a smooth transition model between kinetic and diffusion equations this paper presents a model which provides a smooth transition between a kinetic and a diffusion domain the idea is to use a buffer zone in which both diffusion and kinetic equations will be solved the solution of the original kinetic equation will be recovered as the sum of the solutions of these two equations we use an artificial connecting function which makes the equation on each domain degenerate at the end of the buffer zone thus no boundary condition is needed at the transition point this model avoids the delicate issue of finding the interface condition or iteration in a typical domain decomposition method that couples a kinetic equation with hydrodynamic equations a new asymptotic preserving method for this model is introduced and numerical examples are used to validate this new model and the new numerical method

an improved algorithm for the multidimensional moment constrained maximum entropy problem the maximum entropy principle is a versatile tool for evaluating smooth approximations of probability density functions with the least bias beyond specified constraints in the recent paper we introduced new computational framework for the moment constrained maximum entropy problem in a multidimensional domain and developed a simple numerical algorithm capable of computing maximum entropy problem in a two dimensional domain with moment constraints of order up to 4 here we design an improved numerical algorithm for computing the maximum entropy problem in a two and higher dimensional domain with higher order moment constraints the algorithm features multidimensional orthogonal polynomial basis in the dual space of lagrange multipliers to achieve numerical stability and rapid convergence of newton iterations the new algorithm is found to be capable of solving the maximum entropy problem in the two dimensional domain with moment constraints of order up to 8 in the three dimensional domain with moment constraints of order up to 6 and in the four dimensional domain with moment constraints of order up to 4 corresponding to the total number of moment constraints of 44 83 and 69 respectively the two and higher dimensional maximum entropy test problems in the current work are based upon long term statistics of numerical simulation of the real world geophysical model for wind stress driven oceanic currents such as the gulf stream and the kuroshio

hausdorff moment problem and maximum entropy a unified approach hausdorff moment problem in the approach of maximum entropy is reconsidered some aspects as existence condition entropy convergence entropy decreasing stability are studied through a unified approach results known in literature and some new ones are illustrated

a domain decomposition method for semilinear hyperbolic systems with two scale relaxations we present a domain decomposition method on a semilinear hyperbolic system with multiple relaxation times in the region where the relaxation time is small an asymptotic equilibrium equation can be used for computational efficiency an interface condition is provided to couple the two systems in a domain decomposition setting a rigorous analysis based on the laplace transform on the l2 error estimate is presented for the linear case which shows how the error of the domain decomposition method depends on the smaller relaxation time and the boundary and interface layer effects the given convergence rate is optimal we present a numerical implementation of this domain decomposition method and present some numerical results in order to study the performance of this method

integrable representations for generalized virasoro toroidal lie algebras abstract in this paper we give the structure of irreducible integrable modules of generalized virasoro toroidal lie algebras with finite dimensional weight spaces and non zero central charge

principal vertex operator representations for toroidal lie algebras we introduce the principal vertex operator representations for the toroidal lie algebras generalizing the construction for the affine kac moody algebras we also represent the derivations of the toroidal algebras and introduce analogs of the sugawara operators

a new family of algebras underlying the rogers ramanujan identities and generalizations abstract the classical rogers ramanujan identities have been interpreted by lepowsky milne and the present authors in terms of the representation theory of the euclidean kac moody lie algebra a1 1 also the present authors have introduced certain vertex differential operators providing a construction of a1 1 on its basic module and kac kazhdan and we have generalized this construction to a general class of euclidean lie algebras starting from this viewpoint we now introduce certain new algebras unk v which centralize the action of the principal heisenberg subalgebra of an arbitrary euclidean lie algebra unk on a highest weight unk module v we state a general tautological rogers ramanujan type identity which by our earlier theorem includes the classical identities and we show that unk v can be used to reformulate the general identity for unk a1 1 we develop the representation theory of unk v in considerable detail allowing us to prove our earlier conjecture that our general rogers ramanujan type identity includes certain identities of gordon andrews and bressoud in the process we construct explicit bases of all of the standard and verma modules of nonzero level for a1 1 with an explicit realization of a1 1 as operators in each case the differential operator constructions mentioned above correspond to the trivial case unk v 1 of the present theory

a kinetic equation with kinetic entropy functions for scalar conservation laws we construct a nonlinear kinetic equation and prove that it is welladapted to describe general multidimensional scalar conservation laws in particular we prove that it is well posed uniformly in e the microscopic scale we also show that the proposed kinetic equation is equipped with a family of kinetic entropy functions analogous to boltzmann s microscopich function such that they recover krushkov type entropy inequality on the macroscopic scale finally we prove by both bv compactness arguments in the multidimensional case and by compensated compactness arguments in the one dimensional case that the local density of kinetic particles admits a continuum limit as it converges strongly with e 0 to the unique entropy solution of the corresponding conservation law

irreducible representations for toroidal lie algebras abstract in this work a large number of irreducible representations with finite dimensional weight spaces are constructed for some toroidal lie algebras to accomplish this we develop a general theory of z n graded lie algebras with polynomial multiplication we construct modules by the standard inducing procedure and study their irreducible quotients using the vertex operator technics

a fast newton algorithm for entropy maximization in phase determination a long standing issue in the bayesian statistical approach to the phase problem in x ray crystallography is to solve an entropy maximization subproblem efficiently in every iteration of phase estimation the entropy maximization problem is a semi infinite convex program and can be solved in a finite dual space by using a standard newton method however the newton method is too expensive for this application since it requires o n3 floating point operations per iteration where n corresponds to the number of phases to be estimated other less expensive methods have been used but they cannot guarantee fast convergence in this paper we present a fast newton algorithm for the entropy maximization problem which uses the sherman morrison woodbury formula and the fast fourier transform to compute the newton step and requires only o n log n floating point operations per iteration yet has the same convergence rate as the standard newton we describe the algorithm and discuss related computational issues numerical results on simple test cases will also be presented to demonstrate the behavior of the algorithm

the multidimensional moment constrained maximum entropy problem a bfgs algorithm with constraint scaling in a recent paper we developed a new algorithm for the moment constrained maximum entropy problem in a multidimensional setting using a multidimensional orthogonal polynomial basis in the dual space of lagrange multipliers to achieve numerical stability and rapid convergence of the newton iterations here we introduce two new improvements for the existing algorithm adding significant computational speedup in situations with many moment constraints where the original algorithm is known to converge slowly the first improvement is the use of the bfgs iterations to progress between successive polynomial reorthogonalizations rather than single newton steps typically reducing the total number of computationally expensive polynomial reorthogonalizations for the same maximum entropy problem the second improvement is a constraint rescaling aimed to reduce relative difference in the order of magnitude between different moment constraints improving numerical stability of iterations due to reduced sensitivity of different constraints to changes in lagrange multipliers we observe that these two improvements can yield an average wall clock time speedup of 5 6 times compared to the original algorithm

a rigorous derivation of the coupling of a kinetic equation and burgers equation in this article we investigate the kinetic fluid coupling on a toy model which we obtain rigorously from a hydrodynamical limit the idea is that at the level of the full kinetic model the coupling is obvious we then investigate the coupling obtained when passing into the limit we show that especially in presence of a shock stuck on the interface the coupling involves a kinetic layer known as the milne problem due to this layer the limit process is quite delicate and some blow up techniques are needed to ensure its strong convergence

a practical computational framework for the multidimensional moment constrained maximum entropy principle the maximum entropy principle is a versatile tool for evaluating smooth approximations of probability density functions with a least bias beyond given constraints in particular the moment based constraints are often a common prior information about a statistical state in various areas of science including that of a forecast ensemble or a climate in atmospheric science with that in mind here we present a unified computational framework for an arbitrary number of phase space dimensions and moment constraints for both shannon and relative entropies together with a practical usable convex optimization algorithm based on the newton method with an additional preconditioning and robust numerical integration routine this optimization algorithm has already been used in three studies of predictability and so far was found to be capable of producing reliable results in one and two dimensional phase spaces with moment constraints of up to order 4 the current work extensively references those earlier studies as practical examples of the applicability of the algorithm developed below

hyperbolic limit of the jin xin relaxation model we consider the special jin xin relaxation model 0 1 u i a u u x u xx u tt we assume that the initial data u 0 u 0 t are sufficiently smooth and close to u 0 in l and have small total variation then we prove that there exists a solution u t u t t with uniformly small total variation for all t 0 and this solution depends lipschitz continuously in the l 1 norm with respect to time and the initial data letting 0 the solution u converges to a unique limit providing a relaxation limit solution to the quasi linear nonconservative system 0 2 u t a u u x 0 these limit solutions generate a lipschitz semigroup s on a domain d containing the functions with small total variation and close to u this is precisely the riemann semigroup determined by the unique riemann solver compatible with 0 1

hyperbolic conservation laws with stiff relaxation terms and entropy we study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms reduced systems inviscid and viscous local conservation laws and weakly nonlinear limits are derived through asymptotic expansions an entropy condition is introduced for n n systems that ensures the hyperbolicity of the reduced inviscid system the resulting characteristic speeds are shown to be interlaced with those of the original system moreover the first correction to the reduced system is shown to be dissipative a partial converse is proved for 2 2 systems this structure is then applied to study the convergence to the reduced dynamics for the 2 2 case 1994 john wiley sons inc

maximum entropy algorithm with inexact upper entropy bound based on fup basis functions with compact support the maximum entropy maxent principle is a versatile tool for statistical inference of the probability density function pdf from its moments as a least biased estimation among all other possible pdf s it maximizes shannon entropy satisfying the moment constraints thus the maxent algorithm transforms the original constrained optimization problem to the unconstrained dual optimization problem using lagrangian multipliers the classic moment problem cmp uses algebraic power moments causing typical conventional numerical methods to fail for higher order moments m 5 10 due to different sensitivities of lagrangian multipliers and unbalanced nonlinearities classic maxent algorithms overcome these difficulties by using orthogonal polynomials which enable roughly the same sensitivity for all lagrangian multipliers in this paper we employ an idea based on different principles using fup n basis functions with compact support which can exactly describe algebraic polynomials but only if the fup order n is greater than or equal to the polynomial s order our algorithm solves the cmp with respect to the moments of only low order fup 2 basis functions finding a fup 2 optimal pdf with better balanced lagrangian multipliers the algorithm is numerically very efficient due to localized properties of fup 2 basis functions implying a weaker dependence between lagrangian multipliers and faster convergence only consequences are an iterative scheme of the algorithm where power moments are a sum of fup 2 and residual moments and an inexact entropy upper bound however due to small residual moments the algorithm converges very quickly as demonstrated on two continuous pdf examples the beta distribution and a bi modal pdf and two discontinuous pdf examples the step and double dirac pdf finally these pdf examples present that fup maxent algorithm yields smaller entropy value than classic maxent algorithm but differences are very small for all practical engineering purposes

central extensions of current groups in this paper we study central extensions of the identity component g of the lie group c m k of smooth maps from a compact manifold m into a lie group k which might be infinite dimensional we restrict our attention to lie algebra cocycles of the form d where y is a symmetric invariant bilinear map on the lie algebra of k and the values of lie in 1 m y dc m y for such cocycles we show that a corresponding central lie group extension exists if and only if this is the case for m 1 if k is finite dimensional semisimple this implies the existence of a universal central lie group extension of g the groups diff m and c m k act naturally on g by automorphisms we also show that these smooth actions can be lifted to smooth actions on the central extension if it also is a central extension of the universal covering group g of g

magnetic hydrodynamics with asymmetric stress tensor in this paper we study equations of magnetic hydrodynamics with a stress tensor we interpret this system as the generalized euler equation associated with an abelian extension of the lie algebra of vector fields with a nontrivial 2 cocycle we use the lie algebra approach to prove the energy conservation law and the conservation of cross helicity

tkk algebras and vertex operator representations let be the jordan algebra constructed from a semilattice s of v 1 see allison et al mem amer math soc 603 1997 let be the lie algebra obtained from the jordan algebra by the tits kantor koecher construction the tkk algebra is defined to be the universal central extension of the lie algebra in this paper we present a complete description of the tkk algebra where s is the smallest possible nonlattice semilattice which then allows us to give a representation to this lie algebra by vertex operators

integrable representations for toroidal extended affine lie algebras abstract let g be any untwisted affine kac moody algebra any fixed complex number and g the corresponding toroidal extended affine lie algebra of nullity two for any k tuple 1 k of weights of g and k tuple a a 1 a k of distinct non zero complex numbers we construct a class of modules v a for the extended affine lie algebra g we prove that the g module v a is completely reducible we also prove that the g module v a is integrable when all weights i in are dominant thus we obtain a new class of irreducible integrable weight modules for the toroidal extended affine lie algebra g

well posedness and singular limit of a semilinear hyperbolic relaxation system with a two scale discontinuous relaxation rate nonlinear hyperbolic systems with relaxations may encounter different scales of relaxation time which is a prototype multiscale phenomenon that arises in many applications in such a problem the relaxation time is of o 1 in part of the domain and very small in the remaining domain in which the solution can be approximated by the zero relaxation limit which can be solved numerically much more efficiently for the jin xin relaxation system in such a two scale setting we establish its wellposedness and singular limit as the smaller relaxation time goes to zero the limit is a multiscale coupling problem which couples the original jin xin system on the domain when the relaxation time is o 1 with its relaxation limit in the other domain through interface conditions which can be derived by matched interface layer analysis as a result we also establish the well posedness and regularity such as boundedness in sup norm with bounded total variation and l1 contraction of the coupling problem thus providing a rigorous mathematical foundation in the general nonlinear setting to the multiscale domain decomposition method for this two scale problem originally proposed in jin et al in math comp 82 749 779 2013

an extension of the korteweg de vries hierarchy arising from a representation of a toroidal lie algebra abstract in this article we show how to construct hierarchies of partial differential equations from the vertex operator representations of toroidal lie algebras in the smallest example rank 2 toroidal cover of sl2 we obtain an extension of the kdv hierarchy we use the action of the corresponding infinite dimensional group to construct solutions for these non linear pdes

elliptic lie algebras and tubular algebras abstract this article is to study relations between tubular algebras of ringel and elliptic lie algebras in the sense of saito yoshii using the explicit structure of the derived categories of tubular algebras given by happel ringel we prove that the elliptic lie algebra of type d 4 1 1 e 6 1 1 e 7 1 1 or e 8 1 1 is isomorphic to the ringel hall lie algebra of the root category of the tubular algebra with the same type as a by product of our proof we obtain a chevalley basis of the elliptic lie algebra following indecomposable objects of the root category of the corresponding tubular algebra this can be viewed as an analogue of the frenkel malkin vybornov theorem in which they described a chevalley basis for each untwisted affine kac moody lie algebra by using indecomposable representations of the corresponding affine quiver

extended affine weyl groups abstract in this paper we study the weyl groups of reduced extended affine root systems the root systems of extended affine lie algebras we start by describing the extended affine weyl group as a semidirect product of a finite weyl group and a heisenberg like normal subgroup this provides a unique expression for the weyl group elements in terms of some naturally arisen transformations which is crucial in the further study of extended affine weyl groups we use this to give a presentation called a presentation by conjugation for an important subclass of extended affine weyl groups using a new notion called the index which is an invariant of the extended affine root systems we show that one of the important features of finite and affine root systems related to weyl group holds for the class of extended affine root systems

representations of the lie algebra of derivations for quantum torus abstract let der c q be the derivation lie algebra of the quantum torus c q we construct a functor from gld modules to der c q modules which generalizes the constructions obtained by shen scientia sinica a 29 1986 570 larsson internat j modern phys a 7 1992 6493 and rao j algebra 182 1996 401 we also give a complete description of the structure of the der c q modules when all the entries of the quantum torus matrix q qij d d are roots of unity

locally extended affine lie algebras abstract we propose a new simplified definition of extended affine lie algebras ealas for short and also discuss a general version of extended affine lie algebras called locally extended affine lie algebras local ealas for short we prove a conjecture by v kac for local ealas it turns out that the root system of a local eala becomes a locally finite version of an extended affine root system several examples of new ealas and local ealas are introduced and finally we classify local ealas of nullity 0 and show the connection to locally finite split simple lie algebras

structure and representation for a class of infinite dimensional lie algebras abstract let a be a commutative associative algebra over the complex field c and g be the complexification of the real lie algebra so 3 for any fixed elements e 1 e 2 e 3 a we define a lie algebra l e 1 e 2 e 3 g a with lie bracket given by 1 2 when the associative algebra a is the laurent polynomial algebra c t 1 1 t 1 we determine its derivation lie algebra der l and universal central extension l we also give a vertex operator representation for the lie algebra l this new class of lie algebras includes the affine lie algebra a 1 1 and the toroidal lie algebras of type a 1 we note that in general this kind of lie algebras is not z graded

maximum entropy in the problem of moments the maximum entropy approach to the solution of underdetermined inverse problems is studied in detail in the context of the classical moment problem in important special cases such as the hausdorff moment problem we establish necessary and sufficient conditions for the existence of a maximum entropy solution and examine the convergence of the resulting sequence of approximations a number of explicit illustrations are presented in addition to some elementary examples we analyze the maximum entropy reconstruction of the density of states in harmonic solids and of dynamic correlation functions in quantum spin systems we also briefly indicate possible applications to the lee yang theory of ising models to the summation of divergent series and so on the general conclusion is that maximum entropy provides a valuable approximation scheme a serious competitor of traditional pade like procedures

bosonic realization of toroidal lie algebras of classical types we use fermionic operators to construct toroidal lie algebras of classical types including in particular symplectic affine algebras which are first realized by fermions

highest weight modules over pre exp polynomial lie algebras abstract in this paper we introduce pre exp polynomial lie algebras which include loop algebras virasoro like algebras and some quantum torus lie algebras we study highest weight representations of these z n 1 graded lie algebras more precisely we show that non graded and graded irreducible highest weight modules with the same highest weight simultaneously have all finite dimensional weight spaces or not and they have all finite dimensional weight spaces if and only if the highest weight is an exp polynomial highest weight we also show that non graded and graded highest weight verma modules with the same highest weight are simultaneously irreducible or not and we determine necessary and sufficient conditions for a verma module to be irreducible

calculation of maximum entropy densities with application to income distribution the maximum entropy approach is a flexible and powerful tool for density approximation this paper proposes a sequential updating method to calculate the maximum entropy density subject to known moment constraints instead of imposing the moment constraints simultaneously the sequential updating method incorporates the moment constraints into the calculation from lower to higher moments and updates the density estimates sequentially the proposed method is employed to approximate the size distribution of u s family income empirical evidence demonstrates the efficiency of this method

irreducible representations of virasoro toroidal lie algebras toroidal lie algebras and their vertex operator representations were introduced in mey and a class of indecomposable modules were investigated in this work we extend the toroidal algebra by the virasoro algebra thus constructing a semi direct product algebra containing the toroidal algebra as an ideal and the virasoro algebra as a subalgebra with the use of vertex operators and certain oscillator representations of the virasoro algebra it is proved that the corresponding fock space gives rise to a class of irreducible modules for the virasoro toroidal algebra

derivations and central extensions of finitely generated graded lie algebras abstract some general results concerning derivations of finitely generated lie algebras are established these are employed in order to determine the derivations and central extensions of kac moody lie algebras

extended diffeomorphism algebras and trajectories in jet space let the dro diffeomorphism reparametrization observer algebra dro n be the extension of diff n diff 1 by its four inequivalent virasoro like cocycles here diff n is the diffeomorphism algebra in n dimensional spacetime and diff 1 describes reparametrizations of trajectories in the space of tensor valued p jets dro n has a fock module for each p and each representation of gl n analogous representations for gauge algebras higher dimensional kac moody algebras are also given the reparametrization symmetry can be eliminated by a gauge fixing procedure resulting in previously discovered modules in this process two dro n cocycles transmute into anisotropic cocycles for diff n thus the fock modules of toroidal lie algebras and their derivation algebras are geometrically explained

vertex representations for n toroidal lie algebras and a generalization of the virasoro algebra vertex representations are obtained for toroidal lie algebras for any number of variables these representations afford representations of certainn variable generalizations of the virasoro algebra that are abelian extensions of the lie algebra of vector fields on a torus

integrable representations for the twisted full toroidal lie algebras the purpose of this work is to classify irreducible integrable modules of the twisted full toroidal lie algebra with finite dimensional weight spaces and non zero central charges there are three families of such modules two of which are classified

lie algebras graded by finite root systems and the intersection matrix algebras of slodowy this paper is about toroidal lie algebras certain intersection matrix lie algebras defined by slodowy and their relationship to one another and to certain lie algebra analogues of steinberg groups the main result of the paper is the identification of the intersection matrix algebras arising from multiply affinized cartan matrices of types a d and e with certain steinberg lie algebras and toroidal lie algebras propositions 5 9 and 5 10 a major part of the paper studies and classifies lie algebras graded by finite root systems these become the princi pal tool in our analysis of intersection matrix algebras each lie algebra graded by a simply laced finite root system of rank 2 has attached to it an algebra which according to the type and rank is either commutative and associative only associative or alternative all these possibilities occur in our description of inter section matrix algebras let r be any associative algebra with identity not necessarily finite dimen sional over a field k of characteristic 0 for each positive integer n the associative algebra m r of n n matrices with coefficients in r forms a lie algebra over k under the commutator product we denote this lie algebra by ol r let eis be the i j matrix unit of m r and assume that n 2 the subalgebra e r of oi r generated by the elements reis r r i 4 j is an ideal of 91 r and is perfect i e it is its own derived algebra now any perfect lie algebra o has a universal central extension also perfect called a universal covering algebra u c a of o ga so in particular e r has a u c a that we will denote by t r we define 112 r by the exact sequence 0 1 0 f2 r t r e r 0 dedicated to our teacher maria j wonenburger both authors gratefully acknowledge the support of the natural sciences and engineering research council of canada

quantifying predictability through information theory small sample estimation in a non gaussian framework many situations in complex systems require quantitative estimates of the lack of information in one probability distribution relative to another in short term climate and weather prediction examples of these issues might involve the lack of information in the historical climate record compared with an ensemble prediction or the lack of information in a particular gaussian ensemble prediction strategy involving the first and second moments compared with the non gaussian ensemble itself the relative entropy is a natural way to quantify the predictive utility in this information and recently a systematic computationally feasible hierarchical framework has been developed in practical systems with many degrees of freedom computational overhead limits ensemble predictions to relatively small sample sizes here the notion of predictive utility in a relative entropy framework is extended to small random samples by the definition of a sample utility a measure of the unlikeliness that a random sample was produced by a given prediction strategy the sample utility is the minimum predictability with a statistical level of confidence which is implied by the data two practical algorithms for measuring such a sample utility are developed here the first technique is based on the statistical method of null hypothesis testing while the second is based upon a central limit theorem for the relative entropy of moment based probability densities these techniques are tested on known probability densities with parameterized bimodality and skewness and then applied to the lorenz 96 model a recently developed toy climate model with chaotic dynamics mimicking the atmosphere the results show a detection of non gaussian tendencies of prediction densities at small ensemble sizes with between 50 and 100 members with a 95 confidence level

convergence to equilibrium for the relaxation approximations of conservation laws we study the cauchy problem for 2 2 semilinear and quasilinear hyperbolic systems with a singular relaxation term special comparison and compactness properties are established by assuming the subcharacteristic condition therefore we can prove the convergence to equilibrium of the solutions of these problems as the singular perturbation parameter tends to 0 this research was strongly motivated by the recent numerical investigations of s jin and z xin on the relaxation schemes for conservation laws 1996 john wiley sons inc

a robust moment method for evaluation of the disappearance rate of evaporating sprays in this paper we tackle a critical issue in the numerical modeling by eulerian moment methods of polydisperse multiphase systems constituted of dispersed particles or droplets a general class of systems which include aerosols their modeling starts at a mesoscopic scale with an equation on the number density function ndf of particles droplets which satisfies a population balance equation pbe also called williams equation in the spray community in order to limit the computational cost moment methods provide a system of conservation equation with an eventual closure problem which can be solved using quadrature methods in order to retrieve the unclosed terms from the considered set of moments however a drift velocity that is the rate of change due to continuous phenomena of the internal coordinate such as the size of the particles has sometimes to be taken into account it can be either positive like molecular growth or negative such as for evaporation of droplets in aerosols or oxidation of soots when negative it leads to the disappearance of droplets particles thus creating a negative flux at zero size its closure requires an evaluation of the reconstructed ndf at zero size from the knowledge of some moments the nature of this information pointwise in internal coordinate and its influence on moment dynamics results in a difficulty from both a modeling and a numerical point of view we obtain in the present contribution a comprehensive solution to this important issue since we introduce some new tools in order to resolve the flux evaluation we also introduce a new eulerian type of description which will combine both the flexibility of eulerian models for which the size phase space is discretized into sections i e size intervals and the efficiency of direct quadrature method of moments dqmom it yields a precise and stable description of moment dynamics with a minimal number of variables which should lead to a low computational cost in multi dimensional configurations