on ordinal sums of triangular norms on bounded lattices ordinal sums have been introduced in many different contexts e g for posets semigroups t norms copulas aggregation operators or quite recently for hoops in this contribution we focus on ordinal sums of t norms acting on some bounded lattice which is not necessarily a chain or an ordinal sum of posets necessary and sufficient conditions are provided for an ordinal sum operation yielding again a t norm on some bounded lattice whereas the operation is determined by an arbitrary selection of subintervals as carriers for arbitrary summand t norms by such also the structure of the underlying bounded lattice is investigated further it is shown that up to trivial cases there are no ordinal sum t norms on product lattices in general

characterizing when an ordinal sum of t norms is a t norm on bounded lattices one of the most important operators in soft computing are the triangular norms t norms as well as the combination among them the ordinal sum of triangular norms on 0 1 has been used to construct other triangular norms however on a bounded lattice an ordinal sum of t norms may not be a t norm several necessary and sufficient conditions are presented in this paper for ensuring whether an ordinal sum on a bounded lattice of arbitrary t norms is in fact a t norm moreover we show that a large set of ordinal sums verify these conditions hence they are very interesting in order to verify whether an ordinal sum on a bounded lattice is a t norm for a particular family of t norms

the distributive equations for idempotent uninorms and nullnorms the distributive equations only involving idempotent uninorms and nullnorms are investigated the following two cases are proved i as for a nullnorm with an idempotent uninorm having distributivity the necessary and sufficient conditions are given ii as for an idempotent uninorm with a nullnorm having distributivity the necessary and sufficient conditions are given also

new construction of t norms and t conorms on bounded lattices abstract this article studies the construction of ordinal sums of triangular norms t norms and triangular conorms t conorms on bounded lattices first we present a new class of t norms and t conorms on an arbitrary bounded lattice some examples are provided to illustrate that our new construction method is different from some existing methods for the construction of ordinal sums of t norms and t conorms on an arbitrary bounded lattice then we illustrate that our new construction method can be generalized by induction to a modified ordinal sum construction for t norms and t conorms on an arbitrary bounded lattice finally for any bounded sublattice s of a bounded lattice l we provide a sufficient condition that does not need s and l to have the same bottom and top elements such that an ordinal sum function is a t norm on l whereas the function is determined by an arbitrary selection of bounded sublattices as carriers for arbitrary summand t norms in particular we give a necessary and sufficient condition that needs s to have the same bottom element as l such that an ordinal sum function is a t norm on l

generalized ordinal sums of aggregation operators on bounded lattices abstract in this paper four kinds of generalized ordinal sums of aggregation operators on bounded lattices are provided and discussed firstly necessary and sufficient conditions for lower resp upper generalized ordinal sum of many summands to be an aggregation operator are provided in which case that the smallest resp greatest aggregation operator that coincides with each summand is given meanwhile the idempotent lower resp upper generalized ordinal sum of idempotent aggregation operators is also discussed then the relationships between these four kinds of generalized ordinal sums are given finally the lattice structures of some sets of special aggregation operators are investigated

a note on the ordinal sum theorem and its consequence for the construction of triangular norms in this paper the well known ordinal sum theorem of semigroups is generalized and applied to construct new families of triangular subnorms and triangular norms t norms among them one can find several new families of left continuous t norms too

a symplectic qr like algorithm for the solution of the real algebraic riccati equation a method is presented to solve the real algebraic riccati equation xnx xa a t x k 0 where k k t and n n t the solution for the corresponding eigenvalue problem mx lambda x where m is a hamiltonian matrix is computed by an algorithm similar to the qr algorithm special symplectic matrices are used for the transformation of m such that the hamiltonian form is preserved during the computations

on some characterizations of pairwise star orthogonality using rank and dagger additivity and subtractivity abstract the main results are that pairwise star orthogonality a i a j 0 and a i a j 0 for all i j where a 1 a k are complex m n matrices is equivalent to i a i a j and to ii a i rs a j and a j a j where i 1 k here the superscript denotes the moore penrose inverse while and rs denote respectively the star and rank subtractivity or minus partial orderings five more characterizations of the pairwise star orthogonality of k complex m n matrices are also presented our characterizations extend earlier results by hartwig and styan 1986 for k 2

triangular norms on partially ordered sets in this paper several methods to construct triangular norms on partially ordered sets are proposed these methods include 1 constructing triangular norms on the poset of closed intervals of a given bounded poset 2 constructing triangular norms via galois connections between bounded posets based on a categorical analysis of the construction of triangular norms on closed intervals via monotone functions and their pseudo inverses 3 constructing triangular norms on function spaces between partially ordered sets

a schur decomposition for hamiltonian matrices abstract a schur type decomposition for hamiltonian matrices is given that relies on unitary symplectic similarity transformations these transformations preserve the hamiltonian structure and are numerically stable making them ideal for analysis and computation using this decomposition and a special singular value decomposition for unitary symplectic matrices a canonical reduction of the algebraic riccati equation is obtained which sheds light on the sensitivity of the nonnegative definite solution after presenting some real decompositions for real hamiltonian matrices we look into the possibility of an orthogonal symplectic version of the qr algorithm suitable for hamiltonian matrices a finite step initial reduction to a hessenberg type canonical form is presented however no extension of the francis implicit shift technique was found and reasons for the difficulty are given

on a new class of uninorms on bounded lattices we study and propose some new construction methods to obtain uninorms on bounded lattices considering an arbitrary bounded lattice l we show the existence of idempotent uninorms on l for any element e l 0 1 playing the role of a neutral element by our construction method we obtain the smallest idempotent uninorm and the greatest idempotent uninorm with the neutral element e l 0 1 we see that the obtained uninorms are conjunctive and disjunctive uninorms respectively on the other hand if l is not a chain we also provide an example of an idempotent uninorm which is neither conjunctive nor disjunctive

the transmission of shifts and shift blurring in the qr algorithm abstract the qr algorithm is one of the most widely used algorithms for calculating the eigenvalues of matrices the multishift qr algorithm with multiplicity m is a version that effects m iterations of the qr algorithm at a time it is known that roundoff errors cause the multishift qr algorithm to perform poorly when m is large in this paper the mechanism by which the shifts are transmitted through the matrix in the course of a multishift qr iteration is identified numerical evidence showing that the mechanism works well when m is small and poorly when m is large is presented when the mechanism works poorly the convergence of the algorithm is degraded proportionately

matrix factorizations for symplectic qr like methods abstract symplectic qr like methods use symplectic or unitary symplectic similarity transformations instead of the usual unitary similarity transformations in the qr process a fundamental problem for the development of such methods is the choice of a suitable type of decomposition a sr corresponding to the qr decomposition where s is symplectic or unitary symplectic decompositions of this type are studied with regard to their application in a qr like process

on some characterizations of the star partial ordering for matrices and rank subtractivity abstract the main result is that drazin s star partial ordering a b holds if and only if a b and b a b a where a b is defined by a a a b and aa ba and where a b denotes rank subtractivity here a and b are m n complex matrices and the superscript denotes the moore penrose inverse several other characterizations of a b are given with particular emphasis on what extra condition must be added in order that rank subtractivity becomes the stronger star order a key tool is a new canonical form for rank subtractivity connections with simultaneous singular value decompositions schur complements and idempotent matrices are also mentioned

linear preservers for matrix inequalities and partial orderings we characterize the linear preservers of minus partial order on matrix algebras the developed approach allows us to classify linear transformations that live fixed several matrix relations arising as extremal cases in some classical matrix inequalities including rank additivity relation and related properties applications to the determinant preservers are considered

a class of hamiltonian symplectic methods for solving the algebraic riccati equation abstract the algebraic riccati equation can be solved by finding a certain invariant subspace of a related hamiltonian matrix g ammar and v mehrmann devised a method for calculating this invariant subspace that exploits the hamiltonian structure of the matrix by using unitary symplectic similarity transformations this paper discusses a class of methods that find the invariant subspace by using symplectic similarity transformations that are not necessarily unitary the method of ammar and mehrmann is a special case

on hamiltonian and symplectic hessenberg forms abstract characterizations are given for the hamiltonian matrices that can be reduced to hamiltonian hessenberg form and the symplectic matrices that can be reduced to symplectic hessenberg form by orthogonal symplectic similarity transformations the reduction to these special hessenberg forms is the missing link in the solution of the open problem of constructing a stable structure preserving qr like method of complexity o n 3 for the computation of invariant subspaces of hamiltonian and symplectic matrices our considerations lead us to propose an approach to the computation of lagrangian invariant subspaces of a hamiltonian or symplectic matrix

lattice based sums in this contribution the well known ordinal sum technique of posets is generalized by allowing for a lattice ordered index set instead of a linearly ordered index set and we argue for the merits of this generalization we will call such a proposed sum type construction a lattice based sum our new approach of lattice based sum extends also the horizontal sum we show that the lattice based sum of posets is again a poset subsequently we apply the results for constructing new lattices by investigating lattice based sums when the summand posets are lattices we show that under certain assumptions the lattice based sum of lattices will be a lattice

archimedean overlap functions the ordinal sum and the cancellation idempotency and limiting properties abstract overlap functions are a particular type of aggregation functions given by increasing continuous commutative bivariate functions defined over the unit square satisfying appropriate boundary conditions overlap functions are applied mainly in classification problems image processing and in some problems of decision making based on some kind of fuzzy preference relations in which the associativity property is not strongly required moreover the class of overlap functions is reacher than the class of t norms concerning some properties like idempotency homogeneity and mainly the self closedness feature with respect to the convex sum and the aggregation by generalized composition of overlap functions this flexibility of overlap functions increases their applicability the aim of this papers is to introduce the concept of archimedean overlap functions presenting a study about the cancellation idempotency and limiting properties and providing a characterization of such class of functions the concept of ordinal sum of overlap functions is also introduced providing constructing representing methods of certain classes of overlap functions related to idempotency cancellation limiting and archimedean properties

automorphisms of mn partially ordered by rank subtractivity ordering let mn be the space of all n n matrices with coefficients in r or c where n 3 the rank subtractivity or minus partial order on mn is defined by a rsb iff rankb ranka rank b a we characterize bijective mappings on mn such that a rsb iff a rs b a b mn

lattice based sum of t norms on bounded lattices abstract the concept of ordinal sums in the sense of clifford have long been blamed for their limitations in constructing new t norms including inability to cope with general bounded lattices motivated by this observation and based on the lattice based sum of lattices that has been recently introduced by el zekey et al we propose a new sum type construction of t norms called a lattice based sum of t norms for building new t norms on bounded lattices from given ones the proposed sum is generalizing the well known ordinal and horizontal sum constructions of t norms by allowing for lattice ordered index sets we demonstrate that like the ordinal sum of t norms the lattice based sum of t norms can be generalized using as summands so called t subnorms still leading to a t norm subsequently we apply the results for constructing several new families of t norms and t subnorms on bounded lattices in the same spirit by the duality we will also introduce lattice based sums of t conorms and t subconorms

on the use of aggregation operations in information fusion processes this position paper discusses the role of the existing body of fuzzy set aggregation operations in various kinds of problems where the process of fusion of items coming from several sources is central several kinds of membership functions can be useful according to the nature of the information to be merged numerical vs ordinal inputs preferences vs uncertain data observations vs constraints in each case some aggregation operations look more plausible or feasible than others the aim of this discussion is to suggest directions for putting at work the results of recent mathematical investigations in the structure of aggregation operations

distributivity and conditional distributivity of semi uninorms over continuous t conorms and t norms the distributivity and conditional distributivity of a uninorm and a continuous t conorm present an open problem recalled by klement in the linz2000 closing session in 2006 ruiz and torrens 14 solved this problem for the most usual known classes of uninorms recently rak 13 solved the same problem for several classes of semi uninorms i e a special class of conjunctive semi uninorms a special class of disjunctive semi uninorms and the class of idempotent semi uninorms in this work we continue to investigate the same topic as the above by focusing on two other classes of semi uninorms i e representable semi uninorms and continuous semi uninorms the obtained results are different from the case of representable uninorms and also different from the cases of other known classes of semi uninorms moreover the dual case of distributivity and conditional distributivity of the above two classes of semi uninorms over continuous t norms is also investigated and similar results are obtained

product eigenvalue problems many eigenvalue problems are most naturally viewed as product eigenvalue problems the eigenvalues of a matrix a are wanted but a is not given explicitly instead it is presented as a product of several factors a a k a k 1 cdots a 1 usually more accurate results are obtained by working with the factors rather than forming a explicitly for example if we want eigenvalues vectors of b t b it is better to work directly with b and not compute the product the intent of this paper is to demonstrate that the product eigenvalue problem is a powerful unifying concept diverse examples of eigenvalue problems are discussed and formulated as product eigenvalue problems for all but a couple of these examples it is shown that the standard algorithms for solving them are instances of a generic gr algorithm applied to a related cyclic matrix

characterization of uninorms on bounded lattices in this paper we propose a characterization of uninorms on a general bounded lattice l by means of a sextuple of a triangular norm a triangular conorm and four aggregation functions on l observe that in the case when l is a chain it is enough to consider a triple consisting of a t norm a t conorm and an averaging aggregation function an example of a proper uninorm which has an absorbing element differing from both 0 and 1 is also shown note that this cannot happen if we consider uninorms acting on a chain two general constructions of uninorms on l are added letting a t norm a t conorm linked to a fixed neutral element e of the constructed uninorm to be free and fixing the remaining functions

a symplectic method for approximating all the eigenvalues of a hamiltonian matrix a fast method for computing all the eigenvalues of a hamiltonian matrix m is given the method relies on orthogonal symplectic similarity transformations which preserve structure and have desirable numerical properties the algorithm is about four times faster than the standard q r algorithm the computed eigenvalues are shown to be the exact eigenvalues of a matrix m e where vert e vert depends on the square root of the machine precision the accuracy of a computed eigenvalue depends on its condition and its magnitude larger eigenvalues typically being more accurate

a numerical method for computing the hamiltonian schur form we derive a new numerical method for computing the hamiltonian schur form of a hamiltonian matrix mathcal m in mathbb r 2n times 2n that has no purely imaginary eigenvalues we demonstrate the properties of the new method by showing its performance for the benchmark collection of continuous time algebraic riccati equations despite the fact that no complete error analysis for the method is yet available the numerical results indicate that if no eigenvalues of mathcal m are close to the imaginary axis then the method computes the exact hamiltonian schur form of a nearby hamiltonian matrix and thus is numerically strongly backward stable the new method is of complexity mathbf o n 3 and hence it solves a long standing open problem in numerical analysis

the pfaff lattice on symplectic matrices the pfaff lattice is an integrable system arising from the sr group factorization in an analogous way to how the toda lattice arises from the qr group factorization in our earlier paper kodama and pierce 2007 int math res not arxiv 0705 0510 we studied the pfaff lattice hierarchy for the case where the lax matrix is defined to be a lower hessenberg matrix in this paper we deal with the case of a symplectic lower hessenberg lax matrix this forces the lax matrix to take a 2 2 block tridiagonal shape we then show that the odd members of the pfaff lattice hierarchy are trivial while the even members are equivalent to the indefinite toda lattice hierarchy defined in kodama and ye 1996 physica d 91 321 39 this is analogous to the case of the toda lattice hierarchy in relation to the kac van moerbeke system in the case with the initial matrix having only real or imaginary eigenvalues the fixed points of the even flows are given by 2 2 block diagonal matrices with zero diagonals we also consider a family of skew orthogonal polynomials with a symplectic recursion relation related to the pfaff lattice and find that they are succinctly expressed in terms of orthogonal polynomials appearing in the indefinite toda lattice