equilibrium points of a singular cooperative system with free boundary abstract in this paper we initiate the study of maps minimising the energy d u 2 2 u d x which due to lipschitz character of the integrand gives rise to the singular euler equations u u u u 0 u u 1 u m our primary goal in this paper is to set up a road map for future developments of the theory related to such energy minimising maps our results here concern regularity of the solution as well as that of the free boundary they are achieved by using monotonicity formulas and epiperimetric inequalities in combination with geometric analysis

twisted modules and co invariants for commutative vertex algebras of jet schemes abstract let z a k be an affine scheme over c and j z its jet scheme it is well known that c j z the coordinate ring of j z has the structure of a commutative vertex algebra this paper develops the orbifold theory for c j z a finite order linear automorphism g of z acts by vertex algebra automorphisms on c j z we show that c j g z where j g z is the scheme of g twisted jets has the structure of a g twisted c j z module we consider spaces of orbifold coinvariants valued in the modules c j g z on orbicurves y g with y a smooth projective curve and g a finite group and show that these are isomorphic to c z g

adaptive robot under fuzzy control an arc welding industrial robot isotron 251 is treated its adaptation is considered as an attribute of the robot to trace an arbitrary pretaught welding seam by means of a noncontact laser sensor the control strategy is proposed to use a fuzzy model describing uncertainty of working conditions the experience shows how the notion of fuzzy sets powers is used for on line estimation prediction and control of robot motions the control algorithm has been tested by a three level microprocessor controller

direct digital control auto tuning and supervision using fuzzy logic abstract we describe algorithms for the design and implementation of fuzzy logic controllers and present results of their application to temperature control of a heated air stream auto tuning methods are studied for adjustment of conventional parameters to compensate for actual values of control variables or of external variables and for improvement of performance from observation of results conditions for verifying the consistency of the linguistic protocol with its implementation are presented in an appendix

on the application of fuzzy sets theory to the optimal flood control problem arising in water resources systems abstract we first survey the application of fuzzy sets theory to various problems occurring in water resources systems the problem of optimal flood control planning by an appropriate integration of structural and non structural measures with the objective of optimizing the flood damage reduction due to recurrent floods is modeled via fuzzy sets methodologies a two level optimization model appropriate for planning decisions first on a regional or local level and then on a national level is proposed a third phase involving coordination is appended in particular a fuzzy optimization model involving an adroit combination of fuzzy dynamic programming type reasoning and a branch and bound type search procedure is offered as a more utilitarian approach than its crisp equivalent the performance of the algorithm is illustrated with an example

a pid type fuzzy controller with self tuning scaling factors by relating to the conventional pid control theory we propose a new fuzzy controller structure namely pid type fuzzy controller in order to improve further the performance of the transient state and the steady state of the pid type controller we develop a method to tune the scaling factors of the pid type fuzzy controller on line simulation of the pid type fuzzy controller with the self tuning scaling factors shows a better performance in the transient and steady state response

experiments with the use of a rule based self organising controller for robotics applications abstract the rule based self organising controller soc has as one of its main advantages the fact that there is no need to have a mathematical description of the system to be controlled in this controller the rules are linguistic statements expressed mathematically through the concepts of fuzzy sets and correspond to the actions a human operator would take when controlling a given process this paper describes several experiments which were performed with soc the control was considered of both simulated linear and non linear processes the performance of the controller was investigated for step inputs as well as for waveforms which the process output had to track the effect of process noise was also considered it was shown that soc had a performance slightly superior to that of a pid controller when noise was present and the process had time varying coefficients as well as pronounced non linear characteristics this paper also reviews some of the successful initial work carried out in connection with the application of soc to a revolute joint robot arm this presented a system with highly non linear dynamics to each of several socs working in paralle the sampling times of only a few milliseconds were short compared with several minutes or even hours associated with previous applications of fuzzy controllers to cement kilns

solution of optimal control problems with lumped parameters having single or multiple goal objectives in fuzzy environment abstract in this paper optimal control problems ocp of both linear and nonlinear type with lumped parameters are considered continuous ocp with single or multiple objectives are transformed to finite dimensional optimization problems through discretization the formulation of goal optimal control problem gocp using goal programming methodology first given by levary 1986 for equal time subintervals is extended for unequal subintervals by a generalized formulation the concept of local and global goals proposed here is very useful for multicriteria dynamic optimization problems in several physical situations ocp involve fuzziness in their goals priorities parameters or relative weights and associated linguistic variables etc a new formulation of ocp in fuzzy environment is given as a fuzzy goal optimal control problem fgocp and the solution procedure is illustrated by linear and nonlinear examples

existence and partial regularity of static liquid crystal configurations we establish the existence and partial regularity for solutions of some boundary value problems for the static theory of liquid crystals some related problems involving magnetic or electric fields are also discussed

a heuristic self tuning fuzzy controller abstract in this paper a self tuning fuzzy controller based on the tuning of scaling factors is presented the tuning schemes are tuning the scaling factors is individual tuning rules the tuning rules are transformed into numeric tuning tables by applying suitable membership functions so that the adjustings of the scaling factors are merely matrix mappings since the tuning rules are constructed with the control signal and the output error and change in output error of the plant as their antecedent parts the tuning processes have the advantage of real time property simulation results with a perturbed servo and the inverted pendulum system as the controlled plants show the performance and adaptivity of the proposed controller

twisted modules over vertex algebras on algebraic curves abstract we extend the geometric approach to vertex algebras developed by the first author to twisted modules allowing us to treat orbifold models in conformal field theory let v be a vertex algebra h a finite group of automorphisms of v and c an algebraic curve such that h aut c we show that a suitable collection of twisted v modules gives rise to a section of a certain sheaf on the quotient x c h we introduce the notion of conformal blocks for twisted modules and analyze them in the case of the heisenberg and affine kac moody vertex algebras we also give a chiral algebra interpretation of twisted modules

expert fuzzy controller abstract we describe a fuzzy controller based on our general purpose fuzzy expert system shell flops we take a decision theoretic view in designing an optimal fuzzy controller the variables in the decision problem are 1 the defuzzification rules 2 the fuzzification rules and 3 the fuzzy control rules the objective function is composed of fuzzy goals specified for each system output we differentiate between decision making under certainty risk or uncertainty depending whether or not random elements are present in the system an example is presented illustrating varying the defuzzification rules to improve system performance

fuzzy modelling and identification with genetic algorithm based learning abstract a ga based learning algorithm is proposed in this paper for the identification of tsk models the algorithm consists of four blocks partition block ga block tuning block and termination block the partition block is to determine an estimated partition of input variables the ga block is to optimise the structure of a tsk model the tuning block is to fine tune the parameters of the tsk model using the gradient descent based approach and the termination block checks that the resultant tsk model is satisfactory the proposed gabl algorithm has the advantage of simplicity flexibility high accuracy and automation the presented numerical examples indicate that the gabl algorithm is effective in constructing a good tsk model for complex nonlinear systems

model reference adaptive fuzzy control a linguistic space approach in this paper a model reference adaptive fuzzy controller with the adaptivity achieved by tuning both the scaling factors and the control rules is presented the proposed tuning strategy includes two parts 1 by referring a reference trajectory created in the linguistic space to tune the scaling factors and 2 by applying an auxiliary correction matrix generated by specific tuning rules to compensate the control rules since only the control signal the output error and change in output error of the plant are needed in the construction of the reference trajectory and the tuning rules the tuning strategy has the advantage of real time property simulation results with a damping process and an oscillating process as the controlled plants show the performance and adaptivity of the proposed controller

derivation and application of an expert fuzzy optimal control system abstract this paper presents derivation of a fuzzy optimal controller which may be used to develop a knowledge based expert control system for supervising and tuning of conventional feedback control loops a general theory based on the conceptual framework for optimization in a fuzzy environment is presented for the development of linear and nonlinear fuzzy optimal control systems this technique is based on integration of the optimal control theorem and fuzzy variables the resulting expert controller is expressed in the form of linguistic fuzzy rules representing the knowledge base for tuning of the feedback control loops tuning inferences are made from the response of local feedback control loops by applying the compositional rule of inference to the knowledge base

fuzzy self organizing controller and its application for dynamic processes abstract this paper deals with a self organizing controller used in real time and its application for dynamic processes the control policy of the controller is able to develop and improve automatically a new self organizing control algorithm which modifies the fuzzy control decision table is presented in this paper the control algorithm needs much less computation time and memory of the computer this paper provides two examples of the fuzzy self organizing control a set of experiment results is described it shows that the controller can be applied to some processes such as processes with time lag and non linearity or one whose parameter or running conditions are varied which were usually not controlled very well with a conventional controller

on pid controller tuning using orthonormal series identification abstract the step response of a closed loop system is identified by means of a laguerre expansion this offers certain advantages over arma models namely lack of bias in the estimates structural flexibility and the ability to precompute the regressors the laguerre coefficients of the retuned system are calculated in terms of the tuning constants and a quadratic performance index is optimized results from simulations and from industrial trials are given

studies on the output of fuzzy controller with multiple inputs abstract in this paper we discuss the output of a fuzzy controller in which an arbitrary number of fuzzy control inputs and an arbitrary number of fuzzy sets employed to fuzzify every linguistic variable are considered in the fuzzy control rules a rule base with a rule mapping concept is considered a defuzzifier with the general t norm and t conorm operators is formulated a triangular family is adopted so that the defuzzifier of the complete rule mapping is represented by a formula consisting of only some rules using a specific logic pair to evaluate values we derive that the output of the fuzzy controller can be simply represented by a linear parametric function of the inputs to the fuzzy controller

a method for closed loop automatic tuning of pid controllers abstract a simple method for the automatic tuning of pid controllers in closed loop is proposed a limit cycle is generated through a nonlinear feedback path from the process output to the controller reference signal the frequency of this oscillation is above the crossover frequency and below the critical frequency of the loop transfer function the amplitude and frequency of the oscillation are estimated and the control parameters are adjusted iteratively such that the closed loop transfer function from the controller reference to the process output attains a specified amplitude at the oscillation frequency

a rule self regulating fuzzy controller abstract the most effective way to improve the performance of a fuzzy controller may be to optimize the fuzzy control rules usually fuzzy control rules are generated by 1 translating the operators experience into fuzzy linguistic form directly 2 monitoring and summarizing the control behaviour of the operators 3 modelling the process to be controlled using fuzzy set theory 4 self organizing in running of the control systems different from the above means the authors in 10 have proposed a method to regulate fuzzy control rules where the control rules are represented by an analytic expression with a regulating factor the authors in this paper employ such a method and use a simplified fuzzy control algorithm which allows fuzzy control rules to be regulated on line thus constructing a rule self regulating fuzzy controller rsfc simulation results demonstrate that the performance of rsfc is better than that of a non rule regulating fuzzy controller

genetic algorithm based optimal fuzzy controller design in the linguistic space in this paper a genetic algorithm ga based optimal fuzzy controller design is proposed the design procedure is accomplished by establishing an index function as the consequent part of the fuzzy control rule the inputs of the controller after scaling are utilized by the index function for computing the output linguistic value this linguistic value can then be used to map the suitable fuzzy control actions this proposed novel fuzzy control rule has crisp input and fuzzified output characteristics the index function plays a role in mapping the desired fuzzy sets for defuzzification resulting in a controlled hypersurface in the linguistic space formed by the input fuzzy variables two types of index functions both linear and nonlinear are introduced for controlling systems with different degrees of nonlinearity the parameters of the index function are obtained by applying a simple ga with a suitable fitness function various controlled systems result in various parameter sets depending on their dynamics under the acquired optimal parameter set the optimal index function can be used to generate the desired control actions several simulation examples are given to verify the performance of the proposed ga based fuzzy controller

an elementary partial regularity proof for vector valued obstacle problems soit r n et m r n des ensembles ouverts bornes on suppose n 3 n 2 et on requiert que la frontiere de m est une hyper surface lisse de classe c 2 l espace de fonction h 1 2 m des fonctions de comparaison consiste en toutes les fonctions de sobolev v h 1 2 r n avec la propriete v x m pour presque tous les points x dans si u h 1 2 m minimise localement l energie alors il y a un sous ensemble ouvert 0 de tel que u c 1 0 r n pour tout 0 1 h n 2 0 0 et x 0 si et seulement si lim 1 0 r 2 n br x du 2 dz 0

a remark on the c 2 cofiniteness condition on vertex algebras we show that a finitely strongly generated non negatively graded vertex algebra is c 2 cofinite if and only if it is lisse in the sense of beilinson et al preprint this shows that the c 2 cofiniteness is indeed a natural finiteness condition

calculus of twisted vertex operators starting from an arbitrary isometry of an arbitrary even lattice twisted and shifted vertex operators are introduced under commutators these operators provide realizations of twisted affine lie algebras this construction generalizing a number of known ones is based on a self contained calculus

generating rules for fuzzy logic controllers by functions abstract in this paper we introduce a method of generating control rules for fuzzy logic controllers called parametric functions method by this method we can obtain control rules by adjusting several parameters

freely generated vertex algebras and non linear lie conformal algebras we introduce the notion of a non linear lie conformal superalgebra and prove a pbw theorem for its universal enveloping vertex algebra we also show that conversely any graded freely generated vertex algebra is the universal enveloping algebra of a unique up to isomorphism non linear lie conformal superalgebra this correspondence will be applied in the subsequent work to the problem of classification of finitely generated simple graded vertex algebras

a self tuning fuzzy controller abstract the aim of a fuzzy controller is to compensate the dynamic characteristics of the controlled system it is the same purpose as for the fuzzy logic controller flc now the best design of the flc is required for achievement of this purpose but such a controller has not yet been completely designed because the fuzzy controller is constructed of control rules which are ambiguously described by conventional controller strategies and expert knowledge and because its structure theory is not well established yet that is in order to construct the best rules a highly skilled technique and a tuning technique using trial and error are required as a way to easily design a flc a method which improves a given standard flc is considered in this paper we propose a self tuning algorithm of the flc it has two functions in adjusting the scaling factors which are the parameters of the flc and in improving the control rules of flc by evaluating the control response at real time and the control results after operations we add these functions to the flc and design the self tuning fuzzy controller stfc this makes the flc a good controller

fuzzy optimal control the problem of fuzzy optimal control of nonlinear systems is formulated and solved on the basis of fuzzy mathematical programming the fuzzy optimal control problem is extended to the case of multicriteria optimal control a particular case of fuzzy optimal control with fuzzy objective function and crisp transversality conditions is also considered

on line off line optimization of complex processes using a linguistic self organized optimizing control scheme abstract chemical processes usually have a complicated structure and are subjected to unknown long term and short term disturbances many process optimization schemes were developed under the assumption of the existence of a model that can predict the process steady state and or dynamics however such a model may not be precise nor even exist a rule based methodology without the need of such a model is hence developed in this work the optimizing control scheme developed here is actually a combination of a modified version of the linguistic control scheme developed by procyk and mamdani and the concept of reduced gradient optimization it is assumed that the objective function and constraints of the optimization problem are functions of manipulated variables and measurements only the optimizing control algorithm is also formulated into a look up table the rule based optimizing control scheme possesses three important properties adapting learning and decision making some numerical examples of multivariable nature and constrained situations are solved by this approach and compared with well known approaches it is shown that the method is superior to the previous methods a process system consisting of a series of cstr has been simulated as an illustrative example the simulation results show that the optimizing control scheme is promising and should be of interest to process industry

a pd like self tuning fuzzy controller without steady state error abstract this paper presents a pd like self tuning fuzzy controller stfc based on the tuning of scaling factors a two stage tuning method including a direct tuning stage and indirect tuning stage is proposed for most linear controlled plants application of the direct tuning stage alone is enough to yield satisfactory system responses the proposed stfc can automatically detect the operating ranges of input variables and then adjust the scaling factors in the indirect tuning stage the back propagation algorithm can be directly applied to fine tune the scaling factors when a more complicated plant is controlled furthermore a simple but efficient method for fully compensating for steady state error which is the main problem in a pd like fuzzy logic control system is proposed so that tuning for fuzzy logic rule bases is unnecessary simulation results with linear time invariant systems and a nonlinear unstable system as the controlled plants show that the proposed technique yields zero steady state error responses very quickly without overshoot or oscillatory behavior

cyclotomic double affine hecke algebras and affine parabolic category o using the orbifold kz connection we construct a functor from an affine parabolic category o of type a to the category o of a cyclotomic rational double affine hecke algebra h then we give several results concerning this functor

conformal blocks verlinde formula and diagram automorphisms abstract the verlinde formula computes the dimension of the space of conformal blocks associated to simple lie algebras and stable pointed curves if a simply laced simple lie algebra admits a nontrivial diagram automorphism then this automorphism acts on the space of conformal blocks naturally we prove an analogue of the verlinde formula for the trace of the diagram automorphism on the space of conformal blocks along the way we get an analogue of the kac walton formula for the trace of the diagram automorphism we also get a twining type formula between the conformal blocks for the pair s l 2 n 1 s p 2 n

fuzzy adaptive control of a first order process abstract fuzzy adaptive control of a first order process with a varying gain and time constant is demonstrated the fuzzy adaptive controller is based on a fuzzy model based controller that uses an explicit fuzzy model of the process adaptation is achieved by the addition of on line identification of the fuzzy process model starting with a crude initial model the adaptive controller matches the model to the process to self tune the controller the model is further modified when the operating point is changed to adapt the controller to the new process conditions two different fuzzy identification algorithms are both shown to provide a successful fuzzy adaptive controller the robustness of the algorithms in the presence of process noise is investigated and improved by the addition of weighting factors

self learning fuzzy controllers based on temporal backpropagation a generalized control strategy that enhances fuzzy controllers with self learning capability for achieving prescribed control objectives in a near optimal manner is presented this methodology termed temporal backpropagation is model sensitive in the sense that it can deal with plants that can be represented in a piecewise differentiable format such as difference equations neural networks gmdh structures and fuzzy models regardless of the numbers of inputs and outputs of the plants under consideration the proposed approach can either refine the fuzzy if then rules of human experts or automatically derive the fuzzy if then rules if human experts are not available the inverted pendulum system is employed as a testbed to demonstrate the effectiveness of the proposed control scheme and the robustness of the acquired fuzzy controller

a stability approach to fuzzy control design for nonlinear systems abstract by the application of fuzzy set theory a method for feedback control of nonlinear systems based on the principle of sliding mode control is developed called fuzzy sliding mode control the control method provides a simple way to achieve asymptotic stability of the systems other attractive features of the method include a minimal realization of the fuzzy controller and robustness against model uncertainties and external disturbances in addition the method is capable of handling the chattering problem inherent to sliding mode control simply and effectively a simulation study of an inverted pendulum system is presented to demonstrate these features of the method also it is indicated that the method is easy to use for practicing control engineers

on simplicity of vacuum modules we find necessary and sufficient conditions of irreducibility of vacuum modules over affine lie algebras and superalgebras from this we derive conditions of simplicity of minimal w algebras moreover in the case of the virasoro and neveu schwarz algebras we obtain explicit formulas for the vacuum determinants

generalized moonshine ii borcherds products the goal of this paper is to construct infinite dimensional lie algebras by using infinite product identities and to use these lie algebras to reduce the generalized moonshine conjecture to a pair of hypotheses about group actions on vertex algebras and lie algebras we expect the lie algebras that we construct to manifest as algebras of physical states in an orbifold conformal field theory yet to be fully constructed with symmetries given by the monster simple group we introduce vector valued modular functions attached to families of modular functions of different levels and we prove infinite product identities for a distinguished class of automorphic functions on a product of two half planes we recast this result using the borcherds harvey moore singular theta lift and show that the vector valued functions attached to completely replicable modular functions with integer coefficients lift to automorphic functions with infinite product expansions at all cusps for each element of the monster simple group we construct an infinite dimensional lie algebra such that its denominator formula is an infinite product expansion of the automorphic function arising from that element s mckay thompson series these lie algebras have the unusual property that their simple roots and all root multiplicities are known we show that under certain hypotheses characters of groups acting on these lie algebras form functions on the upper half plane that are either constant or invariant under a genus zero congruence group