linearity or affinity for either f or g which shows that the concept about banach operator pairs is potentially useful in the study of common fixed points

koszul duality patterns in representation theory the aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of koszul duality to repre sentation theory the paper consists of three parts relatively independent of each other the first part gives a reasonably selfcontained introduction to koszul rings and koszul duality koszul rings are certain z graded rings with particularly nice homological properties which involve a kind of duality thus to a koszul ring one associates naturally the dual koszul ring the second part is devoted to an application to representation theory of semisimple lie algebras we show license or copyright restrictions may apply to redistribution see http www ams org journal terms of use koszul duality patterns 527 that the block of the bernstein gelfand gelfand category o that corresponds to any fixed central character is governed by the koszul ring moreover the dual of that ring governs a certain subcategory of the category o again this generalizes the selfduality theorem conjectured by beilinson and ginsburg in 1986 and proved by soergel in 1990 in the third part we study certain cate gories of mixed perverse sheaves on a variety stratified by affine linear spaces we provide a general criterion for such a category to be governed by a koszul ring in the flag variety case this reduces to the setup of part two in the more general case of affine flag manifolds and affine grassmannians the criterion should yield interesting results about representations of quantum groups and affine lie algebras department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address sasha math mit edu department of mathematics the university of chicago chicago illinois 60637 e mail address ginzburg math uchicago edu max planck institut fur mathematik gottfried claren strase 26 d 53 bonn 3 germany current address mathematisches institut universitat freiburg albertstrase 23b d 79104 freiburg germany e mail address soergel sun1 mathematik uni freiburg de license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

common fixed point theorems of gregus type for weakly compatible mappings satisfying contractive conditions of integral type abstract we prove a common fixed point theorem of gregus type for four mappings satisfying a contractive condition of integral type in metric spaces using the concept of weak compatibility which generalizes theorem 2 of a djoudi l nisse gregus type fixed points for weakly compatible mappings bull belg math soc 10 2003 369 378 and other papers we prove also common fixed point theorems of gregus type using a strict contractive condition of integral type a property e a and a common property e a introduced by m aamri d el moutawakil some new common fixed point theorems under strict contractive conditions j math anal appl 270 2002 181 188 and w liu j wu z li common fixed points of single valued and multi valued maps int j math math sci 19 2005 3045 3055 respectively

common fixed point results for noncommuting mappings without continuity in generalized metric spaces using the setting of a generalized metric space a fixed point theorem is proved for one map and several fixed point theorems are proved for two maps these results generalize several well known comparable results in the literature

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

common fixed point results for noncommuting mappings without continuity in cone metric spaces the existence of coincidence points and common fixed points for mappings satisfying certain contractive conditions without appealing to continuity in a cone metric space is established these results generalize several well known comparable results in the literature

common fixed point theorems in menger spaces with common property e a in this paper we introduce the notion of common property e a in menger spaces besides proving a result interrelating the property e a with common property e a thereafter using the common property e a some common fixed point theorems are proved for self mappings in menger spaces which include results involving quasi contraction as well as f type contraction our results generalize several known results in menger as well as metric spaces some related results are also derived besides furnishing an illustrative example

common fixed points in best approximation for banach operator pairs with ciric type i contractions abstract the common fixed point theorems similar to those of ciric lj b ciric on a common fixed point theorem of a gregus type publ inst math beograd n s 49 1991 174 178 lj b ciric on diviccaro fisher and sessa open questions arch math brno 29 1993 145 152 lj b ciric on a generalization of gregus fixed point theorem czechoslovak math j 50 2000 449 458 fisher and sessa b fisher s sessa on a fixed point theorem of gregus internat j math math sci 9 1986 23 28 jungck g jungck on a fixed point theorem of fisher and sessa internat j math math sci 13 1990 497 500 and mukherjee and verma r n mukherjee v verma a note on fixed point theorem of gregus math japon 33 1988 745 749 are proved for a banach operator pair as applications common fixed point and approximation results for banach operator pair satisfying ciric type contractive conditions are obtained without the assumption of linearity or affinity of either t or i our results unify and generalize various known results to a more general class of noncommuting mappings

selection of polynomial chaos bases via bayesian model uncertainty methods with applications to sparse approximation of pdes with stochastic inputs generalized polynomial chaos gpc expansions allow us to represent the solution of a stochastic system using a series of polynomial chaos basis functions the number of gpc terms increases dramatically as the dimension of the random input variables increases when the number of the gpc terms is larger than that of the available samples a scenario that often occurs when the corresponding deterministic solver is computationally expensive evaluation of the gpc expansion can be inaccurate due to over fitting we propose a fully bayesian approach that allows for global recovery of the stochastic solutions in both spatial and random domains by coupling bayesian model uncertainty and regularization regression methods it allows the evaluation of the pc coefficients on a grid of spatial points via 1 the bayesian model average bma or 2 the median probability model and their construction as spatial functions on the spatial domain via spline interpolation the former accounts for the model uncertainty and provides bayes optimal predictions while the latter provides a sparse representation of the stochastic solutions by evaluating the expansion on a subset of dominating gpc bases moreover the proposed methods quantify the importance of the gpc bases in the probabilistic sense through inclusion probabilities we design a markov chain monte carlo mcmc sampler that evaluates all the unknown quantities without the need of ad hoc techniques the proposed methods are suitable for but not restricted to problems whose stochastic solutions are sparse in the stochastic space with respect to the gpc bases while the deterministic solver involved is expensive we demonstrate the accuracy and performance of the proposed methods and make comparisons with other approaches on solving elliptic spdes with 1 14 and 40 random dimensions

fixed point theorems in fuzzy metric spaces using property e a abstract we prove two common fixed point theorems for a pair of weakly compatible maps in fuzzy metric spaces both in the sense of kramosil and michalek and in the sense of george and veeramani by using e a property

approximation of common fixed points of weakly compatible pairs using the jungck iteration abstract we show that the jungck iteration scheme can be used to approximate the common fixed points of some weakly compatible pairs of generalized quasicontractive operators defined on metric spaces the existence of coincidence points are also discussed for those pair of maps the results are generalizations of well known results of the convergence of picard iterations for single self maps of banach spaces in particular the results improve generalize and extend the recent results of berinde v berinde a common fixed point theorem for compatible quasi contractive self mappings in metric spaces applied mathematics and computation 213 2009 348 354 and answers the open question posed in the paper

best simultaneous approximations asymptotically nonexpansive mappings and variational inequalities in banach spaces abstract the existence of common fixed points is established for three mappings where t is either generalized f g nonexpansive or asymptotically f g nonexpansive on a nonempty subset of a banach space as applications the invariant best simultaneous approximation results are proved and the existence of solution of variational inequalities is obtained our results unify and substantially improve several recent results existing in the current literature

fixed point theorems for families of weakly non expansive maps abstract in this paper we present some fixed point theorems for families of weakly non expansive maps under some relatively weaker and more general conditions our results generalize and improve several results due to jungck g jungck fixed points via a generalized local commutativity int j math math sci 25 8 2001 497 507 jachymski j jachymski a generalization of the theorem by rhoades and watson for contractive type mappings math japon 38 6 1993 1095 1102 guo c guo an extension of fixed point theorem of krasnoselski chinese j math p o c 21 1 1993 13 20 rhoades b e rhoades a comparison of various definitions of contractive mappings trans amer math soc 226 1977 257 290 and others

some common fixed point theorems for banach operator pairs with applications in best approximation abstract the ordered pair t i of two self maps of a metric space x d is called a banach operator pair if the set f i of fixed points of i is t invariant i e t f i f i some common fixed point theorems for a banach operator pair and the existence of common fixed points of best approximation are presented in this paper the results prove generalize and extend some results of al thagafi m a al thagafi common fixed points and best approximation j approx theory 85 1996 318 323 carbone a carbone applications of fixed point theorems jnanabha 19 1989 149 155 chen and li j chen z li common fixed points for banach operator pairs in best approximations j math anal appl 336 2007 1466 1475 habiniak l habiniak fixed point theorems and invariant approximation j approx theory 56 1989 241 244 jungck and sessa g jungck s sessa fixed point theorems in best approximation theory math japon 42 1995 249 252 sahab khan and sessa s a sahab m s khan s sessa a result in best approximation theory j approx theory 55 1988 349 351 shahzad n shahzad invariant approximations and r subweakly commuting maps j math anal appl 257 2001 39 45 and of few others

coincidence and fixed points in symmetric spaces under strict contractions abstract some common fixed point theorems due to aamri and el moutawakil m aamri d el moutawakil some new common fixed point theorems under strict contractive conditions j math anal appl 270 2002 181 188 and pant and pant r p pant v pant common fixed points under strict contractive conditions j math anal appl 248 2000 327 332 proved for strict contractive mappings in metric spaces are extended to symmetric semi metric spaces under tight conditions some related results are derived besides discussing illustrative examples which establish the utility of results proved in this note

coincidence and fixed points for hybrid strict contractions abstract we define the property e a for single valued and multivalued mappings and introduce the notion of t weak commutativity for a hybrid pair f t of single valued and multivalued maps we obtain some coincidence and fixed point theorems for this class of maps and derive as application an approximation theorem

an iterative polynomial chaos approach for solution of structural mechanics problem with gaussian material property abstract a new approach for solution of stochastic structural mechanics problem with random coefficient in the framework of polynomial chaos pc expansion is proposed the basic strategy of the proposed method is to iteratively construct a pc expansion with reduced numbers of unknown coefficients the method is based on orthogonal expansion of stochastic responses and generation of an iterative pc based on the responses of the previous iteration the polynomials are evaluated using gram schmidt orthogonalization process the number of random variables in pc expansion is reduced by considering only the dominant components of the response characteristics which is evaluated using karhunen loeve kl expansion in case of random material field problem kl expansion is used to discretize the random field the usefulness of the proposed method in terms of accuracy and computational efficiency is examined the results of linear static structural mechanics problems are compared with mcs and pc method the proposed method is observed to be computationally more efficient and accurate than pc method

noncommuting selfmaps and invariant approximations abstract common fixed point results are established for a new class of noncommuting selfmaps we apply them to obtain several invariant approximation results which unify extend and complement well known results

polynomial chaos expansion with latin hypercube sampling for estimating response variability a computationally efficient procedure for quantifying uncertainty and finding significant parameters of uncertainty models is presented to deal with the random nature of input parameters of structural models several efficient probabilistic methods are investigated specifically the polynomial chaos expansion with latin hypercube sampling is used to represent the response of an uncertain system latin hypercube sampling is employed for evaluating the generalized fourier coefficients of the polynomial chaos expansion because the key challenge in uncertainty analysis is to find the most significant components that drive response variability analysis of variance is employed to find the significant parameters of the approximation model several analytical examples and a large finite element model of a joined wing are used to verify the effectiveness of this procedure

common fixed points for banach operator pairs with applications the existence of common fixed point results for a banach operator pair under certain generalized contractions is established the invariant best approximation results are proved as applications and the existence of solutions of variational inequalities is obtained we also study the solution of functional equations arising from dynamic programming 2007 elsevier ltd all rights reserved

weighted discrete least squares polynomial approximation using randomized quadratures we discuss the problem of polynomial approximation of multivariate functions using discrete least squares collocation the problem stems from uncertainty quantification uq where the independent variables of the functions are random variables with specified probability measure we propose to construct the least squares approximation on points randomly and uniformly sampled from tensor product gaussian quadrature points we analyze the stability properties of this method and prove that the method is asymptotically stable provided that the number of points scales linearly up to a logarithmic factor with the cardinality of the polynomial space specific results in both bounded and unbounded domains are obtained along with a convergence result for chebyshev measure numerical examples are provided to verify the theoretical results

banach operator pair and common fixed points for nonexpansive maps abstract let x be a normed space and t and i be two nonexpansive self maps on the closed convex subset c x it is proved that if i t is a banach operator pair and t c is compact then f i t 0 a family of nonexpansive maps is also investigated the well known de marr s fixed point theorem is extended to the noncommuting case by introducing a notion of banach operator family as an application of the theorem the problem for the common fixed point in invariant approximation for convex set is solved directly in a quite different way from the others before

numerical methods for differential equations in random domains physical phenomena in domains with rough boundaries play an important role in a variety of applications often the topology of such boundaries cannot be accurately described in all of its relevant detail due to either insufficient data or measurement errors or both this topological uncertainty can be efficiently handled by treating rough boundaries as random fields so that an underlying physical phenomenon is described by deterministic or stochastic differential equations in random domains to deal with this class of problems we propose a novel computational framework which is based on using stochastic mappings to transform the original deterministic stochastic problem in a random domain into a stochastic problem in a deterministic domain the latter problem has been studied more extensively and existing analytical numerical techniques can be readily applied in this paper we employ both a stochastic galerkin method and monte carlo simulations to solve the transformed stochastic problem we demonstrate our approach by applying it to an elliptic problem in single and double connected random domains and comment on the accuracy and convergence of the numerical methods

compatible maps and invariant approximations the existence of invariant best approximations for compatible maps is proved our results unify and generalize various known results to a more general class of noncommuting mappings

a rapid and efficient isogeometric design space exploration framework with application to structural mechanics abstract in this paper we present an isogeometric analysis framework for design space exploration while the methodology is presented in the setting of structural mechanics it is applicable to any system of parametric partial differential equations the design space exploration framework elucidates design parameter sensitivities used to inform initial and early stage design moreover this framework enables the visualization of a full system response including the displacement and stress fields throughout the domain by providing an approximation to the system solution vector this is accomplished through a collocation like approach where various geometries throughout the design space under consideration are sampled the sampling scheme follows a quadrature rule while the physical solutions to these sampled geometries are obtained through an isogeometric method a surrogate model to the design space solution manifold is constructed through either an interpolating polynomial or pseudospectral expansion examples of this framework are presented with applications to the scordelis lo roof a flat l bracket and an nrel 5 mw wind turbine blade

representation type of the blocks of category os in types f4 and g2 abstract in this paper a complete classification of the representation type of the infinitesimal blocks of the parabolic category o s for the complex simple lie algebras of types f 4 and g 2 is given

approximation of fixed points of uniformly r subweakly commuting mappings we introduce a new class of uniformly r subweakly commuting mappings and then using this class study the problem of approximation of common fixed points of asymptotically s nonexpansive mappings in a banach space with uniformly g teaux differentiable norm

on polynomial chaos expansion via gradient enhanced l 1 minimization gradient enhanced uncertainty quantification uq has received recent attention in which the derivatives of a quantity of interest qoi with respect to the uncertain parameters are utilized to improve the surrogate approximation polynomial chaos expansions pces are often employed in uq and when the qoi can be represented by a sparse pce 1 minimization can identify the pce coefficients with a relatively small number of samples in this work we investigate a gradient enhanced 1 minimization where derivative information is computed to accelerate the identification of the pce coefficients for this approach stability and convergence analysis are lacking and thus we address these here with a probabilistic result in particular with an appropriate normalization we show the inclusion of derivative information will almost surely lead to improved conditions e g related to the null space and coherence of the measurement matrix for a successful solution recovery further we demonstrate our analysis empirically via three numerical examples a manufactured pce an elliptic partial differential equation with random inputs and a plane poiseuille flow with random boundaries these examples all suggest that including derivative information admits solution recovery at reduced computational cost an approach for sparse polynomial chaos pc expansions including derivative information is presented method relies on 1 minimization to solve for the pc coefficients formal analysis is presented to guarantee the stability and convergence of gradient enhanced 1 minimization three examples presented to illustrate cost accuracy improvements achieved by gradient enhanced 1 minimization

least squares polynomial chaos expansion a review of sampling strategies abstract as non intrusive polynomial chaos expansion pce techniques have gained growing popularity among researchers we here provide a comprehensive review of major sampling strategies for the least squares based pce traditional sampling methods such as monte carlo latin hypercube quasi monte carlo optimal design of experiments ode gaussian quadratures as well as more recent techniques such as coherence optimal and randomized quadratures are discussed we also propose a hybrid sampling method dubbed alphabetic coherence optimal that employs the so called alphabetic optimality criteria used in the context of ode in conjunction with coherence optimal samples a comparison between the empirical performance of the selected sampling methods applied to three numerical examples including high order pce s high dimensional problems and low oversampling ratios is presented to provide a road map for practitioners seeking the most suitable sampling technique for a problem at hand we observed that the alphabetic coherence optimal technique outperforms other sampling methods specially when high order ode are employed and or the oversampling ratio is low

common fixed points for p operator pair with applications the existence of common fixed point results for new classes of noncommuting selfmaps satisfying generalized i contraction or i nonexpansive type conditions is established as applications several invariant best approximation results are proved which unify extend and complement the well known results

category o gradings and translation functors abstract in this article we consider a graded version of category o we reprove some results of beilinson et al j amer math soc 9 1996 473 527 using a different approach furthermore we define a graded version of translation functors and duality this provides the construction of various graded modules on the other hand we describe how to get modules which are not gradable

reweighted l1 minimization method for stochastic elliptic differential equations abstract we consider elliptic stochastic partial differential equations spdes with random coefficients and solve them by expanding the solution using generalized polynomial chaos gpc under some mild conditions on the coefficients the solution is sparse in the random space i e only a small number of gpc basis makes considerable contribution to the solution to exploit this sparsity we employ reweighted l 1 minimization to recover the coefficients of the gpc expansion we also combine this method with random sampling points based on the chebyshev probability measure to further increase the accuracy of the recovery of the gpc coefficients we first present a one dimensional test to demonstrate the main idea and then we consider 14 and 40 dimensional elliptic spdes to demonstrate the significant improvement of this method over the standard l 1 minimization method for moderately high dimensional 10 problems the combination of chebyshev measure with reweighted l 1 minimization performs well while for higher dimensional problems reweighted l 1 only is sufficient the proposed approach is especially suitable for problems for which the deterministic solver is very expensive since it reuses the sampling results and exploits all the information available from limited sources