existence uniqueness and l q estimates for the stokes problem in an exterior domain as is well known the study of the properties of solutions to the stokes system plays an essential role in the mathematical theory of viscous fluid flows governed by the navier stokes equations it is not art overstatement to say that existence uniqueness and a priori estimates in sobolev spaces for the stokes system are fundamental in every basic question related to the navier stokes equations such as existence regularity asymptotic behavior in time etc see e g ladyzhenskaya 1969 solonni ov 1977 and te am 1977 so far however the stokes system has been fully analyzed only for bounded domains see sobolevskn 1960 ladvznenskaya 1959 solonnu ov 1960 vorovlch yot oowcr 1961 cattabrma 1961 in particular in the paper of cattabriga a complete treatment is given in the sobotev spaces w l 1 1 q c for space dimension n 3 cattabriga s results essentially state that if 2 is a bounded domain in r a of class c m m max l 2 2 l 1 then given f e wt q y2 3 gc wt l q o q e wl 2 1 q q 8o a 1 q there exists one and only one distributional solution v c w1 z q q 3 p e wt l q o to the stokes system

fuzzy scores of technical efficiency abstract this paper suggests to extend scores of technical efficiency as used in for example data envelopment analysis dea to efficiency scores defined as so called fuzzy intervals fuzzy scores allow the decision maker to use scores of technical efficiency in combination with other sources of available performance information e g expert opinions key figures etc in this sense fuzzy scores become formal images of the evaluation otherwise done implicitly by the decision maker and may therefore enhance a more consistent evaluation it is shown how to rank a set of fuzzy scores

integrable symplectic maps abstract in this paper we first give a terse survey of symplectic maps their canonical formulation and integrability then we introduce a rigorous procedure to construct integrable symplectic maps starting from integrable evolution equations on lattices a number of illustrative examples are provided

a new family of four dimensional symplectic and integrable mappings abstract we investigate the generalisations of the quispel roberts and thompson qrt family of mappings in the plane leaving a rational quadratic expression invariant to the case of four variables we assume invariance of the rational expression under a cyclic permutation of variables and we impose a symplectic structure with poisson brackets of the weyl type all mappings satisfying these conditions are shown to be integrable either as four dimensional mappings with two explicit integrals which are in involution with respect to the symplectic structure and which can also be inferred from the periodic reductions of the double discrete versions of the modified korteweg devries mkdv and sine gordon sg equations or by reduction to two dimensional mappings with one integral of the symmetric qrt family

fuzzy multiattribute grey related analysis using dea qualitative input is encountered in many real decision making domains often such domains include tradeoffs among multiple attributes and estimates of parameters are often expressed with some degree of uncertainty grey related analysis has been proposed as a means to use interval fuzzy representation of data when dealing with multiple criteria data model parameters that can involve uncertainty include both performances of alternatives on attributes as well as attribute weights dea is proposed as an objective way to derive weights this paper presents a grey related fuzzy set methodology incorporating data envelopment analysis as a way to more objectively rank alternatives the purpose is to demonstrate the method the focus is on identifying alternatives performing the most efficiently with respect to the decision maker s preference the method is demonstrated on a multiattribute siting problem simulation is applied to validate model efficiency

generalization of solutions of the jacobi pdes associated to time reparametrizations of poisson systems the determination of solutions of the jacobi partial differential equations pdes for finite dimensional poisson systems is considered in particular a novel procedure for the construction of solution families is developed such a procedure is based on the use of time reparametrizations preserving the existence of a poisson structure as a result a method which is valid for arbitrary values of the dimension and the rank of the poisson structure under consideration is obtained in this article two main families of time reparametrizations of this kind are characterized in addition these results lead to a novel application which is also developed namely the global and constructive determination of the darboux canonical form for poisson systems of arbitrary dimension and rank two thus improving the local result provided by darboux theorem for such a case

new solution family of the jacobi equations characterization invariants and global darboux analysis a new family of skew symmetric solutions of the jacobi partial differential equations for finite dimensional poisson systems is characterized and analyzed such family has some remarkable properties firstly it is defined for arbitrary values of the dimension and the rank secondly it is described in terms of arbitrary differentiable functions namely it is not limited to a given degree of nonlinearity additionally it is possible to determine explicitly the fundamental properties of those solutions such as their casimir invariants and the algorithm for the reduction to the darboux canonical form which have been reported only for a very limited sample of finite dimensional poisson structures moreover such analysis is carried out globally in phase space thus improving the usual local scope of the darboux theorem

traces for homogeneous sobolev spaces in infinite strip like domains abstract in this paper we construct a trace operator for homogeneous sobolev spaces defined on infinite strip like domains we identify an intrinsic seminorm on the resulting trace space that makes the trace operator bounded and allows us to construct a bounded right inverse the intrinsic seminorm involves two features not encountered in the trace theory of bounded lipschitz domains or half spaces first due to the strip like structure of the domain the boundary splits into two infinite disconnected components the traces onto each component are not completely independent and the intrinsic seminorm contains a term that measures the difference between the two traces second as in the usual trace theory there is a term in the seminorm measuring the fractional sobolev regularity of the trace functions with a difference quotient integral however the finite width of the strip like domain gives rise to a screening effect that bounds the range of the difference quotient perturbation the screened homogeneous fractional sobolev spaces defined by this screened seminorm on arbitrary open sets are of independent interest and we study their basic properties we conclude the paper with applications of the trace theory to partial differential equations

performance evaluation an integrated method using data envelopment analysis and fuzzy preference relations in a multi attribute decision making madm context the decision maker needs to provide his preferences over a set of decision alternatives and constructs a preference relation and then use the derived priority vector of the preference to rank various alternatives this paper proposes an integrated approach to rate decision alternatives using data envelopment analysis and preference relations this proposed approach includes three stages first pairwise efficiency scores are computed using two dea models the ccr model and the proposed cross evaluation dea model second the pairwise efficiency scores are then utilized to construct the fuzzy preference relation and the consistent fuzzy preference relation third by use of the row wise summation technique we yield a priority vector which is used for ranking decision making units dmus for the case of a single output and a single input the preference relation can be directly obtained from the original sample data the proposed approach is validated by two numerical examples

new global solutions of the jacobi partial differential equations abstract a new family of solutions of the jacobi partial differential equations for finite dimensional poisson systems is investigated this family is mathematically remarkable as the functional dependences of the solutions appear to be associated to the distinguished invariants of the solutions themselves this kind of poisson structure termed distinguished solutions or d solutions is defined for every nontrivial combination of values of the dimension and the rank and is also determined in terms of functions of arbitrary nonlinearity properties usually not present simultaneously in the already known solution families in addition d solutions display several properties allowing the generation of an infinity of d solutions from a given one which is an uncommon feature in the framework of the jacobi equations furthermore a special family of d solutions complying with the previous requirements is constructively characterized and analyzed examples are discussed focusing on physical implications and including an application for the global construction of the darboux canonical form

the dirichlet problem for nonlocal operators in this note we set up the elliptic and the parabolic dirichlet problem for linear nonlocal operators as opposed to the classical case of second order differential operators here the boundary data are prescribed on the complement of a given bounded set we formulate the problem in the classical framework of hilbert spaces and prove unique solvability using standard techniques like the fredholm alternative

coadjoint orbits vortices and clebsch variables for incompressible fluids this paper is a study of incompressible fluids especially their clebsch variables and vortices using symplectic geometry and the lie poisson structure on the dual of a lie algebra following ideas of arnold and others it is shown that euler s equations are lie poisson equations associated to the group of volume preserving diffeomorphisms the dual of the lie algebra is seen to be the space of vortices and kelvin s circulation theorem is interpreted as preservation of coadjoint orbits in this context clebsch variables can be understood as momentum maps the motion of n point vortices is shown to be identifiable with the dynamics on a special coadjoint orbit and the standard canonical variables for them are a special kind of clebsch variables point vortices with cores vortex patches and vortex filaments can be understood in a similar way this leads to an explanation of the geometry behind the hald beale majda convergence theorems for vorticity algorithms symplectic structures on the coadjoint orbits of a vortex patch and filament are computed and shown to be closely related to those commonly used for the kdv and the schrodinger equations respectively

a paradifferential approach for well posedness of the muskat problem we study the muskat problem for one fluid or two fluids with or without viscosity jump with or without rigid boundaries and in arbitrary space dimension d of the interface the muskat problem is scaling invariant in the sobolev space h s c mathbb r d where s c 1 frac d 2 employing a paradifferential approach we prove local well posedness for large data in any subcritical sobolev spaces h s mathbb r d s s c moreover the rigid boundaries are only required to be lipschitz and can have arbitrarily large variation the rayleigh taylor stability condition is assumed for the case of two fluids with viscosity jump but is proved to be automatically satisfied for the case of one fluid the starting point of this work is a reformulation solely in terms of the drichlet neumann operator the key elements of proofs are new paralinearization and contraction results for the drichlet neumann operator in rough domains

a geometric treatment of reduction of order of ordinary difference equations we generalize the theory of lie symmetries of ordinary difference equations to the nonautonomous case a coordinate invariant treatment in which solutions are sections of a fibre bundle is employed it is shown that lie symmetries of difference equations must be projectable which is not the case for differential equations in fact this result extends to partial difference equations we also show that a time like symmetry can be used to reduce a nonautonomous difference equation to the autonomous case examples are given of this process and of the reduction of order of a nonautonomous system

higher dimensional integrable mappings we construct higher dimensional analogues of two dimensional integrable mappings preserving one biquadratic integral we also provide their non autonomous forms

discrete versions of some classical integrable systems and factorization of matrix polynomials discrete versions of several classical integrable systems are investigated such as a discrete analogue of the higher dimensional force free spinning top euler arnold equations the heisenberg chain with classical spins and a new discrete system on the stiefel manifold the integrability is shown with the help of a lax pair representation which is found via a factorization of certain matrix polynomials the complete description of the dynamics is given in terms of abelian functions the flow becomes linear on a prym variety corresponding to a spectral curve the approach is also applied to the billiard problem in the interior of ann dimensional ellipsoid

explicit auto transformations of integrable chains a construction scheme for explicit auto transformations of integrable discrete differential equations chains is presented these transformations are rather convenient to obtain the exact solutions for chains as well as associated partial differential systems on the other hand they exemplify new integrable discrete mappings their group properties are also of great interest the scheme is illustrated by several examples of integrable systems which contains the nonlinear schrodinger system and the landau lifshits model

integrable two component systems of difference equations we present two lists of two component systems of integrable difference equations defined on the edges of the z2 graph the integrability of these systems is manifested by their lax formulation whic

a procedure for ranking efficient units in data envelopment analysis data envelopment analysis dea evaluates the relative efficiency of decision making units dmus but does not allow for a ranking of the efficient units themselves a modified version of dea based upon comparison of efficient dmus relative to a reference technology spanned by all other units is developed the procedure provides a framework for ranking efficient units and facilitates comparison with rankings based on parametric methods

multidimensional maps of qrt type we consider a class of multidimensional maps which naturally generalize the qrt map of the plane our 2n dimensional maps are volume preserving and have n rational invariants but we do not generally have a symplectic form however many specializations and reductions are integrable some of which we present included in these are some new four dimensional generalizations of the mcmillan map

darboux transformations finite reduction groups and related yang baxter maps in this paper we construct yang baxter yb maps using darboux matrices which are invariant under the action of finite reduction groups we present six dimensional yb maps corresponding to darboux transformations for the nonlinear schrodinger nls equation and the derivative nonlinear schrodinger dnls equation these yb maps can be restricted to four dimensional yb maps on invariant leaves the former are completely integrable and they also have applications to a recent theory of maps preserving functions with symmetries fordy a and kassotakis a 2013 j phys a math theor 46 205201 we give a six dimensional yb map corresponding to the darboux transformation for a deformation of the dnls equation we also consider vector generalizations of the yb maps corresponding to the nls and dnls equations

integrable mappings derived from soliton equations abstract we derive a hierarchy of ibtegrable mappings integrable ordinary difference equations corresponding to solutions of the initial value problem of an integrable partial difference equation with periodic initial data for each n n this hierarchy contains at least one integrable mapping r n r n the integrals of these mappings are constructed using the lax pair of the underlying partial difference equation our approach is illustrated for the integrable partial difference analogues of the sine gordon and the modified korteweg de vries equations

trace maps as 3d reversible dynamical systems with an invariant one link between the theory of quasicrystals and the theory of nonlinear dynamics is provided by the study of so called trace maps a subclass of them are mappings on a one parameter family of 2d surfaces that foliate 3 and also 3 they are derived from transfer matrix approaches to properties of 1d quasicrystals in this article we consider various dynamical properties of trace maps we first discuss the fibonacci trace map and give new results concerning boundedness of orbits on certain subfamilies of its invariant 2d surfaces we highlight a particular surface where the motion is integrable and semiconjugate to an anosov system i e the mapping acts as a pseudo anosov map we identify properties of symmetry and reversibility time reversal symmetry in the fibonacci trace map dynamics and discuss the consequences for the structure of periodic orbits we show that a conservative period boubling sequence can be identified when moving through the one parameter family of 2d surfaces by using generator trace maps in terms of which all trace maps obtained from invertible two letter substitution rules can be expressed we show that many features of the fibonacci trace map hold in general the role of the fricke character hat i x y z x 2 y 2 z 2 2xyz 1 its symmetry group and reversibility for the nielsen trace maps are described algebraically finally we outline possible higher dimensional generalizations

direct method to construct integrals for nth order autonomous ordinary difference equations a direct method to construct integrals for an nth order autonomous ordinary difference equation oe wnnf wn wnn1 is presented as an illustration we first consider third order autonomous oe wn3

yang baxter maps with first degree polynomial 2 2 lax matrices a family of non parametric yang baxter yb maps is constructed by re factorization of 2 2 matrix polynomials of first degree these maps are poisson with respect to the sklyanin bracket for each casimir function a parametric poisson yb map is generated by reduction on the corresponding level set by considering a complete set of casimir functions symplectic multi parametric yb maps are derived these maps are quadrirational with explicit formulae in terms of matrix operations their lax matrices are by construction 2 2 first degree polynomial in the spectral parameter and are classified by the jordan normal form of the leading term degenerate parametric yb maps constructed as limits of the non degenerate ones are connected to known integrable systems on quadgraphs

a message negotiation approach to e services by utility function and multi criteria decision analysis in modern society information and internet techniques have changed human lifestyles service providers provide innovative e services to attract and retain their customers the popularity of e services has made selecting a good e service which will also provide reasonable composite e service solutions an important issue therefore this work proposes a message negotiation approach to e services to assist consumers in acquiring a reasonable composite e service solution an e service composition involves a complete e service process including a series of steps e service formalization an e service utility model and a multi criteria decision analysis are used to determine the optimal selection order of e services for each specific e service composition step then the selected e service for each step is composed through a message negotiation process the result is presented to the consumer as a reasonable composite e service solution this work uses an ingredient procurement use case of property management in taiwan to demonstrate that the proposed approach is effective

new four dimensional solutions of the jacobi equations for poisson structures a four dimensional family of skew symmetric solutions of the jacobi equations for poisson structures is characterized as a consequence previously known types of poisson structures found in a diversity of physical situations appear to be obtainable as particular cases of new family of solutions additionally it is possible to apply constructive methods for the explicit determination of fundamental properties of those solutions such as their casimir invariants symplectic structure and the algorithm for the reduction to the darboux canonical form which have been reported only for a limited sample of known finite dimensional poisson structures moreover the results developed are valid globally in phase space thus ameliorating the usual scope of darboux theorem which is of local nature

traces of weighted sobolev spaces old and new abstract we give short simple proofs of uspenski s results characterizing besov spaces as trace spaces of weighted sobolev spaces we generalize uspenski s results and prove the optimality of these generalizations we next show how classical results on the functional calculus in the besov spaces can be obtained as straightforward consequences of the theory of weighted sobolev spaces

the construction of a poisson structure out of a symmetry and a conservation law of a dynamical system a method to construct hamiltonian theories for systems of both ordinary and partial differential equations is presented the knowledge of a lagrangian is not at all necessary to achieve the result the only ingredients required for the construction are one solution of the symmetry perturbation equation and one constant of motion of the original system it turns out that the poisson bracket structure for the dynamical variables is far from becoming uniquely determined by the differential equations of motion examples in classical mechanics as well as in field theory are presented

simulation of fuzzy multiattribute models for grey relationships multiattribute decision making involves tradeoffs among alternative performances over multiple attributes the accuracy of performance measures are usually assumed to be accurate most multiattribute models also assume given values for the relative importance of weights for attributes however there is usually some uncertainty involved in both of these model inputs outranking multiattribute methods have always provided fuzzy input for performance scores many analysts have also recognized that weight estimates also involve some imprecision either through individual decision maker uncertainty or through aggregation of diverging group member preferences many fuzzy multiattribute models have been proposed but they have focused on identifying the expected value solution or extreme solutions this paper demonstrates how simulation can be used to reflect fuzzy inputs which allows more complete probabilistic interpretation of model results

global dynamics of the rossler system with conserved quantities the subject of this paper concerns with a class of rossler systems that admits conserved quantities for this class of systems a complete description of the global dynamics in the poincare sphere is provided

k integrals and k lie symmetries in discrete dynamical systems abstract we generalize the concept of symplectic maps to that of k symplectic maps maps whose kth iterates are symplectic similarly k symmetries and k integrals are symmetries resp integrals of the kth iterate of the map it is shown that k symmetries and k integrals are related by the k symplectic structure as in the k 1 continuous case noether s theorem examples are given of k integrals and their related k symmetries for k 1 4

integrable mappings via rational elliptic surfaces we present a geometric description of the qrt map which is an integrable mapping introduced by quispel roberts and thompson in terms of the addition formula of a rational elliptic surface by this formulation we classify all the cases when the qrt map is periodic and show that its period is 2 3 4 5 or 6 a generalization of the qrt map which acts birationally on a pencil of k3 surfaces or calabi yau manifolds is also presented

on formulae decoupling the total variation of bv functions abstract in this paper we prove several formulae that enable one to capture the singular portion of the measure derivative of a function of bounded variation as a limit of non local functionals one special case shows that rescalings of the fractional laplacian of a function u s b v converge strictly to the singular portion of d u

mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions in this paper we study the linear bond based nonlocal peridynamic models with a particular focus on problems associated with nonstandard nonlocal displacement loading conditions both stationary and time dependent problems are considered for a one dimensional scalar equation defined on a finite bar and for a two dimensional system defined on a square the related peridynamic operators and associated functional spaces are analyzed applications to the numerical analysis of the finite dimensional approximations to peridynamic models are also illustrated

nonlinear differential difference equations as backlund transformations shows that the best known nonlinear differential difference equations associated with the discrete schrodinger spectral problem and also with the discrete zakharov shabat spectral problem can be interpreted as backlund transformations for some continuous nonlinear evolution equations

on well posedness of the muskat problem with surface tension abstract we consider the muskat problem with surface tension for one fluid or two fluids with or without viscosity jump with infinite depth or lipschitz rigid boundaries and in arbitrary dimension d of the interface the problem is nonlocal quasilinear and to leading order is scaling invariant in the sobolev space h s c r d with s c 1 d 2 we prove local well posedness for large data in all subcritical sobolev spaces h s r d s s c allowing for initial interfaces whose curvatures are unbounded and furthermore when d 1 not locally square integrable to the best of our knowledge this is the first large data well posedness result that covers all subcritical sobolev spaces for the muskat problem with surface tension we reformulate the problem in terms of the dirichlet neumann operator and use a paradifferential approach to reduce the problem to an explicit parabolic equation which is of independent interest

dual models of interval dea and its extension to interval data abstract the purpose of this paper is to develop a new dea with an interval efficiency an original dea model is to evaluate each dmu optimistically there is another model called inverted dea to evaluate each dmu pessimistically but there are no relations essentially between dea and inverted dea thus we formulate a dea model with an interval efficiency which consists of efficiencies obtained from the optimistic and pessimistic viewpoints thus two end points can construct an interval efficiency with the same idea we deal with the interval inefficiency model which is inverse to interval efficiency finally we extend the proposed dea model to interval data and fuzzy data

analysis and approximation of nonlocal diffusion problems with volume constraints a recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on bounded domains in mbrn the nonlocal vector calculus also enables striking analogies to be drawn between the nonlocal model and classical models for diffusion including a notion of nonlocal flux the ubiquity of the nonlocal operator in applications is illustrated by a number of examples ranging from continuum mechanics to graph theory in particular it is shown that fractional laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion that we consider the numerous applications elucidate different interpretations of the operator and the associated governing equations for example a probabilistic perspective explains that the nonlocal spatial operator appearing in our model corresponds to the infinitesimal generator for a symmetric jump process sufficie

new solutions of the jacobi equations for three dimensional poisson structures a systematic investigation of the skew symmetric solutions of the three dimensional jacobi equations is presented as a result three disjoint and complementary new families of solutions are characterized such families are very general thus unifying many different and well known poisson structures seemingly unrelated which now appear embraced as particular cases of a more general solution this unification is not only conceptual but allows the development of algorithms for the explicit determination of important properties such as the symplectic structure the casimir invariants and the darboux canonical form which are known only for a limited sample of poisson structures these common procedures are thus simultaneously valid for all the particular cases which can now be analyzed in a unified and more economic framework instead of using a case by case approach in addition the methods developed are valid globally in phase space thus ameliorating the usual scope of darboux reduction which is only of

a multiple criteria approach to data envelopment analysis in this paper we present a multiple criteria data envelopment analysis mcdea model which can be used to improve discriminating power of dea methods and also effectively yield more reasonable input and output weights without a priori information about the weights in the proposed model several different efficiency measures including classical dea efficiency are defined under the same constraints each measure serves as a criterion to be optimized efficiencies are then evaluated under the framework of multiple objective linear programming molp the method is illustrated through three examples in which data sets are taken from previous research on dea s discriminating power and weight restriction

sufficient conditions for dynamical systems to have pre symplectic or pre implectic structures abstract we present a number of alternative sufficient conditions for the existence of pre symplectic or pre implectic poisson structures for both ordinary differential ode and ordinary difference o e equations four alternative sets of conditions are obtained for odes and o es in n dependent variables 1 a vector field in involution with the ode and an integral or two symmetries for an o e imply a pre implectic structure 2 volume preservation and n 2 symmetries imply a pre symplectic structure 3 volume preservation and n 2 integrals imply a pre implectic structure 4 complex implectic structure implies infinitely many real implectic structures in all but the first case the methods can give a set of distinct structures

entwining yang baxter maps related to nls type equations we construct birational maps that satisfy the parametric set theoretical yang baxter equation and its entwining generalisation for this purpose we employ darboux transformations related to integrable nonlinear schrodinger type equations and study the refactorisation problems of the product of their associated darboux matrices additionally we study various algebraic properties of the derived maps such as invariants and associated symplectic or poisson structures and we prove their complete integrability in the liouville sense

integrable maps which preserve functions with symmetries we consider maps which preserve functions which are built out of the invariants of some simple vector fields we give a reduction procedure which can be used to derive commuting maps of the plane which preserve the same symplectic form and first integral we show how our method can be applied to some maps which have recently appeared in the context of yang baxter maps