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

distributivity equations in the class of semi t operators recently the distributivity equations have been discussed in families of certain operations e g triangular norms conorms uninorms and nullnorms in this study we present the solutions to distributivity between semi t operators previous results related to the distributivity between nullnorms and between semi t operators and semi nullnorms can be obtained as simple corollaries

derivation of the time dependent propagator for the three dimensional schrodinger equation with one point interaction an explicit formula for the time dependent propagator of the schrodinger equation with one point interaction in three dimensions is given the derivation is based on the inverse laplace transformation applied to the corresponding resolvent

spectral analysis of a flat plasma sheet model the spectral analysis of the electromagnetic field on the background of an infinitely thin flat plasma layer is carried out this model loosely imitates a single base plane from graphite and it is of interest for theoretical studies of fullerenes by making use of the hertz potentials the solutions to maxwell equations with the appropriate matching conditions at the plasma layer are derived and on this basis the spectrum of electromagnetic oscillations is determined the model is naturally split into the te sector and tm sector both the sectors have positive continuous spectra but the tm modes have in addition a bound state namely the surface plasmon this analysis relies on the consideration of the scattering problem in the te and tm sectors the spectral zeta function and integrated heat kernel are constructed for different branches of the spectrum in an explicit form as a preliminary the rigorous procedure of integration over the continuous spectra is formulated by introducing the spectral density in terms of the scattering phase shifts the asymptotic expansion of the integrated heat kernel at small values of the evolution parameter is derived by making use of the technique of integral equations developed earlier by the same authors the local heat kernel green s function or fundamental solution is also constructed as a by product a new method is demonstrated for deriving the fundamental solution to the heat conduction equation or to the schrodinger equation on an infinite line with the like source in particular for the heat conduction equation on an infinite line with the source a nontrivial counterpart is found namely a spectral problem with point interaction which possesses the same integrated heat kernel while the local heat kernels fundamental solutions in these spectral problems are different

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

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

common nature of learning between back propagation and hopfield type neural networks for generalized matrix inversion with simplified models in this paper two simple structure neural networks based on the error back propagation bp algorithm i e bp type neural networks bpnns are proposed developed and investigated for online generalized matrix inversion specifically the bpnn l and bpnn r models are proposed and investigated for the left and right generalized matrix inversion respectively in addition for the same problem solving task two discrete time hopfield type neural networks hnns are developed and investigated in this paper similar to the classification of the presented bpnn l and bpnn r models the presented hnn l and hnn r models correspond to the left and right generalized matrix inversion respectively comparing the bpnn weight updating formula with the hnn state transition equation for the specific i e left or right generalized matrix inversion we show that such two derived learning expressions turn out to be the same in mathematics although the bp and hopfield type neural networks are evidently different from each other a great deal in terms of network architecture physical meaning and training patterns numerical results with different illustrative examples further demonstrate the efficacy of the presented bpnns and hnns for online generalized matrix inversion and more importantly their common natures of learning

restricted distributivity for aggregation operators with absorbing element the problem of restricted distributivity i e a form of relaxed distributivity on the restricted domain plays an important role in many different fields such as utility theory and integration theory this paper considers the following two cases of restricted distributivity i a continuous nullnorm with respect to a continuous t conorm and ii a continuous nullnorm with respect to a uninorm of the form u m i n or u m a x with continuous underlying t norm and t conorm

on distributivity equations for uninorms over semi t operators recently dryga generalized nullnorms and t operators and introduced semi t operators by eliminating commutativity from the axioms of t operators distributivity equations were investigated in families of certain operations e g triangular norms conorms uninorms and nullnorms or called t operators in this paper we give out the solutions of distributivity equations for uninorms over semi t operators previous results about distributivity equations for uninorms over nullnorms can be obtained as corollaries

explicit derivation of the propagator for a dirac delta potential an explicit derivation of the propagator is given for a particle in a dirac delta potential lambda delta x by making a direct application of a variational technique followed by a systematic use of laplace transforms although the solution of the corresponding schrodinger equation has been known for years the solution of the propagator has been obtained only recently

a characterization of a class of uninorms with continuous underlying operators in this paper all uninorms locally internal in the region a e given by the complement in 0 1 2 of 0 e 2 e 1 2 where e is the neutral element of the uninorm having continuous underlying operators are studied and characterized by distinguishing some cases when the underlying t norm and t conorm are not given by ordinal sums it is proved that uninorms locally internal in a e are in fact all possible uninorms with these underlying operators except when both the t norm and the t conorm are strict in which case there is also the class of representable uninorms leading to a finite number of possibilities when at least one of the continuous underlying operators is given by an ordinal sum again there are other possible uninorms than those that are locally internal in a e but all uninorms with this property are also characterized in this case infinitely many possibilities can appear depending on the set of idempotent elements of the uninorm

distributivity and conditional distributivity for uni nullnorms abstract the distributivity and conditional distributivity of a uninorm over a continuous t conorm is an open problem posed by klement in the linz seminar 2000 closing session after this the study of distributivity and conditional distributivity between two aggregation operators become specially interesting in this work we continue investigating the same topic for uni nullnorms then the full characterization of all pairs f g satisfying distributivity and conditional distributivity is given where f is a uni nullnorm g is a t norm or a t conorm or a uninorm from u min and u max

distributivity and conditional distributivity for s uninorms abstract the distributivity condition plays an essential role in the framework of fuzzy connectives this paper studies both regular and conditional distributivity equations for s uninorms and gives the characterizations of distributivity equations between some binary aggregation operators and s uninorms in u min moreover we describe t norms t uninorms and semi t operators that are conditional distributive over s uninorms

representation of nullnorms on bounded lattices abstract recently nullnorms on bounded lattices have been frequently investigated by many researchers in this paper we give the full characterization for quite a general class of nullnorms on bounded lattices and represent them by their underlying t norms and t conorms an example is given to show the relationship between each block of the nullnorms we also fully determine the values of arbitrary nullnorms when only one of the two variables is comparable with the absorbing elements for the rest cases we give some necessary conditions

some implications of the restricted distributivity of aggregation operators with absorbing elements for utility theory the issue of restricted distributivity conditional distributivity i e distributivity on the relaxed domain is crucial for many different areas such as utility theory and integration theory this paper considers restricted distributivity of a continuous nullnorm with respect to a continuous t conorm and some applications of the obtained results in utility theory

on the associativity functional equation abstract let a b be any bounded closed real interval the class of all continuous nondecreasing associative functions m a b 2 a b fulfilling the boundary conditions m a a a and m b b b is described

corrigendum corrigendum to the distributivity condition for uninorms and t operators fuzzy sets and systems 128 2002 209 225 in this paper the study of the distributivity equation involving uninorms in u given in fuzzy sets and systems 128 2002 209 225 is revised two wrong propositions in the mentioned reference are corrected and their right versions are proved

the distributivity equation for uninorms revisited abstract the distributivity equation has been widely studied involving different classes of aggregation functions from t norms and t conorms to uninorms nullnorms and generalizations of them it is important in the framework of logical connectives because of its applications in fuzzy logic and approximate reasoning as well as in image processing since uninorms have been used in these topics the study of the distributivity between two uninorms becomes specially interesting in a recent paper by the same authors the already known solutions were compiled and completed when the first uninorm is in any of the most studied classes of uninorms and the second uninorm is anyone in this paper we want to achieve this study by focusing on the reverse direction that is for the cases when the second uninorm lies in any of the most studied classes of uninorms and the first one is any uninorm we show along the paper that this new point of view leads to many new solutions

left and right distributivity equations for semi t operators and uninorms abstract recently the distributivity equation for different kind of operators has become a focus of research as the crucial tool in the numerous applications as utility and optimization theory or integration theory the main purpose of this paper is to study the distributivity equations between two special classes of aggregation operators namely semi t operators and uninorms without the commutativity assumption of semi t operators it is necessary to consider the left and the right distributivity conditions separately in both cases the obtained solutions are significantly different

active identification for discrete time nonlinear control ii strict feedback systems the problem of controlling nonlinear systems with unknown parameters has received a great deal of attention in the continuous time case in contrast its discrete time counterpart remains largely unexplored primarily due to the difficulties associated with utilizing lyapunov design techniques in a discrete time framework existing results impose restrictive growth conditions on the nonlinearities to yield global stability in this paper we propose a novel approach which removes this obstacle and yields global stability and tracking for systems that can be transformed into an output feedback canonical form in which the nonlinearities depend only on the measured output but are otherwise arbitrary the main novelties of our design are i the temporal and algorithmic separation of the parameter estimation task from the control task and ii the development of an active identification procedure which uses the control input to actively drive the system state to points in the state space that allow the orthogonalized projection estimator to acquire all the necessary information about the unknown parameters we prove that our algorithm guarantees complete for control purposes identification in a finite time interval whose maximum length we compute thus the traditional structure of concurrent online estimation and control is replaced by a two phase control strategy first use active identification and then utilize the acquired parameter information to implement any control strategy as if the parameters were known

on line parameter estimator derived from modified matrix pseudoinverse abstract a method based on the matrix pseudoinverse is considered for on line parameter identification of discrete time nonlinear systems a recursive scheme was previously reported but assuming some restrictions on the input output data by resorting to the formulae developed for modified matrix pseudoinverse an iterative algorithm is derived in this work this algorithm applies to the general class of proposed systems while keeping well known desired properties for parameter identification

on the conditional distributivity of nullnorms over uninorms conditional distributivity of nullnorm over uninorm with continuous operators is discussed the structure of uninorms with idempotent operators is discussed the problem of conditional distributivity for different two aggregation operators plays an important role in many different fields such as utility theory and integration theory this paper deals with the conditional distributivity of a continuous nullnorm with respect to a uninorm with the continuous underlying t norm and t conorm

the distributivity equations for semi t operators over uninorms recently the distributivity equation was discussed in families of certain operations e g triangular norms conorms uninorms and nullnorms or called t operators in this paper we describe the solutions of distributivity for semi t operators over uninorms previous results about distributivity for nullnorms over uninorms can be obtained as a corollary

distributivity and conditional distributivity of a uninorm with continuous underlying operators over a continuous t conorm this paper deals with the distributivity and conditional distributivity of uninorms with continuous underlying operators over continuous triangular conorms in particular the involved triangular conorm is either maximum operator or an ordinal sum with only one summand in which the corresponding triangular conorm is strict from the obtained results it is deduced that distributivity and conditional distributivity are equivalent moreover we obtain the full characterization of some cases of this class of uninorms of which either the underlying triangular norm or triangular conorm is strict

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

on the conditional distributivity of continuous semi t operators over uninorms abstract the issue of conditional distributivity or how it is also called restricted distributivity which is a form of relaxed distributivity on the restricted domain is crucial for many different areas such as utility theory and integration theory the focus of this paper is on this specific form of distributivity for a continuous semi t operator with respect to a continuous t conorm and for a continuous semi t operator with respect to a uninorm of the form u min or u max with continuous underlying t norm and t conorm

distributivity and conditional distributivity for t uninorms abstract aggregation operators play an important role in many theoretical and practical aspects of applied mathematics recently the focus is on operators with an annihilator therefore the topic of this paper is distributivity both conditional and regular for certain classes of aggregation operators with this property the characterization of all pairs f g of aggregation operators that are satisfying distributivity law on both whole and restricted domain where f is a t uninorm in u max and g is a t conorm or a uninorm from u min or u max is given

on some solutions of the distributivity equation abstract the aim of this paper is to present some interesting results on the distributivity equation for which several pairs of binary operations on 0 1 are considered moreover different methods to solve distributivity equations are used in this work

left and right distributivity equations in the class of semi t operators abstract the issue of distributivity is crucial for many different theoretical areas such as the utility theory and the integration theory also this problem is of high interest for modelling some practical situations from e g fields of economics and social sciences thus recently the distributivity equations have been studied for different classes of aggregation operators by a significant number of researchers the focus of this paper in particular is on solving the left and the right distributivity equations between semi t operators

on fractional powers of singular perturbations of the laplacian abstract we qualify a relevant range of fractional powers of the so called hamiltonian of point interaction in three dimensions namely the singular perturbation of the negative laplacian with a contact interaction supported at the origin in particular we provide an explicit control of the domain of such a fractional operator and of its decomposition into regular and singular parts we also qualify the norms of the resulting singular fractional sobolev spaces and their mutual control with the corresponding classical sobolev norms

distributivity and conditional distributivity of a uninorm and a continuous t conorm the open problem recalled by klement in the linz2000 closing session related to distributivity and conditional distributivity of a uninorm and a continuous t conorm is solved for the most usual known classes of uninorms from the obtained results it is deduced that distributivity and conditional distributivity are equivalent for these cases it is remarkable that solutions appear involving not only strict t conorms but also ordinal sums of the maximum with a strict t conorm conversely the distributivity of a t conorm over a uninorm is also studied leading only to already known solutions moreover the dual case of distributivity and conditional distributivity involving uninorms and continuous t norms is also solved proving again the equivalence of both kinds of distributivities

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

on the distributivity property for uninorms abstract distributivity between two operations is a property that was already posed many years ago and that is especially interesting in the framework of logical connectives for this reason the distributivity property has been extensively studied for several families of operations like triangular norms and conorms some kinds of uninorms and nullnorms also called t operators and even for some generalizations of them in this paper we investigate the distributivity equation involving two uninorms lying in any one of the most studied classes of uninorms leading to many new solutions

nullnorms on bounded lattices derived from t norms and t conorms abstract nullnorms with an annihilator a in any point of a bounded lattice are generalizations and unifications of t norms and t conorms this study continues to investigate the construction of nullnorms on bounded lattices we propose some methods to construct nullnorms derived from t norms and t conorms on bounded lattices where some sufficient and necessary conditions on theirs annihilator are required as a by product of these constructions idempotent nullnorms on bounded lattices are obtained further we provide some illustrative examples of the new classes of nullnorms idempotent nullnorms on bounded lattices

fractional powers and singular perturbations of quantum differential hamiltonians we consider the fractional powers of singular point like perturbations of the laplacian and the singular perturbations of fractional powers of the laplacian and we compare two such constructions focusing on their perturbative structure for resolvents and on the local singularity structure of their domains in application to the linear and non linear schrodinger equations for the corresponding operators we outline a programme of relevant questions that deserve being investigated we consider the fractional powers of singular point like perturbations of the laplacian and the singular perturbations of fractional powers of the laplacian and we compare two such constructions focusing on their perturbative structure for resolvents and on the local singularity structure of their domains in application to the linear and non linear schrodinger equations for the corresponding operators we outline a programme of relevant questions that deserve being investigated

time dependent propagator with point interaction we compute the time dependent schrodinger propagator with a point interaction in dimension n or 3 including the new cases of n 2 and the most general interaction supported by a point for n 1 we also give the small time asymptotics for n or 3 the case n 2 has the peculiarity of involving logarithmic terms in the expansion

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

conditional distributivity of continuous semi t operators over disjunctive uninorms with continuous underlying t norms and t conorms abstract the issue of conditional distributivity which is also known as restricted distributivity is crucial for many different areas such as utility theory and integration theory in the present study we focus on this specific form of distributivity for continuous semi t operators with respect to disjunctive uninorms with continuous underlying t norms and t conorms this study extends previous research by focusing on a much wider class of uninorms

explicit time dependent schrodinger propagators the authors compute explicitly the time dependent schrodinger and heat propagator for the potentials lambda 2m m 1 cosh2 lambda x a delta function potential several cases of periodic delta function potentials and a delta function potential in a square well they also discuss the semiclassical approximation for the delta function potential

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

distributivity between semi t operators and semi nullnorms recently the distributivity equation was discussed in families of certain operations e g triangular norms conorms uninorms and nullnorms in this paper we describe the solutions of distributivity between semi t operators and semi nullnorms previous results about distributivity between nullnorms can be obtained as simple corollaries