formulation and analysis of a parameter free stabilized finite element method we present and study a parameter free stabilized finite element method for the scalar advection diffusion equation edge element lifting of diffusive edge fluxes defines the stabilization the amount of edge diffusion varies with the edge peclet number and adapts to solution features without mesh dependent parameters we prove that the method is first order accurate in the advective limit numerical studies confirm the theoretical estimates

tropical linear algebra with the łukasiewicz t norm the max ukasiewicz semiring is defined as the unit interval 0 1 equipped with the arithmetics a b max a b and a b max 0 a b 1 linear algebra over this semiring can be developed in the usual way we observe that any problem of the max ukasiewicz linear algebra can be equivalently formulated as a problem of the tropical max plus linear algebra based on this equivalence we develop a theory of the matrix powers and the eigenproblem over the max ukasiewicz semiring

normal forms in bl algebra and their contribution to universal approximation of functions this paper continues the investigation of approximating properties of generalized normal forms in fuzzy logic the problem is formalized and solved algebraically normal forms are considered in two variants infinite and finite it is proved that infinite normal forms are universal representation formulas whereas finite normal forms are universal approximation formulas for any l valued function where l is a support set of any complete bl algebra the estimation of the quality of approximation is suggested moreover functions which can be precisely represented by the discrete normal forms are considered

bases in max algebra for n tuples over the algebraic system r r max concepts such as linear dependence space and basis may be defined by analogy with classical linear algebra whenever a space is finitely generated it possesses a basis and all its bases are trivially related and therefore have the same cardinality however for any given n 2 spaces with bases of arbitrary cardinality may be constructed as well as spaces with no basis

towards a higher degree f transform the aim of this study is to show how the f transform technique can be generalized from the case of constant components to the case of polynomial components after a general presentation of an f m transform m 0 a detailed characterization of the f 1 transform is given we apply a numeric integration technique in order to simplify the computation of f 1 transform components the inverse f m transform m 0 is defined similarly to the ordinary inverse f transform the quality of approximation using the inverse f m transform increases with an increase in m

max algebra the linear algebra of combinatorics abstract let a b max a b a b a b for a b r r by max algebra we understand the analogue of linear algebra developed for the pair of operations extended to matrices and vectors max algebra which has been studied for more than 40 years is an attractive way of describing a class of nonlinear problems appearing for instance in machine scheduling information technology and discrete event dynamic systems this paper focuses on presenting a number of links between basic max algebraic problems like systems of linear equations eigenvalue eigenvector problem linear independence regularity and characteristic polynomial on one hand and combinatorial or combinatorial optimisation problems on the other hand this indicates that max algebra may be regarded as a linear algebraic encoding of a class of combinatorial problems the paper is intended for wider readership including researchers not familiar with max algebra

di uniformities and hutton uniformities the authors characterize di uniformities on a texture s s in the sense of ozcag and brown di uniform texture spaces appl gen top 4 1 2003 157 192 in terms of functions on the texturing s this characterization enables quasi uniformities in the sense of hutton uniformities on fuzzy topological spaces j math anal appl 58 1977 559 571 to be regarded as di uniformities on the corresponding hutton texture thereby revealing di uniformities as a generalization of hutton quasi uniformities the effect of imposing a complementation on s s is also considered and several important results established

the cardinality of bases in semilinear spaces over commutative semirings abstract this paper investigates the cardinality of a basis in semilinear spaces of n dimensional vectors over commutative semirings it first discusses the cardinality of a basis and gives a necessary and sufficient condition that each basis has the same number of elements which is then used to present the characterizations of bases by the way it obtains an equivalent description of an invertible matrix it finally shows a necessary and sufficient condition that each basis has the same number of elements

algebraic analysis of fuzzy systems in this paper we have developed an algebraic theory suitable for the analysis of fuzzy systems we have used the notions of semiring and semimodule introduced the notion of semilinear space have given numerous examples of them and defined also the notions of linear dependence and independence we have shown that the composition operation which plays an essential role in the analysis of fuzzy systems because of its role in the compositional rule of inference can be interpreted as a homomorphism between special semimodules consequently this operation is in a certain sense a linear operation this property formally explains why fuzzy systems are attractive for applications

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

interpretation of af c algebras in łukasiewicz sentential calculus on donne un critere pour la non simplicite d une af c algebre 7b u en termes de proprietes de la theorie de la recursivite du groupe dimension k 0 7b u

indices and periods of incline matrices abstract inclines are the additively idempotent semirings in which products are less than or equal to factors thus inclines generalize boolean algebra fuzzy algebra and distributive lattice and the boolean matrices the fuzzy matrices and the lattice matrices are the prototypical examples of the incline matrices i e the matrices over inclines this paper studies the power sequence of incline matrices in detail a necessary and sufficient condition for the incline matrix to have index is given and the indices of some incline matrices with indices are estimated it is proved that the period of n n incline matrix with index is a divisor of n and that the set of periods of n n incline matrices with indices is not bounded from above in the sense of a power of n for all n an equivalent condition and some sufficient conditions for the incline matrix to converge in finite steps are established the stability of the orbits of an incline matrix is considered and a theorem in fuzzy sets and systems 81 1996 227 is pointed out being false the results in the present paper include some previous results in the literatures which were obtained for the boolean matrices the fuzzy matrices and the lattice matrices among their special cases

stability and performance robustness for multivariable linear systems abstract this paper deals with both aspects of stability and performance robustness of linear systems in the time domain a linear time invariant feedback control law which yields the largest set of parameter variations is first determined it is then used as an initial guess for designing a robust control law which produces satisfactory performance for each of several operating points of the system given a priori the stability robustness properties of the resulting control law is also studied the proposed approach is applied to an example yielding larger stability bounds including satisfactory performance in comparison with recently reported results

resolution of fuzzy relational equations method algorithm and software with applications analytical methods and algorithms for inverse problem resolution of fuzzy linear systems of equations in some bl algebras godel algebra in case of max min and min max compositions and goguen algebra in case of max product composition are presented algorithms with software realization for solving fuzzy linear systems of equations are proposed applications include fuzzy optimization with fuzzy linear systems of equation constraint fuzzy machines and covering problem

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

computing behavior of finite fuzzy machines algorithm and its application to reduction and minimization we define finite fuzzy machines and investigate their behavior algorithm and software are proposed for computing behavior for establishing equivalence and redundancy of states and for solving reduction and minimization problems computational complexity of the algorithm is discussed testing examples are supplied the results are valid for finite max min min max and max product fuzzy machines

a new control volume finite element method for the stable and accurate solution of the drift diffusion equations on general unstructured grids abstract we present a new control volume finite element method with multi dimensional scharfetter gummel upwinding cvfem sg for the drift diffusion equations the method combines a conservative formulation of the carrier density continuity equations with an edge element lifting of the one dimensional scharfetter gummel edge currents into curl conforming elemental currents these elemental currents combine the upwind effect from all element edges and enable accurate computation of the flux on arbitrary surfaces inside the elements in so doing we obtain a formulation that is stable and accurate on general unstructured finite element grids this approach sets our formulation apart from other methods which require the control volumes to be topologically dual to the primal grid numerical studies of the cvfem sg for a suite of scalar advection diffusion test problems confirm the accuracy and the robustness of the new formulation simulations of a pn diode and an n channel mosfet device demonstrate the performance of the method for the fully coupled drift diffusion system

improved measures of stability robustness for linear state space models in this paper the aspect of stability robustness of linear systems is analyzed in the time domain a bound on the structured perturbation of an asymptotically stable linear system is obtained to maintain stability using a lyapunov matrix equation solution the resulting bound is shown to be an improved bound over the ones recently reported in the literature also special cases of the nominal system matrix are considered for which the bound is given in terms of the nominal matrix thereby avoiding the solution of the lyapunov matrix equation examples given include comparison of the proposed approach with the recently reported results

note on the free sets and free subsemimodules in semimodules abstract some conditions that a finite set in an l semimodule is free are established this gives an answer to an open problem raised by tan 2016 in his work 7

signatures definitions operators and applications to fuzzy modelling this paper presents a new framework for the symbolic representation of data which is referred to as signatures the definitions of signatures and of signature trees are first given original operators on signatures are next presented i e contraction extension pruning addition multiplication and grafting attractive applications of signatures related to the modelling of fuzzy inference systems are suggested and discussed an example is included to accompany the theoretical results

exponentially fitted shape functions for advection dominated flow problems in two dimensions we sketch out the basic properties of three novel exponentially fitted shape functions that generalize to two dimensional triangular elements the one dimensional functions used in the well known scharfetter gummel method this scheme is widely employed in the finite element approximation of the drift diffusion equations arising in semiconductor device modelling

logical structure of fuzzy if then rules abstract this paper provides a logical basis for manipulation with fuzzy if then rules our theory is wide enough and it encompasses not only finding a conclusion by means of the compositional rule of inference due to lotfi a zadeh but also other kinds of approximate reasoning methods e g perception based deduction provided that there exists a possibility to characterize them within a formal logical system in contrast with other approaches employing variants of multiple valued first order logic the approach presented here employs fuzzy type theory of v novak which has sufficient expressive power to present the essential concepts and results in a compact elegant and justifiable form within the effectively formalized representation developed here based on a complete logical system it is possible to reconstruct numerous well known properties of cri related fuzzy inference methods albeit not from the analytic point of view as usually presented but as formal derivations of the logical system employed the authors are confident that eventually all relevant knowledge about fuzzy inference methods based on fuzzy if then rule bases will be represented formalized and backed up by proof within the well founded logical representation presented here an immediate positive consequence of this approach is that suddenly all elements of a fuzzy inference method based on fuzzy if then rules are first class citizens of the representation there are clear logically founded definitions for fuzzy if then rule bases to be consistent complete or independent

bases in semilinear spaces over zerosumfree semirings this paper investigates the cardinality of a basis and the characterizations of a basis in semilinear space of n dimensional vectors over zerosumfree semirings first it discusses the cardinality of a basis and gives some necessary and sufficient conditions that each basis has the same number of elements then it presents some conditions that a set of vectors is a basis and that a set of linearly independent vectors can be extended to a basis in the end it shows a necessary and sufficient condition that two semilinear spaces are isomorphic

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

a novel exponentially fitted triangular finite element method for an advection diffusion problem with boundary layers in this paper we develop an exponentially fitted finite element method for a singularly perturbed advection diffusion problem with a singular perturbation parameter this finite element method is based on a set of novel piecewise exponential basis functions constructed on unstructured triangular meshes the basis functions can not be expressed explicitly but the values of each of them and its associated flux at a point are determined by a set of two point boundary value problems which can be solved exactly a method for evaluating elements of the stiffness matrix is also proposed for the case that is small numerical results presented to validate the method show that the method is stable for a large range of it is also shown by the numerical results that the rate of convergence of the method in an energy norm is of orderh1 2when is small

perturbation bounds for robust stability in linear state space models abstract this paper addresses the aspect of stability robustness of linear systems in the time domain upper bounds on the linear perturbation of an asymptotically stable linear system are obtained to maintain stability both for structured as well as unstructured perturbations using the lyapunov approach for structured perturbation the resulting bound is such that it garners the structural information about the nominal as well as the perturbation matrix into a single unified expression in the case of unstructured perturbations special features of the nominally stable matrix are exploited resulting in simpler expressions for the bound without the need to solve the lyapunov equation improvement of the proposed measures is illustrated with the help of examples

solution sets of finite fuzzy relation equations with sup inf composition over bounded brouwerian lattices this paper considers the resolution of finite fuzzy relation equations with sup inf composition over a bounded brouwerian lattice the solution sets of finite fuzzy relation equations on a bounded brouwerian lattice are described in a similar way as those of linear spaces of n dimensional vectors in linear algebra

characterization of allowable perturbations for robust stability the problem of determining stability for a closed loop linear multivariable process in the presence of ill defined process perturbations is considered it is shown that m matrices may be used to characterize a class of allowable perturbations

bases in semilinear spaces over join semirings abstract this paper investigates the cardinality of a basis in semilinear spaces of n dimensional vectors over join semirings first it introduces the notion of an irredundant decomposition of an element in a join semiring then discusses the cardinality of a basis and proves that the cardinality of each basis is n if and only if the multiplicative identity element 1 is join irreducible if 1 is not a join irreducible element then each basis need not have the same number of elements in semilinear spaces of n dimensional vectors over join semirings this gives an answer to an open problem raised by di nola et al in their work algebraic analysis of fuzzy systems fuzzy sets and systems 158 2007 1 22

towards multiscale functions enriching finite element spaces with local but not bubble like functions in this paper we propose a novel way via finite elements to treat problems that can be singular perturbed a reaction diffusion equation in our case we enrich the usual piecewise linear or bilinear finite element trial spaces with local solutions of the original problem as in the residual free bubble rfb setting but do not require these functions to vanish on each element edge a departure from the rfb paradigm such multiscale functions have an analytic expression for triangles and rectangles bubbles are the choice for the test functions allowing static condensation thus our method is of petrov galerkin type we perform several numerical validations which confirm the good performance of the method 2004 elsevier b v all rights reserved

eigenvalues and eigen functionals of diagonally dominant endomorphisms in min max analysis the so called min analysis may be viewed as an extension to the continuous case and to functional spaces of shortest path algebras in graphs we investigate here min max analysis which extends in some similar way minimum spanning tree problems and maximum capacity path problems in graphs an endomorphisms a of the functional min max semi module acts on any functional to produce a where a x miny max a x y y we present here a complete characterization of eigenvalues and eigen functionals of diagonally dominant endomorphisms i e such that x y a x x 0a a x y 0a it is shown in particular that any real value 0a is an eigenvalue and that the associated eigen semi module has a unique minimal generator

residuation in fuzzy algebra and some applications abstract the system 0 1 max min has a number of well known applications many of which involve the inversion of linear relations we show the advantage of residuation theory for this process and apply it to the analysis of the problem of chebychev approximation

robustness results in linear quadratic gaussian based multivariable control designs the robustness of control systems with respect to model uncertainty is considered using simple frequency domain criteria available and new results are derived under a common framework in which the minimum singular value of the return differences transfer matrix is the key quantity in particular robustness results associated with multivariable control systems designed on the basis of linear quadratic lq and the linear quadratic gaussian lqg design methodologies are presented

linear independence in bottleneck algebras abstract let r be a linearly ordered set a bottleneck algebra is the set r together with two operations a b max a b and a b min a b in this paper we suggest various definitions of linear independence of sets of n tuples over r study their characteristics and interrelations

standard orthogonal vectors in semilinear spaces and their applications abstract this paper investigates the standard orthogonal vectors in semilinear spaces of n dimensional vectors over commutative zerosumfree semirings first we discuss some characterizations of standard orthogonal vectors then as applications we obtain some necessary and sufficient conditions that a set of vectors is a basis of a semilinear subspace which is generated by standard orthogonal vectors prove that a set of linearly independent nonstandard orthogonal vectors cannot be orthogonalized if it has at least two nonzero vectors and show that the analog of the kronecker capelli theorem is valid for systems of equations

invertible matrices and semilinear spaces over commutative semirings this paper deals with invertible matrices and semilinear spaces over commutative semirings some necessary and sufficient conditions for matrices to be invertible over commutative semirings are obtained the condition under which the cardinality of each basis of semilinear space of n dimensional vectors is n is given in the end the factor rank of matrices over semirings is investigated

a new family of mixed finite elements in ir 3 we introduce two families of mixed finite element on conforming inh div and one conforming inh curl these finite elements can be used to approximate the stokes system

on invertible matrices over antirings abstract an antiring is a semiring which is zerosumfree i e a b 0 implies a b 0 for any a b in this semiring in this paper the complete description of the invertible matrices over a commutative antiring is given and some necessary and sufficient conditions for a matrix over a commutative antiring to be invertible are obtained also cramer s rule over commutative antirings is presented the main results in this paper generalize and develop the corresponding results for the boolean matrices the fuzzy matrices the lattice matrices and the incline matrices

note on cardinality of bases in semilinear spaces over zerosumfree semirings abstract a necessary and sufficient condition under which all bases in a semilinear space of n dimensional vectors over zerosumfree semirings have exactly n elements is established

inverses of l fuzzy matrices abstract inverses and generalized inverses of matrices over a brouwerian lattice are considered necessary and sufficient conditions for the existence of inverses g inverses thierrin vagner inverses minimum norm g inverses least squares g inverses and moore penrose inverses of matrices are obtained algorithms for finding all these types of generalized inverses are given

determinantal identities over commutative semirings abstract we present a development of determinantal identities over commutative semirings this includes a generalization of the cauchy binet and laplace theorems as well as results on compound matrices and adjoints it is further shown that laplace s theorem is a special case of the grade s adjoint identity

dioids and semirings links to fuzzy sets and other applications besides the classical algebraic structures of groups rings and fields which have long been the almost exclusive reference concepts used in mathematical modelling other algebraic structures such as dioids and semirings have emerged in the last two or three decades in connection with modelling and solving a rich variety of non classical problems e g in decision analysis fuzzy set theory operations research automatic control and mathematical physics the present paper aims at providing an overview of applications of dioid and semiring structures stressing links with fuzzy sets and emphasizing linear algebraic problems solving linear systems computing eigenvalues and eigenvectors non classical path finding problems using algebras of endomorphisms and connections between dioid structure and nonlinear analysis in view of solving problems in mathematical physics

fuzzy function as an approximate solution to a system of fuzzy relation equations this paper is mostly focused on the problem of approximate solvability of a system of fuzzy relation equations however we put a new light on this problem connecting it with the interpolation of a fuzzy function we introduce a notion of fuzzy function and its representation by fuzzy relation we demonstrate how problems of interpolation and approximation of fuzzy functions are connected with solvability of systems of fuzzy relation equations first we explain the general framework and then we prove some results related to the problem of the best approximation in particular we have shown that fuzzy relations introduced by sanchez and mamdani are the best approximations in certain approximation spaces

two dimensional exponential fitting and applications to drift diffusion models a class of methods is introduced that generalizes the so called exponential fitting method to two dimensional problems and more specifically to equations of the type operatorname div underset raise0 3em hbox smash scriptscriptstyle nabla u u underset raise0 3em hbox smash scriptscriptstyle nabla psi f with psi given and underset raise0 3em hbox smash scriptscriptstyle nabla psi big in a part of the domain the basic idea is the following i write the equation as operatorname div e psi underset raise0 3em hbox smash scriptscriptstyle nabla rho f through the change of variables u rho e psi ii discretize the symmetric equation by means of a method that uses some kind of harmonic average for the coefficient e psi mixed hybrid etc iii write the discrete scheme in terms of the nodal values of u e psi rho this produces an m matrix

fuzzy relational inequalities and equations fuzzy quasi orders closures and openings of fuzzy sets the paper deals with systems of fuzzy relation equations known in the literature as eigen fuzzy set equations as well as with systems of related fuzzy relation inequalities our main results are algorithms for computing the greatest and the smallest solutions to the considered systems of fuzzy relation inequalities and algorithms for computing the greatest solutions to the considered systems of fuzzy relation equations the systems are studied in the framework of complete residuated lattices as the structure of membership values by means of fuzzy quasi orders and closures and openings on a collection of fuzzy sets

lyapunov stability robust analysis and robustness design for linear continuous time systems the linear continuous time systems to be discussed are described by state space models with structured time varying uncertainties first the explicit maximal perturbation bound for maintaining quadratic lyapunov stability of the closed loop systems is presented then a robust design method is proposed given an open loop nominal system with information of the uncertainty structure the proposed robust design method produces a linear constant output feedback control law which maximizes the perturbation bound for maintaining quadratic lyapunov stability both the optimal results for robust analysis and robustness design can be obtained by using an efficient numerical algorithm e g minimax of matlab with analytic gradients given in this paper examples are investigated in some detail to show the improvements and advantages of the proposed methods over previous ones

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100