the structure of max plus hyperplanes a max plus hyperplane briefly a hyperplane is the set of all points x x1 xn rmaxn satisfying an equation of the form a1x1 anxn an 1 b1x1 bnxn bn 1 that is max a1 x1 an xn an 1 max b1 x1 bn xn bn 1 with ai bi rmax i 1 n 1 where each side contains at least one term and where ai bi for at least one index i we show that the complements of max plus semispaces at finite points z rn are building blocks for the hyperplanes in rmaxn recall that a semispace at z is a maximal with respect to inclusion max plus convex subset of rmaxn z namely observing that up to a permutation of indices we may write the equation of any hyperplane h in one of the following two forms a1x1 apxp ap 1xp 1 aqxq a1x1 apxp aq 1xq 1 amxm an 1 where 0 p q m n and all ai i 1 m n 1 are finite or a1x1 apxp ap 1xp 1 aqxq an 1 a1x1 apxp aq 1xq 1 amxm an 1 where 0 p q m n and all ai i 1 m are finite and an 1 is either finite or we give a formula that expresses a nondegenerate strictly affine hyperplane i e with m n and an 1 as a union of complements of semispaces at a point z rn called the center of h with the boundary of a union of complements of other semispaces at z using this formula we obtain characterizations of nondegenerate strictly affine hyperplanes with empty interior we give a description of the boundary of a nondegenerate strictly affine hyperplane with the aid of complements of semispaces at its center and we characterize the cases in which the boundary bd h of a nondegenerate strictly affine hyperplane h is also a hyperplane next we give the relations between nondegenerate strictly affine hyperplanes h their centers z and their coefficients ai in the converse direction we show that any union of complements of semispaces at a point z rn with the boundary of any union of complements of some other semispaces at that point z is a nondegenerate strictly affine hyperplane we obtain a formula for the total number of strictly affine hyperplanes we give complete lists of all strictly affine hyperplanes for the cases n 1 and n 2 we show that each linear hyperplane h in rmaxn i e with an 1 can be decomposed as the union of four parts where each part is easy to describe in terms of complements of semispaces some of them in a lower dimensional space

convergence rate for a curse of dimensionality free method for a class of hjb pdes in previous work of the first author and others max plus methods have been explored for solution of first order nonlinear hamilton jacobi bellman partial differential equations hjb pdes and corresponding nonlinear control problems although max plus basis expansion and max plus finite element methods can provide substantial computational speed advantages they still generally suffer from the curse of dimensionality here we consider hjb pdes where the hamiltonian takes the form of a pointwise maximum of linear quadratic forms the approach to solution will be rather general but in order to ground the work we consider only constituent hamiltonians corresponding to long run average cost per unit time optimal control problems for the development we consider a previously obtained numerical method not subject to the curse of dimensionality the method is based on construction of the dual space semigroup corresponding to the hjb pde this dual space semigroup is constructed from the dual space semigroups corresponding to the constituent linear quadratic hamiltonians the dual space semigroup is particularly useful due to its form as a max plus integral operator with kernel obtained from the originating semigroup one considers repeated application of the dual space semigroup to obtain the solution although previous work indicated that the method was not subject to the curse of dimensionality it did not indicate any error bounds or convergence rate here we obtain specific error bounds

a curse of dimensionality free numerical method for solution of certain hjb pdes in previous works of the author and others max plus methods have been explored for the solution of first order nonlinear hamilton jacobi bellman partial differential equations hjb pdes and corresponding nonlinear control problems these methods exploit the max plus linearity of the associated semigroups in particular although the problems are nonlinear the semigroups are linear in the max plus sense these methods have been used successfully to compute solutions although they provide certain computational speed advantages they still generally suffer from the curse of dimensionality here we consider hjb pdes in which the hamiltonian takes the form of a pointwise maximum of linear quadratic forms the approach to the solution will be rather general but in order to ground the work we consider only constituent hamiltonians corresponding to long run average cost per unit time optimal control problems for the development we obtain a numerical method not subject to the curse of dimensionality the method is based on construction of the dual space semigroup corresponding to the hjb pde this dual space semigroup is constructed from the dual space semigroups corresponding to the constituent linear quadratic hamiltonians the dual space semigroup is particularly useful due to its form as a max plus integral operator with a kernel obtained from the originating semigroup one considers repeated application of the dual space semigroup to obtain the solution

weak dual residuations applied to tropical linear equations an extension to an algorithm of r a cuninghame green and k zimmermann for solving equations with residuated functions is presented this extension relies on the concept of weak residuation and in the so called strong property it is shown that a contextualization of this method to tropical linear equations which will be denoted as primal method in contrast with the dual method another algorithm described in literature generates a non decreasing sequence which converges to the smallest solution in a special semimodule it is also shown the connections of this method with previously published works

free sets and free subsemimodules in a semimodule abstract in this paper we investigate the free sets and the free subsemimodules in a semimodule over a commutative semiring s first we discuss some properties of the free sets and give a sufficient condition for a nonempty finite set to be free in a finitely generated free s semimodule and obtain a relation between free set and linear independent set in an s semimodule then we consider the free subsemimodules and prove that the rank of any free subsemimodule of a finitely generated s semimodule m does not exceed that of m also we give some equivalent descriptions for a commutative semiring s to have the property that all nonzero subsemimodules of any finitely generated free s semimodule are free partial results obtained in the paper develop and generalize the corresponding results for modules over rings and linear spaces over fields

on the separation of convex sets in some idempotent semimodules abstract two linear maps are usually needed to separate disjoint convex subsets of an idempotent semimodule in the context of max plus convexity separation can be achieved by a single map if one considers linear maps with values in a linearly ordered semimodule whose construction is given here which is not the max plus semiring r

universal approximation using radial basis function networks there have been several recent studies concerning feedforward networks and the problem of approximating arbitrary functionals of a finite number of real variables some of these studies deal with cases in which the hidden layer nonlinearity is not a sigmoid this was motivated by successful applications of feedforward networks with nonsigmoidal hidden layer units this paper reports on a related study of radial basis function rbf networks and it is proved that rbf networks having one hidden layer are capable of universal approximation here the emphasis is on the case of typical rbf networks and the results show that a certain class of rbf networks with the same smoothing factor in each kernel node is broad enough for universal approximation

weighted digraphs and tropical cones this paper is about the combinatorics of finite point configurations in the tropical projective space or dually of arrangements of finitely many tropical hyperplanes moreover arrangements of finitely many tropical halfspaces can be considered via coarsenings of the resulting polyhedral decompositions of rd this leads to natural cell decompositions of the tropical projective space tpmind 1 our method is to employ a known class of ordinary convex polyhedra naturally associated with weighted digraphs this way we can relate to and use results from combinatorics and optimization one outcome is the solution of a conjecture of develin and yu 2007

bases in semimodules over commutative semirings abstract in this paper the bases of a semimodule over a commutative semiring r are investigated some properties and characterizations of the bases are discussed and some equivalent conditions for a basis to be a free basis in a finitely generated free semimodule over r are given the different possible cardinalities for a basis in a finitely generated free semimodule over r are considered and some equivalent descriptions are obtained for a commutative semiring r satisfying the property that any two bases for a finitely generated free r semimodule have the same cardinality partial results obtained in the paper develop and generalize the corresponding results for commutative join semirings and for commutative zerosumfree semirings

on the regularity of weak solutions of a one dimensional two phase degenerate stefan problem with convection we shall be concerned with the following one dimensional two phase stefan problem let t 0 and q 0 1 x 0 t find a pair of functions u x t s t which are defined on qt and 0 t respectively and which satisfy al u uxx f u u in x t o x s l o t t l la az ub4 u xx in x t s t x l o tdt l lb u s t r 0 o t t l lc i t u s t t u s t t o t t lid u 0 t 1 06t t l le u 1 t 1 obt t l lf u x 0 u x o x l 1 k s 0 so e 0 1 l lh here so is a given number and the ar a f and u are known functions of their arguments problem 1 1 may be viewed as a model of heat conduction in a region involving phase change an example situation is an ice water mix in which circulation of the water is present then the term f u u represents the contribution of the field velocity generated by the water flow in the liquid phase a more detailed discussion on the formulation of 1 1 can be found elsewhere see e g s

inclines and incline matrices a survey inclines are additively idempotent semirings in which products are less than or equal to either factor thus they generalize boolean and fuzzy algebra we outline results obtained in a book with cao cao kim roush incline algebra and applications john wiley sons new york 1984 on algebraic structure of inclines themselves matrices over inclines topological and convergence results and some applications this survey paper represents our talk at the 10th annual ilas conference at auburn university june 2002

tropical implementation of the analytical hierarchy process decision method abstract we apply methods and techniques of tropical optimization to develop a new theoretical and computational framework for the implementation of the analytic hierarchy process in multi criteria problems of rating alternatives from pairwise comparison data in this framework we first consider the minimax chebyshev approximation of pairwise comparison matrices by consistent matrices in the logarithmic scale recasting this approximation problem as a problem of tropical pseudo quadratic programming we then write out a closed form solution to it this solution might be either a unique score vector up to a positive factor or a set of different score vectors to handle the problem when the solution is not unique we develop tropical optimization techniques of maximizing and minimizing the hilbert seminorm to find those vectors from the solution set that are the most and least differentiating between the alternatives with the highest and lowest scores and thus are well representative of the entire solution set

extended valued topical and anti topical functions on semimodules in the papers 16 and 17 we have studied functions defined on a b complete idempotent semimodule x over a b complete idempotent semifield k k with values in k where k may or may not contain a greatest element supk and the residuation x y is not defined for x x and y infx in the present paper we assume that k has no greatest element then adjoin to k an outside greatest element supk and extend the operations and from k to k k so as to obtain a meaning also for x infx for any x x and study functions with values in k in fact we consider two different extensions of the product from k to k denoted by and respectively and use them to give characterizations of topical i e increasing homogeneous defined with the aid of and anti topical i e decreasing anti homogeneous defined with the aid of functions in terms of some inequalities next we introduce and study for functions f x k their conjugates and biconjugates of fenchel moreau type with respect to the coupling functions x y x y x y x and x y d inf x y d x y x d k and use them to obtain characterizations of topical and anti topical functions in the subsequent sections we consider for the coupling functions and some concepts that have been studied in rubinov and singer 2001 11 and singer 2004 15 for the so called additive min type coupling functions rmaxn rmaxn rmax and an an a respectively where a is a conditionally complete lattice ordered group and x y inf1 i n xi yi x y rmaxn or an thus we study the polars of a set g x for the coupling functions and and we consider for a function f x k the notion of support set of f with respect to the set t of all elementary topical functions t y x x y x x y x infx and two concepts of support set of f at a point x0 x the main differences between the properties of the conjugations with respect to the coupling functions and and between the properties of the polars of a set g with respect to the coupling functions and are caused by the fact that while is symmetric with values only in rmax resp a and are not symmetric and take values also outside rmax resp a

tropical polar cones hypergraph transversals and mean payoff games we discuss the tropical analogues of several basic questions of convex duality in particular the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy we characterize the extreme rays of the polar in terms of certain minimal set covers which may be thought of as weighted generalizations of minimal transversals in hypergraphs we also give a tropical analogue of farkas lemma which allows one to check whether a linear inequality is implied by a finite family of linear inequalities here the certificate is a strategy of a mean payoff game we discuss examples showing that the number of extreme rays of the polar of the tropical cyclic polyhedral cone is polynomially bounded and that there is no unique minimal system of inequalities defining a given tropical polyhedral cone

a linear system theoretic view of discrete event processes and its use for performance evaluation in manufacturing a discrete event system is a system whose behavior can be described by means of a set of time consuming activities performed according to a prescribed ordering events correspond to starting or ending some activity an analogy between linear systems and a class of discrete event systems is developed following this analogy such discrete event systems can be viewed as linear in the sense of an appropriate algebra the periodical behavior of closed discrete event systems i e involving a set of repeatedly performed activities can be totally characterized by solving an eigenvalue and eigenvector equation in this algebra this problem is numerically solved by an efficient algorithm which basically consists of finding the shortest paths from one node to all other nodes in a graph the potentiality of this approach for the performance evaluation of flexible manufacturing systems is emphasized the case of a flowshop like production process is analyzed in detail

sequential subspace robustness assessment and sensitivity analysis to avoid an overconservative design and to ensure desired performance in an optimal way the product quality and robustness are considered in terms of the product performance mean and variance in this paper to facilitate robust design exploration under uncertainty a new sequential subspace robustness assessment method is presented to assess not only the mean and variance of performance but also their sensitivities with respect to design parameters the proposed method is based on the computational framework that integrates the univariate revolving integration and surrogate modeling of univariate integral functions the proposed framework enables consideration of bivariate interaction effects approximately by the aggregation of multiple revolving integration terms in a partial set of bivariate subspaces it is found that the proposed method provides better accuracy with comparable computational cost in assessing the statistical moments and sensitivities of product performance than existing methods suc

extending conceptualisation modes for generalised formal concept analysis formal concept analysis fca is an exploratory data analysis technique for boolean relations based on lattice theory its main result is the existence of a dual order isomorphism between two set lattices induced by a binary relation between a set of objects and a set of attributes pairs of dually isomorphic sets of objects and attributes called formal concepts form a concept lattice but actually model only a conjunctive mode of conceptualisation in this paper we augment this formalism in two ways first we extend fca to consider different modes of conceptualisation by changing the basic dual isomorphism in a modal logic motivated way this creates the three new types of concepts and lattices of extended fca viz the lattice of neighbourhood of objects that of attributes and the lattice of unrelatedness second we consider incidences with values in idempotent semirings concretely the completed max plus or schedule algebra email protected m a x and focus on generalising fca to try and replicate the modes of conceptualisation mentioned above to provide a concrete example of the use of these techniques we analyse the performance of multi class classifiers by conceptually analysing their confusion matrices

scheduling with and or precedence constraints in many scheduling applications it is required that the processing of some job be postponed until some other job which can be chosen from a pregiven set of alternatives has been completed the traditional concept of precedence constraints fails to model such restrictions therefore the concept has been generalized to so called and or precedence constraints which can cope with this kind of requirement in the context of traditional precedence constraints feasibility transitivity and the computation of earliest start times for jobs are fundamental well studied problems the purpose of this paper is to provide efficient algorithms for these tasks for the more general model of and or precedence constraints we show that feasibility as well as many questions related to transitivity can be solved by applying essentially the same linear time algorithm in order to compute earliest start times we propose two polynomial time algorithms to cope with different classes of time distances between jobs

on injectivity of semimodules over additively idempotent division semirings and chain mv semirings abstract in this paper we give a characterization of injective semimodules over additively idempotent semirings consequently we provide a complete description of injective semimodules over the semifield of tropical integers and give an explicit construction of the injective hulls of semimodules over chain division semirings we also give a criterion for self injective mv semirings with an atomic boolean center and describe the structure of finitely generated injective semimodules over finite mv semirings as well as we show that every complete mv semiring with an atomic boolean center is an exact semiring which is defined by a hahn banach type separation property on semimodules arising in the tropical case from the phenomenon of tropical matrix duality moreover we show that complete boolean algebras are precisely the mv semirings in which every principal ideal is injective

first order saddlepoint approximation for reliability analysis in the approximation methods of reliability analysis nonnormal random variables are transformed into equivalent standard normal random variables this transformation tends to increase the nonlinearity of a limit state function and hence results in less accurate reliability approximation the first order saddlepoint approximation for reliability analysis is proposed to improve the accuracy of reliability analysis by the approximation of a limit state function at the most likelihood point in the original random space and employment of the accurate saddlepoint approximation the proposed method reduces the chance of an increase in the nonlinearity of the limit state function this approach generates more accurate reliability approximation than the first order reliability method without an increase in the computational effort the effectiveness of the proposed method is demonstrated with two examples and is compared with the first and second order reliability methods

enhanced two point diagonal quadratic approximation methods for design optimization abstract based on two point diagonal quadratic approximation referred to as tdqa developed by kim et al min soo kim jong rip kim jae young jeon dong hoon choi efficient mechanical system optimization using two point diagonal approximation in the nonlinear intervening variable space j ksme 15 2001 1257 126 enhanced two point approximation methods are proposed in this paper the suggested methods reinforce tdqa with new quadratic correction terms using the concept of tana 3 these methods overcome the disadvantage of tdqa when the derivatives at two design points have the same sign in addition the values and derivatives of the proposed approximation functions are completely equal to those of an original function at the two design points whether the derivatives at those points have the same sign or not several examples show the numerical performance and accuracy of the proposed methods compared to previous work and optimization examples show that these methods can successfully reach the optimum via sequential approximation optimization

on a degenerate diffusion equation of the form c z t ϑ zx x with application to population dynamics 51 here l u and v are strictly positive constants m 1 and i c s for 60 for s o 1 2 with c 0 the initial data z0 will always belong to w 3m q and often to c d and will always have zb 0 zo l u and z l v problem i arises in a theory developed in 173 for two interacting biological species that disperse in response to population pressure this theory is based on the assumption that the individual dispersal velocities are proportional to u u with u and v the corresponding densities if we assume that the habitat q is isolated then the resulting initial value problem has

development of a simple and efficient method for robust optimization robust optimization problems are newly formulated and an efficient computational scheme is proposed both design variables and design parameters are considered as random variables about their nominal values to ensure the robustness of objective performance we introduce a new performance index bounding the performance together with a constraint limiting the performance variation the constraint variations are regulated by considering the probability of feasibility each probability constraint is transformed into a sub optimization problem by the advanced first order second moment afosm method for computational efficiency the proposed robust optimization method has the advantages that the mean value and the variation of the performance function are controlled simultaneously and rationally and the second order sensitivity information is not required even in case of gradient based optimization process the suggested method is examined by solving three examples and the results are compared with those for the deterministic case and those available in the literature copyright 2002 john wiley sons ltd

topical functions on semimodules and generalizations we give first some characterizations of strongly supertopical respectively topical that is increasing strongly superhomogeneous respectively increasing homogeneous functions on a b complete semimodule x over a b complete idempotent semiring respectively semifield k k with values in k that improve and complement the main result of 12 for example we show that if k is a semifield and e and e denote the neutral elements of k for and respectively then every strongly supertopical function f x k satisfying f infx e is topical and that if and only if k e e then every strongly supertopical function f x k is topical we also give characterizations of strongly topical and topical functions with the aid of some inequalities next generalizing 10 we introduce elementary affine functions f x k and we apply them to obtain characterizations and a representation of topical functions as a consequence we obtain some characterizations of downward sets in x with the aid of elementary affine functions next we extend a result on topical functions f rn r r max given in 12 to functions f ri r where i is an arbitrary index set finally we give characterizations of subtopical i e increasing subhomogeneous functions f x k encompassing results of 10 11 our main tool is residuation theory

pure dimension and projectivity of tropical polytopes abstract we study how geometric properties of tropical convex sets and polytopes which are of interest in many application areas manifest themselves in their algebraic structure as modules over the tropical semiring our main results establish a close connection between pure dimension of tropical convex sets and projectivity in the sense of ring theory these results lead to a geometric understanding of idempotency for tropical matrices as well as their direct interest our results suggest that there is substantial scope to apply ideas and techniques from abstract algebra in particular ring theory in tropical geometry

inner products on semimodules over a commutative semiring abstract in this paper the inner products on the semimodules over a commutative semiring are investigated some characterizations for inner products and for orthogonal sets in the semimodules are given and standard orthogonal basis and orthogonal subsemimodule are discussed in particular an equivalent description is obtained for a semifield s satisfying the property that every standard orthogonal set in a finitely generated semimodule m over s can be extended to a standard orthogonal basis for m also the analogue of the kronecker capelli theorem is proved to be valid for a matrix equation over a commutative semiring partial results obtained in this paper generalize and develop corresponding results for semilinear spaces of n dimensional vectors over commutative zerosumfree semirings

max plus definite matrix closures and their eigenspaces in this paper we introduce the definite closure operation for matrices with finite permanent reveal inner structures of definite eigenspaces and establish some facts about hilbert distances between these inner structures and the boundary of the definite eigenspace

tropical linear maps on the plane abstract in this paper we fully describe all tropical linear maps in the tropical projective plane tp 2 that is maps from the tropical plane to itself given by tropical multiplication by a real 3 3 matrix a the map f a is continuous and piecewise linear in the classical sense in some particular cases the map f a is a parallel projection onto the set spanned by the columns of a in the general case after a change of coordinates the map collapses at most three regions of the plane onto certain segments called antennas and is a parallel projection elsewhere theorem 3 in order to study f a we may assume that a is normal i e i a 0 up to changes of coordinates a given matrix a admits infinitely many normalizations our approach is to define and compute a unique normalization for a which we call lower canonical normalization theorem 1 and then always work with it due both to its algebraic simplicity and its geometrical meaning on r n any n n some aspects of tropical linear maps have been studied in 6 we work in tp 2 adding a geometric view and doing everything explicitly we give precise pictures inspiration for this paper comes from 3 5 6 8 12 26 we have tried to make it self contained our preparatory results present noticeable relationships between the algebraic properties of a given matrix a idempotent normal matrix permutation matrix etc and classical geometric properties of the points spanned by the columns of a classical convexity and others see theorem 2 and corollary 1 as a by product we compute all the tropical square roots of normal matrices of a certain type see corollary 4 this is perhaps a curious result in tropical algebra our final aim is however to give a precise description of the map f a tp 2 tp 2 this is particularly easy when two tropical triangles arising from a denoted t a and t a fit as much as possible then the action of f a is easily described on the closure of each cell of the cell decomposition c a see theorem 3 normal matrices play a crucial role in this paper the tropical powers of normal matrices of size n n satisfy a n 1 a n a n 1 this statement can be traced back at least to 26 and appears later many times such as 1 2 6 9 10 in lemma 1 we give a direct proof of this fact for n 3 but now the equality a 2 a 3 means that the columns of a 2 are three fixed points of f a and in fact any point spanned by the columns of a 2 is fixed by f a among 3 3 normal matrices the idempotent ones i e those satisfying a a 2 are particularly nice we prove that the columns of such a matrix tropically span a set which is classically compact connected and convex lemma 2 and corollary 1 in our terminology it is a good tropical triangle

duality between invariant spaces for max plus linear discrete event systems we extend the notions of conditioned and controlled invariant spaces to linear dynamical systems over the max plus or tropical semiring we establish a duality theorem relating both notions which we use to construct dynamic observers these are useful in situations in which some of the system coefficients may vary within certain intervals the results are illustrated by an application to a manufacturing system

regularity results for a one dimensional two phase degenerate stefan problem l lh 40 30 l li where a u 0 and ai 0 for i 1 2 in the classical stefan problem the specific heat e u is required to be strictly positive where a u ml u if u is in q and 11 if u is in q however in more general free boundary problems various types of degeneracy in a can occur in the electro chemical machining process the specific heat is zero in both phases resulting in elliptic differential equations but with a stefan type boundary condition see 3 for a description of this model in the study of saturated unsaturated flow in

the perron frobenius theorem for homogeneous monotone functions if a is a nonnegative matrix whose associated directed graph is strongly connected the perron frobenius theorem asserts that a has an eigenvector in the positive cone r n we associate a directed graph to any homogeneous monotone function f r n r n and show that if the graph is strongly connected then f has a nonlinear eigenvector in r n several results in the literature emerge as corollaries our methods show that the perron frobenius theorem is really about the boundedness of invariant subsets in the hilbert projective metric they lead to further existence results and open problems

elements d algebre lineaire tropicale abstract we define a formal framework for the study of algebras of type max plus min plus tropical algebras and more generally algebras over a commutative idempotent semi field this work is motivated by the increasingly diversified use of these algebras which occur also in control theory automata theory as well as in algebraic geometry and in more specific ways in other parts of mathematics such as the theory of monoids in this first article we expecially re examine linear algebra over idempotent semi fields the most delicate but undoubtedly the most interesting point is the notion of a singular point seen as a generalization of the notion of zero we thus rediscover many notions of regularity already introduced for matrices and this allows us to define further notions new in this context such as that of the kernel of a linear form and to apply duality to obtain a good notion of tropical dimension of a submodule

distributed dynamic programming for discrete time stochastic control and idempotent algorithms idempotent methods have been found to be extremely fast for the solution of dynamic programming equations associated with deterministic control problems the original methods exploited the idempotent e g max plus linearity of the associated semigroup operator it is now known that curse of dimensionality free idempotent methods do not require this linearity instead it is sufficient that certain solution forms are retained through application of the associated semigroup operator here we see that idempotent methods may be used to solve some classes of stochastic control problems the key is the use of the idempotent distributive property we demonstrate this approach for a class of nonlinear discrete time stochastic control problems

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

duality and separation theorems in idempotent semimodules abstract we consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition we introduce a nonlinear projection on subsemimodules the projection of a point is the maximal approximation from below of the point in the subsemimodule we use this projection to separate a point from a convex set we also show that the projection minimizes the analogue of hilbert s projective metric we develop more generally a theory of dual pairs for idempotent semimodules we obtain as a corollary duality results between the row and column spaces of matrices with entries in idempotent semirings we illustrate the results by showing polyhedra and half spaces over the max plus semiring

duality and interval analysis over idempotent semirings abstract in this paper semirings with an idempotent addition are considered these algebraic structures are endowed with a partial order this allows to consider residuated maps to solve systems of inequalities a x b see 3 the purpose of this paper is to consider a dual product denoted and the dual residuation of matrices in order to solve the following inequality a x x b x sufficient conditions ensuring the existence of a non linear projector in the solution set are proposed the results are extended to semirings of intervals such as they were introduced in 25

best approximation in max plus semimodules abstract we establish new results concerning projectors on max plus spaces as well as separating half spaces and derive an explicit formula for the distance in hilbert s projective metric between a point and a half space over the max plus semiring as well as explicit descriptions of the set of minimizers as a consequence we obtain a cyclic projection type algorithm to solve systems of max plus linear inequalities

on the existence of the weak solution and the regularity of the free boundary to a one dimensional two phase degenerate stefan problem existence and regularity of the free boundary s t are demonstrated for the weak solution u u x t and s s t of the degenerate stefan problem 1 u ut uxx 0 0 0 t t u 1 t 2 t 0 0 t t u x 0 u0 x 0 x 1 u s t t 0 0 t t t ux s t 0 t ux s t 0 t 0 t t s 0 s0 where i u 0 and i 0 0 i 1 2

elementary topical functions on b complete semimodules over b complete idempotent semifields we extend some results of rubinov and singer 13 and rubinov et al 14 on topical functions on rn and downward subsets of rn to the case where rn is replaced by a b complete semimodule x over a b complete idempotent semifield k to this end using residuation we introduce elementary topical functions on x and we show that they are the building blocks of the set of all topical functions on x

k formal concept analysis as linear algebra over idempotent semifields abstract we report on progress in characterizing k valued fca in algebraic terms where k is an idempotent semifield in this data mining inspired approach incidences are matrices and sets of objects and attributes are vectors the algebraization allows us to write matrix calculus formulae describing the polars and the fixpoint equations for extents and intents adopting also the point of view of the theory of linear operators between vector spaces we explore the similarities and differences of the idempotent semimodules of extents and intents with the subspaces related to a linear operator in standard algebra this allows us to shed some light into formal concept analysis from the point of view of the theory of linear operators over idempotent semimodules in the opposite direction we state the importance of fca related concepts for dual order homomorphisms of linear spaces over idempotent semifields specially congruences the lattices of extents intents and formal concepts

which semifields are exact abstract every left linear function on a subspace of a finite dimensional vector space over a skew field can be extended to a left linear function on the whole space this paper explores the extent to what this basic fact of linear algebra is applicable to more general structures semifields with a similar property imposed on linear functions are called left exact and we present a complete description of such semifields namely we show that a semifield s is left exact if and only if s is either a skew field or an idempotent semiring

characterization of tropical hemispaces by p r decompositions abstract we consider tropical hemispaces defined as tropically convex sets whose complements are also tropically convex and tropical semispaces defined as maximal tropically convex sets not containing a given point we introduce the concept of p r decomposition this yields to our knowledge a new kind of representation of tropically convex sets extending the classical idea of representing convex sets by means of extreme points and rays we characterize tropical hemispaces as tropically convex sets that admit a p r decomposition of certain kind in this characterization with each tropical hemispace we associate a matrix with coefficients in the completed tropical semifield satisfying an extended rank one condition our proof techniques are based on homogenization lifting a convex set to a cone and the relation between tropical hemispaces and semispaces

regularity results for an elliptic parabolic free boundary problem on etudie un probleme aux limites libre elliptique parabolique a une dimension d espace on donne plusieurs resultats de regularite pour la solution faible et la frontiere libre on donne des conditions pour que la frontiere libre soit une courbe c 1

effective two point function approximation for design optimization anew two pointapproximationofafunctionthatisverystraightforwardtobuildandthatprovidesaccurateand stable approximation is presented earlier developments of the two point approximations had either incomplete matches at two data points or needed the solution of additional equations to get all of the parameters the present two point approximation is an incomplete second order taylor series expansion in terms of intervening variables thehessianmatrixhasonlydiagonalelements anditdependsondesignvariables theexponentofeachintervening design variable and the unknown constant of the second order terms are evaluated by matching the derivatives and the value of the approximation with the previous data point gradients and the value of the original function respectively all of theunknownsareidentie ed in aclosedform both thefunctionandthegradientof thetwo point approximation are equal to those of the original function at two data points several examples are given to show the accuracy and efe ciency of this method

nonstationary filtration in partially saturated porous media from the mathematical formulation of a one dimensional flow through a partially saturated porous medium we arrive at a nonlinear free boundary problem the boundary being between the saturated and the unsaturated regions in the medium in particular we obtain an equation which is parabolic in the unsaturated part of the domain and elliptic in the saturated part

the structure of max min hyperplanes abstract in this article continuing 12 13 further contributions to the theory of max min convex geometry are given the max min semiring is the set r r endowed with the operations max min in r a max min hyperplane briefly a hyperplane is the set of all points x x 1 x n r n satisfying an equation of the form a 1 x 1 a n x n a n 1 b 1 x 1 b n x n b n 1 with a i b i r i 1 n 1 where each side contains at least one term and where a i b i for at least one index i the main result is a description of a hyperplane in terms of simple polyhedral blocks as an application one shows that the separation of max min closed convex sets by max min hyperplanes is not possible in general

permuted max algebraic eigenvector problem is np complete abstract let a b max a b and a b a b for a b r r and extend these operations to matrices and vectors as in conventional linear algebra the following eigenvector problem has been intensively studied in the past given a r n n find all x r n x t eigenvectors such that a x x for some r the present paper deals with the permuted eigenvector problem given a r n n and x r n is it possible to permute the components of x so that it becomes a max algebraic eigenvector of a using a polynomial transformation from bandwidth we prove that the integer version of this problem is np complete

the minkowski theorem for max plus convex sets abstract we establish the following max plus analogue of minkowski s theorem any point of a compact max plus convex subset of r n can be written as the max plus convex combination of at most n 1 of the extreme points of this subset we establish related results for closed max plus cones and closed unbounded max plus convex sets in particular we show that a closed max plus convex set can be decomposed as a max plus sum of its recession cone and of the max plus convex hull of its extreme points

the tropical analogue of polar cones we study the max plus or tropical analogue of the notion of polar the polar of a cone represents the set of linear inequalities satisfied by its elements we establish an analogue of the bipolar theorem which characterizes all the inequalities satisfied by the elements of a tropical convex cone we derive this characterization from a new separation theorem we also establish variants of these results concerning systems of linear equalities

improved two point function approximations for design optimization two point approximations are developed by utilizing both the function and gradient information of two data points the objective of this work is to build a high quality approximation to realize computational savings in solving complex optimization and reliability analysis problems two developments are proposed in the new twopoint approximations 1 calculation of a correction term by matching with the previous known function value and supplementing it to the first order approximation for including the effects of higher order terms and 2 development of a second order approximation without the actual calculation of second order derivatives by using an approximate hessian matrix several highly nonlinear functions and structural examples are used for demonstrating the new two point approximations that improved the accuracy of existing first order methods evelopmentof new and improved constraint approximations has been an active area of research in the mathematical optimization community over the last two decades 1 3 more accurate function approximations reduce the finite element analysis cost and increase the search domain in each iteration of an optimization run recent developments in constraint approximations addressed methods for eigenvalues dynamic response static forces structural reliability etc some of the typical methods are summarized in the following kirsch4 developed a scaling factor for a stiffness matrix and approximated the displacements stresses and forces with respect to sizing and topology variables vanderplaats and salajegheh5 used two levels of approximations for discrete sizing and shape design of structures first for solving the continuous problem and next the discrete one using a dual theory canfield6 developed an approach for eigenvalues by approximating the modal strain and kinetic energies independently thomas et al 7 presented an approximation for the frequency response of damped structures by using the concept of ref 6 a multivariate spline approximation was developed by wang and grandhi8 making use of the previously generated exact finite element analyses similarly a hermite approximation for n dimensional problems used several data points and had the property of reproducing the function and gradient values at the known data points 9 canfield10 used multipoint data in building an approximate hessian matrix and constructed a second order approximation for improving the accuracy toropov et al 11 approximated functions as a linear combination or products of variables and the coefficients of the polynomial were computed by applying a least squared approach on multiple data points fadel et al 12 considered intermediate variables in terms of exponentials and they were computed by matching the approximate function gradients with the previous point s exact values this approach did not compare the function values of the previous point wang and grandhi13 used a single exponential for all of the variables and calculated it by matching the approximate function with the previous point exact value and the gradient information of the previous point was not utilized the proposed paper develops an improved two point approximation utilizing both function and gradient information of two data points in the new two point approximation two developments are proposed 1 calculation of a correction term based on two function values and

exact rings and semirings abstract we introduce and study an abstract class of semirings which we call exact semirings defined by a hahn banach type separation property on modules our motivation comes from the tropical semiring and in particular a desire to understand the often surprising extent to which it behaves like a field the definition of exactness abstracts an elementary property common to both fields and the tropical semiring which we believe is fundamental to explaining this similarity the class of exact semirings turns out to include many other important examples of both rings proper quotients of principal ideal domains matrix rings and finite group rings over these and over fields and semirings the boolean semiring generalisations of the tropical semiring matrix semirings and group semirings over these

characterizations of minimal elements of topical functions on semimodules with applications abstract in this paper we first give characterizations of the superdifferential of extended valued topical functions defined on a semimodule with values in a semifield next we characterize minimal elements of the upper support set of extended valued topical functions finally as an application we present a necessary and sufficient condition for global maximum of the difference of two strictly topical functions defined on a semimodule

idempotent and tropical mathematics complexity of algorithms and interval analysis abstract a very brief introduction to tropical and idempotent mathematics is presented tropical mathematics can be treated as a result of a dequantization of the traditional mathematics as the planck constant tends to zero taking imaginary values in the framework of idempotent mathematics constructions and algorithms are usually simpler than their traditional analogs we examine in particular algorithms of tropical idempotent mathematics generated by a collection of basic semiring or semifield operations and other good operations every algorithm of this type has an interval version the complexity of this interval version coincides with the complexity of the initial algorithm the interval version of an algorithm of this type gives exact interval estimates for the corresponding output data algorithms of linear algebra over idempotent semirings are examined in this case basic algorithms like their interval versions are polynomial this situation is very different from the traditional linear algebra case where basic algorithms are polynomial but the corresponding interval versions are np hard and interval estimates are not exact

what shape is your conjugate a survey of computational convex analysis and its applications computational convex analysis algorithms have been rediscovered several times in the past by researchers from different fields to further communications between practitioners we review the field of computational convex analysis which focuses on the numerical computation of fundamental transforms arising from convex analysis current models use symbolic numeric and hybrid symbolic numeric algorithms our objective is to disseminate widely the most efficient numerical algorithms useful for applications in image processing computing the distance transform the generalized distance transform and mathematical morphology operators partial differential equations solving hamilton jacobi equations and using differential equations numerical schemes to compute the convex envelope max plus algebra computing the equivalent of the fast fourier transform multifractal analysis etc the fields of applications include among others computer vision robot navigation thermodynamics electrical networks medical imaging and network communication

a new fundamental solution for differential riccati equations arising in control the matrix differential riccati equation dre is ubiquitous in control and systems theory the presence of the quadratic term implies that a simple linear systems fundamental solution does not exist of course it is well known that the bernoulli substitution may be applied to obtain a linear system of doubled size here however tools from max plus analysis and semiconvex duality are brought to bear on the dre we consider the dre as a finite dimensional solution to a deterministic linear quadratic control problem taking the semiconvex dual of the associated semigroup one obtains the solution operator as a max plus integral operator with quadratic kernel the kernel is equivalently represented as a matrix using the semigroup property of the dual operator one obtains a matrix operation whereby the kernel matrix propagates as a semigroup the propagation forward is through some simple matrix operations this time indexed family of matrices forms a new fundamental solution for the dre solution for any initial condition is obtained by a few matrix operations on the fundamental solution and the initial condition in analogy with standard algebra linear systems the fundamental solution can be viewed as an exponential form over a certain idempotent semiring this fundamental solution has a particularly nice control interpretation and might lead to improved dre solution speeds

the spectra of irreducible matrices over completed idempotent semifields motivated by some spectral results in the characterization of concept lattices we investigate the spectra of reducible matrices over complete idempotent semifields in the framework of naturally ordered semirings or dioids we find non null eigenvectors for every non null element in the semifield and conclude that the notion of spectrum has to be refined to encompass that of the incomplete semifield case so as to include only those eigenvalues with eigenvectors that have finite coordinates considering special sets of eigenvectors brings out finite complete lattices in the picture and we contend that such structure may be more important than standard eigenspaces for matrices over completed idempotent semifields

max plus summation of fenchel transformed semigroups for solution of nonlinear bellman equations abstract max plus methods have been explored for solution of first order nonlinear hamilton jacobi bellman partial differential equations hjb pdes and corresponding nonlinear control problems these methods exploit the max plus linearity of the associated semigroups in particular although the problems are nonlinear the semigroups are linear in the max plus sense these methods have been used successfully to compute solutions although they provide certain advantages they still generally suffer from the curse of dimensionality the natural analog to the laplace transform in ordinary spaces is the fenchel transform over max plus spaces the range space being referred to as the dual space one can transform the semigroup operators into operators on the dual space there are natural operations on the transformed operators which may be used to construct solutions of nonlinear control problems natural building blocks correspond to transforms of operators for linear quadratic problems in this paper a method for exploiting operations in the fenchel transform space is used to develop a method for certain problems such that the computational growth in the most time consuming portion of the computations can be hugely reduced although the curse of dimensionality is not entirely avoided the computational cost reductions are very high for some classes of problems

convexity of tropical polytopes abstract we study the relationship between min plus max plus and euclidean convexity for subsets of r n we introduce a construction which associates to any max plus convex set with compact projectivisation a canonical matrix called its dominator the dominator is a kleene star whose max plus column space is the min plus convex hull of the original set we apply this to show that a set which is any two of i a max plus polytope ii a min plus polytope and iii a euclidean polytope must also be the third in particular these results answer a question of sergeev schneider and butkovic 16 and show that row spaces of tropical kleene star matrices are exactly the polytropes studied by joswig and kulas 13