fuzzy system reliability analysis based on level λ 1 interval valued fuzzy numbers this study uses level 1 interval valued fuzzy numbers to examine the fuzzy reliability of a serial system and a parallel system and obtain the estimated reliability of both systems in the fuzzy sense

least squares estimation of transformation parameters between two point patterns in many applications of computer vision the following problem is encountered two point patterns sets of points x sub i and x sub i i 1 2 n are given in m dimensional space and the similarity transformation parameters rotation translation and scaling that give the least mean squared error between these point patterns are needed recently k s arun et al 1987 and b k p horn et al 1987 presented a solution of this problem their solution however sometimes fails to give a correct rotation matrix and gives a reflection instead when the data is severely corrupted the proposed theorem is a strict solution of the problem and it always gives the correct transformation parameters even when the data is corrupted

universal presheaves on group geometries and modular representations abstract we extend the ideas of sheaf homology on buildings m a ronan and s d smith j algebra 96 1985 319 346 to more general geometries with transitive automorphism groups the main technical result is the construction of a universal extension of certain partial sheaves this guarantees a rich supply of sheaves on such geometries computation of zero homology then affords representations of the group we consider various applications of the technique for example to modular irreducibles for sporadic simple groups

reduced fusion systems over 2 groups of small order we prove when s is a 2 group of order at most 29 that each reduced fusion system over s is the fusion system of a finite simple group and is tame it then follows that each saturated fusion system over a 2 group of order at most 29 is realizable what is most interesting about this result is the method of proof we show that among 2 groups with order in this range the ones which can be sylow 2 subgroups of finite simple groups are almost completely determined by criteria based on bender s classification of groups with strongly 2 embedded subgroups

a riemannian framework for tensor computing tensors are nowadays a common source of geometric information in this paper we propose to endow the tensor space with an affine invariant riemannian metric we demonstrate that it leads to strong theoretical properties the cone of positive definite symmetric matrices is replaced by a regular and complete manifold without boundaries null eigenvalues are at the infinity the geodesic between two tensors and the mean of a set of tensors are uniquely defined etc we have previously shown that the riemannian metric provides a powerful framework for generalizing statistics to manifolds in this paper we show that it is also possible to generalize to tensor fields many important geometric data processing algorithms such as interpolation filtering diffusion and restoration of missing data for instance most interpolation and gaussian filtering schemes can be tackled efficiently through a weighted mean computation linear and anisotropic diffusion schemes can be adapted to our riemannian framework through partial differential evolution equations provided that the metric of the tensor space is taken into account for that purpose we provide intrinsic numerical schemes to compute the gradient and laplace beltrami operators finally to enforce the fidelity to the data either sparsely distributed tensors or complete tensors fields we propose least squares criteria based on our invariant riemannian distance which are particularly simple and efficient to solve

fuzzy system reliability analysis by interval of confidence abstract the use of fuzzy number over interval of confidence instead of probability and possibilitic considerations for evaluating the range value and interval of the fuzzy reliability is proposed this approach of evaluating fuzzy system reliability by interval arithmetic and cuts has a considerable advantage we can calculate reliability problems easily by the use of derive and mathcad packages through theoretical analysis an illustrative example and computational results we show that the proposed approach is more general and simplified

parameter estimations of normal fuzzy variables abstract although fairly much attention has been paid to theoretical applications of nahmias fuzzy variables little effort has been focused on the related parameter estimation issues this keeps down the practical use of fuzzy variables in this note we make a preliminary discussion on parameter estimation of normal fuzzy variables with parameters a b and distinguish three situations estimation of a with b known estimation of b with a known and estimation of unknown a and b formulae for calculating parameter estimates by use of practical collected data are presented

street lighting lamps replacement a fuzzy viewpoint abstract campbell s statement on a street lighting system is expressed as that n street lighting posts have to be provided continually with lamps in this paper this statement is fuzzified such that the system failure is defined in a fuzzy way then a concept of fuzzy reliability is introduced in the context of the transition from the system fuzzy success state to the system fuzzy failure state where a transition between two fuzzy states is just a fuzzy event with the concept of fuzzy reliability we make a comparison between two replacement policies viz the block replacement policy in a non fuzzy environment and the periodic relacement policy without repair at lamp failures in a fuzzy environment and comclude that a complex system may be as reliable as a simple one in view of the system fuzzy reliability

existence and uniqueness of linking systems chermak s proof via obstruction theory we present a version of a proof by andy chermak of the existence and uniqueness of centric linking systems associated with arbitrary saturated fusion systems this proof differs from the one in ch2 in that it is based on the computation of derived functors of certain inverse limits this leads to a much shorter proof but one which is aimed mostly at researchers familiar with homological algebra

control of fixed points and existence and uniqueness of centric linking systems a chermak has recently proved that to each saturated fusion system over a finite p group there is a unique associated centric linking system b oliver extended chermak s proof by showing that all the higher cohomological obstruction groups relevant to unique existence of centric linking systems vanish both proofs indirectly assume the classification of finite simple groups we show how to remove this assumption thereby giving a classification free proof of the martino priddy conjecture concerning the p completed classifying spaces of finite groups our main tool is a 1971 result of the first author on control of fixed points by p local subgroups this result is directly applicable for odd primes and we show how a slight variation of it allows applications for p 2 in the presence of offenders

a characteristic subgroup for fusion systems abstract as a counterpart for the prime 2 to glauberman s zj theorem stellmacher proves that any nontrivial 2 group s has a nontrivial characteristic subgroup w s with the following property for any finite 4 free group g with s a sylow 2 subgroup of g and with o 2 g self centralizing the subgroup w s is normal in g we generalize stellmacher s result to fusion systems a similar construction of w s can be done for odd primes and gives rise to a glauberman functor

an efficient approach for design optimization of structures involving fuzzy variables abstract we propose a new approach for solving design optimization problems of structures involving fuzzy variables first the possibilistic safety model of structures with fuzzy variables is investigated a possibilistic safety index psi is presented to compare different designs this paper then presents the psi based design optimization psibdo model for structures with fuzzy variables the optimization problem is a triple loop nested problem and the computational cost is high to reduce the computation time a technique called target performance based design approach tpbda is proposed to overcome difficulties encountered by the psibdo with the proposed tpbda the triple loop nested problem is reduced to a double loop one and the computational time is considerably reduced several examples are given to demonstrate the efficiency of the proposed approach

all p local finite groups of rank two for odd prime p in this paper we give a classification of the rank two p local finite groups for odd p this study requires the analysis of the possible saturated fusion systems in terms of the outer automorphism group of the possible f radical subgroups also for each case in the classification either we give a finite group with the corresponding fusion system or we check that it corresponds to an exotic p local finite group getting some new examples of these for p 3

fuzzy system reliability analysis using fuzzy number arithmetic operations abstract this paper presents a new method to analyze fuzzy system reliability using fuzzy number arithmetic operations where the reliability of each system component is represented by a triangular fuzzy number because the proposed method uses simplified fuzzy arithmetic operations of fuzzy numbers rather than the complicated interval arithmetic operations of fuzzy numbers mentioned by cheng and mon in fuzzy sets and systems 56 1993 29 35 or the complicated extended algebraic operations of fuzzy numbers mentioned by singer in fuzzy sets and systems 34 1990 145 155 its execution is faster than the ones presented by cheng and mon and singer

fusion category algebras the fusion system f on a defect group p of a block b of a finite group g over a suitable p adic ring o does not in general determine the number l b of isomorphism classes of simple modules of the block we show that conjecturally the missing information should be encoded in a single second cohomology class of the constant functor with value k on the orbit category f c of f centric subgroups q of p of b which glues together the second cohomology classes q of autf q with values in k in k ulshammer puig 13 1 8 we show that if exists there is a canonical quasi hereditary k algebra f b such that alperin s weight conjecture becomes equivalent to the equality l b l f b by work of broto levi oliver 3 the existence of a classifying space of the block b is equivalent to the existence of a certain extension category l of fc by the center functor z if both invariants l exist we show that there is an o algebra l b associated with b having f b as quotient such that alperin s weight conjecture becomes again equivalent to the equality l b l l b furthermore if b has an abelian defect group l b is isomorphic to a source algebra of the brauer correspondent of b

fuzzy system reliability analysis by the use of t ω the weakest t norm on fuzzy number arithmetic operations abstract in general the sup min convolution has been used for fuzzy arithmetic to analyze fuzzy system reliability where the reliability of each system component is represented by fuzzy numbers it is well known that t based addition preserves the shape of l r type fuzzy numbers in this paper we show t based multiplication also preserves the shape of l r type fuzzy numbers we then apply t based arithmetic operations to fuzzy system reliability analysis in fact we show that we can simplify fuzzy arithmetic operations and even get the exact solutions for l r type fuzzy system reliability while others singer fuzzy sets syst 34 1990 145 cheng and mon fuzzy sets syst 56 1993 29 chen fuzzy sets syst 64 1994 31 have got the approximate solutions using sup min convolution for evaluating fuzzy system reliability

the geometric median on riemannian manifolds with application to robust atlas estimation one of the primary goals of computational anatomy is the statistical analysis of anatomical variability in large populations of images the study of anatomical shape is inherently related to the construction of transformations of the underlying coordinate space which map one anatomy to another it is now well established that representing the geometry of shapes or images in euclidian spaces undermines our ability to represent natural variability in populations in our previous work we have extended classical statistical analysis techniques such as averaging principal components analysis and regression to riemannian manifolds which are more appropriate representations for describing anatomical variability in this paper we extend the notion of robust estimation a well established and powerful tool in traditional statistical analysis of euclidian data to manifold valued representations of anatomical variability in particular we extend the geometric median a classic robust estimator of centrality for data in euclidean spaces we formulate the geometric median of data on a riemannian manifold as the minimizer of the sum of geodesic distances to the data points we prove existence and uniqueness of the geometric median on manifolds with non positive sectional curvature and give sufficient conditions for uniqueness on positively curved manifolds generalizing the weiszfeld procedure for finding the geometric median of euclidean data we present an algorithm for computing the geometric median on an arbitrary manifold we show that this algorithm converges to the unique solution when it exists in this paper we exemplify the robustness of the estimation technique by applying the procedure to various manifolds commonly used in the analysis of medical images using this approach we also present a robust brain atlas estimation technique based on the geometric median in the space of deformable images

color perception of diffusion tensor images using hierarchical manifold learning abstract the color perception of diffusion tensor images dti by using voxel based statistical analysis suffers from high computational cost and vague regional structure to address these issues we therefore propose a novel approach for color perception of dti based on hierarchical manifold learning first the selection of the representative nodes as seeds within similar region to build them into the bottom to up hierarchical structure is derived from the algebraic multigrid and multi scale graph partitioning next the low dimensional coordinates of the top layer seeds are calculated using manifold based techniques with a new distance metric and mapping of these coordinates into the rgb color space last the color perception of dti is obtained through interpolating the seeds to the bottom layer of all nodes the experimental results demonstrate that the proposed algorithm can reduce the computation complexity from o n 3 based on algorithms in the literature ghassan et al 2011 9 to o n 2 and highlight the different regional structures of the brain via color perception of variation

fusion systems and amalgams we study reduced fusion systems from the point of view of their essential subgroups using the classification by goldschmidt and fan of amalgams of prime index to analyze certain pairs of such subgroups our results are applied here to study reduced fusion systems over 2 groups of order at most 64 and also reduced fusion systems over 2 groups having abelian subgroups of index two more applications are given in later papers

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

perception based visualization of manifold valued medical images using distance preserving dimensionality reduction a method for visualizing manifold valued medical image data is proposed the method operates on images in which each pixel is assumed to be sampled from an underlying manifold for example each pixel may contain a high dimensional vector such as the time activity curve tac in a dynamic positron emission tomography dpet or a dynamic single photon emission computed tomography dspect image or the positive semi definite tensor in a diffusion tensor magnetic resonance image dtmri a nonlinear mapping reduces the dimensionality of the pixel data to achieve two goals distance preservation and embedding into a perceptual color space we use multidimensional scaling distance preserving mapping to render similar pixels e g dt or tac pixels with perceptually similar colors the 3d cielab perceptual color space is adopted as the range of the distance preserving mapping with a final similarity transform mapping colors to a maximum gamut size similarity between pixels is either determined analytically as geodesics on the manifold of pixels or is approximated using manifold learning techniques in particular dissimilarity between dtmri pixels is evaluated via a log euclidean riemannian metric respecting the manifold of the rank 3 second order positive semi definite dts whereas the dissimilarity between tacs is approximated via isomap we demonstrate our approach via artificial high dimensional manifold valued data as well as case studies of normal and pathological clinical brain and heart dtmri dpet and dspect images our results demonstrate the effectiveness of our approach in capturing in a perceptually meaningful way important features in the data

some homotopy equivalences for sporadic geometries a previous work rsy90 established the projectivity of the reduced lefschetz modules of certain sporadic group geometries and the present paper continues that work in a wider context recent developments in sporadic group cohomology include some applications of rsy90 which in turn suggested treatment of a broader class of geometries recurring similarities in the proofs also led to a more unified treatment establishing the stronger result of homotopy equivalence of thep local geometry with the usual elementary poset ap g one equivalence method proceeds by means of a new closed set in a standard technique of quillen it was further observed that the larger list of simple groups now treated essentially coincides with those of characteristicp type suggesting another equivalence method via the poset bp g of radical or stubborn p subgroups in particular one finds that these sporadic groups satisfy an analogue of the borel tits theorem that normalizers ofp groups lie in simplex stabilizers still further intriguing coincidences remain to be explained

extensions of p local finite groups a p local finite group consists of a finite p group s together with a pair of categories which encode conjugacy relations among subgroups of s and which are modelled on the fusion in a sylow p subgroup of a finite group it contains enough information to define a classifying space which has many of the same properties as p completed classifying spaces of finite groups in this paper we study and classify extensions of p local finite groups and also compute the fundamental group of the classifying space of a p local finite group

products in fusion systems abstract we revisit the notion of a product of a normal subsystem with a p subgroup as defined by aschbacher 2011 asc11 chapter 8 in particular we give a previously unknown more transparent construction

subgroup families controlling p local finite groups a p local finite group consists of a finite p group s together with a pair of categories which encode conjugacy relations among subgroups of s and which are modelled on the fusion in a sylow p subgroup of a finite group it contains enough information to define a classifying space which has many of the same properties as p completed classifying spaces of finite groups in this paper we examine which subgroups control this structure more precisely we prove that the question of whether an abstract fusion system mathcal f over a finite p group s is saturated can be determined by just looking at smaller classes of subgroups of s we also prove that the homotopy type of the classifying space of a given p local finite group is independent of the family of subgroups used to define it in the sense that it remains unchanged when that family ranges from the set of mathcal f centric mathcal f radical subgroups at a minimum to the set of mathcal f quasicentric subgroups at a maximum finally we look at constrained fusion systems analogous to p constrained finite groups and prove that they in fact all arise from groups

posbist reliability behavior of typical systems with two types of failure abstract the posbist reliability theory is based on the possibility assumption and the binary state assumption in this paper we study the posbist reliability behavior in the presence of two failure modes for various classes of typical systems including series systems parallel systems k out of n systems series parallel systems and parallel series systems it is shown that for each typical system the posbist system reliability is bounded by the minimum and the maximum of the posbist component reliabilities

homotopy classification of self maps of bg via g actions to complete the classification of self maps of bg theorem 4 2 it remains to prove theorem 4 1 i e to show that the higher limits of certain functors n 1h 2 f p g z p wall vanish and that h0 mp g f i has the expected value in this section these limits h mp g hi will be calculated in the easy cases when g is a simply connected classical group i e one of the groups su n sp n or spin n or when i 2 the computation of h mp g hi for the simply connected exceptional lie groups will be put off until section 6 we begin with some general properties of higher limits let 6 be any small discrete category it is convenient when dealing with higher limits of functors on to let 4 moldenote the category of contravariant functors from v to ad then for any f e w muok we can identify

wreath products and representations of p local finite groups abstract given two finite p local finite groups and a fusion preserving morphism between their sylow subgroups we study the question of extending it to a continuous map between their classifying spaces the results depend on the construction of the wreath product of p local finite groups which is also used to study p local permutation representations

trustworthy dimension reduction for visualization different data sets abstract a new nonlinear dimension reduction dr method which is called trustworthy stochastic proximity embedding tspe is introduced in this paper to visualize different types of data sets tspe overcome the main shortcomings of the dr by sending the false neighbour points to the correct locations and preserving the neighbourhood relation to the true neighbours which are inside the local neighbourhood the visualization of our proposed method displays the trustworthy useful and meaningful colours where the objects of the image can be easily distinguished the performances of tspe and 20 dimension reduction methods are compared and the efficiency of the proposed method in both visualization accuracy and computational cost is shown the results showed the ability of our method in preserving neighbourhood relation where they revealed more interested information in real data set the efficiency of the visualization of tensor images data sets by tspe might help the specialist to make a good decision about a patient s treatment the comparison with experimental data set as three dimensions of curved cylinder showed the ability of tspe to unfold this complex data set efficiently whilst preserving most information of the original data set

reliability and mean time to failure of unrepairable systems with fuzzy random lifetimes in practice the lifetimes of unrepairable systems contain both randomness and fuzziness so it is appropriate to assume the lifetimes of systems to be fuzzy random variables in this paper the definitions of reliability and mean time to failure mttf of unrepairable systems with fuzzy random lifetimes are given respectively then basic mathematical models of unrepairable systems with fuzzy random lifetimes are established furthermore the reliability and mttf of series systems parallel systems series parallel systems parallel series systems and cold standby systems are discussed respectively finally some numerical examples are presented to illustrate how to calculate the reliability and mttf of unrepairable systems with fuzzy random lifetimes

a note on oliver s p group conjecture abstract let s be a finite p group for an odd prime p oliver proposed the conjecture that the thompson subgroup j s is always contained in the oliver subgroup x s that means he conjectured that j s x s x s 1 let x 1 s be a subgroup of s such that x 1 s x s is the center of s x s in this short note we prove that j s x s if and only if j s x 1 s as an easy application we prove that j s x s x s p

amalgams blocks weights fusion systems and finite simple groups abstract we associate an iterated amalgam of finite groups to a certain class of fusion systems on finite p groups p a prime in such a way that the p group of the fusion system is a maximal finite p subgroup of the resulting group unique up to conjugacy and furthermore the conjugation action of the resulting usually infinite group on p subgroups induces the original fusion system on the p group 1 in view of earlier work of puig and of broto castellana grodal levi and oliver c broto n castellana j grodal r levi r oliver subgroup families controlling p local finite groups proc london math soc 3 91 2005 325 354 the fusion systems we deal with include all saturated fusion systems the resulting amalgam has free normal subgroups of finite index and we examine the images of the group by its maximal free normal subgroups of finite index these images all contain isomorphic copies of the original p group and are generated by the images of the finite groups used in the amalgamation if there is no non trivial normal p subgroup of the fusion system equivalently if the iterated amalgam constructed has no non trivial normal p subgroup then the generalised fitting subgroup of each of these homomorphic images is a direct product of non abelian simple groups each of order divisible by p and if there is no proper non trivial strongly closed p subgroup in the fusion system then the generalised fitting subgroup of any of these finite groups is characteristically simple we note that the finite dimensional representation theory of this amalgam is almost by construction p locally determined in the case of the fusion system associated to a p block of a finite group we suggest strong links between block theoretic invariants of the above finite epimorphic images of the associated amalgam and of the original block we believe that the results of this paper offer the prospect of at least the beginnings of a structural explanation for some of the current numerical conjectures in block theory e g j l alperin weights for finite groups in proc sympos pure math vol 47 amer math soc providence 1987 pp 369 379 e c dade counting characters in blocks i invent math 109 1 1992 187 210

bilateral filtering of diffusion tensor magnetic resonance images we extend the well known scalar image bilateral filtering technique to diffusion tensor magnetic resonance images dtmri the scalar version of bilateral image filtering is extended to perform edge preserving smoothing of dt field data the bilateral dt filtering is performed in the log euclidean framework which guarantees valid output tensors smoothing is achieved by weighted averaging of neighboring tensors analogous to bilateral filtering of scalar images the weights are chosen to be inversely proportional to two distance measures the geometrical euclidean distance between the spatial locations of tensors and the dissimilarity of tensors we describe the noniterative dt smoothing equation in closed form and show how interpolation of dt data is treated as a special case of bilateral filtering where only spatial distance is used we evaluate different dt tensor dissimilarity metrics including the log euclidean the similarity invariant log euclidean the square root of the j divergence and the distance scaled mutual diffusion coefficient we present qualitative and quantitative smoothing and interpolation results and show their effect on segmentation for both synthetic dt field data as well as real cardiac and brain dtmri data

the classification of p local finite groups over the extraspecial group of order p 3 and exponent p in this paper we classify the p local finite groups over p1 2 the extraspecial group of order p3 and exponent p for odd p this study reduces to the classification of the saturated fusion systems over p1 2 which will be characterized by the outer automorphism group the number of open image in new window radical subgroups and the automorphism group of each nontrivial open image in new window radical subgroup as part of this classification we obtain three new exotic 7 local finite groups

representations of categories and their applications we define for each small category c a category algebra rc over a base ring r and study its representations when c is an ei category we develop a theory of vertices and sources for rc mod which parameterizes the indecomposable rc modules as a main application we use our theory to find formulas for computing higher inverse limits over c

the gluing problem in the fusion systems of the symmetric alternating and linear groups a solution to linckelmann s gluing problem in all fusion systems of blocks yields an alternative formulation of alperin s weight conjecture using bredon equivariant cohomology techniques we solve the gluing problem in fusion systems of blocks which are isomorphic to the fusion systems of the symmetric groups or the alternating groups or gld q or sld q at the defining characteristic

extensions of linking systems with p group kernel we study extensions of p local finite groups where the kernel is a p group in particular we construct examples of saturated fusion systems mathcal f which do not come from finite groups but which have normal p subgroups a vartriangleleft mathcal f such that mathcal f a is the fusion system of a finite group one of the tools used to do this is the concept of a transporter system which is modelled on the transporter category of a finite group and is more general than a linking system

automorphisms of p local compact groups abstract self equivalences of classifying spaces of p local compact groups are well understood by means of the algebraic structure that gives rise to them but explicit descriptions are lacking in this paper we use robinson s construction of an amalgam g realising a given fusion system to produce a split epimorphism from the outer automorphism group of g to the group of homotopy classes of self homotopy equivalences of the classifying space of the corresponding p local compact group

minami webb splittings and some exotic p local finite groups we give a new proof of the minami webb formula for classifying spaces of finite groups by exploiting symonds s resolution of webb s conjecture the methods are applicable to obtain a stable decomposition of minami s type for the classifying spaces of the three exotic p local finite groups which were introduced by ruiz and viruel at the prime 7 we obtain a similar decomposition for the classifying spaces of a family of exotic p local finite groups which were constructed by broto levi and oliver

parameter estimations of mutually t related fuzzy variables abstract cai fuzzy sets and systems 55 1993 179 185 discussed parameter estimation methods for normal fuzzy variables in nahmias sense in this paper we investigate more the parameter estimations of general t related not necessarily normal fuzzy variables

fuzzy variables as a basis for a theory of fuzzy reliability in the possibility context abstract conventional reliability theory is based on the binary state assumption and the probability assumption namely 1 the systems are precisely defined as functioning or failed 2 the system behavior is fully characterized in the context of probability measures however the probability assumption is no longer reasonable in the case of small size of samples since the term probability stems from the sample generality in this way possibility measures should be used to characterize the sample particularity in this paper the binary state assumption is reserved and the possibility assumption is taken in place of the probability assumption thus a concept of fuzzy reliability i e posbist reliability is introduced and a theory of fuzzy reliability i e posbist reliability theory is established various conclusions which are contrary to statements in conventional reliability theory i e probist reliability theory are then drawn

a general formal approach for fuzzy reliability analysis in the possibility context abstract in this paper we focus on the fuzzy reliability analysis in the posbist reliability theory in accordance with this theory the system failure state is defined precisely and the system failure and repair behaviour is fully characterized in the context of possibility measures to formalize the reliability analysis in the posbist theory we propose a general approach on the basis of a system of functional equations the solution of the system allows us to obtain fuzzy reliability characteristics of the arbitrary repairable and unrepairable systems to illustrate the construction and application of the system of functional equations various examples are given

homotopy equivalences of p completed classifying spaces of finite groups we study homotopy equivalences of p completions of classifying spaces of finite groups to each finite group g and each prime p we associate a finite category p c g with the following properties two p completed classifying spaces bg p and bg p have the same homotopy type if and only if the associated categories p c g and p c g are equivalent furthermore the topological monoid aut bg p of self equivalences is determined by the self equivalences of the associated category p c g

alperin s weight conjecture in terms of equivariant bredon cohomology alperin s weight conjecture 1 admits a reformulation in terms of the cohomology of a functor on a category obtained from a subdivision construction applied to a centric linking system 7 of a fusion system of a block which in turn can be interpreted as the equivariant bredon cohomology of a certain functor on the g poset of centric brauer pairs the underlying general constructions of categories and functors needed for this reformulation are described in 1 and 2 respectively and 3 provides a tool for computing the cohomology of the functors arising in 2 taking as starting point the alternating sum formulation of alperin s weight conjecture by knorr robinson 11 the material of the previous sections is applied in 4 to interpret the terms in this alternating sum as dimensions of cohomology spaces of appropriate functors using further work of robinson 15 16 17

the homotopy theory of fusion systems the main goal of this paper is to identify and study a certain class of spaces which in many ways behave like p completed classifying spaces of finite groups these spaces occur as the classifying spaces of certain algebraic objects which we call p local finite groups a p local finite group consists roughly speaking of a finite p group s and fusion data on subgroups of s encoded in a way explained below our starting point is our earlier paper blo on p completed classifying spaces of finite groups together with the axiomatic treatment by lluis puig pu pu2 of systems of fusion among subgroups of a given p group the p completion of a space x is a space xp which isolates the properties of x at the prime p and more precisely the properties which determine its mod p cohomology for example a map of spaces x f y induces a homotopy equivalence xp y p if and only if f induces an isomorphism in mod p cohomology and h xp fp h x fp in favorable cases if x is p good when g is a finite group the p completion bgp of its classifying space encodes many of the properties of g at p for example not only the mod p cohomology of bg but also the sylow p subgroup of g together with all fusion among its subgroups are determined up to isomorphism by the homotopy type of bgp our goal here is to give a direct link between p local structures and homotopy types which arise from them this theory tries to make explicit the essence of what it means to be the p completed classifying space of a finite group and at the same time yields new spaces which are not of this type but which still enjoy most of the properties a space of the form bgp would have we hope that the ideas presented here will have further applications and generalizations in algebraic topology but this theory also fits well with certain aspects of modular representation theory in particular it may give a way of constructing classifying spaces for blocks in the group ring of a finite group over an algebraically closed field of characteristic p a saturated fusion system f over a p group s consists of a set homf p q of monomorphisms for each pair of subgroups p q s which form a category under composition include all monomorphisms induced by conjugation in s and satisfy certain other axioms formulated by puig definitions 1 1 and 1 2 below in particular these axioms are satisfied by the conjugacy homomorphisms in a finite group we refer to pu and pu2 for more details of puig s work on saturated

parametric nonlinear programming approach for a repairable system with switching failure and fuzzy parameters this paper proposes a procedure to construct the membership functions of the system characteristics of a repairable system with two primary units one standby unit and one repair facility when switching to standby may fail failure times of primary and standby units are assumed to follow fuzzified exponential distributions and repair times of the failed units are also assumed to have a fuzzified exponential distribution the a cut approach is used to extract from the fuzzy repairable system from a family of conventional crisp intervals for the desired system characteristics determined with a set of parametric nonlinear programs using their membership functions a numerical example is solved successfully to illustrate the practicality of the proposed approach because the system characteristics are governed by the membership functions more information is provided for use by designers and practitioners the successful extension of the parameter spaces to fuzzy environments permits the repairable system to have wider practical applications