automata theory based on quantum logic some characterizations automata theory based on quantum logic abbr l valued automata theory may be viewed as a logical approach of quantum computation in this paper we characterize some fundamental properties of l valued automata theory and discover that some properties of the truth value lattices of the underlying logic are equivalent to certain properties of automata more specifically i the transition relations of l valued automata are extended to describe the transitions enabled by strings of input symbols and particularly these extensions depend on the distributivity of the truth value lattices proposition 3 1 ii some properties of the l valued successor and source operators and l valued subautomata are demonstrated to be equivalent to a property of the truth value lattices which is exactly equivalent to the distributive law proposition 4 3 and corollary 4 4 this is a new characterization of boolean algebras in the framework of l valued automata theory iii we verify that the intersection of two l valued subautomata is still an l valued subautomaton if and only if the multiplication is distributive over the union in the truth value lattices proposition 4 5 which is strictly weaker than the usual distributivity iv we show that some topological characterizations in terms of the l valued successor and source operators also rely on the distributivity of truth value lattices theorem 5 6 finally we address some related topics for further study

pade numerical schemes for the sine gordon equation abstract the sine gordon equation turn up in several problems in science and engineering although it is integrable in practical applications its numerical solution is powerful and versatile four novel implicit finite difference methods based on q s pade approximations with q s th order in space have been developed and analyzed for this equation all share the same treatment for the nonlinearity and integration in time concretely 0 4 2 2 2 4 and 4 4 pade methods additionally the energy conserving strauss vazquez scheme has been considered in a 0 2 pade implementation these methods have been compared among them for both the kink antikink and breather solutions in terms of global error computational cost and energy conservation the 0 4 and 2 4 pade methods are the most cost effective ones for small and large global error respectively our results indicate that spatial order of accuracy is more relevant to effectiveness of a method than energy conservation even in very long time integrations

backward error analysis for multi symplectic integration methods a useful method for understanding discretization error in the numerical solution of odes is to compare the system of odes with the modified equations obtained through backward error analysis and using symplectic integration for hamiltonian odes provides more incite into the modified equations in this paper the ideas of symplectic integration are extended to hamiltonian pdes and this paves the way for the development of a local modified equation analysis solely as a useful diagnostic tool for the study of these types of discretizations in particular local conservation laws of energy and momentum are not preserved exactly when symplectic integrators are used to discretize but the modified equations are used to derive modified conservation laws that are preserved to higher order along the numerical solution these results are also applied to the nonlinear wave equation

conditions for minimal fuzzy deterministic finite automata via brzozowski s procedure this paper deals with the application of brzozowski s minimization procedure to fuzzy finite automata with truth values in a complete residuated zero divisor free lattice for a given fuzzy finite automaton mathcal a the procedure computes the automaton d r d r mathcal a where d mathcal a is a fuzzy determinization of mathcal a and r mathcal a is the reverse automaton of mathcal a it is observed that the size of the resulting automaton is strongly dependent on the determinization method used in the procedure consequently this paper studies necessary and sufficient conditions for obtaining minimal fuzzy deterministic finite automata via brzozowski s procedure the study is accomplished for determinization methods based on a generalized notion of accessible fuzzy subset construction the obtained conditions determine that brzozowski s procedure returns a minimal fuzzy deterministic finite automaton when the determinization method does not produce proportional fuzzy states this is the reason to obtain minimal fuzzy deterministic finite automata using brzozowski s procedure for fuzzy finite automata with truth values in the product structure when the determinization method is the method based on factorization of fuzzy states

two energy conserving numerical schemes for the sine gordon equation two explicit conservative numerical schemes for the sine gordon equation are proposed their stability and convergence are proved numerical simulation of a sine gordon soliton shows that the new schemes are very accurate and fast

on the first and second variations of a nonlocal isoperimetric problem we consider a nonlocal perturbation of an isoperimetric variational prob lem the problem may be viewed as a mathematical paradigm for the ubiquitous phenom enon of energy driven pattern formation associated with competing short and long range interactions in particular it arises as a g limit of a model for microphase separation of diblock copolymers in this article we establish precise conditions for criticality and stability i e we explicitly compute the first and second variations we also present some applications

preserving energy resp dissipation in numerical pdes using the average vector field method we give a systematic method for discretizing hamiltonian partial differential equations pdes with constant symplectic structure while preserving their energy exactly the same method applied to pdes with constant dissipative structure also preserves the correct monotonic decrease of energy the method is illustrated by many examples in the hamiltonian case these include the sine gordon korteweg de vries nonlinear schrodinger linear time dependent schrodinger and maxwell equations in the dissipative case the examples are the allen cahn cahn hilliard ginzburg landau and heat equations

representation of occurrences for road vehicle traffic our 3d model based computer vision subsystem extracts vehicle trajectories from monocular digitized videos recording road vehicles in inner city traffic steps are documented which import these quantitative geometrical results into a conceptual representation based on a fuzzy metric temporal horn logic fmthl see k h schafer unscharfe zeitlogische modellierung von situationen und handlungen in bildfolgenauswertung und robotik dissertation 1996 the facts created by this import step can be understood as verb phrases which describe elementary actions of vehicles in image sequences of road traffic scenes the current contribution suggests a complete conceptual representation of elementary vehicle actions and reports results obtained by an implementation of this approach from real world traffic videos

quantum automata and quantum grammars abstract to study quantum computation it might be helpful to generalize structures from language and automata theory to the quantum case to that end we propose quantum versions of finite state and push down automata and regular and context free grammars we find analogs of several classical theorems including pumping lemmas closure properties rational and algebraic generating functions and greibach normal form we also show that there are quantum context free languages that are not context free so qcfl cfl

equilibrium configurations of epitaxially strained elastic films second order minimality conditions and qualitative properties of solutions we consider a variational model introduced in the physical literature to describe the epitaxial growth of an elastic film over a thick flat substrate when a lattice mismatch between the two materials is present we study quantitative and qualitative properties of equilibrium configurations that is of local and global minimizers of the free energy functional more precisely we determine analytically the critical threshold for the local minimality of the flat configuration and we also prove several results concerning its global minimality the non occurrence of singularities in non flat global minimizers is also addressed one of the main results of the paper is a new sufficient condition for local minimality which provides the first extension of the classical criteria based on the positivity of the second variation to the context of functionals with bulk and surface energies

some mathematical aspects of the micro phase separation in diblock copolymers abstract a free energy functional of nonlocal type is considered that was originally introduced to describe the micro phase separation of diblock copolymer a mathematical framework is given to the issues of the scaling law for stationary states and the governing equation of morphology the scaling law is equivalent to finding a nice rescaling in order to have a well defined limiting interfacial problem which is free from the interfacial thickness and the total chain length of copolymer and the associated stationary problem becomes a morphology equation that governs the configuration of final patterns although the steady patterns become finer and finer in our scaling limit due to the mesoscopic nature a possible rigorous approach is presented to stability and morphological selection problems for them

multi symplectic runge kutta collocation methods for hamiltonian wave equations a number of conservative pdes like various wave equations allow for a multi symplectic formulation which can be viewed as a generalization of the symplectic structure of hamiltonian odes we show that gauss legendre collocation in space and time leads to multi symplectic integrators i e to numerical methods that preserve a symplectic conservation law similar to the conservation of symplecticity under a symplectic method for hamiltonian odes we also discuss the issue of conservation of energy and momentum since time discretization by a gauss legendre method is computational rather expensive we suggest several semi explicit multi symplectic methods based on gauss legendre collocation in space and explicit or linearly implicit symplectic discretizations in time

characterizations of complete residuated lattice valued finite tree automata this paper deals with the concept of complete residuated lattice valued referred to as l valued finite tree automata in this regard we first define an l valued regular tree language and then we prove a necessary and sufficient condition for the regularity of an l valued tree language furthermore we generalize the pumping lemma for l valued finite tree automata l fta afterwards the behavior of l fta is addressed and some theorems are provided moreover the existence of the minimal form of an l fta is considered finally a minimization algorithm of the l fta is presented and its time complexity is analyzed

symplectic runge kutta and related methods recent results symplectic algorithms are numerical integrators for hamiltonian systems that preserve the symplectic structure in phase space in long time integrations these algorithms tend to perform better than their nonsymplectic counterparts some symplectic algorithms are derived by explicitly finding a generating function other symplectic algorithms are members of standard families of methods such as runge kutta methods that just turn out to preserve the symplectic structure here we survey what is known about the second type of symplectic algorithms

weak bisimulations for fuzzy automata abstract forward and backward bisimulations have been introduced recently by ciric et al 2012 9 as a means for modeling the equivalence between states of fuzzy automata and approximating the language equivalence as well as for use in the state reduction of fuzzy automata the main aim of the present paper is to introduce two new kinds of bisimulations weak forward and backward bisimulations which provide better approximations of the language equivalence than forward and backward bisimulations and when employed in the state reduction they provide better reductions we give procedures for deciding whether there exist weak forward and backward simulations and bisimulations and for computing the greatest ones whenever they exist using weak bisimulations in conjunction with the concept of a uniform fuzzy relation we determine necessary and sufficient conditions under which two fuzzy automata are weak bisimulation equivalent we also characterize uniform weak backward and forward bisimulations between two fuzzy automata in terms of isomorphisms between their nerode and reverse nerode automata

a comment on automata theory based on complete residuated lattice valued logic pushdown automata automata theory based on complete residuated lattice valued logic has been first established by qiu and then has been systematically studied by qiu and others the definition of l valued chomsky normal form in xing and qiu automata theory based on complete residuated lattice valued logic pushdown automata fuzzy sets and systems 160 2009 1125 1140 is somewhat different from that in xing and qiu pumping lemma in context free grammar theory based on complete residuated lattice valued logic fuzzy sets and systems 160 2009 1141 1151 in this note we give a more general l valued chomsky normal form to unify the two definitions we mainly show that for an l valued context free grammar an l valued greibach normal form can be equivalently constructed

numerical solution of the sine gordon equation two new difference schemes are proposed for the sine gordon equation they have the advantage that there is a discrete energy which is conserved their stability and convergence is proved also we show some numerical computations using one of these schemes

on periodic critical points and local minimizers of the ohta kawasaki functional abstract in this paper we collect some new observations about periodic critical points and local minimizers of a nonlocal isoperimetric problem arising in the modeling of diblock copolymers in the main result by means of a purely variational procedure we show that it is possible to construct locally minimizing periodic critical points whose shape resemble that of any given strictly stable constant mean curvature periodic hypersurface along the way we establish several auxiliary results of independent interest

determinization of fuzzy automata with membership values in complete residuated lattices in this paper we introduce a new method for determinization of fuzzy finite automata with membership values in complete residuated lattices in comparison with the previous methods developed by belohlavek r belohlavek determinism and fuzzy automata information sciences 143 2002 205 209 and li and pedrycz y m li w pedrycz fuzzy finite automata and fuzzy regular expressions with membership values in lattice ordered monoids fuzzy sets and systems 156 2005 68 92 our method always gives a smaller automaton and in some cases when the previous methods result in infinite automata our method can result in a finite one we also show that determinization of fuzzy automata is closely related to fuzzy right congruences on a free monoid and fuzzy automata associated with them and in particular to the concept of the nerode s fuzzy right congruence of a fuzzy automaton which we introduce and study here

remark on a nonlocal isoperimetric problem abstract we consider isoperimetric problem with a nonlocal repulsive term given by the newtonian potential we prove that regular critical sets of the functional are analytic this optimal regularity holds also for critical sets of the ohta kawasaki functional we also prove that when the strength of the nonlocal part is small the ball is the only possible stable critical set

automata theory based on complete residuated lattice valued logic reduction and minimization automata theory based on complete residuated lattice valued logic called l valued finite automata l vfas has been established by qiu recently in this paper we define a kind of mealy type of l vfas mlfas a generalization of l vfas two kinds of statewise equivalence relations are introduced and a minimal form is defined we study the existence of the minimal form of an mlfa then we show that any two states can be distinguished by some word with finite length also a minimization algorithm of the mlfas is presented in addition we obtain a minimization algorithm for l vfas as well finally we define l valued languages l vls and l valued regular languages l vrls recognized by l vfas and provide some properties of l vrls

consequences and conjectures in preordered sets in a preordered set or preset consequence operators in the sense of tarski defined on families of subsets are introduced from them the corresponding sets of conjectures hypotheses speculations and refutations are considered studying the relationships between these sets and those previously defined on ortholattices all the concepts introduced are illustrated with three particular consequence operators whose behavior is studied in detail the results obtained are applied to the case of fuzzy sets endowed with the usual pointwise ordering

pumping lemma in context free grammar theory based on complete residuated lattice valued logic residuated lattices are important algebras and have close links with various important algebras automata theory based on complete residuated lattice valued logic called l valued automata has been established by the second author in 2001 and 2002 as a continuation of automata theory based on complete residuated lattice valued logic in this paper we mainly deal with the problem concerning pumping lemma in l valued context free languages l cfls as a generalization of the notion in the theory of formal grammars the definition of l valued context free grammars l cfgs is introduced we also discuss a special case of l cfgs l right or left linear grammars and show the equivalence between l linear grammars and l regular grammars this result shows that we generalize the pumping lemma in l valued regular languages l rls more recently established by the second author

numerical methods for hamiltonian pdes the paper provides an introduction and survey of conservative discretization methods for hamiltonian partial differential equations the emphasis is on variational symplectic and multi symplectic methods the derivation of methods as well as some of their fundamental geometric properties are discussed basic principles are illustrated by means of examples from wave and fluid dynamics

on conjectures in orthocomplemented lattices abstract a mathematical model for conjectures in orthocomplemented lattices is presented after defining when a conjecture is a consequence or a hypothesis some operators of conjectures consequences and hypotheses are introduced and some properties they show are studied this is the case for example of being monotonic or non monotonic operators as orthocomplemented lattices contain orthomodular lattices and boolean algebras they offer a sufficiently broad framework to obtain some general results that can be restricted to such particular but important lattices this is for example the case of the structure s theorem for hypotheses some results are illustrated by examples of mathematical or linguistic character and an appendix on orthocomplemented lattices is included

stability of spot and ring solutions of the diblock copolymer equation the convergence theory shows that under certain conditions the diblock copolymer equation has spot and ring solutions we determine the asymptotic properties of the critical eigenvalues of these solutions in order to understand their stability in two dimensions a threshold exists for the stability of the spot solution it is stable if the sample size is small and unstable if the sample size is large the stability of the ring solutions is reduced to a family of finite dimensional eigenvalue problems in one study no two interface ring solutions are found by the convergence method if the sample is small a stable two interface ring solution exists if the sample size is increased it becomes unstable if the sample size is increased further

backward error analysis for numerical integrators backward error analysis has become an important tool for understanding the long time behavior of numerical integration methods this is true in particular for the integration of hamiltonian systems where backward error analysis can be used to show that a symplectic method will conserve energy over exponentially long periods of time such results are typically based on two aspects of backward error analysis i it can be shown that the modified vector fields have some qualitative properties which they share with the given problem and ii an estimate is given for the difference between the best interpolating vector field and the numerical method these aspects have been investigated recently for example by benettin and giorgilli in j statist phys 74 1994 pp 1117 1143 by hairer in ann numer math 1 1994 pp 107 132 and by hairer and lubich in numer math 76 1997 pp 441 462 in this paper we aim at providing a unifying framework and a simplification of the existing results and corresponding proofs our approach to backward error analysis is based on a simple recursive definition of the modified vector fields that does not require explicit taylor series expansion of the numerical method and the corresponding flow maps as in the above cited works as an application we discuss the long time integration of chaotic hamiltonian systems and the approximation of time averages along numerically computed trajectories

a note on trillas chc models trillas et al e trillas s cubillo e castineira on conjectures in orthocomplemented lattices artificial intelligence 117 2000 255 275 recently proposed a mathematical model for conjectures hypotheses and consequences abbr chcs and with this model we can execute certain mathematical reasoning and reformulate some important theorems in classical logic we demonstrate that the orthomodular condition is not necessary for holding watanabe s structure theorem of hypotheses and indeed in some orthocomplemented but not orthomodular lattices this theorem is still valid we use the chc operators to describe the theorem of deduction the theorem of contradiction and the lindenbaum theorem of classical logic and clarify their existence in the chc models a number of examples is presented and we re define the chc operators in residuated lattices and particularly reveal the essential differences between the chc operators in orthocomplemented lattices and residuated lattices

invariant conserving finite difference algorithms for the nonlinear klein gordon equation a formalism for systematicaiiy deriving second order accurate finite difference algorithms which conserve certain invariant quantities in the original nonlinear pdes is presented three algorithms are derived for the nonlinear klein gordon equation nlkge based on the proposed formalism the local conservation laws of the nlkge form the basic starting point in our derivation which hinges essentially on the commutativity of certain finite difference operators such commutativity in the discrete approximations allows a preservation of the derivation properties of the continuous counterparts at the pde level with appropriate boundary conditions the proposed algorithms preserve in the discrete sense either the total system energy or the system s linear momentum several variants of the present algorithms and their relation to previously proposed algorithms are discussed an analysis of the accuracy and stability is conducted to compare the different variants of the proposed algorithms the preservation of energy of the present algorithms for the nlkge can also be viewed as providing a method of stabilization for conditionally stable algo thms for the linear wave equation the computer impiementation of the proposed algorithms with the treatment of the boundary conditions is presented in detail numerical examples are given concerning soliton collisions in the sine gordon equation the double sine gordon equation and the cb phi four equation the numerical results demonstrate that the present algorithms can preserve accurately up to 10 decimal digits the total system energy for a very coarse grid reliable algorithms for josephson junction models which contain dissipation damping mechanisms and driving bias current are obtained as direct by products of the proposed invariant conserving algorithms for the nlkge even though presented mainly for the 1 d case the proposed algorithms are generalizable to the 2 d and 3 d cases and to the case of complex valued nlkge

high order multisymplectic runge kutta methods we study the spatial semidiscretizations obtained by applying runge kutta rk and partitioned runge kutta prk methods to multisymplectic hamiltonian pdes these methods can be regarded as multisymplectic hp finite element methods for wave equations all the methods we consider are multisymplectic we determine their properties with regard to existence of solutions dispersion and order the lobatto iiia iiib prk method can lead to explicit odes and exhibits a surprisingly high order of 2 higher than the number of degrees of freedom per cell

automata theory based on complete residuated lattice valued logic pushdown automata automata theory based on complete residuated lattice valued logic called l valued automata has been proposed by qiu automata theory based on complete residuated latticed valued logic sci china ser f 44 6 2001 419 429 automata theory based on complete residuated latticed valued logic ii sci china ser f 45 6 2002 442 452 in this paper we discuss some properties of l valued context free grammars languages and pushdown automata we show that for such grammars chomsky and greibach normal forms can be equivalently constructed and we also prove that the languages accepted by final states and by empty stack in l valued pushdown automata are equivalent in particular we prove the equivalence between l valued context free grammars and l valued pushdown automata

reflections on an old problem that of preserving the logical forms and symmetry abstract this paper is devoted to analyze and discuss some of the new aspects about the preservation of the logical forms appearing in ordinary reasoning and fuzzy logic considering a basic fuzzy algebra bfa as the basic formal model for representing imprecise predicates several classical properties of logical forms which can be not valid depending on the representation of the logical connectives in a bfa are studied special emphasis is put on the general symmetry property or its absence in some logical forms that in some particular cases and in some sense can become symmetrical as they are for instance the linguistic negation the law of perfect repartition or the additive law of measures in any case the difference between what is supposed in a mathematical model and what can be checked in the reality cannot be forgotten since any reality and the study of the ordinary reasoning in natural language is a clear example of that is but an observed reality anyway if what is in a mathematical fuzzy model is not always symmetrical and since these models represent linguistic statements affected by imprecision and uncertainty it can be concluded that in general neither plain language nor commonsense or ordinary reasoning show the same level of symmetry exhibited by the specialized language and the deductive reasoning allowing mathematical proofs

minimality via second variation for a nonlocal isoperimetric problem we discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts we show that critical configurations with positive second variation are local minimizers of the nonlocal area functional and in fact satisfy a quantitative isoperimetric inequality with respect to sets that are l 1 close the link with local minimizers for the diffuse interface ohta kawasaki energy is also discussed as a byproduct of the quantitative estimate we get new results concerning periodic local minimizers of the area functional and a proof via second variation of the sharp quantitative isoperimetric inequality in the standard euclidean case as a further application we address the global and local minimality of certain lamellar configurations

on the reducibility of hypotheses and consequences this paper only tries to shed some additional light on the concepts of hypotheses and consequences introduced by trillas et al e trillas s cubillo and e castineira on conjectures in orthocomplemented lattices artificial intelligence 117 2 2000 255 275 as particular cases of the more general concept of conjecture the definitions of reducible hypothesis and reducible consequence are presented and it is shown that in orthomodular lattices all hypotheses and consequences are reducible but that this is not necessarily the case in proper ortholattices in addition the sets of reducible hypotheses and consequences are characterized both for general ortholattices as well as for the particular case of orthomodular lattices

energy conservation issues in the numerical solution of the semilinear wave equation in this paper we discuss energy conservation issues related to the numerical solution of the semilinear wave equation as is well known this problem can be cast as a hamiltonian system that may be autonomous or not depending on the prescribed boundary conditions we relate the conservation properties of the original problem to those of its semi discrete version obtained by the method of lines subsequently we show that the very same properties can be transferred to the solutions of the fully discretized problem obtained by using energy conserving methods in the hbvms hamiltonian boundary value methods class similar arguments hold true for different types of hamiltonian partial differential equations e g the nonlinear schrodinger equation

automata theory based on complete residuated lattice valued logic turing machines automata theory based on complete residuated lattice valued logic called l valued finite automata l vfas has been established by the second author in 2001 in view of the importance of turing machines in this paper we establish a theory of turing machines based on complete residuated lattice valued logic which is a continuation of l vfas first we give the definition of l valued nondeterministic turing machines l ntms and observe that the multitape l ntms have the same language recognizing power as the single tape l ntms we give some related properties of l valued turing machines and discuss computing with fuzzy letters via l valued turing machines second we introduce the concepts of l valued recursively enumerable languages and l valued recursive languages and obtain some equivalent relations some results concerning the characterization of n recursively enumerable sets are given and the super computing power of l valued turing machines is investigated we also prove that l valued deterministic turing machines and l ntms are not equivalent in the sense of recognizing or deciding languages finally we show that there is no universal l valued turing machine however a universal l valued turing machine exists if the membership degrees of l valued sets are restricted to a finite complete residuated lattice with universal bounds 0 and 1

a logic for default reasoning the need to make default assumptions is frequently encountered in reasoning about incompletely specified worlds inferences sanctioned by default are best viewed as beliefs which may well be modified or rejected by subsequent observations it is this property which leads to the non monotonicity of any logic of defaults in this paper we propose a logic for default reasoning we then specialize our treatment to a very large class of commonly occuring defaults for this class we develop a complete proof theory and show how to interface it with a top down resolution theorem prover finally we provide criteria under which the revision of derived beliefs must be effected

droplet minimizers of an isoperimetric problem with long range interactions we give a detailed description of the geometry of single droplet patterns in a non local isoperimetric problem in particular we focus on the sharp interface limit of the ohta kawasaki free energy for diblock copolymers regarded as a paradigm for those energies modeling physical systems characterized by a competition between short and a long range interactions exploiting fine properties of the regularity theory for minimal surfaces we extend previous partial results in different directions and give robust tools for the geometric analysis of more complex patterns where u is the order parameter of a two phases system confined in r n and and m are two nonnegative numerical parameters the two terms in the energy mimic attractive short range and repulsive long range energies between the phases more precisely the first term is local it favors minimal interface area and drives the system toward a partition into few pure phases while the second term involving a coulomb like kernel g is non local and favor a fine mixing of the phases a detailed description of the energy is given in 1 the competition between these two terms is expected to induce the formation of highly regular mesoscopic patterns whose geometry strongly depends on the choice of the parameters and m e g spherical spots cylinders gyroids and lamellae 0 1 the ohta kawasaki functional for diblock copolymers the model we consider arises as a simplification of a ginzburg landau functional proposed by ohta and kawasaki in their pioneering paper 41 as a possible description of a diblock copolymers dbc system even though it is questionable whether such an energy actually describes dbcs see choksi and ren 15 muratov 39 and niethammer and oshita 40 nevertheless it is a first and mathematically non trivial attempt to capture some of the main features of these systems for such a reason it deserved over the last twenty years great attention from both the mathematical and the physical

lattice theoretic models of conjectures hypotheses and consequences trillas cubillo and castineira artificial intelligence 117 2000 255 275 defined several interesting operators in orthocomplemented lattices these operators give a quite general algebraic model for conjectures consequences and hypotheses we present some properties of conjectures consequences and hypotheses in orthocomplemented lattices which complement or improve the results by trillas cubillo and castineira furthermore we introduce the graded versions of these notions in the setting of residuated lattices and derive some of their properties these graded notions provide certain mathematical tools for modelling conjectures consequences and hypotheses in the environment where uncertain and vague information is involved

lattice valued fuzzy turing machines computing power universality and efficiency in this paper we study fuzzy turing machines with membership degrees in distributive lattices which we called them lattice valued fuzzy turing machines first we give several formulations of lattice valued fuzzy turing machines including in particular deterministic and non deterministic lattice valued fuzzy turing machines l dtmcs and l ntms we then show that l dtmcs and l ntms are not equivalent as the acceptors of fuzzy languages this contrasts sharply with classical turing machines second we show that lattice valued fuzzy turing machines can recognize n r e sets in the sense of bedregal and figueira the super computing power of fuzzy turing machines is established in the lattice setting third we show that the truth valued lattice being finite is a necessary and sufficient condition for the existence of a universal lattice valued fuzzy turing machine for an infinite distributive lattice with a compact metric we also show that a universal fuzzy turing machine exists in an approximate sense this means for any prescribed accuracy there is a universal machine that can simulate any lattice valued fuzzy turing machine on it with the given accuracy finally we introduce the notions of lattice valued fuzzy polynomial time bounded computation lp and lattice valued non deterministic fuzzy polynomial time bounded computation lnp and investigate their connections with p and np we claim that lattice valued fuzzy turing machines are more efficient than classical turing machines

analogies and differences between quantum and stochastic automata abstract we analyze some features of the behaviour of quantum automata providing analogies and differences with the corresponding stochastic models in particular we prove there is a quantum automaton where the change of state depends on unitary transformations defined by matrices with nonnull amplitudes that accepts a non regular language with cut point zero and inverse error polynomially bounded stochastic automata with matrices having nonnull elements and with polynomial bounds on the inverse error recognize only regular languages the class of stochastic languages contains the class of quantum languages quantum languages are empty or contain an infinite number of words the class of quantum languages is not closed under complementation

bisimulations for fuzzy automata bisimulations have been widely used in many areas of computer science to model equivalence between various systems and to reduce the number of states of these systems whereas uniform fuzzy relations have recently been introduced as a means to model the fuzzy equivalence between elements of two possible different sets here we use the conjunction of these two concepts as a powerful tool in the study of equivalence between fuzzy automata we prove that a uniform fuzzy relation between fuzzy automata a and b is a forward bisimulation if and only if its kernel and co kernel are forward bisimulation fuzzy equivalence relations on a and b and there is a special isomorphism between factor fuzzy automata with respect to these fuzzy equivalence relations as a consequence we get that fuzzy automata a and b are ufb equivalent i e there is a uniform forward bisimulation between them if and only if there is a special isomorphism between the factor fuzzy automata of a and b with respect to their greatest forward bisimulation fuzzy equivalence relations this result reduces the problem of testing ufb equivalence to the problem of testing isomorphism of fuzzy automata which is closely related to the well known graph isomorphism problem we prove some similar results for backward forward bisimulations and we point to fundamental differences because of the duality with the studied concepts backward and forward backward bisimulations are not considered separately finally we give a comprehensive overview of various concepts on deterministic nondeterministic fuzzy and weighted automata which are related to bisimulations

a new class of energy preserving numerical integration methods the first ever energy preserving b series numerical integration method for ordinary differential equations is presented and applied to several hamiltonian systems related novel lie algebraic results are also discussed

automata theory based on quantum logic reversibilities and pushdown automata automata theory based on quantum logic called l valued finite automata l vfas may be viewed as a logical approach to quantum computing this work is mainly divided into two parts one part deals with reversibility of l vfas and the other establishes a basic framework of l valued pushdown automata l vpdas first we provide some preliminaries concerning quantum logic and l vfas and we prove a useful property of l valued successor and source operators then we clarify the relationships between various reversibilities closely related to quantum finite automata in the literature in particular we define a reversibility of l vfas which is termed as retrievability and we clarify the relationships between a number of different fashions regarding retrievability of l vfas we prove that some of them are equivalent but for the others to be equivalent the truth value set is required to satisfy a certain condition this is an essential difference from the classical situation afterwards we introduce l vpdas and show that the class of the languages accepted by l vpdas by empty stack coincides with that accepted by l vpdas by final state finally we provide some examples of l vfas and conclude with some remarks

finite difference calculus invariant structure of a class of algorithms for the nonlinear klein gordon equation in a previous work the authors have presented a formalism for deriving systematically invariant symmetric finite difference algorithms for nonlinear evolution differential equations that admit conserved quantities this formalism is herein cast in the context of exact finite difference calculus the algorithms obtained from the proposed formalism are shown to derive exactly from discrete scalar potential functions using finite difference calculus in the same sense as that of the corresponding differential equation being derivable from its associated energy function a conserved quantity a clear ramification of this result is that the derived algorithms preserve certain discrete invariant quantities which are the consistent counterpart of the invariant quantities in the continuous case results on the nonlinear stability of a class of algorithms that are derived using the proposed formalism and that preserve energy or linear momentum are discussed in the context of finite difference calculus some

a selection principle for the sharp quantitative isoperimetric inequality we introduce a new variational method for the study of isoperimetric inequalities with quantitative terms the method is general as it relies on a penalization technique combined with the regularity theory for quasiminimizers of the perimeter two notable applications are presented first we give a new proof of the sharp quantitative isoperimetric inequality in mathbb r n second we positively answer a conjecture by hall concerning the best constant for the quantitative isoperimetric inequality in mathbb r 2 in the small asymmetry regime

symplectic integration of hamiltonian wave equations the numerical integration of a wide class of hamiltonian partial differential equations by standard symplectic schemes is discussed with a consistent hamiltonian approach we discretize the hamiltonian and the poisson structure separately then form the the resulting ode s the stability accuracy and dispersion of different explicit splitting methods are analyzed and we give the circumstances under which the best results can be obtained in particular when the hamiltonian can be split into linear and nonlinear terms many different treatments and examples are compared

automata theory based on complete residuated lattice valued logic a categorical approach automata theory based on complete residuated lattice valued logic called l valued automata l vas has been primarily established by qiu in 2001 and 2002 in this paper we consider the l vas that have l valued initial and final states we study the categorical issue of l vas the main technical contributions include 1 we investigate the relationship between the category of l vas and the category of non deterministic automata ndas also we study the relationship between the category of generalized l vas and the category of ndas 2 we prove the existence of isomorphisms between the category of l vas and the subcategory of generalized l vas and between the category of l vas and the category of sets of ndas 3 finally we further investigate some specific relationships between the output l valued subsets of generalized l vas and the output l valued subsets of ndas

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

fuzzifying topology based on complete residuated lattice valued logic i abstract we greatly extend fuzzifying topology by introducing a unary fuzzy predicate interpreted as the property to be fuzzifying topological spaces on the class of all so called fuzzifying pretopological spaces and by adopting the semantic method of complete residuated lattice valued logic

space time spectral method for the two dimensional generalized sine gordon equation abstract in this paper we propose a space time spectral method for solving the generalized two dimensional sine gordon equation with nonhomogeneous dirichlet boundary conditions and initial conditions the proposed method is based on the legendre galerkin spectral method in space and the spectral collocation method or block spectral collocation method in time we derive a priori error estimates in both l 2 and h 1 norms for the semidiscrete formulation our formulation has spectral accuracy in both space and time our numerical results confirm the exponential convergence of the proposed method in both space and time

a note on the efficient implementation of hamiltonian bvms we discuss the efficient implementation of hamiltonian bvms hbvms a recently introduced class of energy preserving methods for canonical hamiltonian systems see brugnano et al 8 and references therein also sketching their blended formulation we also discuss the case of separable problems for which the structure of the problem can be exploited to gain efficiency

the solution of the two dimensional sine gordon equation using the method of lines the method of lines is used to transform the initial boundary value problem associated with the two dimensional sine gordon equation in two space variables into a second order initial value problem the finite difference methods are developed by replacing the matrix exponential term in a recurrence relation with rational approximants the resulting finite difference methods are analyzed for local truncation error stability and convergence to avoid solving the nonlinear system a predictor corrector scheme using the explicit method as predictor and the implicit as corrector is applied numerical solutions for cases involving the most known from the bibliography line and ring solitons are given

a fourth order avf method for the numerical integration of sine gordon equation abstract in this paper a new scheme which has energy preserving property is proposed for solving the sine gordon equation with periodic boundary conditions it is obtained by the fourier pseudo spectral method and the fourth order average vector field method in numerical experiments the new high order energy preserving scheme is compared with a number of existing numerical schemes for the one dimensional sine gordon equation the new high order energy preserving scheme for the two dimensional sine gordon equation is also investigated numerical results are addressed to further illustrate the conservation of energy and the evolutional behaviors of solitons