the trace of spatial brownian motion is capacity equivalent to the unit square we show that with probability 1 the trace b 0 1 of brownian motion in space has positive capacity with respect to exactly the same kernels as the unit square more precisely the energy of occupation measure on b 0 1 in the kernel f x y is bounded above and below by constant multiples of the energy of lebesgue measure on the unit square the constants are random but do not depend on the kernel as an application we give almost sure asymptotics for the probability that an stable process approaches within of b 0 1 conditional on b 0 1

moderate deviations and law of the iterated logarithm for intersections of the ranges of random walks let s 1 n s p n be independent symmetric random walks in z d we establish moderate deviations and law of the iterated logarithm for the intersection of the ranges s l 0 n s p 0 n in the case d 2 p 2 and the case d 3 p 2

on finity countability cardinalities and cylindric extensions of type 2 fuzzy sets in linguistic summarization of databases the discussion in this paper is closely related to our idea of a type 2 linguistic summary of a database presented by niewiadomski a niewiadomski a type 2 fuzzy approach to linguistic summarization of data ieee trans fuzzy syst vol 16 no 1 pp 198 213 feb 2008 which is intended to be an efficient tool of knowledge discovery from large databases in that approach we put emphasis on algorithms and applications of type 2 fuzzy sets in summarizing databases hence we have implicitly assumed and considered linguistic expressions represented by sets in discrete or even finite universes of discourse now in this paper we discuss more thoroughly some of the properties and formal aspects of both discrete and continuous type 2 fuzzy sets that represent the elements of linguistic summaries i e quantifiers summarizers and or qualifiers we underline differences between type 2 fuzzy sets in finite and in infinite usually continuous universes of discourse and henceforth between their cardinalities and similar measures which are used in evaluating the goodness quality of type 2 linguistic summaries in addition we define new imprecision measures of linguistic expressions represented by type 2 fuzzy sets and propose a definition of the cylindric extension of a type 2 fuzzy set finally we apply these new concepts to generalize algorithms of type 2 linguistic summarization and propose some new features in comparison to our previous approach a niewiadomski a type 2 fuzzy approach to linguistic summarization of data ieee trans fuzzy syst vol 16 no 1 pp 198 213 feb 2008

a note on the lattice structure for subalgebras of the algebra of truth values of type 2 fuzzy sets abstract the algebra of truth values of type 2 fuzzy sets is not a lattice but some of its subalgebras may be lattices in this paper we give a necessary and sufficient condition under which subalgebras of the truth value algebra of type 2 fuzzy sets form lattices further we prove that if a certain subalgebra is a lattice then it is isomorphic to an appropriate subalgebra whose all elements are convex functions with the same height based on these results if a certain subalgebra is a lattice the equivalent characterizations of the partial order in it are given

aggregation operators on type 2 fuzzy sets abstract cubillo et al in 2015 established the axioms that an operation must fulfill to be an aggregation operator on a bounded poset partially ordered set in particular on m set of fuzzy membership degrees of t2fss which are the functions from 0 1 to 0 1 previously takac in 2014 had applied zadeh s extension principle to obtain a set of operators on m which are under some conditions aggregation operators on l the set of strongly normal and convex functions of m in this paper we introduce a more general set of operators on m than were given by takac and we study among other properties the conditions required to satisfy the axioms of the aggregation operator on l set of normal and convex functions on m which includes the set l

strong invariance for local times let y 1 y 2 be a sequence of i i d random variables with distribution p y 1 k p k k 1 2 e y 1 0 e y 1 2 2 put t n y 1 y n and n x n k 0 k n t k x extending the result of revesz 1981 it is shown that for appropriate skorohod construction we have 1 mathop text sup limits x in mathbb z text l x n sigma 2 text sigma text 2 n x n o n 1 mathord left vphantom 1 4 varepsilon right kern nulldelimiterspace 4 varepsilon text a text s text provided all moments e y 1 m m 0 exists where l is the local time of a wiener process certain rate of convergence is given also under weaker conditions and for l x n 2 2 n x n too when x is fixed

on the fock space representation of functionals of the occupation field and their renormalization on fournit une representation du champ d occupation et de ses puissances renormalisees dans un bon espace de fock

lattices of convex normal functions the algebra of truth values of type 2 fuzzy sets is the set of all functions from the unit interval into itself with operations defined in terms of certain convolutions of these functions with respect to pointwise max and min this algebra has been studied rather extensively both from a theoretical and from a practical point of view it has a number of interesting subalgebras and this paper is about the subalgebra of all convex normal functions and closely related ones these particular algebras are de morgan algebras and our concern is principally with their completeness as lattices a special feature of our treatment is a representation of these algebras as monotone functions with pointwise order making the operations more intuitive

note on the absorption laws in the algebra of truth values of type 2 fuzzy sets abstract this note gives a necessary and sufficient condition under which the absorption laws in the algebra of truth values of type 2 fuzzy sets are true

notes on aggregation of fuzzy truth values takac recently introduced the notion of aggregation operations on fuzzy truth values called ftv aggregation operations and investigated extended aggregation operations on the algebra of convex normal fuzzy truth values with their left and right parts in this paper it is pointed out that the left resp right part of an arbitrary convex normal fuzzy truth value is not necessarily its increasing resp decreasing part moreover an arbitrary convex normal fuzzy truth value is not equivalent to either intersection or union of its left and right parts the conclusions of extended aggregation operations are revalidated in our paper which were proven with uncorrect assertions about convex normal fuzzy truth values

self intersection gauge for random walks and for brownian motion a class of random fields associated with multiple points of a random walk in the plane is studied it is proved that these fields converge in distribution to analogous fields measuring self intersections of the planar brownian motion the concluding section contains a survey of literature on intersection local times and their renormalizations a brief look through the first pages of this section could provide the reader with additional motivation for the present work

cyclic cellular automata and related processes abstract a cyclic cellular automaton is a discrete time process defined on the state space 0 1 n 1 zd starting from a random initial condition at each time step each site looks at its nearest neighbors if the state at site x is i then site x will change state to i 1 mod n if it locates state i 1 mod n among its neighbors otherwise site x will remain in state i cyclic cellular automata have properties akin to cyclic particle systems introduced by bramson and griffeath which are continuous time markov processes on 0 1 n 1 zd with a stochastic evolution rather than a deterministic one in one dimension some clustering results about cyclic cellular automata are presented the analogous problems in the context of cyclic particle systems are still unresolved also we discuss the consequences of these clustering results in the context of one dimensional systems of particles with various interaction mechanisms finally we touch upon the work currently being done on two dimensional cyclic particle systems and cellular automata

a large deviation result for the range of random walk and for the wiener sausage let s n be a random walk on d and let r n be the number of different points among 0 s1 s n 1 we prove here that if d 2 then x lim n 1 n logp r n nx exists for x 0 and establish some convexity and monotonicity properties of x the one dimensional case will be treated in a separate paper

renormalization of a self avoiding brownian motion in two dimensions abstract we obtain a measure defined on the space of continuous sample paths which describes a weakly self avoiding brownian motion in a 2 dimensional euclidean space our method is also applicable to a brownian motion with the dirichlet boundary conditions in a 2 dimensional domain for the new measures an inequality of simon lieb type is also obtained

the range of stable random walks limit theorems are proved for the range of d dimensional random walks in the domain of attraction of a stable process of index p3 the range rn is the number of distinct sites of zd visited by the random walk before time n our results depend on the value of the ratio 3 d the most interesting results are obtained for 2 3 1 which can only occur if d 1 we prove the convergence in distribution of rn e rn toward some constant times the lebesgue measure of the range of the limiting stable process some of our results require regularity assumptions on the characteristic function of x

fluctuation results for the wiener sausage on demontre des resultats de fluctuation pour le volume de la saucisse de wiener associe au mouvement brownien a d dimensions et a un ensemble compact de capacite positive

large deviations for local times and intersection local times of fractional brownian motions and riemann liouville processes in this paper we prove exact forms of large deviations for local times and intersection local times of fractional brownian motions and riemann liouville processes we also show that a fractional brownian motion and the related riemann liouville process behave like constant multiples of each other with regard to large deviations for their local and intersection local times as a consequence of our large deviation estimates we derive laws of iterated logarithm for the corresponding local times the key points of our methods 1 logarithmic superadditivity of a normalized sequence of moments of exponentially randomized local time of a fractional brownian motion 2 logarithmic subadditivity of a normalized sequence of moments of exponentially randomized intersection local time of riemann liouville processes 3 comparison of local and intersection local times based on embedding of a part of a fractional brownian motion into the reproducing kernel hilbert space of the riemann liouville process

a renormalization result for the intersection local time of lattice random walks ind 3 dimensions in this paper we derive some general conditions on stable walks inz d under which the central limit theorem holds for their normalized intersection local time in particular we prove that the process given by the normalized intersection local time of the simple random walk inz d withd 3 is weakly convergent to the standard brownian motion

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

on the extension of nullnorms and uninorms to fuzzy truth values abstract in this work by applying zadeh s extension principle we extend type 1 proper nullnorms and proper uninorms to fuzzy truth values the definitions of type 2 nullnorms and uninorms are introduced we mainly discuss in which subalgebras of fuzzy truth values extended proper nullnorms and proper uninorms are type 2 nullnorms and uninorms respectively

extended triangular norms the paper is devoted to classical t norms extended to operations on fuzzy quantities in accordance with the generalized zadeh extension principle such extended t norms are used for calculating intersection of type 2 fuzzy sets analytical expressions for membership functions of some extended t norms are derived assuming special classes of fuzzy quantities i e fuzzy truth intervals or fuzzy truth numbers the possibility of applying these results in the construction of type 2 adaptive network fuzzy inference systems is illustrated on several examples

large deviations estimates for self intersection local times for simple random walk in mathbb z 3 we obtain large deviations estimates for the self intersection local times for a simple random walk in dimension 3 also we show that the main contribution to making the self intersection large in a time period of length n comes from sites visited less than some power of log n this is opposite to the situation in dimensions larger or equal to 5 finally we present an application of our estimates to moderate deviations for random walk in random sceneries

large deviations for renormalized self intersection local times of stable processes we study large deviations for the renormalized self intersection local time of d dimensional stable processes of index 2d 3 d we find a difference between the upper and lower tail in addition we find that the behavior of the lower tail depends critically on whether d or d

fuzzy decision making based on likelihood based comparison relations in this paper we present a new fuzzy decision making method which is based on likelihood based comparison relations first we introduce the concepts of likelihood based comparison relations for intervals then we propose the concept of likelihood based comparison relations for type 1 fuzzy sets and interval type 2 fuzzy sets then we present a new method to rank fuzzy sets by using fuzzy targets based on the proposed likelihood based comparison relations for fuzzy sets finally we present a new fuzzy decision making method based on the proposed likelihood based comparison relations for fuzzy sets and the proposed fuzzy ranking method the proposed fuzzy decision making method has the advantage that the evaluated values can either be represented by crisp values intervals type 1 fuzzy sets or interval type 2 fuzzy sets it can overcome the drawbacks of huynh et al s method due to the fact that huynh et al s method cannot deal with the ranking of interval type 2 fuzzy sets for fuzzy decision making and cannot distinguish the ranking order between the alternatives in some situations

proprietes d intersection des marches aleatoires i convergence vers le temps local d intersection we study intersection properties of multi dimensional random walks letx andy be two independent random walks with values in d d 3 satisfying suitable moment assumptions and letin denote the number of common points to the paths ofx andy up to timen the sequence in suitably normalized is shown to converge in distribution towards the intersection local time of two independent brownian motions results are applied to the proof of a central limit theorem for the range of a two dimensional recurrent random walk thus answering a question raised by n c jain and w e pruitt

note on on the extension of nullnorms and uninorms to fuzzy truth values fuzzy sets syst 352 2018 92 118 abstract xie recently extended nullnorms and uninorms to fuzzy truth values and investigated whether the extended nullnorms and extended uninorms form respectively type 2 nullnorms and type 2 uninorms on the algebra of fuzzy truth values in this note we show by counterexamples that there exist some flaws in a previous paper by xie fuzzy sets syst 352 2018 92 118 such as corollaries 2 1 3 1 4 1 and 4 2 lemmas 3 3 3 4 3 5 3 6 3 7 and 3 8 propositions 3 4 3 5 3 10 3 11 4 1 and 4 4 theorems 3 1 3 3 4 1 and 4 3 and then we provide the correct versions in addition proposition 3 7 in that paper is correct but its proof contains a flaw we point out the flaw and give a more general result than proposition 3 7 in that paper

the asymptotics of stable sausages in the plane in this paper we develop an asymptotic expansion for the e neighborhood of the symmetric stable process of order 3 1 f 2 our expansion is in powers of 82 with the nth coefficient related to n fold self intersections of our stable process 1 introduction in this paper we will develop asymptotic expansions for the area of the neighborhood of x the symmetric stable process of order f3 1 f3 2 in r2 our asymptotic expansion as 8 0 is in powers 82 n with coefficients yn which are random variables related to n fold self intersections of x the 8 neighborhood of x known as the stable sausage is defined as

general type 2 fuzzy c means algorithm for uncertain fuzzy clustering pattern recognition in real world data is subject to various sources of uncertainty that should be appropriately managed the focus of this paper is the management of uncertainty associated with parameters of fuzzy clustering algorithms type 2 fuzzy sets t2 fss have received increased research interest over the past decade primarily due to their potential to model various uncertainties however because of the computational intensity of the processing of general t2 fuzzy sets gt2 fss only their constrained version i e the interval t2 it2 fss were typically used fortunately the recently introduced concepts of planes and zslices allow for efficient representation and computation with gt2 fss following this recent development this paper presents a novel approach for uncertain fuzzy clustering using the general type 2 fuzzy c means gt2 fcm algorithm the proposed method builds on top of the previously published it2 fcm algorithm which is extended via the planes representation theorem the fuzzifier parameter of the fcm algorithm can be expressed using linguistic terms such as small or high which are modeled as t1 fss this linguistic fuzzifier value is then used to construct the gt2 fcm cluster membership functions the linguistic uncertainty is transformed into uncertain fuzzy positions of the extracted clusters the gt2 fcm algorithm was found to balance the performance of t1 fcm algorithms in various uncertain pattern recognition tasks and to provide increased robustness in situations where noisy or insufficient training data are present

random walks and intersection local time with each random walk on z 2 we associate a functional related to the number of steps which the walk spends in sites occupied at least k times we show that if our random walk is in the domain of attraction of a stable process of order greater than 2 2k 2 2k 1 then our functional converges to the intersection local time of the limiting process

large deviations and laws of the iterated logarithm for the local times of additive stable processes we study the upper tail behaviors of the local times of the additive stable processes let x 1 t x p t be independent d dimensional symmetric stable processes with stable index 0 2 and consider the additive stable process x t 1 t p x 1 t 1 x p t p under the condition d ap we obtain a precise form of the large deviation principle for the local time x 0 t p t 0 t 0 x x 1 s 1 x p sp ds 1 ds p of the multiparameter process x t 1 t p and for its supremum norm sup x r d x 0 t p our results apply to the law of the iterated logarithm and our approach is based on fourier analysis moment computation and time exponentiation

the algebra of fuzzy truth values the purpose of this paper is to give a straightforward mathematical treatment of algebras of fuzzy truth values for type 2 fuzzy sets

an almost sure invariance principle for the range of planar random walks for a symmetric random walk in z 2 with 2 moments we represent r n the cardinality of the range in terms of an expansion involving the renormalized intersection local times of a brownian motion we show that for each k 1 logn k 1 n r n 1 j 1 2 logn c x j j n 0 a s where w t is a brownian motion w n t w nt n j n is the renormalized intersection local time at time 1 for w n and c x is a constant depending on the distribution of the random walk

stochastic heat equation driven by fractional noise and local time the aim of this paper is to study the d dimensional stochastic heat equation with a multiplicative gaussian noise which is white in space and has the covariance of a fractional brownian motion with hurst parameter h 0 1 in time two types of equations are considered first we consider the equation in the ito skorohod sense and later in the stratonovich sense an explicit chaos expansion for the solution is obtained on the other hand the moments of the solution are expressed in terms of the exponential moments of some weighted intersection local time of the brownian motion

growth behavior of helical cellular automata a helical cellular automata hca model constructed on a two dimensional grid of cells with a helical structure is presented and the pattern formation of this model studied by numerous computer simulations it is found that the evolutions of the hca are sensitive to the circumference of the helix p with various p the initial growth of the model generates various patterns ranging from sierpinski triangle gasket complex textured pattern to lateral quasiperiodic structure a sudden transition from regular fractal to compact pattern occurs near the point where p is equal to a positive integer power of 2 with increasing height of the patterns increasing growth time the model also exhibits different growth behaviors in the vertical direction for various p including the formation of regular periodic patterns and the evolution from initial regular patterns to eventual random structures fractal dimension analysis is used to characterize these different evolution processes quantitatively

an extension of a result of burdzy and lawler it is shown that for all mean zero finite variance random walks the critical non intersection exponents are equal to those for brownian motion the method uses the local time of intersection

self intersection local times of random walks exponential moments in subcritical dimensions fix p 1 not necessarily integer with p d 2 0 that are bounded from above possibly tending to zero the speed is identified in terms of mixed powers of t and t and the precise rate is characterized in terms of a variational formula which is in close connection to the gagliardo nirenberg inequality as a corollary we obtain a large deviation principle for lt p trt for deviation functions rt satisfying t r t gg mathbb e ell t p informally it turns out that the random walk homogeneously squeezes in a t dependent box with diameter of order t1 d to produce the required amount of self intersections our main tool is an upper bound for the joint density of the local times of the walk

exact calculations of extended logical operations on fuzzy truth values in this paper we propose computationally simple pointwise formulas for extended t norms and t conorms on fuzzy truth values the complex convolutions of the extended operations are shown to be equivalent to simple pointwise expressions for several special cases linear fuzzy truth values are defined and it is shown that the extended lukasiewicz operations preserve linearity since linear fuzzy truth values are common in representing linguistic modifiers the results simplify fuzzy truth value based reasoning methods the results can also be applied immediately to type 2 fuzzy set operations

toward general type 2 fuzzy logic systems based on zslices higher order fuzzy logic systems flss such as interval type 2 flss have been shown to be very well suited to deal with the high levels of uncertainties present in the majority of real world applications general type 2 flss are expected to further extend this capability however the immense computational complexities associated with general type 2 flss have until recently prevented their application to real world control problems this paper aims to address this problem by the introduction of a complete representation framework which is referred to as zslices based general type 2 fuzzy systems the proposed approach will lead to a significant reduction in both the complexity and the computational requirements for general type 2 flss while it offers the capability to represent complex general type 2 fuzzy sets as a proof of concept application we have implemented a zslices based general type 2 fls for a two wheeled mobile robot which operates in a real world outdoor environment we have evaluated the computational performance of the zslices based general type 2 fls which is suitable for multiprocessor execution finally we have compared the performance of the zslices based general type 2 fls against type 1 and interval type 2 flss and a series of results is presented which is related to the different levels of uncertainty handled by the different types of flss

large and moderate deviations for intersection local times we study the large and moderate deviations for intersection local times generated by respectively independent brownian local times and independent local times of symmetric random walks our result in the brownian case generalizes the large deviation principle achieved in mansmann 1991 for the l 2 norm of brownian local times and coincides with the large deviation obtained by csorgo shi and yor 1991 for self intersection local times of brownian bridges our approach relies on a feynman kac type large deviation for brownian occupation time certain localization techniques from donsker varadhan 1975 and mansmann 1991 and some general methods developed along the line of probability in banach space our treatment in the case of random walks also involves rescaling spectral representation and invariance principle the law of the iterated logarithm for intersection local times is given as an application of our deviation results

law of the iterated logarithm for the range of random walks in two and three dimensions let s n be a random walk in z d and let r n be the range of s n we prove an almost sure invariance principle for r n when d 3 and a law of the iterated logarithm for r n when d 2

local times on curves and uniform invariance principles sufficient conditions are given for a family of local times l t ofd dimensional brownian motion to be jointly continuous as a function oft and then invariance principles are given for the weak convergence of local times of lattice valued random walks to the local times of brownian motion uniformly over a large family of measures applications included some new results for intersection local times for brownian motions on 2 and 2

wiener sausage and self intersection local times abstract let b bt t 0 be a standard brownian motion in r 2 for every e 0 and every compact subset k of r 2 the wiener sausage of radius e associated with k is defined as the union of the sets bs ek s 0 1 the present paper gives full asymptotic expansions for the area of the wiener sausage when the radius e goes to 0 the kth term of the expansion is of order log e k and involves a random variable which measures the number of k multiple self intersections of the process such random variables are called renormalized self intersection local times and have been recently introduced and studied by e b dynkin a self contained construction of these local times is given together with a number of new approximations

number of distinct sites visited by a random walk with internal states in the classical paper of dvoretzky and erd s proceedings of the 2nd berkeley symposium on mathematical statistics and probability pp 353 367 1951 asymptotics for the expected value and the variance of the number of distinct sites visited by a simple symmetric random walk were calculated here these results are generalized for random walks with internal states moreover both weak and strong laws of large numbers are proved as a tool for these results the error term of the local limit theorem in kramli and szasz zeitschrift wahrscheinlichkeitstheorie verw gebiete 63 85 95 1983 is also estimated

integral representation of renormalized self intersection local times in this paper we apply clark ocone formula to deduce an explicit integral representation for the renormalized self intersection local time of the d dimensional fractional brownian motion with hurst parameter h 0 1 as a consequence we derive the existence of some exponential moments for this random variable

a useful estimate in the multidimensional invariance principle an estimate of the convergence speed in the multidimensional invariance principle is obtained using this estimate we can prove strong invariance principles for partial sums of independent not necessarily identically distributed multidimensional random vectors

polylinear additive functionals of superprocesses abstract letxbe a superdiffusion in a domaineof r d a polylinear additive functional ofxcorresponding to a positive borel function is given by the formula a b b dt 1 dt k e k t 1 z 1 t k z k x t 1 dz 1 x t k dz k by a passage to the limit we extend this definition to a certain class of generalized functions more precisely we associate an additive functionala of xt p with every measure subject to the condition dt 1 dz 1 dt k dz k dt 1 dz 1 dt k dz k emsp q t 1 z 1 t k dz k t 1 z 1 t k dz k for a certain kernelq we prove that if t ti s 0 fori 1 kand for alls then measurea has a s the same property this result is applicable in particular to self intersection local times ofx

collective behavior of random activated mobile cellular automata abstract dynamical properties of 2d cellular automata with mobile elements are examined qualitatively results show that a system containing elements with local interactions but with no fixed connections due to movement and connection breaking are able to display periodic oscillations by collective synchronization of non periodical randomly activated elements the system studied is found to be robust spatial dynamics is shown to generate interesting spatial structures suggesting the presence of self organization moreover maximum lyapunov exponents and fractal dimension of attractors have been calculated in order to show that the dynamics of interaction among elements is chaotic

the fluctuation result for the multiple point range of two dimensional recurrent random walks we study the fluctuation problem for the multiple point range of random walks in the two dimensional integer lattice with mean 0 and finite variance the p multiple point range means the number of distinct sites with multiplicity p of random walk paths before time n the suitably normalized multiple point range is proved to converge to a constant which is independent of the multiplicity multiple of the renormalized self intersection local time of a planar brownian motion

self intersection local time critical exponent large deviations and laws of the iterated logarithm if t is renormalized self intersection local time for planar brownian motion we characterize when ee 1 is finite or infinite in terms of the best constant of a gagliardo nirenberg inequality we prove large deviation estimates for 1 and 1 we establish lim sup and lim inf laws of the iterated logarithm for t as t

brownian intersection local times upper tail asymptotics and thick points we equip the intersection of p independent brownian paths in r d d 2 with the natural measure l defined by projecting the intersection local time measure via one of the brownian motions onto the set of intersection points given a bounded domain u c r d we show that as a the probability of the event l u a decays with an exponential rate of a 1 p where is described in terms of a variational problem in the important special case when u is the unit ball in r 3 and p 2 we characterize in terms of an ordinary differential equation we apply our results to the problem of finding the hausdorff dimension spectrum for the thick points of the intersection of two independent brownian paths in r 3

some properties of the fractional ornstein uhlenbeck process we consider the fractional analogue of the ornstein uhlenbeck process i e the solution of the langevin equation driven by a fractional brownian motion in place of the usual brownian motion we establish some properties of these processes we show that the process is local nondeterminism for a two dimensional process we show that its renormalized self intersection local time exists in l2 if and only if

type 2 implications on non interactive fuzzy truth values in this paper we consider algebras of fuzzy truth values equipped with an implication operation generalizing the definition of type 1 fuzzy implications we discuss extended fuzzy s implications and coimplications and also extended residual implications and coimplications in particular to the extended residual of minimum we study the algebraic properties of these operations and their relationships to extended t norms and t conorms in particular to meet and join these investigations are intended to provide a theoretical background for type 2 approximate reasoning applications

random walk in random scenery and self intersection local times in dimensions d 5 let s k k 0 be a symmetric random walk on mathbb z d and eta x x in mathbb z d an independent random field of centered i i d random variables with tail decay p eta x t approx exp t alpha we consider a random walk in random scenery that is x n eta s 0 dots eta s n we present asymptotics for the probability over both randomness that x n n for 1 2 and 1 to obtain such asymptotics we establish large deviations estimates for the self intersection local times process sum x l n 2 x where l n x is the number of visits of site x up to time n

geometric type 1 and type 2 fuzzy logic systems this paper presents a novel approach to the representation of type 1 and type 2 fuzzy sets utilising computational geometry to achieve this our approach borrows ideas from the field of computational geometry and applies these techniques in the novel setting of fuzzy logic we provide new algorithms for various operations on type 1 and type 2 fuzzy sets and for defuzzification results of experiments indicate that this approach reduces the execution speed of these operations