computing with words via turing machines a formal approach computing with words cw as a methodology means computing and reasoning by the use of words in place of numbers or symbols which may conform more to humans perception when describing real world problems in this paper as a continuation of a previous paper we aim to develop and deepen a formal aspect of cw according to the previous paper the basic point of departure is that cw treats certain formal modes of computation with strings of fuzzy subsets instead of symbols as their inputs specifically 1 we elaborate on cw via turing machine tm models showing the time complexity is at least exponential if the inputs are strings of words 2 a negative result of 6 not holding is verified which indicates that the extension principle for cw via tms needs to be re examined 3 we discuss cw via context free grammars and regular grammars and the extension principles for cw via these formal grammars are set up 4 some equivalences between fuzzy pushdown automata respectively fuzzy finite state automata fuzzy context free grammars respectively fuzzy regular grammars are demonstrated in the sense that the inputs are instead strings of words 5 some instances are described in detail summarily formal aspect of cw is more systematically established more deeply dealt with while some new problems also emerge

adaptive mesh redistribution for a boundary element panel method in this paper we extend the concept of a grading function as introduced by carey and dinh2 in a one dimensional context to curved contours we then apply the idea to adaptive grid redistribution for a boundary element method to solve a model boundary value problem numerical results are included for a potential problem to demonstrate the effect of the redistribution scheme on the solution and the geometry of the domain

determinization of weighted finite automata over strong bimonoids we consider weighted finite automata over strong bimonoids where these weight structures can be considered as semirings which might lack distributivity then in general the well known run semantics initial algebra semantics and transition semantics of an automaton are different we prove an algebraic characterization for the initial algebra semantics in terms of stable finitely generated submonoids moreover for a given weighted finite automaton we construct the nerode automaton and myhill automaton both being crisp deterministic which are equivalent to the original automaton with respect to the initial algebra semantics respectively the transition semantics we prove necessary and sufficient conditions under which the nerode automaton and the myhill automaton are finite and we provide efficient algorithms for their construction also for a given weighted finite automaton we show sufficient conditions under which a given weighted finite automaton can be determinized preserving its run semantics

brzozowski type determinization for fuzzy automata abstract in this paper we adapt the well known brzozowski determinization method to fuzzy automata this method gives better results than all previously known methods for determinization of fuzzy automata developed by b lohlavek 4 li and pedrycz 20 ignjatovic et al 15 and jancic et al 18 namely as in the case of ordinary nondeterministic automata brzozowski type determinization of a fuzzy automaton results in a minimal crisp deterministic fuzzy automaton equivalent to the starting fuzzy automaton and we show that there are cases when all previous methods result in infinite automata while brzozowski type determinization results in a finite one the paper deals with fuzzy automata over complete residuated lattices but identical results can also be obtained in a more general context for fuzzy automata over lattice ordered monoids and even for weighted automata over commutative semirings

theh p version of the finite element method for elliptic equations of order2m the problems of elliptic partial differential equations stemming from engineering problems are usually characterized by piecewise analytic data it has been shown in 3 4 5 that the solutions of the second order and fourth order equations belong to the spacesb 1 where the weighted sobolev norms of thek th derivatives are bounded bycd k l k l k l l 2 wherec andd are constants independent ofk in this case theh p version of the finite element method leads to an exponential rate of convergence measured in the energy norm 6 12 13 theh p version was implemented in the code probe1 18 and has been very successfully used in the industry we will discuss in this paper the generalization of these results for problems of order2m we will show also that the exponential rate can be achieved if the exact solution belongs to the spacesb 1 where the weighted sobolev norm of thek th derivatives is bounded bycd k l k l k l m 1 c andd are independent ofk in addition if the data is piecewise analytic then in fact the exact solution belongs to the spacesb m 1 problems of this type are related obviously to many engineering problems such as problems of plates and shells and are also important in connection with well known locking problems

p adaptive boundary elements this paper presents the implementation of an adaptive philosophy to plane potential problems using the direct boundary element method after some considerations about the state of the art and a discussion of the standard approach features the possibility of separately treating the modelling of variables and their interpolation through hierarchical shape functions is analysed then the proposed indicators and estimators are given followed by a description of a small computer program written for an ibm pc finally some examples show the kind of results to be expected

adaptive h p and hp versions for boundary integral element methods recently very promising results in a so called hp version of the finite element method have been obtained the basic idea is a balanced combination of mesh refinement and increase of the polynomial degree of the shape functions this idea is applied to a boundary collocation method in this paper the new method is compared with adaptive h and p versions and it is shown in numerical examples that even in the presence of singularities in the exact solution exponential convergence is obtained

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

an improved algorithm for determinization of weighted and fuzzy automata for a given weighted finite automaton over a strong bimonoid we construct its reduced nerode automaton which is crisp deterministic and equivalent to the original weighted automaton with respect to the initial algebra semantics we show that the reduced nerode automaton is even smaller than the nerode automaton which was previously used in determinization related to this semantics we determine necessary and sufficient conditions under which the reduced nerode automaton is finite and provide an efficient algorithm which computes the reduced nerode automaton whenever it is finite in determinization of weighted finite automata over semirings and fuzzy finite automata over lattice ordered monoids this algorithm gives smaller crisp deterministic automata than any other known determinization algorithm

grading functions and mesh redistribution we consider the problem of determining an appropriate grading of a mesh for piecewise polynomial interpolation and for approximate solution of two point boundary value problems by finite difference or finite element methods an analysis of optimality of the mesh and optimal grading functions in various norms and seminorms for the interpolation problem leads to the formulation of an adaptive mesh redistribution algorithm for boundary value problems in one dimension error estimates are given and numerical results presented to demonstrate the performance of the scheme and compare alternative redistribution criteria

a method of grid optimization for finite element methods a computationally useful criterion for grid optimization is derived based on a measure of the interpolation error associated with the finite element model the result is intended to be used to improve the quality of finite element solutions by changing the location of the nodes within a fixed number of degrees of freedom examples of the application of the criterion are provided the finite element method has become an important tool of engineering analysis and design much of the popularity of the method is due to the freedom that it allows in the construction of the discretized model it has long been realized however that this freedom must be carefully exercised since the quality of the finite element solution greatly depends on how the discretization is performed several methods have been proposed to improve the discretized model in an iterative manner one such method is presented in this paper a natural way of improving the quality of finite element solutions is to increase the number of degrees of freedom the process is normally performed after an initial solution is already available several schemes have been devised to introduce the new degrees of freedom in a selective manner in order to produce the greatest possible improvement of the previous solution this calls for the definition of criteria to identify the regions of the domain where the finite element approximation is poorer the new degrees of freedom are added in these regions by either increasing the order of polynomial approximation inside elements the so called p method or by subdivision of elements or h method the process is continued until a specified accuracy is achieved the quality of the finite element solution may be improved also by optimizing the disposition of the nodes analysts often rely on their experience to construct grids that make an efficient use of the available degrees of freedom it is also possible to improve the quality of existing meshes iteratively using predefined guidelines for the redistribution of the nodes this paper considers the development of such guidelines

fuzzy tree language recognizability a fuzzy tree language with membership grades in an arbitrary set is syntactically recognizable s recognizable if its syntactic algebra is finite the equality problem for such languages is decidable and their syntactic algebra can be effectively constructed provided that they are s recognizable linear but non arbitrary tree homomorphisms preserve s recognizability tree automata whose transitions are weighted over the unit interval and whose behavior is computed with respect to a pair made of a t norm distributive over a t conorm have the syntactic recognition power and thus their equivalence problem is decidable however s recognizability is more powerful when dealing with non distributive pairs of such operations

solution of elasticity problems by a self adaptive mesh refinement technique for boundary element computation a self adaptive mesh refinement technique is developed for boundary element solutions of the two dimensional navier s equation the method is based on error reduction and applied on the element and global level to estimate the residual error associated with each mesh results for several problems employing constant two dimensional isotropic elastostatic boundary elements are presented

syntactic recognizability of graphs with fuzzy attributes abstract we present a syntactic recognition theory for graphs with fuzzy edges and fuzzy nodes by employing the algebraic structure of magmoids several examples are illustrated and it is proved that the cut of a recognizable set of fuzzy graphs is not always recognizable as a set of crisp graphs the described theory is then applied for the recognition of fuzzy cognitive maps and the recognizability of interesting examples of sets of fuzzy cognitive maps is examined

feynman graph representation of the perturbation series for general functional measures abstract a representation of the perturbation series of a general functional measure is given in terms of generalized feynman graphs and rules the graphical calculus is applied to certain functional measures of levy type a graphical notion of wick ordering is introduced and is compared with orthogonal decompositions of the wiener ito segal type it is also shown that the linked cluster theorem for feynman graphs extends to generalized feynman graphs we perturbatively prove existence of the thermodynamic limit for the free energy density and the moment functions the results are applied to the gas of charged microscopic or mesoscopic particles neutral in average in d 2 dimensions generating a static field with quadratic energy density giving rise to a pair interaction the pressure function for this system is calculated up to fourth order we also discuss the subtraction of logarithmically divergent self energy terms for a gas of only one particle type by a local counterterm of first order

fuzzy tree automata in this paper we map an arbitrary algebra to a completely distributive lattice and define algebras of fuzzy sets when the algebra is a term algebra we obtain fuzzy tree languages after defining fuzzy equational sets fuzzy rational sets and fuzzy recognizable tree languages we derive a kleene theorem for fuzzy tree languages

complex velocity transformations and the bisognano wichmann theorem for quantum fields acting on krein spaces it is proven that in indefinite metric quantum field theory there exists a dense set of analytic vectors for the generator of the one parameter group of x0 x1 velocity transformations this makes it possible to define complex velocity transformations also for the indefinite metric case in combination with the results of bros epstein and moschella commun math phys 196 535 570 1998 proving bisognano wichmann bw analyticity within the linear program one then obtains a suitable generalization of the bw theorem for local relativistic quantum fields acting on krein spaces quantum fields with indefinite metric

grid optimization for the boundary element method the quality of solution obtained using the boundary element method bem is dependent on how the boundary is discretized this is particularly true in domains of complex geometry a rule for grid optimization for the bem is derived on the bases of an asymptotic measure of the boundary element error that preserves the number of elements degrees of freedom three example problems are provided to show the advantages of grid optimization in terms of accuracy and cost

on the recognizability of fuzzy languages ii it is shown that a fuzzy language is recognized by a finite monoid iff it has finitely many right derivatives iff it has finitely many left derivatives finally we show that the behaviors of lukasiewicz automata have the above properties

perturbative quantum field theory at positive temperatures an axiomatic approach it is shown that the perturbative expansions of the correlation functions of a relativistic quantum field theory at finite temperature are uniquely determined by the equations of motion and standard axiomatic requirements including the kms condition an explicit expression as a sum over generalized feynman graphs is derived the canonical formalism is not used and the derivation proceeds from the beginning in the thermodynamic limit no doubling of fields is invoked an unsolved problem concerning existence of these perturbative expressions is pointed out

error indicators for adaptive mesh refinement in the boundary element method a new approach in this paper a new approach to the development of a posteriori error indicators for the direct boundary element method bem is presented it takes advantage of a unique opportunity offered by the collocation bem that is the possibility of obtaining more than one numerical solution from a given boundary element mesh unlike other approaches the error indicator introduced in the present paper does not require any explicit evaluation of the residual of the approximate solution in order to test the capability of the proposed error indicators to actually control a selective mesh refinement process an h adaptive refinement strategy is also described together with the criteria for refinement and for the termination of the adaptive process full details on the whole numerical implementation for two dimensional problems are provided in particular there is described a direct method for the numerical evaluation of cauchy principal value integrals numerical examples show that pointwise convergence is always obtained in a few steps and the final meshes look nearly as if they were designed by an expert analyst

algebraic recognizability of regular tree languages we propose a new algebraic framework to discuss and classify recognizable tree languages and to characterize interesting classes of such languages our algebraic tool called preclones encompasses the classical notion of syntactic algebra or minimal tree automaton but adds new expressivity to it the main result in this paper is a variety theorem a la eilenberg but we also discuss important examples of logically defined classes of recognizable tree languages whose characterization and decidability was established in recent papers by benedikt and segoufin and by bojanczyk and walukiewicz and can be naturally formulated in terms of pseudovarieties of preclones finally this paper constitutes the foundation for another paper by the same authors where first order definable tree languages receive an algebraic characterization

perturbation theory of wightman functions a perturbative expansion of the wightman functions and more generally of vacuum expectation values of products of time ordered and anti time ordered products is derived for 4 4 field theory the result is expressed as a sum over generalized feynman graphs the derivation is based exclusively on the equation of motion and the wightman axioms neither canonical commutation relations nor asymptotic conditions are needed at any point in the zero mass case the individual graphs are infrared divergent but the sum over all graphs of a given order is convergent

an adaptive h r boundary element algorithm for the laplace equation in this paper the combination of the h method mesh refinement and the r method mesh redistribution is employed to solve the laplace equation using the boundary element procedure the key in this approach is to derive on upper bound for the residual associated with the boundary element solution and minimize this bound with respect to an unknown grading function the latter part is achieved by employing techniques of the calculus of variations

pumping lemma in automata theory based on complete residuated lattice valued logic a note abstract automata theory based on complete residuated lattice valued logic called l valued automata was established in qiu automata theory based on completed residuated lattice valued logic i science in china f 44 6 2001 419 429 qiu automata theory based on completed residuated lattice valued logic ii science in china f 45 6 2002 442 452 in this note we deal with the pumping lemma in l valued automata theory after recalling some preliminaries related to complete residuated lattices and l valued automata we define a number of l valued accepting predicates in particular the pumping lemma for l valued automata theory is set up we show that if those related l valued predicates are defined by using connective instead of then the pumping lemma may not hold again furthermore we investigate the l valued automata with e transitions and present the equivalence between the l valued automata without e transitions and those with e transitions finally a number of related questions is addressed

characterizations of fuzzy finite automata in this paper we establish some important concepts in fuzzy finite automata ffas with bifuzzy property and clarify their essential relationships first we present a number of basic definitions and properties in ffas we then define a bifuzzy family of subautomata bifuzzy source and successor operators as well as demonstrate the bifuzzy topological characterization of ffas also we clarify the relationships between our results and previous ones and thus derive a relation figure visualizing their connections afterwards we investigate further the bifuzzy separability retrievability and homomorphism fuzzy continuous mapping and open mapping are also incorporated in particular we prove a number of equivalent characterizations for these concepts and expound the relationships amongst them this also concludes that both the bifuzzy source and successor operators are fuzzy closure operators we discover that our investigation generalizes to some extent the algebraic fuzzy automata elaborated by malik mordeson and others finally the main results obtained are summarized the potential applications are indicated and a number of related problems for further study are addressed

adaptive boundary element methods for strongly elliptic integral equations integral operators are nonlocal operators the operators defined in boundary integral equations to elliptic boundary value problems however are pseudo differential operators on the boundary and therefore provide additional pseudolocal properties these allow the successful application of adaptive procedures to some boundary element methods in this paper we analyze these methods for general strongly elliptic integral equations and obtain a posteriori error estimates for boundary element solutions we also apply these methods to nodal collocation with odd degree splines some numerical examples show that these adaptive procedures are reliable and effective