upward separations and weaker hypotheses in resource bounded measure we consider resource bounded measure in double exponential time complexity classes in contrast to complexity class separation translating downwards we show that measure separation translates upwards for example m p np email protected email protected e ne email protected email protected e x p nexp 0 we also show that if ne does not have e measure 0 then the np machine hypothesis holds we give oracles relative to which the converses of these statements do not hold therefore the hypothesis on the e measure of ne is relativizably weaker than the often investigated p measure hypothesis on np but it has many of the same consequences

a note on measuring in p we revisit the problem of generalising lutz s resource bounded measure rbm to small complexity classes and propose a definition of a random based rbm on p k n dtime o nk which we argue as being a good generalisation to p of lutz s rbm we cannot unconditionally prove the existence of such a measure but we give sufficient and necessary conditions for its existence we also revisit an rbm for p defined by strauss inform comput 136 1 1997 1 and correct an erroneous claim concerning the relations between and random sets a correction to this mistake is then proposed which is a less powerful but accurate relation between and random sets in order to obtain these results we introduce a mathematical structure called a measuring system which is a general setting that can be used to compare different rbms on any fixed complexity class through a partial ordering relation

dynamics of a free boundary problem with curvature modeling electrostatic mems the dynamics of a free boundary problem for electrostatically actuated microelectromechanical systems mems is investigated the model couples the electric potential to the deformation of the membrane the deflection of the membrane being caused by application of a voltage difference across the device more precisely the electrostatic potential is a harmonic function in the angular domain that is partly bounded by the deformable membrane the gradient trace of the electric potential on this free boundary part acts as a source term in the governing equation for the membrane deformation the main feature of the model considered herein is that unlike most previous research the small deformation assumption is dropped and curvature for the deformation of the membrane is taken into account what leads to a quasilinear parabolic equation the free boundary problem is shown to be well posed locally in time for arbitrary voltage values and globally in time for small voltages values furthermore existence of asymptotically stable steady state configurations is proved in case of small voltage values as well as non existence of steady states if the applied voltage difference is large finally convergence of solutions of the free boundary problem to the solutions of the well established small aspect ratio model is shown

on the partial differential equations of electrostatic mems devices iii refined touchdown behavior abstract this paper is a continuation of n ghoussoub y guo on the partial differential equations of electrostatic mems devices stationary case siam j math anal 38 2007 1423 1449 and n ghoussoub y guo on the partial differential equations of electrostatic mems devices ii dynamic case nodea nonlinear differential equations appl 2008 in press where we analyzed nonlinear parabolic problem u t u f x 1 u 2 on a bounded domain of r n with dirichlet boundary conditions this equation models a simple electrostatic micro electromechanical system mems device consisting of a thin dielectric elastic membrane with boundary supported at 0 above a rigid ground plate located at 1 here u is modeled to describe dynamic deflection of the elastic membrane when a voltage represented here by is applied the membrane deflects towards the ground plate and a snap through touchdown must occur when it exceeds a certain critical value pull in voltage creating a so called pull in instability which greatly affects the design of many devices in an effort to achieve better mems design the material properties of the membrane can be technologically fabricated with a spatially varying dielectric permittivity profile f x in this work some a priori estimates of touchdown behavior are established based on which the refined touchdown profiles are obtained by adapting self similar method and center manifold analysis applying various analytical and numerical techniques some properties of touchdown set such as compactness location and shape are also discussed for different classes of varying permittivity profiles

generic density and small span theorem we refine the genericity concept of ambos spies by assigning a real number in 0 1 to every generic set called its generic density we construct sets of generic density any e computable real in 0 1 and show a relationship between generic density and lutz resource bounded dimension we also introduce strong generic density and show that it is related to packing dimension we show that all four notions are different we show that whereas dimension notions depend on the underlying probability measure generic density does not which implies that every dimension result proved by generic density arguments simultaneously holds under any biased coin based probability measure we prove such a result we improve the small span theorem of juedes and lutz to the packing dimension setting for k bounded truth table reductions under any biased coin probability measure

complete blow up after tmax for the solution of a semilinear heat equation soit un sous ensemble ouvert borne de r n a frontiere lisse on considere le probleme u t u f u dans o t u 0 sur o t u x o u 0 x x ou f r r est localement de lipschitz non decroissante et f o 0

elicitation and fine tuning of fuzzy control rules using symbiotic evolution this paper exploits the ability of symbiotic evolution se as a generic methodology to elicit a fuzzy rule base of the mamdani type almost all fuzzy rule base generation algorithms produce rule bases with redundant and overlapped membership functions that limit their interpretability elegance in their application we address this problem by applying an algorithm to merge any similar membership functions it is shown that our proposed algorithm leads generally to a more transparent and more interpretable rule base with a minimum number of membership functions and a reduced number of rules in addition a new post processing approach is proposed for recovering any probable performance lost after membership functions merging the proposed methodology has been applied successfully for the design of an active control suspension system using a non linear bond graphs bg based half car model with parameters that relate to a ford fiesta mk2

self similar solutions for a quenching problem with spatially dependent nonlinearity abstract this paper is devoted to the study of the self similar solutions for a semilinear parabolic equation with spatially dependent nonlinearity which arises in the model of micro electro mechanical system we first provide a result on the non existence of slow orbit for a certain range of parameters next we prove the existence of backward solutions with the desired polynomial growth condition at infinity to the associated equation by a fixed point argument then we give a detailed analysis of the behavior of global solutions at the origin finally as an application of the above results we prove a uniqueness theorem

a fuzzy logic based approach to qualitative modeling this paper discusses a general approach to quali tative modeling based on fuzzy logic the method of qualitative modeling is divided into two parts fuzzy modeling and linguistic approximation it proposes to use a fuzzy clustering method fuzzy c means method to identify the structure of a fuzzy model to clarify the advantages of the proposed method it also shows some examples of modeling among them a model of a dynamical process and a model of a human operator s control action

adaptive fuzzy inference neural network an adaptive fuzzy inference neural network afinn is proposed in this paper it has self construction ability parameter estimation ability and rule extraction ability the structure of afinn is formed by the following four phases 1 initial rule creation 2 selection of important input elements 3 identification of the network structure and 4 parameter estimation using lms least mean square algorithm when the number of input dimension is large the conventional fuzzy systems often cannot handle the task correctly because the degree of each rule becomes too small afinn solves such a problem by modification of the learning and inference algorithm

continuation of blowup solutions of nonlinear heat equations in several space dimensions the possible continuation of solutions of the nonlinear heat equation in r n r ut u m u p with m 0 p 1 after the blowup time is studied and the different continuation modes are discussed in terms of the exponents m and p thus for m p 2 we find a phenomenon of nontrivial continuation where the regionfx u x t 1g is bounded and propagates with finite speed this we call incomplete blowup for n 3 and p m n 2 n 2 we find solutions that blow up at finite t t and then become bounded again for t t otherwise we find that blowup is complete for a wide class of initial data in the analysis of the behavior for large p a list of critical exponents appears whose role is described we also discuss a number of related problems and equations we apply the same technique of analysis to the problem of continuation after the onset of extinction for example for the equation ut u m u p m 0

martingale families and dimension in p we introduce a new measure notion on small complexity classes called f measure based on martingale families that gets rid of some drawbacks of previous measure notions it can be used to define dimension because martingale families can make money on all strings and it yields random sequences with an equal frequency of 0 s and 1 s on larger complexity classes e and above f measure is equivalent to lutz resource bounded measure as applications to f measure we answer a question raised in e allender m strauss measure on small complexity classes with application for bpp in proc of the 35th ann ieee symp on found of comp sci 1994 pp 807 818 by improving their result to for almost every language a decidable in subexponential time p a bpp a we show that almost all languages in pspace do not have small non uniform complexity we compare f measure to previous notions and prove that martingale families are strictly stronger than c measure e allender m strauss measure on small complexity classes with application for bpp in proc of the 35th ann ieee symp on found of comp sci 1994 pp 807 818 we also discuss the limitations of martingale families concerning finite unions we observe that all classes closed under polynomial many one reductions have measure zero in exp iff they have measure zero in subexp we use martingale families to introduce a natural generalization of lutz resource bounded dimension j h lutz dimension in complexity classes in proceedings of the 15th annual ieee conference on computational complexity 2000 pp 158 169 on p which meets the intuition behind lutz s notion we show that p dimension lies between finite state dimension and dimension on e we prove an analogue of a theorem of eggleston in p i e the class of languages whose characteristic sequence contains 1 s with frequency a has dimension the shannon entropy of a in p

measure stochasticity and the density of hard languages the main theorem of this paper is that for every real number alpha 1 e g alpha 0 99 only a measure 0 subset of the languages decidable in exponential time are leq p n alpha tt reducible to languages that are not exponentially dense thus every leq p n alpha tt hard language for e is exponentially dense this strengthens watanabe s 1987 result that every leq p o log n tt hard language for e is exponentially dense the combinatorial technique used here the sequentially most frequent query selection also gives a new simpler proof of watanabe s result the main theorem also has implications for the structure of np under strong hypotheses ogiwara and watanabe 1991 have shown that the hypothesis p ne np implies that every leq p btt hard language for np is nonsparse i e not polynomially sparse their technique does not appear to allow significant relaxation of either the query bound or the sparseness criterion it is shown here that a stronger hypothesis namely that np does not have measure 0 in exponential time implies the stronger conclusion that for every real alpha 1 every leq p n alpha tt hard language for np is exponentially dense evidence is presented that this stronger hypothesis is reasonable the proof of the main theorem uses a new very general weak stochasticity theorem ensuring that almost every language in e is statistically unpredictable by feasible deterministic algorithms even with linear nonuniform advice

a neural fuzzy modelling framework based on granular computing concepts and applications fuzzy and neural fuzzy systems have successfully and extensively applied to solve problems in many research areas such as those associated with industrial medical and academic applications however recent trends reveal a demand for a workflow with a particular emphasis on transparency simplicity system interpretability as well as on a strong link with human cognition such requirement is mainly driven by research areas where expert knowledge is of very high importance and any new proposed modelling system falls under the interpretability scrutiny of experts in order to confirm the system s validity the relatively recent paradigm of granular computing grc offers an ideal opportunity for a transparent knowledge discovery methodology to be combined with fuzzy logic thereby towards a systematic modelling framework with a focus on the overall transparency of the system such transparency in the workflow allows for better interaction between the expert process knowledge and the system design which translates into a better performing system in this paper a systematic modelling approach using granular computing grc and neural fuzzy modelling is presented in this research study a grc algorithm is used to extract relational information and data characteristics out of an initial database the extracted knowledge and granular features are then translated into a linguistic rule base of a fuzzy system this rule base is finally elicited and optimised via a neural fuzzy modelling structure during the various steps of this methodology the transparency features are highlighted and it is shown here how the system designer can take advantage of such features to enhance the system the proposed modelling framework is applied to a multi dimensional and complex data set consisting of measurements of mechanical properties of heat treated steel the data set is collected from a real industrial process and the measurements are dictated by customer production demands and the data set is very sparse with many discontinuities the proposed framework successfully models the mechanical properties of heat treated steel and it further improves upon the performance of previously established modelling structures

weakly hard problems a weak completeness phenomenon is investigated in the complexity class rm e rm dtime 2 rm linear according to standard terminology a language h is leq rm p m hard for e if the set rm p m h consisting of all languages a leq rm p m h contains the entire class e a language c is leq rm p m complete for e if it is leq rm p m hard for e and is also an element of e generalizing this a language h is weakly leq rm p m hard for e if the set rm p m h does not have measure 0 in e a language c is weakly leq rm p m complete for e if it is weakly leq rm p m hard for e and is also an element of e the main result of this paper is the construction of a language that is weakly leq rm p m complete but not leq rm p m complete for e the existence of such languages implies that previously known strong lower bounds on the complexity of weakly leq rm p m hard problems for e given by work of lutz mayordomo and juedes are indeed more general than the corresponding bounds for leq rm p m hard problems for e the proof of this result introduces a new diagonalization method called martingale diagonalization using this method one simultaneously develops an infinite family of polynomial time computable martingales betting strategies and a corresponding family of languages that defeat these martingales prevent them from winning too much money while also pursuing another agenda martingale diagonalization may be useful for a variety of applications

computation times of np sets of different densities abstract in this paper we study the computational complexity of sets of different densities in np we show that the deterministic computation time for sets in np can depend on their density if and only if there is a collapse or partial collapse of the corresponding higher nondeterministic and deterministic time bounded complexity classes we show also that for np sets of different densities there exist complete sets of the corresponding density under polynomial time turing reductions finally we show that these results can be interpreted as results about the complexity of theorem proving and proof presentation in axiomatized mathematical systems this interpretation relates fundamental questions about the complexity of our intellectual tools to basic structural problems about p np conp and pspace discussed in this paper

a hierarchical fuzzy clustering approach to fuzzy modeling abstract this paper introduces a new method for fuzzy modeling based on a hierarchical fuzzy clustering scheme the method consists of a sequence of steps aiming towards developing a takagi sugeno ts fuzzy model of optimal structure where the fuzzy sets in the premise part are of gaussian type starting from an initial ordinary fuzzy partition of the input space the algorithm performs a nearest neighbor search and groups the original input training data into a number of clusters the centers of these clusters are further processed using an optimal fuzzy clustering technique which is based on the weighted fuzzy c means algorithm the resulted optimal fuzzy partition defines the number of fuzzy rules and provides an initial estimation for the system parameters which in a next step are fine tuned using the well known gradient descend algorithm the proposed method is successfully applied to three test examples where the produced fuzzy models prove to be very accurate as well as compact in size

a parabolic free boundary problem modeling electrostatic mems the evolution problem for a membrane based model of an electrostatically actuated microelectromechanical system is studied the model describes the dynamics of the membrane displacement and the electric potential the latter is a harmonic function in an angular domain the deformable membrane being a part of the boundary the former solves a heat equation with a right hand side that depends on the square of the trace of the gradient of the electric potential on the membrane the resulting free boundary problem is shown to be well posed locally in time furthermore solutions corresponding to small voltage values exist globally in time while global existence is shown not to hold for high voltage values it is also proven that for small voltage values there is an asymptotically stable steady state solution finally the small aspect ratio limit is rigorously justified

the size of spp derandomization techniques are used to show that at least one of the following holds regarding the size of the counting complexity class spp 1 p spp 0 2 ph spp in other words spp is small by being a negligible subset of exponential time or large by containing the entire polynomial time hierarchy this addresses an open problem about the complexity of the graph isomorphism problem it is not weakly complete for exponential time unless ph is contained in spp it is also shown that the polynomial time hierarchy is contained in sppnp if np does not have p measure 0

some singular equations modeling mems in the past fifteen years mathematical models for microelectromechanical systems mems have been the subject of several studies in particular due to the interesting qualitative properties they feature still most research is devoted to an illustrative but simplified model which is deduced from a more complex model when the aspect ratio of the device vanishes the so called vanishing or small aspect ratio model the analysis of the aforementioned complex model involving a moving boundary has started only recently and an outlook of the results obtained so far in this direction is provided in this survey

the zero one law holds for bpp abstract we show that bpp has p measure zero if and only if bpp differs from exp the same holds when we replace bpp by any complexity class c that contains bpp and is closed under tt reductions the zero one law for each of these classes c follows within exp c has either measure zero or else measure one

input selection for nonlinear regression models a simple and effective method for the selection of significant inputs in nonlinear regression models is proposed given a set of input output data and an initial superset of potential inputs the relevant inputs are selected by checking whether after deleting a particular input the data set is still consistent with the basic property of a function in order to be able to handle real valued and noisy data in a sensible manner fuzzy clustering is first applied the obtained clusters are compared by using a similarity measure in order to find inconsistencies within the data several examples using simulated and real world data sets are presented to demonstrate the effectiveness of the algorithm

graph nonisomorphism has subexponential size proofs unless the polynomial time hierarchy collapses traditional hardness versus randomness results focus on time efficient randomized decision procedures we generalize these trade offs to a much wider class of randomized processes we work out various applications most notably to derandomizing arthur merlin games we show that every language with a bounded round arthur merlin game has subexponential size membership proofs for infinitely many input lengths unless exponential time coincides with the third level of the polynomial time hierarchy and hence the polynomial time hierarchy collapses since the graph nonisomorphism problem has a bounded round arthur merlin game this provides the first strong evidence that graph nonisomorphism has subexponential size proofs we also establish hardness versus randomness trade offs for space bounded computation

knowledge discovery by a neuro fuzzy modeling framework in this paper a neuro fuzzy modeling framework is proposed which is devoted to discover knowledge from data and represent it in the form of fuzzy rules the core of the framework is a knowledge extraction procedure that is aimed to identify the structure and the parameters of a fuzzy rule base through a two phase learning of a neuro fuzzy network in order to obtain reliable and readable knowledge two further stages are integrated with the knowledge extraction procedure a pre processing stage performing variable selection on the available data to obtain simpler and more reliable fuzzy rules and a post processing stage that granulates outputs of the extracted fuzzy rules so as to provide a validity range of estimated outputs moreover the framework can address complex multi input multi output problems in such case two distinct modeling strategies can be followed with the opportunity of producing both a single mimo model or a collection of miso models the proposed framework is verified on a real world case study involving prediction of chemical composition of ashes produced by combustion processes carried out in thermo electric generators located in italy

baire categories on small complexity classes and meager comeager laws we introduce two resource bounded baire category notions on small complexity classes such as p quasipoly subexp and pspace and on probabilistic classes such as bpp which differ on how the corresponding finite extension strategies are computed we give an alternative characterization of small sets via resource bounded banach mazur games as an application of the first notion we show that for almost every language a i e all except a meager class computable in subexponential time p a bpp a we also show that almost all languages in pspace do not have small nonuniform complexity we then switch to the second baire category notion called locally computable and show that the class sparse is meager in p we show that in contrast to the resource bounded measure case meager comeager laws can be obtained for many standard complexity classes relative to locally computable baire category on bpp and pspace another topic where locally computable baire categories differ from resource bounded measure is regarding weak completeness we show that there is no weak completeness notion in p based on locally computable baire categories i e every p weakly complete set is complete for p we also prove that the class of complete sets for p under turing logspace reductions is meager in p if p is not equal to dspace logn and that the same holds unconditionally for quasipoly finally we observe that locally computable baire categories are incomparable with all existing resource bounded measure notions on small complexity classes which might explain why those two settings seem to differ so fundamentally

normal numbers and sources for bpp abstract in 10 lutz proposed a notion of source a nonrandom sequence that can substitute in a certain way for the random bits used by bounded error probabilistic machines he showed that almost every sequence in dspace 2 polynomial is a source we improve this abundance result to pspace by first showing that the sources are exactly the classical normal numbers or normal sequences of borel there are sequences clearly in p that have long been known to be normal and we go on to show there are sources in ac 0 this suggests that alternate notions of source should be explored

a new model for electrostatic mems with two free boundaries abstract a moving boundary problem with two free boundaries modeling a two dimensional idealized mems device with pull in instability is discussed we use a fixed point argument to show that the model possesses stationary solutions for small source voltages we also give a rigorous evidence that solutions of the model converge towards solutions of the associated small aspect ratio equation in the vanishing aspect ratio limit

compressibility and resource bounded measure we give a new definition of resource bounded measure based on compressibility of infinite binary strings we prove that the new definition is equivalent to the one commonly used this new characterization offers us a different way to look at resource bounded measure shedding more light on the meaning of measure zero results and providing one more tool to prove such results the main contribution of the paper is the new definition and the proofs leading to the equivalence result we then show how this new characterization can be used to prove that the class of linear autoreducible sets has p measure 0 we also prove that the class of sets that are truth table reducible to a p selective set has p measure 0 and that the class of sets that turing reduce to a subpolynomial dense set has p measure 0 this strengthens various results

quenching rate for a nonlocal problem arising in the micro electro mechanical system abstract in this paper we study the quenching rate of the solution for a nonlocal parabolic problem which arises in the study of the micro electro mechanical system this question is equivalent to the stabilization of the solution to the transformed problem in self similar variables first some a priori estimates are provided in order to construct a lyapunov function due to the lack of time monotonicity property we then derive some very useful and challenging estimates by a delicate analysis finally with this lyapunov function we prove that the quenching rate is self similar which is the same as the problem without the nonlocal term except the constant limit depends on the solution itself

on a hyperbolic equation arising in electrostatic mems abstract we consider the initial boundary value problem of a damped wave equation with singular nonlinearity which describes an electrostatic micro electro mechanical system mems device the results of the pull in voltage being the critical threshold for global existence and quenching are obtained if the applied voltage then the equation admits a unique global small solution that exponentially converges to the minimal steady state while large solution may quench in finite time if then any solution quenches in finite time finally in the sense of the viscosity dominated limit the asymptotic relation of solutions between the hyperbolic equation and the parabolic one is investigated also the related error estimates in arbitrary order are derived

on the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term we study questions of existence uniqueness and asymptotic behaviour for the solutions of u x t of the problem begin gathered text u t delta u lambda e u text lambda text 0 t 0 text x text varepsilon b hfill p text u x 0 u 0 x text x text varepsilon b hfill text u x t 0 text on text partial b times 0 infty hfill end gathered where b is the unit ball x varepsilon r n x text leqq text 1 text and n geqq 3 our interest is focused on the parameter 0 2 n 2 for which p admits a singular stationary solution of the form s x 2log x

on approximation algorithms for p the theme of this paper is to investigate to what extent approximation possibly together with randomization can reduce the complexity of problems in valiant s class p in general any function in p can be approximated to within any constant factor by a function in the class delta 3 p of the polynomial time hierarchy relative to a particular oracle delta 3 p cannot be replaced by delta 2 p in this result another part of the paper introduces a model of random sampling where the size of a set x is estimated by checking for various sample sets s whether or not s intersects x for various classes of sample sets upper and lower bounds on the number of samples required to estimate the size of x are discussed this type of sampling is motivated by particular problems in p such as computing the size of a backtrack search tree in the case of backtrack search trees a sample amounts to checking whether a certain path exists in the tree one of the lower bounds suggests that such tests

measure on p we give a notion of measure at p in a paradigm that differs somewhat from the standard theory our new notion overcomes some limitations of earlier formulations specifically concerning closure of null sets under union first we analyze formally some of the difficulties in defining measure at p we then present the new definitions and determine the basic properties of the notion including the density and immunity characteristics of a random language we argue that these results are parallel to previous measure results at exponential time

global solutions of singular parabolic equations arising from electrostatic mems abstract we study dynamic solutions of the singular parabolic problem p u t u x 1 u 2 x t b 0 u x 0 u 0 x 0 x b u x t 0 x b where 0 and 0 are two parameters and b is the unit ball x r n x 1 with n 2 our interest is focussed on p with 2 3 n 4 9 for which p admits a singular stationary solution in the form s x 1 x 2 3 this equation models dynamic deflection of a simple electrostatic micro electro mechanical system mems device under the assumption u 0 s x we address the existence uniqueness regularity stability or instability and asymptotic behavior of weak solutions for p given 4 6 n 3 6 n 2 4 in particular we show that if either n 8 and or 2 n 7 then the minimal compact stationary solution u of p is stable and while s x is unstable however for n 8 and 0 p has no compact minimal solution and s x is an attractor from below not from above furthermore the refined asymptotic behavior of global solutions for p is also discussed for the case where n 8 and 0 which is given by a certain matching of different asymptotic developments in the large outer region closer to the boundary and the thin inner region near the singularity

p printable sets p printable sets arise naturally in the studies of generalized kolmogorov complexity and data compression as well as in other areas we present new characterizations of the p printable sets and present necessary and sufficient conditions for the existence of sparse sets in p that are not p printable as a corollary to one of our results we show that the class of sets of small generalized kolmogorov complexity is exactly the class of sets which are p isomorphic to a tally language

input selection and partition validation for fuzzy modelling using neural network a simple and effective method for selecting significant input variables and determining optimal number of fuzzy rules when building a fuzzy model from data is proposed in contrast to the existing clustering based methods in this approach both input selecting and partition validating are determined on the basis of a class of sub clusters created by a self organising network instead of on the data the important input variables which independently and significantly influence the system output can be extracted by a fuzzy neural network on the other hand the optimal number of fuzzy rules can be determined separately via the fuzzy c means algorithm with a modified fuzzy entropy as the criterion of cluster validation the simulation results show that the proposed method can provide good model structures for fuzzy modelling and has high computing efficiency

category and measure in complexity classes this paper presents resource bounded category and resource bounded measure two new tools for computational complexity theory and some applications of these tools to the structure theory of exponent

a zero one law for rp and derandomization of am if np is not small we show that if rp does not have p measure zero then zpp exp as corollaries we obtain a zero one law for rp in exp and that both probabilistic classes zpp and rp have the same measure in exp we also prove that if np does not have p measure zero then np am

compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity we study the branch of semistable and unstable solutions i e those whose morse index is at most 1 of the dirichlet boundary value problem u f x 1 u 2 on a bounded domain n which models among other things a simple electrostatic microelectromechanical system mems device we extend the results of 11 relating to the minimal branch by obtaining compactness along unstable branches for 1 n 7 on any domain and for a large class of permittivity profiles f we also show the remarkable fact that powerlike profiles f x x can push back the critical dimension n 7 of this problem by establishing compactness for the semistable branch on the unit ball also for n 8 and as long as as a byproduct we are able to follow the second branch of the bifurcation diagram and prove the existence of a second solution for in a natural range in all these results the conditions on the space dimension and on the power of the profile are essentially sharp 2007 wiley periodicals inc

construction of fuzzy models through clustering techniques abstract in this note we will discuss fuzzy models designed with the aid of fuzzy clustering the clustering method utilizes a generalized objective function involving a collection of linear varieties in this way the model is distributed and consists of a series of local linear type models based on the revealed clusters in comparison to the algorithms existing in the literature and producing function like models the proposed one is of a relational character allowing for multidirectional accessibility a detailed identification algorithm is given and illustrated with the aid of numerical examples

on pseudorandomness and resource bounded measure abstract in this paper we extend a key result of nisan and wigderson j comput system sci 49 1994 149 167 to the nondeterministic setting for all 0 we show that if there is a language in e dtime 2 o n that is hard to approximate by nondeterministic circuits of size 2 n then there is a pseudorandom generator that can be used to derandomize bp np in symbols bp np np by applying this extension we are able to answer some open questions in lutz theory comput systems 30 1997 429 442 regarding the derandomization of the classes bp p k and bp p k under plausible measure theoretic assumptions as a consequence if p 2 does not have p measure 0 then am coam is low for p 2 thus in this case the graph isomorphism problem is low for p 2 by using the nisan wigderson design of a pseudorandom generator we unconditionally show the inclusion ma zpp np and that ma coma is low for zpp np