semilinear copulas a family of copulas called semilinear is constructed starting with some assumptions about the linearity of the copulas along some segments of the unit square this family contains some other known families of copulas e g cuadras auge frechet and has a nice statistical interpretation several construction methods are provided especially concerning aggregation of semilinear copulas and a special form of ordinal sum construction is introduced some results about related families of quasi copulas and semicopulas are hence given

the pseudo linear semantics of interval valued fuzzy logics triangle algebras are equationally defined structures that are equivalent with certain residuated lattices on a set of intervals which are called interval valued residuated lattices ivrls triangle algebras have been used to construct triangle logic tl a formal fuzzy logic that is sound and complete w r t the class of ivrls in this paper we prove that the so called pseudo prelinear triangle algebras are subdirect products of pseudo linear triangle algebras this can be compared with mtl algebras prelinear residuated lattices being subdirect products of linear residuated lattices as a consequence we are able to prove the pseudo chain completeness of pseudo linear triangle logic ptl an axiomatic extension of tl introduced in this paper this kind of completeness is the analogue of the chain completeness of monoidal t norm based logic mtl this result also provides a better insight in the structure of triangle algebras it enables us amongst others to prove properties of pseudo prelinear triangle algebras more easily it is known that there is a one to one correspondence between triangle algebras and couples l a in which l is a residuated lattice and a an element in that residuated lattice we give a schematic overview of some properties of pseudo prelinear triangle algebras and a number of others that can be imposed on a triangle algebra and the according necessary and sufficient conditions on l and a

local extinction versus local exponential growth for spatial branching processes let x be either the branching diffusion corresponding to the operator lu u2 u on d rd where x 0 and 0 is bounded from above or the superprocess corresponding to the operator lu u u2 on d rd with 0 and is bounded from above but no restriction on its sign let c denote the generalized principal eigenvalue for the operator l on d we prove the following dichotomy either c 0 and x exhibits local extinction or c 0 and there is exponential growth of mass on compacts of d with rate c for superdiffusions this completes the local extinction criterion obtained by pinsky ann probab 24 1996 237 267 and a recent result on the local growth of mass under a spectral assumption given by englander and turaev ann probab 30 2002 683 722 the proofs in the above two papers are based on pde techniques however the proofs we offer are probabilistically conceptual for the most part they are based on spine decompositions or immortal particle representations along with martingale convergence and the law of large numbers further they are generic in the sense that they work for both types of processes

the standard completeness of interval valued monoidal t norm based logic in this paper we prove the strong standard completeness of interval valued monoidal t norm based logic ivmtl and some of its extensions for other extensions we show that they are not strong standard complete we also give a local deduction theorem for ivmtl and other extensions of interval valued monoidal logic similar results are obtained for interval valued fuzzy logics expanded with baaz s delta

the ł and ł propositional and predicate logics abstract this paper has two main goals the first goal is to show a different axiomatic system of the and 1 2 propositional logics these propositional logics were introduced in esteva et al arch math logic to appear and they are the combinations of the ukasiewicz and the product logic together with the constant 1 2 in case of the 1 2 logic the second goal is to show an axiomatic system and the completeness theorem of a predicate version of the and 1 2 propositional logics it will be shown that godel product and ukasiewicz predicate logics are contained in the logic

a note on invariance of the eigenvalues singular values and norms of matrix products involving generalized inverses abstract it is shown that the set of singular values of the product ab c as well as any norm of this product is invariant with respect to the choice of a generalized inverse b if and only if ab c is invariant itself this is in contrast with the corresponding invariance property of the set of eigenvalues of ab c for which the invariance of ab c is not necessary

aggregation functions construction methods conjunctive disjunctive and mixed classes in this second part of our state of the art overview on aggregation theory based again on our recent monograph on aggregation functions we focus on several construction methods for aggregation functions and on special classes of aggregation functions covering the well known conjunctive disjunctive and mixed aggregation functions some fields of applications are included

aggregation based extensions of fuzzy measures we present a method extending fuzzy measures on n 1 n represented as boolean utility functions to n ary aggregation functions utility functions by means of a suitable n ary aggregation function and the mobius transform of the considered fuzzy measure the method generalizes the well known lovasz and owen extensions of nondecreasing pseudo boolean functions linked to fuzzy measures all n ary aggregation functions suitable for the proposed construction are completely characterized including among others all n ary copulas associative extended aggregation functions applicable in the case of an arbitrary arity are also completely characterized

mathematical aspects of fuzzy sets and fuzzy logic the paper considers some of the main trends of the development of mathematical fuzzy logic and of formalized fuzzy set theory which came out in the history of fuzzy sets and which aim to give solid theoretical foundations for fuzzy set theory

2 increasing binary aggregation operators abstract in this work we investigate the class of binary aggregation operators agops satisfying the 2 increasing property obtaining some characterizations for agops having other special properties e g quasi arithmetic mean choquet integral based modularity and presenting some construction methods in particular the notion of p increasing function is used in order to characterize the composition of 2 increasing agops the lattice structure with respect to the pointwise order of some subclasses of 2 increasing agops is presented finally a method is given for constructing copulas beginning from 2 increasing and 1 lipschitz agops

a new axiomatization for involutive monoidal t norm based logic on the real unit interval the notion of a girard monoid coincides with the notion of a t norm based residuated lattice with strong induced negation a geometrical approach toward these girard monoids based on the notion of rotation invariance is turned in an adequate axiomatization for the involutive monoidal t norm based residuated logic

absolutely continuous copulas with given sub diagonal section abstract recently durante and jaworski 2008 6 have proved that the class of absolutely continuous copulas with a given diagonal section is non empty in case that the diagonal function is such that the set of points where this coincides with the identity function has null measure in this paper we show that if we consider sub diagonals or super diagonals then the framework changes concretely for each sub diagonal function there exists an absolutely continuous copula having this function as sub diagonal section

an almost sure scaling limit theorem for dawson watanabe superprocesses abstract we establish a scaling limit theorem for a large class of dawson watanabe superprocesses whose underlying spatial motions are symmetric hunt processes where the convergence is in the sense of convergence in probability when the underling process is a symmetric diffusion with c b 1 coefficients or a symmetric levy process on r d whose levy exponent is bounded from below by c for some c 0 and 0 2 when is large a stronger almost sure limit theorem is established for the superprocess our approach uses the principal eigenvalue and the ground state for some associated schrodinger operator the limit theorems are established under the assumption that an associated schrodinger operator has a spectral gap

on the conditioned exit measures of super brownian motion in this paper we present a martingale related to the exit measures of super brownian motion by changing measure with this martingale in the canonical way we have a new process associated with the conditioned exit measure this measure is shown to be identical to a measure generated by a non homogeneous branching particle system with immigration of mass an application is given to the problem of conditioning the exit measure to hit a number of specified points on the boundary of a domain the results are similar in flavor to the immortal particle picture of conditioned super brownian motion but more general as the change of measure is given by a martingale which need not arise from a single harmonic function

aggregation functions means this two part state of the art overview on aggregation theory summarizes the essential information concerning aggregation issues an overview of aggregation properties is given including the basic classification on aggregation functions in this first part the stress is put on means i e averaging aggregation functions both with fixed arity n ary means and with multiple arities extended means

two representations of a conditioned superprocess we consider a class of measure valued markov processes constructed by taking a superprocess over some underlying markov process and conditioning it to stay alive forever we obtain two representations of such a process the first representation is in terms of an immortal particle that moves around according to the underlying markov process and throws off pieces of mass which then proceed to evolve in the same way that mass evolves for the unconditioned superprocess as a consequence of this representation we show that the tail field of the conditioned superprocess is trivial if the tail field of the underlying process is trivial the second representation is analogous to one obtained by legall in the unconditioned case it represents the conditioned superprocess in terms of a certain process taking values in the path space of the underlying process this representation is useful for studying the transience and recurrence properties of the closed support process

the matrix inequality m b mb abstract the loewner partial ordering between nonnegative definite matrices m and b mb is considered the main result obtained generalizes a number of results known in the literature the rank subtractivity partial ordering and the drazin partial ordering between m and b mb are also discussed

on a family of multivariate copulas for aggregation processes we introduce a family of multivariate copulas a special type of n ary aggregation operations depending on a univariate function this family is used in the construction of a special aggregation operation that satisfies a lipschitz condition several examples are provided and some statistical properties are studied

further results on invariance of the eigenvalues of matrix products involving generalized inverses abstract it is shown that if m n n n and p m complex matrices a b and c do not satisfy at least one of the range inclusions r a c r b and r ca r b then for each complex number there is a choice of a generalized inverse b such that this number is an eigenvalue of the product ab c combined with criteria derived by baksalary and puntanen 1990 and by baksalary and pukkila 1992 this result implies that the invariance of the magnitude largest eigenvalue and or the magnitude smallest nonzero eigenvalue of ab c respect to the choice of b is equivalent to the invariance of the set of all eigenvalues

migrative t norms with respect to continuous ordinal sums abstract in this paper we study and characterize continuous migrative t norms t with respect to a continuous t norm t0 depending on whether is an idempotent element of t0 or not the t0 migrative property restricts the ordinal sum structure of t especially locally i e at or around it outside this well defined neighbourhood of the t norm t can be arbitrary under the only condition of keeping it continuous the investigation exploits the ordinal sum structure of continuous t norms and our former results related to the migrative property

lipschitzian de morgan triplets of fuzzy connectives after studying lipschitzian strong negations lipschitzian de morgan triplets t s n where t is a triangular norm s a triangular conorm and n a strong negation are investigated the relationships between the best lipschitzian constants of functions t s and n are given several examples are included

on triangular norm based propositional fuzzy logics abstract fuzzy logics based on triangular norms and their corresponding conorms are investigated an affirmative answer to the question whether in such logics a specific level of satisfiability of a set of formulas can be characterized by the same level of satisfiability of its finite subsets is given tautologies contradictions and contigencies with respect to such fuzzy logics are studied in particular for the important cases of min max and ukasiewicz logics finally fundamental t norm based fuzzy logics are shown to provide a gradual transition between min max and ukasiewicz logics

on continuous triangular norms that are migrative in this paper we completely describe all continuous migrative triangular norms since the migrative property excludes both idempotent and nilpotent classes the characterization and construction is carried out by solving a functional equation for additive generators of strict t norms we also study cases when the construction results in a smooth generator

limit theorems for branching markov processes abstract we establish almost sure limit theorems for a branching symmetric hunt process in terms of the principal eigenvalue and the ground state of an associated schrodinger operator here the branching rate and the branching mechanism can be state dependent in particular the branching rate can be a measure belonging to a certain kato class and is allowed to be singular with respect to the symmetrizing measure for the underlying hunt process x the almost sure limit theorems are established under the assumption that the associated schrodinger operator of x has a spectral gap such an assumption is satisfied if the underlying process x is a brownian motion a symmetric stable like process on r n or a relativistic symmetric stable process on r n

supermigrative semi copulas and triangular norms a semi copula s 0 1 2 0 1 is called supermigrative if it is commutative and satisfies s ax y s x ay for all email protected 0 1 and for all x email protected 0 1 such that y

a scaling limit theorem for a class of superdiffusions consider the finite measure valued diffusion corresponding to the evolution equation u t lu x u f x u where f x u x u 2 e ku 1 k u n x dk and n is a smooth kernel satisfying an integrability condition we assume that c r d with 0 1 and a 0 under appropriate spectral theoretical assumptions we prove the existence of the random measure e c t x t dx with respect to the vague topology where c is the generalized principal eigenvalue of l on r d and it is assumed to be finite and positive completing a result of pinsky on the expectation of the rescaled process moreover we prove that this limiting random measure is a nonnegative nondegenerate random multiple of a deterministic measure related to the operator l when is bounded from above x is finite measure valued in this case under an additional assumption on l we can actually prove the existence of the previous limit with respect to the weak topology as a particular case we show that if l corresponds to a positive recurrent diffusion y and is a positive constant then e t x t dx exists and equals a nonnegative nondegenerate random multiple of the invariant measure for y taking l 1 2 on r and replacing by 0 super brownian motion with a single point source we prove a similar result with c replaced by 1 2 and with the deterministic measure e x dx giving an answer in the affirmative to a problem proposed by englander and fleischmann stochastic process appl 88 2000 37 58 the proofs are based upon two new results on invariant curves of strongly continuous nonlinear semigroups

lowner partial ordering and space preordering of hermitian non negative definite matrices abstract if a matrix a is below a matrix b with respect to the lowner partial ordering within the set of hermitian non negative definite matrices then the columns of a are linear combinations of the columns of b in this note the class of all such combinations providing a complete characterization of the lowner partial ordering is presented in addition new results on partial ordering of squares of hermitian non negative definite matrices are given

rank invariance criterion and its application to the unified theory of least squares abstract necessary and sufficient conditions are established for the product ab c to have its rank invariant with respect to the choice of a generalized inverse b in particular cases these conditions coincide with the results of mitra they are discussed also in the statistical context of the unified theory of least squares introduced by rao

on α β homogeneous copulas we introduce and characterize the family of a b homogeneous copulas and study some of its salient properties such as the greatest and the lowest copulas measures of association and the relationships with the a migrativity and invariant copulas

shift invariant binary aggregation operators when applying an information fusion method usually the output results are required to be a scalar value in multiple situations the application of the fusion method to input values shifted by a constant should result in an output that is shifted likewise in this paper we present a study of the shift invariance property of aggregation operators the relationship between shift invariant binary aggregation operators and 1 lipschitz aggregation operators is shown and a full description of shift invariant aggregation operators is given

on copulas and their diagonals we formulate the necessary and sufficient conditions for a function d 0 1 0 1 to be the diagonal section of a multivariate absolutely continuous copula moreover we provide some simple analytic formulas for copulas having given diagonal section or given distribution functions of order statistics

vines a new graphical model for dependent random variables a new graphical model called a vine for dependent random variables is introduced vines generalize the markov trees often used in modelling high dimensional distributions they differ from markov trees and bayesian belief nets in that the concept of conditional independence is weakened to allow for various forms of conditional dependence

on representations of 2 increasing binary aggregation functions in this paper we provide two different representations of 2 increasing binary aggregation functions by means of their lower and upper margins and a suitable copula

on the class of bivariate archimax copulas under constraints abstract the class of bivariate archimax copulas spans a broad range of dependence however its size is much reduced when the degree of association is prespecified the extent of this reduction is studied for a large set of measures of association the cases of exchangeable and non exchangeable archimax copulas are both considered

a note on the convex combinations of triangular norms we consider a property on binary operations which we call a migrativity and we use it to obtain t norms by means of convex combinations of two t norms one of them being discontinuous

asymmetric linkages maxmin vs reflected maxmin copulas abstract in this paper we introduce some new copulas emerging from shock models it was shown in 21 that reflected maxmin copulas rmm for short are not just some specific singular copulas they contain many important absolutely continuous copulas including the negative quadrant dependent half of the eyraud farlie gumbel morgenstern class the main goal of this paper is to develop the rmm copulas with dependent endogenous shocks and give evidence that rmm copulas may exhibit some characteristics better than the original maxmin copulas mm for short 1 an important evidence for that is the iteration procedure of the rmm transformation which we prove to be always convergent and we give many properties of it that are useful in applications 2 using this result we find also the limit of the iteration procedure of the mm transformation thus answering a question proposed in 10 3 we give the multivariate dependent rmm copula that compares to the mm version given in 10 in all our copulas the idiosyncratic and systemic shocks are combined via asymmetric linking functions as opposed to marshall copulas where symmetric linking functions are used

left star and right star partial orderings abstract two partial orderings in the set of complex matrices are introduced by combining each of the conditions a a a b and aa ba which define the star partial ordering with one of the conditions m a m b and m a m b which define the space preordering several properties of these orderings are examined with main emphasis on comparing the new orderings with the star ordering the minus ordering and other related partial orderings moreover some further characterizations of partial orderings in terms of inclusions of appropriate classes of generalized inverses are derived with the main emphasis on characterizations involving reflexive generalized inverses

on the lattice structure of the set of supermodular quasi copulas abstract we study the relationship between the poset of supermodular n quasi copulas and the posets of n quasi copulas and n copulas it is known that for n 3 the poset of n quasi copulas is not order isomorphic to the dedekind macneille completion of the poset of n copulas we complement the latter result by showing that for n 3 the structure of the poset of n quasi copulas is more closely related to that of the poset of supermodular n quasi copulas than that of the poset of n copulas

spectrum and trace invariance criterion and its statistical applications abstract necessary and sufficient conditions are established for the set of nonzero eigenvalues of the product b a to be invariant with respect to the choice of a generalized inverse b it turns out that the same conditions constitute a criterion for the invariance of trace b a the algebraic result derived is then applied to investigate some properties of canonical correlations in the context of the general gauss markov model m y x 2 v in particular the results of watson 1967 and puntanen 1987 are generalized and a new characterization of the best linear unbiased estimator of x under the model m is established

on the construction and support properties of measure valued diffusions on d subseteq mathbb r d with spatially dependent branching in this paper we construct a measure valued diffusion on d r d whose underlying motion is a diffusion process with absorption at the boundary corresponding to an elliptic operator l 1 2 a b on d r d and whose spatially dependent branching term is of the form x z x z 2 x d where satisfies a very general condition and 0 in the case that and are bounded from above we show that the measure valued process can also be obtained as a limit of approximating branching particle systems we give criteria for extinction survival recurrence transience of the support compactness of the support compactness of the range and local extinction for the measure valued diffusion we also present a number of examples which reveal that the behavior of the measure valued diffusion may be dramatically different from that of the approximating particle systems

new constructions of diagonal patchwork copulas we present a method for constructing symmetric copulas which generalizes the diagonal patchwork construction of copulas procedure we also show how it is related to a new construction of a generalized farlie gumbel morgenstern distribution and to the copula transforms

quasi copulas with a given sub diagonal section abstract as is well known the frechet hoeffding bounds are the best possible for both copulas and quasi copulas for every quasi copula q max x y 1 0 q x y min x y for all x y 0 1 sharper bounds hold when the quasi copulas take prescribed values e g along their diagonal or horizontal resp vertical sections here we pursue two goals first we investigate construction methods for quasi copulas with a given sub diagonal section i e with prescribed values along the straight line segment joining the points x 0 0 and 1 1 x 0 for x 0 0 1 then we determine the best possible bounds for sets of quasi copulas with a given sub diagonal section

spines skeletons and the strong law of large numbers for superdiffusions consider a supercritical superdiffusion x t t 0 on a domain d r d with branching mechanism x z x z x z 2 0 e zy 1 zy x dy the skeleton decomposition provides a pathwise description of the process in terms of immigration along a branching particle diffusion we use this decomposition to derive the strong law of large numbers slln for a wide class of superdiffusions from the corresponding result for branching particle diffusions that is we show that for suitable test functions f and starting measures f x t p f x t w p almost surely as t where w is a finite non deterministic random variable characterized as a martingale limit our method is based on skeleton and spine techniques and offers structural insights into the driving force behind the slln for superdiffusions the result covers many of the key examples of interest and in particular proves a conjecture by fleischmann and swart stochastic process appl 106 2003 141 165 for the super wright fisher diffusion

reflected maxmin copulas and modeling quadrant subindependence abstract copula models have become popular in different applications including modeling shocks in view of their ability to describe better the dependence concepts in stochastic systems the class of maxmin copulas was recently introduced by omladic and ru ic 21 it extends the well known classes of marshall olkin and marshall copulas by allowing the external shocks to have different effects on the two components of the system by a reflection flip in one of the variables we introduce a new class of bivariate copulas called reflected maxmin rmm copulas we explore their properties and show that symmetric rmm copulas relate to general rmm copulas similarly as semilinear copulas relate to marshall copulas we transfer that relation also to maxmin copulas we also characterize possible diagonal functions of symmetric rmm copulas

the logic of tied implications part 1 properties applications and representation a conjunction t ties an implication operator a if the identity a a a b z a t a b z holds a a abdel hamid n n morsi associatively tied implications fuzzy sets and systems 136 2003 291 311 we study the class of tied adjointness algebras which are five connective algebras on two partially ordered sets in which the implications are tied by triangular norms this class contains besides residuated implications several other implications employed in fuzzy logic nevertheless we show that the algebraic inequalities of residuated algebras remain true for our tied implications but in forms that distribute roles over the five connectives of the algebra we apply the properties of tied implications to a generalized modus ponens inference scheme with two successive rules we prove its equivalence to a scheme with one compound rule when both schemata are interpreted by the compositional rule of inference and all connectives are taken from one tied adjointness algebra then we quote another application of this rich theory a notion of many valued rough sets which exhibit the basic mathematical behaviour of the rough sets of pawlak a comparator h is said to be prelinear if it satisfies h y z h z y 1 for all y z hajek we introduce prelinear tied adjointness algebras in which two comparators are prelinear we provide a representation of those algebras as subdirect products of tied adjointness chains on the lines of hajek s representation of bl algebras but our representations are more economical because we employ minimal prime filters on residuated lattices only rather than all prime filters

a note on archimax copulas and their representation by means of conic copulas abstract the aim of this note is to prove the equivalence between the class of conic copulas and the class of archimax copulas based on the lower frechet hoeffding bound w we demonstrate the transition from arbitrary conic copula to appropriate pickands dependence function and vice versa our results show that archimax copulas are isomorphic transforms of either ev copulas or conic copulas

kernel aggregation operators and their marginals binary kernel aggregation operators that is monotone binary operators on 0 1 with the chebyshev norm equal to 1 are studied marginal functions of such operators are shown to be 1 lipschitz several constructions of kernel aggregation operators with fixed marginal functions are given including maximal and minimal kernel aggregation operators with prescribed marginals finally kernel aggregation operators uniquely determined by their marginals are characterized

mathematical fuzzy logic as a tool for the treatment of vague information the paper considers some of the main trends of the recent development of mathematical fuzzy logic as an important tool in the toolbox of approximate reasoning techniques particularly the focus is on fuzzy logics as systems of formal logic constituted by a formalized language by a semantics and by a calculus for the derivation of formulas besides this logical approach also a more algebraic approach is discussed and the paper ends with some hints toward applications which are based upon these theoretical considerations

d dimensional dependence functions and archimax copulas we introduce a new class of dependence functions and method for constructing tail and pickands dependence functions finally we conjecture the structure of d dimensional archimax copulas and discuss some classes of d dimensional archimax copulas

kernel aggregation functions on finite scales constructions from their marginals abstract the study of discrete aggregation functions those defined on a finite chain with some kind of smoothness has been extensively developed in last years many different kinds of aggregation functions have been characterized in this context in this paper discrete aggregation functions with the kernel property which implies the smoothness property are investigated some properties and characterizations as well as some construction methods for this kind of discrete aggregation functions are studied it is also investigated when the marginal functions of a discrete kernel aggregation function fully determine it

trimmed trees and embedded particle systems in a supercritical branching particle system the trimmed tree consists of those particles which have descendants at all times we develop this concept in the superprocess setting for a class of continuous superprocesses with feller underlying motion on compact spaces we identify the trimmed tree which turns out to be a binary splitting particle system with a new underlying motion that is a compensated h transform of the old one we show how trimmed trees may be estimated from above by embedded binary branching particle systems

multivariate archimedean copulas d monotone functions and l1 norm symmetric distributions it is shown that a necessary and sufficient condition for an archimedean copula generator to generate a d dimensional copula is that the generator is a d monotone function the class of d dimensional archimedean copulas is shown to coincide with the class of survival copulas of d dimensional l 1 norm symmetric distributions that place no point mass at the origin the d monotone archimedean copula generators may be characterized using a little known integral transform of williamson duke math j 23 1956 189 207 in an analogous manner to the well known bernstein widder characterization of completely monotone generators in terms of the laplace transform these insights allow the construction of new archimedean copula families and provide a general solution to the problem of sampling multivariate archimedean copulas they also yield useful expressions for the d dimensional kendall function and kendall s rank correlation coefficients and facilitate the derivation of results on the existence of densities and the description of singular components for archimedean copulas the existence of a sharp lower bound for archimedean copulas with respect to the positive lower orthant dependence ordering is shown

measures for branching exit markov systems and their applications to differential equations semilinear equations lu u where l is an elliptic differential operator and is a positive function can be investigated by using l superdiffusions in a special case u u2 a powerful probabilistic tool the brownian snake introduced by le gall was successfully applied by him and his school to get deep results on solutions of this equation some of these results but not all of them were extended by dynkin and kuznetsov to general equations by applying superprocesses an important role in the theory of the brownian snake and its applications is played by measures open image in new windowx on the space of continuous paths our goal is to introduce analogous measures related to superprocesses and to general branching exit markov systems they are defined on the space of measures and we call them open image in new window measures using open image in new window measures allows to combine some advantages of brownian snakes and of superprocesses as tools for a study of semilinear pdes

on the α migrativity of semicopulas quasi copulas and copulas in this paper we address the problem of a migrativity for a fixed a for semicopulas copulas and quasi copulas we introduce the concept of an a sum of semicopulas this new concept allows us to completely characterize a migrative semicopulas and copulas moreover a sums also provide a means to obtain a partial characterization of a migrative quasi copulas

on the relationship between modular functions and copulas copulas are nothing else but supermodular functions with absorbing element 0 and neutral element 1 although a copula c cannot be modular on the unit square itself it is effectively so on every rectangle with zero c volume in this paper we show that an appropriate modular function on the unit square can be transformed into a copula by performing a simple truncation operation either from below or above to ensure the compatibility with the frechet hoeffding bounds modular functions on the unit square admit an additive representation in terms of two univariate functions we specifically focus on cases where these univariate functions are sections horizontal vertical diagonal of a given copula ample illustrations are provided

semi copulas capacities and families of level sets we point out how capacities non additive measures can be used to provide explanations about the concept of bivariate ageing function the latter had been introduced to describe the family of the level sets of a survival function and motivated the interest toward the concept of semi copula

fuzzy class theory the paper introduces a simple yet powerful axiomatization of zadeh s notion of fuzzy set based on formal fuzzy logic the presented formalism is strong enough to serve as foundations of a large part of fuzzy mathematics its essence is elementary fuzzy set theory cast as two sorted first order theory over fuzzy logic which is generalized to simple type theory we show a reduction of the elementary fuzzy set theory to fuzzy propositional calculus and a general method of fuzzification of classical mathematical theories within this formalism in this paper we restrict ourselves to set relations and operations that are definable without any structure on the universe of objects presupposed however we also demonstrate how to add structure to the universe of discourse within our framework

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

growth of levy trees we construct random locally compact real trees called levy trees that are the genealogical trees associated with continuous state branching processes more precisely we define a growing family of discrete galton watson trees with i i d exponential branch lengths that is consistent under bernoulli percolation on leaves we define the levy tree as the limit of this growing family with respect to the gromov hausdorff topology on metric spaces this elementary approach notably includes supercritical trees and does not make use of the height process introduced by le gall and le jan to code the genealogy of sub critical continuous state branching processes we construct the mass measure of levy trees and we give a decomposition along the ancestral subtree of a poisson sampling directed by the mass measure

orthogonal grid constructions of copulas we establish a general framework for constructing copulas that can be regarded as a patchwork like assembly of arbitrary copulas with nonoverlapping rectangles as patches we derive a family of construction methods that require the choice of a single background copula when this background copula is the greatest copula we retrieve the well known ordinal sum construction while in the case of the smallest copula we obtain a construction method that is dual to the ordinal sum construction also nonsingular background copulas lead to suitable construction methods

on the construction of radially symmetric copulas in higher dimensions abstract we prove a representation theorem for copulas that are simultaneously symmetric and radially symmetric we use this representation theorem to propose a method to construct an n ary symmetric function that is radially symmetric starting from an n 1 dimensional copula and an n ary auxiliary function we study the necessary and sufficient conditions on this auxiliary function that guarantee our construction method to result in a symmetric and radially symmetric n dimensional copula we examine several options for defining the auxiliary function in the trivariate case inspired by the nesting of copulas the lifting of copulas and product like extensions for each choice of auxiliary function we provide several examples for different families of copulas

some remarks on partial orderings of nonnegative definite matrices abstract baksalary and pukelsheim 1990 investigated partial orderings of nonnegative definite matrices some additional remarks are given in this note

transience recurrence and local extinction properties of the support for supercritical finite measure valued diffusions whose log laplace equation is associated with the semilinear equation ut lu flu au2 where a 8 0 and l 1 ed j l aij d92 xxi dxj ed 1 bi d 8xi a path x is said to survive if x t 0 for all t 2 0 since 0 p x q survives 0 for all 0 w u e4 rd where rd denotes the space of finite measures on rd we define transience recurrence and local extinction for the support of the supercritical measure valued diffusion starting from a finite measure as follows the support is recurrent if p x t b 0 for some t 2 0 i x survives 1 for every 0 u ej rd and every open set b c rd for d 2 the support is transient if p x t b 0 for some t 2 0 i x survives 1 for every ut ej rd and bounded b c rd which satisfy supp u n b 0 a similar definition taking into account the topology of r1 is given for d 1 the support exhibits local extinction if for each t e 4 rd and each bounded b c rd there exists a p l almost surely finite random time b such that x t b 0 for all t 2 b criteria for transience recurrence and local extinction are developed in this paper also studied is the

resolution of a system of the max product fuzzy relation equations using lou factorization in this paper the lu factorization is extended to the fuzzy square matrix with respect to the max product composition operator called l u factorization equivalently we will find two fuzzy lower and upper triangular matrices l and u for a fuzzy square matrix a such that a l u where is the max product composition an algorithm is presented to find the matrices l and u furthermore some necessary and sufficient conditions are proposed for the existence and uniqueness of the l u factorization for a given fuzzy square matrix a an algorithm is also proposed to find the solution set of a square system of fuzzy relation equations fre using the l u factorization the algorithm finds the solution set without finding its minimal solutions and maximum solution it is shown that the two algorithms have a polynomial time complexity as o n 3 since the determination of the minimal solutions is an np hard problem the algorithm can be very important from the practical point of view

interacting fisher wright diffusions in a catalytic medium we study the longtime behaviour of interacting systems in a randomly fluctuating space time medium and focus on models from population genetics there are two prototypes of spatial models in population genetics spatial branching processes and interacting fisher wright diffusions quite a bit is known on spatial branching processes where the local branching rate is proportional to a random environment catalytic medium

uniform convergence of martingales in the branching random walk in a discrete time supercritical branching random walk let z n be the point process formed by the n th generation let m lambda be the laplace transform of the intensity measure of z 1 then w n lambda int e lambda x z n dx m lambda n which is the laplace transform of z n normalized by its expected value forms a martingale for any lambda with m lambda finite but nonzero the convergence of these martingales uniformly in lambda for lambda lying in a suitable set is the first main result of this paper this will imply that on that set the martingale limit w lambda is actually an analytic function of lambda the uniform convergence results are used to obtain extensions of known results on the growth of z n nc d with n for bounded intervals d and fixed c this forms the second part of the paper where local large deviation results for z n which are uniform in c are considered finally similar results both on martingale convergence and uniform local large deviations are also obtained for continuous time models including branching brownian motion

some new characterizations and properties of quasi copulas in this paper we characterize the concept of quasi copula in three different ways as a special subclass of aggregation operators in terms of its associated volume and for the bivariate case in terms of non increasing tracks on 0 1 2 we also provide sufficient conditions and new properties for quasi copulas

conic aggregation functions inspired by the notion of conic t norms we introduce conic aggregation functions such aggregation functions are completely characterized by their zero set special classes of binary conic aggregation functions such as conic quasi copulas and conic copulas are considered we provide the necessary and sufficient conditions on the boundary curve of the zero set of a conic aggregation function to obtain a conic quasi copula and conclude that the class of conic copulas is a proper subclass of the class of conic quasi copulas moreover we characterize the class of singular conic copulas we investigate some aggregations of conic quasi copulas some examples are also provided

filters of residuated lattices and triangle algebras an important concept in the theory of residuated lattices and other algebraic structures used for formal fuzzy logic is that of a filter filters can be used amongst others to define congruence relations specific kinds of filters include boolean filters and prime filters in this paper we define several different filters of residuated lattices and triangle algebras and examine their mutual dependencies and connections triangle algebras characterize interval valued residuated lattices

on some construction methods for 1 lipschitz aggregation functions flipping method for constructing copulas from a given copula is extended to aggregation functions aggregation functions admitting flipping method are characterized and studied moreover the algebraic structure of several kinds of dualities on aggregation functions including the flipping method the reverse method and the standard duality is classified based on composition of construction methods several new construction methods for 1 lipschitz aggregation functions are introduced and their algebraic relationships are studied

pseudo bck algebras as partial algebras it is well known that the representation of several classes of residuated lattices involves lattice ordered groups an often applicable method to determine the representing group or groups from a residuated lattice is based on partial algebras the monoidal operation is restricted to those pairs that fulfil a certain extremality condition and else left undefined the subsequent construction applied to the partial algebra is easy transparent and leads directly to the structure needed for representation in this paper we consider subreducts of residuated lattices the monoidal and the meet operation being dropped the resulting algebras are pseudo bck semilattices assuming divisibility we can pass on to partial algebras also in this case to reconstruct the underlying group structure from this partial algebra if applicable is again straightforward we demonstrate the elegance of this method for two classes of pseudo bck semilattices semilinear divisible pseudo bck algebras and cone algebras

some further results on hermitian matrix inequalities abstract some new results on the lowner partial ordering between certain sums products and direct products of matrices are derived proofs of these results are based on a necessary and sufficient condition for the nonnegative definiteness of the difference of two hermitian matrices which follows from the criterion for the nonnegative definiteness of a partitioned hermitian matrix due to albert 1969

bivariate copulas generated by perturbations in this paper we provide a family of bivariate copulas based on a perturbation of a given copula by a factor term the new class generalizes well known families of copulas and allows to describe a wide range of possible dependencies

on the α migrativity of multivariate semi copulas in this paper we introduce and study the a migrative property for the class of multivariate semi copulas in particular a characterization of a migrative copulas is provided

flipping and cyclic shifting of binary aggregation functions we introduce two types of transformations of random variables called flipping and cyclic shifting as these transformations preserve monotonicity at the level of univariate cumulative distribution functions they can be used to develop corresponding coordinate wise transformations of binary aggregation functions we lay bare the admissibility of these transformations i e the necessary and sufficient conditions under which they result in a binary aggregation function we investigate which additional properties such as the 1 lipschitz property and 2 increasingness entail these admissibility conditions moreover we point out which of these properties are preserved under flipping and or cyclic shifting interestingly quasi copulas remain quasi copulas under flipping while copulas remain copulas under flipping as well as under cyclic shifting

extremal lipschitz continuous aggregation functions with a given diagonal section abstract in this paper we study the smallest and the greatest m lipschitz continuous n ary aggregation functions with a given diagonal section we show that several properties that were studied for the smallest and the greatest 1 lipschitz continuous binary aggregation functions with a given diagonal section extend naturally to higher dimensions while considering different lipschitz constants just as in the binary case we show that the smallest n quasi copula with a given diagonal section coincides with the smallest 1 lipschitz n ary aggregation function with that diagonal section additionally we show that the smallest n quasi copula with a given diagonal section called the bertino n quasi copula is supermodular for any n 2

monoidal t norm based logic towards a logic for left continuous t norms hajek s bl logic is the fuzzy logic capturing the tautologies of continuous t norms and their residua in this paper we investigate a weaker logic mtl which is intended to cope with the tautologies of left continuous t norms and their residua the corresponding algebraic structures mtl algebras are defined and completeness of mtl with respect to linearly ordered mtl algebras is proved besides several schematic extensions of mtl are also considered as well as their corresponding predicate calculi