distributivity equation in the class of 2 uninorms this paper is mainly devoted to solving the functional equation of distributivity between aggregation operators with 2 neutral element our investigations are motivated by the couple of distributive logical connectives and their generalizations used in fuzzy set theory e g triangular norms conorms uninorms nullnorms and implications one of the recent generalizations covering both uninorms and nullnorms are 2 uninorms which form a class of commutative associative and increasing operators on the unit interval with an absorbing element separating two subintervals having their own neutral elements in this work the distributivity of two binary operators from the class of 2 uninorms is considered in particular all possible solutions of the distributivity equation for the three defined subclasses of these operators depending on the position of its zero and neutral elements are characterized

some results about nullnorms on bounded lattices abstract nullnorms as a generalization of the concepts of t norms and t conorms admit a zero element to be any element in a bounded lattice in this paper we discuss the characterization of idempotent nullnorms on bounded lattices and investigate some of the main properties of them we propose some new methods yielding nullnorms in particular idempotent nullnorms on bounded lattices with the fixed zero element under some additional constraints moreover some illustrative examples are presented to demonstrate the feasibility of the proposed methods

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

medians and nullnorms on bounded lattices nullnorms are generalizations of triangular norms and triangular conorms with a zero element in the interior of the unit interval in this paper we work on nullnorms which are defined on an arbitrary bounded lattice we introduce a general median based method for constructing nullnorms by means of triangular norms and triangular conorms furthermore we highlight a significant difference between the existence and representation of idempotent nullnorms on chains distributive bounded lattices and general bounded lattices

construction of nullnorms on bounded lattices and an equivalence relation on nullnorms abstract in this study we present two methods for the construction of nullnorms on bounded lattices based on the existence of t norms as well as t conorms on an arbitrary bounded lattice l we demonstrate that these methods differ from the existing approaches mainly because of their more generalized structures an equivalence relation is also defined based on a class of nullnorms and some properties of this relation are examined

on the distributivity of uninorms over nullnorms this paper investigates the functional equation of distributivity of uninorms over nullnorms we consider the case where the uninorm is continuous in 0 1 2 or is representable it has been proved that the absorbing element of the nullnorm is an idempotent element of the uninorm if the distributive equation holds and then combining the structures of the uninorm and the nullnorm we give the characterization of the pair of the functions of the uninorm and the nullnorm moreover when the nullnorm is continuous we obtain the sufficient and necessary conditions under which the distributive equation holds

quasi static crack growth in hydraulic fracture abstract we present a variational model for the quasi static crack growth in hydraulic fracture in the framework of the energy formulation of rate independent processes the cracks are assumed to lie on a prescribed plane and to satisfy a very weak regularity assumption

idempotent nullnorms on bounded lattices abstract nullnorms are generalizations of triangular norms and triangular conorms with a zero element to be an arbitrary point from a bounded lattice in this paper we study and discuss the existence of idempotent nullnorms on bounded lattices considering an arbitrary distributive bounded lattice l we show that there exists a unique idempotent nullnorm on l we prove that an idempotent nullnorm may not always exist on an arbitrary bounded lattice furthermore we propose a construction method to obtain idempotent nullnorms on a bounded lattice l with an additional constraint on a for the given zero element a l 0 1

self similar solutions for elastohydrodynamic cavity flow the enlargement of a lens shaped cavity lying in a plane of cleavage between two elastic half spaces and filling with viscous fluid from a source on the axis of symmetry is considered the internal flow is modelled by lubrication theory which gives a nonlinear partial differential equation connecting the pressure to the cavity shape and the same two quantities are also related by the singular integral equation of linear elasticity if the total volume of fluid q t in the cavity at time t is proportional either to t or to exp t the resulting boundary value problem can be reduced to a self similar form in which time does not appear explicitly the solution in non dimensional terms depends on a single parameter which may be interpreted as the stress intensity factor k at the tip calculations have been made for the two dimensional version of the problem for a range of values of and for a range of stress intensities the numerical method is to expand the cavity height in a chebyshev series the coefficients being found by a nonlinear optimization technique to yield a least squares fit to the reynolds equation these lead to expressions for the rate of cavity growth and other quantities of physical interest

constructing uninorms via closure operators on a bounded lattice abstract in this paper via closure resp interior operators on a bounded lattice l we propose a rather effective method to construct uninorms on l with a given triangular norm t resp triangular conorm s on the subinterval 0 e resp e 1 of l our method encompasses as special cases the construction methods introduced by f karacal and r mesiar 2015 16 as well as by g d cayli et al 2016 5 we also exploit a pair of novel closure and interior operators on complete lattices and utilize this pair of operators to construct uninorms several interesting examples are included to illustrate our method

the modularity condition for uninorms and t operators in this work we study the functional equation given by the modularity condition f x g y z g f x y z for all z x where the unknown functions f g are uninorms and or t operators we characterize all the solutions in the four possible cases i when f g are t operators ii when f is a t operator and g a uninorm iii when f is a uninorm and g a t operator and finally iv when f g are uninorms

semi t operators on bounded lattices abstract we study semi t operators on bounded lattices and show the existence of semi t operators using the fact that there exist some pseudo t norms and pseudo t conorms on the arbitrary fixed bounded lattice l as a by product some constructive approaches to semi t operators are discussed using these approaches we obtain the least semi t operators and the greatest semi t operators on bounded lattices a partial order relation is also defined based on the semi t operator and some properties of this relation are examined

coupling schemes for modeling hydraulic fracture propagation using the xfem abstract we describe coupled algorithms that use the extended finite element method xfem to solve the elastic crack component of the elasto hydrodynamic equations that govern the propagation of hydraulic fractures in an elastic medium with appropriate enrichment the xfem resolves the neumann to dirichlet nd map for crack problems with o h 2 accuracy and the dirichlet to neumann dn map with o h accuracy for hydraulic fracture problems with a lag separating the fluid front from the fracture front we demonstrate that the finite pressure field makes it possible to use a scheme based on the o h 2 xfem solution to the nd map to treat problems in which there is a coalescence of the fluid and fracture fronts resulting in singular tip pressures we developed a novel mixed algorithm that combines the tip width asymptotic solution with the o h 2 xfem solution of the nd map away from the tips enrichment basis functions required for these singular pressure fields correspond to width power law indices 1 2 which are different from the index 1 2 of linear elastic fracture mechanics the solutions obtained from the new coupled xfem schemes agree extremely well with those of published reference solutions

a representation of nullnorms on a bounded lattice in terms of beam operations abstract we introduce three broad classes of nullnorms on a bounded lattice and lay bare the structure of their members for that purpose we introduce particular subsets of a bounded lattice called upper resp lower beams and appropriate associative operations on them called beam resp dual beam operations which conveniently generalize triangular norms resp triangular conorms it is shown that nullnorms in the first resp second class are characterized by a triangular conorm resp triangular norm and a beam operation resp dual beam operation while nullnorms in the third class are characterized by a triangular conorm and a triangular norm we also discuss the relationships among these three classes

separators in infinite words generated by morphisms resume nous appelons separateur d un mot x a l indice n e n le plus petit facteur de x dont l indice du debut de la premiere occurrence est n notons sx n la longueur de ce separateur lorsqu il existe et posons sx n 0 sinon nous montrons que l application sx ainsi definie est q reguli e re au sens d allouche et shallit lorsque x est un mot infini engendre par iteration d un morphisme q uniforme et que x est circulaire de maniere equivalente cela revient a dire que la serie formelle dont les coefficients sont les valeurs sx n en base q est n rationnelle salomaa soittola

on the structure of semi t operators on bounded lattices abstract in this study we show some construction methods to obtain a semi t operator from a given pseudo t conorm and pseudo t norm on bounded lattices using these methods we prove that the known methods of constructing semi t operators on bounded lattices proposed by fang and hu are incorrect and present the correct forms at the same time furthermore some other construction methods for semi t operators on bounded lattices are also added

aggregation functions on the set of discrete fuzzy numbers defined from a pair of discrete aggregations abstract in this paper we propose a method to construct aggregation functions on the set of discrete fuzzy numbers whose support is a set of consecutive natural numbers from a couple of discrete aggregation functions the interest on these discrete fuzzy numbers lies on the fact that they can be interpreted as linguistic expert valuations that increase the flexibility of the elicitation of qualitative information based on linguistic terms finally a linguistic decision making model based on a pair of aggregation functions defined on discrete fuzzy numbers is given

the distributivity condition for uninorms and t operators in this work we study the functional equation given by the distributivity condition f x g y z g f x y f x z for all x y z 0 1 where the unknown functions f g are uninorms and or t operators we characterize all the solutions in the four possible cases i when f g are t operators ii when f is a t operator and g a uninorm iii when f is a uninorm and g a t operator and finally iv when f g are uninorms

properties of words and recognizability of fixed points of a substitution resume soit u un mot infini non periodique point fixe d une substitution primitive nous montrons que pour n assez grand u ne contient aucune puissance n eme de mot nous donnons ensuite une condition necessaire et suffisante pour que soit reconnaissable au sens de host et queffelec et nous montrons qu ill suffit de bilateraliser cette definition pour obtenir une propriete verifiee cette fois par toutes les substitutions primitives a point fixe non periodique

the ring of k regular sequences ii in this paper we continue our study of k regular sequences begun in 1992 we prove some new results give many new examples from the literature and state some open problems

distributivity between uninorms and nullnorms recently the distributivity equation was investigated in families of certain operations e g triangular norms conorms uninorms and nullnorms generally the assumption of associativity is not necessary in consideration of distributivity equation moreover if we omit commutativity assumption consideration of the left and right distributivity conditions is reasonable previous results about distributivity between uninorms and nullnorms can be obtained as simple corollaries

multi polar t conorms and uninorms abstract transformation of uninorms defined on the unit interval to the bipolar scale yields bipolar t conorms and transformation of nullnorms on the unit interval yields bipolar t norms we use this relation in the investigation of the structure of bipolar and multi polar t norms and t conorms we use the same transformation of the unit interval into the bipolar scale in order to obtain bipolar aggregation operators from ordinal sum t norms and t conorms the extensions of these special bipolar aggregation operators generalize both multi polar t norms and t conorms and therefore we call them multi polar uninorms several examples of multi polar t conorms and uninorms are also presented

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 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

on the t partial order and properties for a t norm t on 0 1 a partial order t was recently defined and studied in this study for any fixed email protected 0 1 we define the set of incomparable elements according to t and this set is deeply investigated by means of the equivalence relation defined in 14 it is shown the set 0 1 denoting the set of all equivalence classes on t norms on 0 1 is uncountably infinite finally with the help of any t norm t on 0 1 it is obtained that the family t l l 0 1 of t norms on 0 1 if t is a divisible t norm then it is obtained that 0 1 t l is a lattice thus we give an answer to an open problem in 10

a model for the quasi static growth of brittle fractures existence and approximation results we give a precise mathematical formulation of a variational model for the irreversible quasi static evolution of brittle fractures proposed by g a francfort and j j marigo and based on griffith s theory of crack growth in the two dimensional case we prove an existence result for the quasi static evolution and show that the total energy is an absolutely continuous function of time although we cannot exclude the possibility that the bulk energy and the surface energy may present some jump discontinuities this existence result is proved by a time discretization process where at each step a global energy minimization is performed with the constraint that the new crack contains all cracks formed at the previous time steps this procedure provides an effective way to approximate the continuous time evolution

uni nullnorms on bounded lattices abstract in this paper we study uni nullnorms with both neutral elements and zero elements on bounded lattices in order to illustrate the existence of uni nullnorms on bounded lattices we present two construction methods for obtaining such a uni nullnorm also we show two additional methods for the construction of idempotent nullnorms on bounded lattices which are generalizations of the one proposed by cayli and karacal furthermore we discuss the presence of idempotent uni nullnorms on bounded lattices and propose two construction methods to obtain an idempotent uni nullnorm on any bounded lattice

approximation of functional depending on jumps by elliptic functional via t convergence we show how it is possible to approximate the mumford shah see 29 image segmentation functional by elliptic functionals defined on sobolev spaces the heuristic idea is to consider functionals h u z with z ranging between 0 and 1 and related to the set k the minimizing zh are near to 1 in a neighborhood of the set k and far from the neighborhood they are very small the neighborhood shrinks as h for a similar approach to the problem compare kulkarni see 25 the approximation of h to takes place in a variational sense the de giorgi f convergence

construction methods for idempotent nullnorms on bounded lattices abstract in this paper we discuss the presence of idempotent nullnorms on bounded lattices and investigate some properties of these operators we introduce two new methods for constructing idempotent nullnorms on bounded lattices for a zero element under a constraint that all elements incomparable with the zero element are comparable with each other in order to stress the role of this constraint satisfying that our constructions yield an idempotent nullnorm some illustrative examples are added

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

a characterization of idempotent nullnorms nullnorms are generalization of triangular norms and conorms with a zero element anywhere in the unit interval nullnorms are built up from triangular norms and triangular conorms if nullnorm is idempotent then it is composed of a minimum and a maximum the idea of this paper is to find minimal assumptions on an operation in order to get an idempotent nullnorm

nullnorms on bounded lattices nullnorms are generalization of triangular norms and triangular conorms with a zero element in the interior of the unit interval in this paper we generalize nullnorms to an arbitrary bounded lattices we show the existence of nullnorms with zero element a for arbitrary element a l 0 1 using the fact that some t norms and t conorms on an arbitrary fixed bounded lattice l always exist as a by product we give a method of constructing nullnorms using this method we obtain also the smallest nullnorm and the greatest nullnorm with given zero element a some other construction methods for nullnorms are also added

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

an implicit algorithm for the propagation of a hydraulic fracture with a fluid lag abstract we present an implicit moving mesh algorithm for the study of the propagation of plane strain hydraulic fracture in an impermeable medium in particular the fluid front is allowed to lag behind the fracture tip the solution expressed in the proper scaling evolves for a given value of a dimensionless toughness from a zero stress zero time self similar solution characterized by a finite lag to a large stress large time self similar solution with zero lag a numerical solution for the transient problem is presented using two distinct meshes stretching at different velocities for the lag and fluid filled part of the crack the lubrication and elasticity equations are solved in a coupled manner using an implicit scheme the lag size also coupled to the preceding equations is obtained via the fracture propagation condition the numerical results are discussed and compared with the known zero lag solution for large stress large time