order equivalent triangular norms in this paper an equivalence relation on the class of t norms induced by a t partial order is provided and discussed the equivalence classes linked to some special t norms are characterized as well as some properties preserved by the introduced equivalence defining the set of incomparable elements with respect to the t partial order this set is deeply investigated finally we give an answer to a recently posed open problem

a simple and accurate complex image interpretation of vertical antennas present in contiguous dielectric half spaces an efficient technique is developed to solve the important problem of vertical antennas located above and penetrating the interface between contiguous dielectric half spaces it begins by deriving the spectral expressions of the vector and scalar potential green functions in terms of the tm wave spectral transmission coefficient the complex image technique is used to derive simple accurate and rapidly convergent expressions of the full wave spatial green functions the spatial green functions are used in the moment method to model vertical thin wire antennas residing above the penetrating the interface separating the contiguous dielectric half spaces with and without loss

constructing nested bases approximations from the entries of non local operators in this article a method for constructing nested bases approximations to large scale fully populated discretizations of integral operators is introduced the scheme uses only few of the matrix entries for approximating the whole matrix in this sense it is similar to the adaptive cross approximation method however its computational complexity is improved

acoustic propagation over an impedance plane a new solution is obtained for the acoustic pressure field generated by a point source and scattered by a locally reacting impedance covered plane the new solution is obtained in the form of an asymptotic series which has been shown to agree well with other studies one of the most important features of this solution is that higher order terms can be calculated from preceding terms in the series by the use of recursion formulas comparing data predicted from this solution with that from a numerical integration of the exact expression shows the asymptotic series to be extremely accurate even for relatively low values of the parameter kr not surprisingly the plane wave solution often shows major deviations from the exact integral solution

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

sound propagation above an impedance boundary an asymptotic solution of the scalar wave field due to a point source above a locally reacting plane surface is obtained by a modified saddle point method numerical calculation indicates that the relatively simple solution is more accurate than the thomasson asymptotic solution and the chien soroka solution furthermore it is shown that another solution simplified still more is also in good agreement with the exact solution

on the fast matrix multiplication in the boundary element method by panel clustering the boundary element method bem leads to a system of linear equations with a full matrix while fem yields sparse matrices this fact seems to require much computational work for the definition of the matrix for the solution of the system and in particular for the matrix vector multiplication which always occurs as an elementary in this paper a method for the approximate matrix vector multiplication is described which requires much less arithmetical work in addition the storage requirements are strongly reduced

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

generalized continuous and left continuous t norms arising from algebraic semantics for fuzzy logics this paper focuses on the issue of how generalizations of continuous and left continuous t norms over linearly ordered sets should be from a logical point of view taking into account recent results in the scope of algebraic semantics for fuzzy logics over chains with a monoidal residuated operation we advocate linearly ordered bl algebras and mtl algebras as adequate generalizations of continuous and left continuous t norms respectively in both cases the underlying basic structure is that of linearly ordered residuated lattices although the residuation property is equivalent to left continuity in t norms continuous t norms have received much more attention due to their simpler structure we review their complete description in terms of ordinal sums and discuss the problem of describing the structure of their generalization to bl chains in particular we show the good behavior of bl algebras over a finite or complete chain and discuss the partial knowledge of rational bl chains then we move to the general non continuous case corresponding to left continuous t norms and mtl chains the unsolved problem of describing the structure of left continuous t norms is presented together with a fistful of construction decomposition techniques that apply to some distinguished families of t norms and finally we discuss the situation in the general study of mtl chains as a natural generalization of left continuous t norms

tt cross approximation for multidimensional arrays abstract as is well known a rank r matrix can be recovered from a cross of r linearly independent columns and rows and an arbitrary matrix can be interpolated on the cross entries other entries by this cross or pseudo skeleton approximation are given with errors depending on the closeness of the matrix to a rank r matrix and as well on the choice of cross in this paper we extend this construction to d dimensional arrays tensors and suggest a new interpolation formula in which a d dimensional array is interpolated on the entries of some tt cross tensor train cross the total number of entries and the complexity of our interpolation algorithm depend on d linearly so the approach does not suffer from the curse of dimensionality we also propose a tt cross method for computation of d dimensional integrals and apply it to some examples with dimensionality in the range from d 100 up to d 4000 and the relative accuracy of order 10 10 in all constructions we capitalize on the new tensor decomposition in the form of tensor trains tt decomposition

problems on triangular norms and related operators abstract a number of open problems on triangular norms and related operators was posed during the 24th linz seminar on fuzzy set theory triangular norms and related operators in many valued logics held in february 2003 they are collected here together with some other open problems in this context and with some problems which were posed earlier and have been solved in the meantime

distributive n ary semicopulas on lattices in the present paper definitions of an n ary semicopula on a bounded lattice l a distributive n ary semicopula on l and an infinite distributive n ary semicopula on l are introduced a relation between the set of all bounded lattices l with a distributive n ary semicopula on l and the set of all bounded lattices l with a distributive m ary semicopula on l where 2 n 2 m n m is obtained the method is given for investigation of the following questions a description of all bounded lattices l such that every n ary semicopula on l is distributive and a description of all complete lattices l such that every n ary semicopula on l is infinite distributive

a note on reflection of spherical waves in 1909 sommerfeld gave an exact solution for the reflection of a spherical wave from a plane surface in terms of an oscillatory integral and also presented an asymptotic solution for the case where both source and receiver are at the boundary weyl 1919 presented an alternative solution and also an asymptotic solution for the case where the source is at the boundary it is known that the general case is solved if a general solution for the case where the source is at the boundary is known here it is demonstrated that it is sufficient to have the general solution for the case where both source and receiver are at the boundary this is mainly of theoretical interest but may have practical applications as an example it is demonstrated that sommerfeld s approximate solution gives ingard s 1951 approximate solution which is valid for arbitrary source and receiver heights

taylor expansion based fast multipole method for 3 d helmholtz equations in layered media abstract in this paper we develop a taylor expansion te based fast multipole method fmm for low frequency 3d helmholtz green s function in layered media two forms of taylor expansions with either non symmetric or symmetric derivatives of layered media green s functions are used for the implementations of the proposed te fmm in the implementation with non symmetric derivatives an algorithm based on discrete complex image approximations and recurrence formulas is shown to be very efficient and accurate in computing the high order derivatives meanwhile the implementation based on symmetric derivatives is more robust and pre computed tables for the high order derivatives in translation operators are used numerical tests in layered media have validated the accuracy and o n complexity of the proposed algorithms

construction of data sparse mathcal h 2 matrices by hierarchical compression hierarchical matrices mathcal h matrices provide an elegant approach to handling large densely populated matrices the matrix is split into a hierarchy of blocks and each block is approximated by a low rank matrix in factorized form it has been demonstrated that this representation can be used to treat integral and partial differential equations solve matrix equations from the field of control theory and evaluate matrix functions efficiently mathcal h 2 matrices use a refined representation that employs a multilevel structure in order to reduce the storage requirements of hierarchical matrices it has been shown that mathcal h 2 matrices can significantly reduce storage requirements for large problems in particular when combined with modern error control schemes until now all algorithms for constructing an efficient approximation of a general matrix by an mathcal h 2 matrix required a representation of the entire original matrix to be kept in storage therefore the storage requirements of mathcal h 2 matrix algorithms could be far larger than those of the final approximation this paper presents a new approach that allows us to construct an mathcal h 2 matrix without storing the entire original matrix the central idea is to approximate submatrices and combine them by an efficient new algorithm to form approximations of larger matrices until the entire matrix has been treated using this new approach many mathcal h matrix algorithms can be refitted easily to compute results in the more efficient representation possible applications include efficient matrix arithmetics for the construction of preconditioners the approximation of matrix functions or solutions of matrix equations or efficient compression schemes based on the popular cross approximation algorithms

a note on the continuity of triangular norms abstract it is shown that for an associative function on the unit interval with certain boundary conditions separate continuity implies joint continuity this answers a question on triangular norms raised by alsina frank and schweizer in 2003

on the direct decomposability of pseudo t norms t norms and implication operators on product lattices in this paper we discuss the direct decomposability of pseudo t norms and t norms on a product lattice characterize the implication operator on a product lattice that is a direct product of two implication operators and study the direct decomposability of s implications r implications n reciprocal r implications and n reciprocal ql implications on a product lattice

approximation of boundary element matrices this article considers the problem of approximating a general asymptotically smooth function in two variables typically arising in integral formulations of boundary value problems by a sum of products of two functions in one variable from these results an iterative algorithm for the low rank approximation of blocks of large unstructured matrices generated by asymptotically smooth functions is developed this algorithm uses only few entries from the original block and since it has a natural stopping criterion the approximative rank is not needed in advance

sound propagation above an inhomogeneous impedance plane abstract boundary integral equation bie methods are described for the prediction of sound propagation in particular from a line source over a flat plane of inhomogeneous impedance approximate methods which satisfy reciprocity for the calculation of the wave field over a two impedance plane are proposed these approximations applied to propagation from a line source give results agreeing well with those of the bie method when they are applied to propagation from a point source agreement with experiment is shown

uncertainty modeling by bilattice based squares and triangles in this paper ginsberg s fitting s theory of bilattices and in particular the associated constructs of bilattice based squares and triangles is introduced as an attractive framework for the representation of uncertain and potentially conflicting information paralleling goguen s l fuzzy set theory we recall some of the advantages of bilattice based frameworks for handling fuzzy sets and systems provide the related structures with adequately defined graded versions of the basic logical connectives and study their properties and relationships

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

constructions of triangular norms on lattices by means of irreducible elements constructions and characterizations of triangular norms t norms have been discussed in many different contexts in this paper we present two methods to construct t norms on lattices from given partial information since the idea is to follow from part to whole we especially consider the lattices with decompositions firstly we investigate the structure of the partially ordered set of irreducible elements then for a given t norm t on the complete distributive lattice 0 1 2 we study the restriction of t to the poset of irreducible elements of 0 1 2 furthermore we give a method for generating t norms on a finite distributive lattice l by means of irreducible elements in l we show that a t norm on a finite distributive lattice is idempotent if and only if it is idempotent on the set of irreducible elements finally we introduce a formula to obtain t norms on l n from given t norms on irreducible elements of l n we present the dual statements for t conorms

on the direct decomposability of strong negations and s implication operators on product lattices in this paper we discuss the direct product of strong negations we characterize strong negations on product lattices that are direct products of strong negations furthermore we present a method for constructing t norms on product lattices which are not direct products this method is quite general and useful to generate a wide class of such t norms and solves an open problem posed in 2003 by jenei and de baets finally we study direct decompositions of s implication operators on product lattices

internal direct product of integral ν distributive binary aggregation functions and the classification of all integral ν distributive binary aggregation functions of length 3 we present some relations between an external direct product and an internal direct product of a family of integral distributive binary aggregation functions relations between a direct decomposition of the unit and a family of pairwise comaximal elements in a lattice with an integral distributive binary aggregation function are investigated using these results the classification of all integral distributive binary aggregation functions on a lattice of length 3 and a description of all lattices of length 3 with an integral distributive binary aggregation function are given

distributive and infinitely distributive t norms on complete lattices abstract a relation between the direct decomposability of an infinitely distributive t norm on a complete lattice l and direct decompositions of the neutral element 1 of l is obtained some useful applications of this relation are given in particularly the description of all t 1 lattices without infinitely distributive t norms on complete lattices and the description of all distributive t norms on lattices of length 2 are given the paper also contains the solutions of two open problems

the complex equivalent source method for sound propagation over an impedance plane the sound field caused by a monopole source above an impedance plane can be calculated by using a superposition of equivalent point sources located along a line in the mirror space below the plane originally such an approach for representing the half space green s function was described by sommerfeld at the beginning of the last century in order to treat half space problems of heat conduction however the representation converges only for masslike impedances and cannot be used for the more important case of reflecting planes with springlike surface impedances the singular part of the line integral can be transformed into a hankel function which shows that surface waves are contained in the whole solution unfortunately this representation suffers from the lack of validity at certain receiver points and from restrictions on wave number and impedance range to ensure the necessary convergence the main idea of the present method is to use also a superposition of equivalent point sources but to allow that these sources can be located at complex source points the corresponding form of the half space green s function is suitable for both masslike and springlike surface impedances and can be used as a cornerstone for a boundary element method

black box approximation of tensors in hierarchical tucker format we derive and analyse a scheme for the approximation of order d tensors a rn1 nd in the hierarchical h tucker format a dimension multilevel variant of the tucker format and strongly related to the tt tensor train format for a fixed rank parameter k the storage complexity of a tensor in h tucker format is odk3 k i 1dni and we present a heuristic algorithm that finds an approximation to a tensor in the h tucker format in odk4 log d k2 i 1dni by inspection of only odk3 log d k2 i 1dni entries under mild assumptions tensors in the h tucker format are reconstructed for general tensors we derive error bounds that are based on the approximability of matrices matricizations of the tensor by few outer products of its rows and columns the construction parallelizes with respect to the order d and we also propose an adaptive approach that aims at finding the rank parameter for a given target accuracy e automatically

on the direct decomposability of t norms on product lattices in this paper a method is presented for constructing t norms on product lattices in other words commutative partially ordered integral monoids over product lattices which are not direct products the method is fairly general and allows to generate a broad class of such t norms this solves an open problem posed in 1999 by de baets and mesiar

near field computation for sound propagation above ground using complex image theory in this paper a very simple method is introduced for the efficient computation of the sound field in the near region above an impedance ground or an extended reaction ground the procedure is based on complex image theory using prony s method the slowly varying factor in the integrand of the sommerfeld integral is approximated by a short series in terms of complex exponential functions the result can be exactly evaluated by the sommerfeld identity a new closed form approximate expression of the sound field is obtained which is represented as the original source its quasistatic image and several images with complex amplitudes and positions avoiding numerical evaluation of the sommerfeld integral makes these closed form expressions computationally very efficient the representation of the field with complex images makes the formula useful in measurement of ground impedance or in other cases requiring repeated mapping of the sound field numerical results show that these closed form green s functions are very accurate for the computation of the near field

infinitely distributive t norms on complete lattices and pseudo complements in this paper we have introduced the concepts of t prime element t prime radical element and t semi prime element and have studied their properties we have investigated the construction methods of new infinitely distributive t norms from given infinitely distributive t norms by means of t prime elements and pseudo complements if every element of l is t prime radical element then we show that l is a complete brouwerian lattice and email protected if l is an algebraic lattice t is a t norm on l and every compact element of l is idempotent then we obtain that email protected we show that l r is a complete lattice and that the restriction of t to l r is an infinitely distributive t norm on l r when t is an infinitely distributive t norm on l and 0 is a t prime we present infinitely distributive t norms on a complete sublattice l and study their properties

broadband impedance boundary conditions for the simulation of sound propagation in the time domain an accurate and practical surface impedance boundary condition in the time domain has been developed for application to broadband frequency simulation in aeroacoustic problems to show the capability of this method two kinds of numerical simulations are performed and compared with the analytical experimental results one is acoustic wave reflection by a monopole source over an impedance surface and the other is acoustic wave propagation in a duct with a finite impedance wall both single frequency and broadband frequency simulations are performed within the framework of linearized euler equations a high order dispersion relation preserving finite difference method and a low dissipation low dispersion runge kutta method are used for spatial discretization and time integration respectively the results show excellent agreement with the analytical experimental results at various frequencies the method accurately predicts both the amplitude and the phase of acoustic pressure and ensures the well posedness of the broadband time domain impedance boundary condition