splitting np complete sets we show that a set is m autoreducible if and only if it is m mitotic this solves a long standing open question in a surprising way as a consequence of this unconditional result and recent work by glaser et al complete sets for all of the following complexity classes are m mitotic mathrm np mathrm conp oplus mathrm p mathrm pspace and mathrm nexp as well as all levels of mathrm ph mathrm modph and the boolean hierarchy over mathrm np in the cases of mathrm np mathrm pspace mathrm nexp and mathrm ph this at once answers several well studied open questions these results tell us that complete sets share a redundancy that was not known before in particular every mathrm np complete set a splits into two mathrm np complete sets a 1 and a 2 we disprove the equivalence between autoreducibility and mitoticity for all polynomial time bounded reducibilities between 3 tt reducibility and turing reducibility there exists a sparse set in mathrm exp that is polynomial time 3 tt autoreducible but not weakly polynomial time t mitotic in particular polynomial time t autoreducibility does not imply polynomial time weak t mitoticity which solves an open question by buhrman and torenvliet

on culler shalen seminorms and dehn filling if f is a finitely generated discrete group and g a complex algebraic lie group the g character variety of r is an affine algebraic variety whose points correspond to characters of representations of r with values in g marc culler and peter shalen developed the theory of sl2 c character varieties of finitely generated groups and applied their results to study the topology of 3 dimensional manifolds in the papers 6 7 8 consider the exterior m of a hyperbolic knot lying in a closed connected orientable 3 manifold the mostow rigidity theorem implies that the holonomy representation p iri m isom h3 psl2 c is unique up to conjugation and taking complex conjugates the orientability of m can be used to show 1 lifts to a representation p e hom ri m sl2 c whose character determines an essentially unique point of xp of x iri m the sl2 c character variety of irl m culler and shalen 8 proved that the component x0 of x x1ri m which contains x is a curve one of their major contributions was to show how x0 determines a norm on h1 wm r which encodes many topological properties of m in particular it provides information on the dehn fillings of m their construction may be applied to arbitrary curves in the sl2 c character variety of a connected compact orientable irreducible 3manifold whose boundary is a torus though in this generality one can only guarantee that it will define a seminorm the first half of this paper is devoted to the development of the general theory of culler shalen seminorms defined for curves of psl2 c characters by working over psl2 c we obtain a theory that is more generally applicable than its sl2 c counterpart while being only mildly more difficult to set up in the second half of this paper we apply the theory of culler shalen seminorms to study the dehn filling operation in particular we examine the relationship between fillings which yield manifolds having a positive dimen

hard promise problems and nonuniform complexity for every recursive set a let pp a denote the following promise problem input x and y promise x a y a property x a we show that if l is a solution of pp a then a p l poly from this result it follows that if a is t p hard for np then all solutions of pp a are hard for np under a reduction that generalizes both t p and t sn specifically if a is np hard then all solutions of pp a are generalized high2 bbs86b the main theorem that leads to this result states that if b is a self reducible set b t p a and a p l poly then 2 p b 2 p l several interesting connections between uniform and nonuniform complexity follow directly from this theorem

separation of np completeness notions we use hypotheses of structural complexity theory to separate various np completeness notions in particular we introduce an hypothesis from which we describe a set in np that is mbox leq rm p rm t complete but not mbox leq rm p tt complete we provide fairly thorough analyses of the hypotheses that we introduce

weak mitoticity of bounded disjunctive and conjunctive truth table autoreducible sets abstract glaser et al siamjcomp 2008 and tcs 2009 2 proved existence of two sparse sets a and b in exp where a is 3 tt truth table polynomial time autoreducible but not weakly polynomial time turing mitotic and b is polynomial time 2 tt autoreducible but not weakly polynomial time 2 tt mitotic we unify and strengthen both of those results by showing that there is a sparse set in exp that is polynomial time 2 tt autoreducible but not even weakly polynomial time turing mitotic all these results indicate that polynomial time autoreducibilities in general do not imply polynomial time mitoticity at all with the only exceptions of the many one and 1 tt reductions on the other hand however we proved that every autoreducible set for the polynomial time bounded disjunctive or conjunctive tt reductions is weakly mitotic for the polynomial time tt reduction that makes logarithmically many queries only this shows that autoreducible sets for reductions making more than one query could still be mitotic in some way if they possess certain special properties

fuzzy programming approach to multi level programming problems multi level programming is characterized as mathematical programming to solve decentralized planning problems we have considered a multi level linear programming problem and applied fuzzy mathematical programming fmp approach to obtain the solution of the system we have suggested fmp method for the minimization of the objectives using linear membership functions fmp is a supervised search procedure supervised by the higher level decision maker dm the higher level dm provides the preferred values of decision variables under his control to enable the lower level dm to search for his optimum in a wider feasible space and the bounds of his objective function to direct the lower level dm to search for his solutions in the right direction

duality theory in fuzzy mathematical programming problems with fuzzy coefficients in this paper the notions of subgradient subdifferential and differential with respect to convex fuzzy mappings are investigated which provides the basis for the fuzzy extremum problem theory we consider the problems of minimizing or maximizing a convex fuzzy mapping over a convex set and develop the necessary and or sufficient optimality conditions furthermore the concept of saddle points and minimax theorems under fuzzy environment is discussed the results obtained are used to formulate the lagrangian dual of fuzzy programming under certain fuzzy convexity assumptions kkt conditions for fuzzy programming are derived and the perturbed convex fuzzy programming is considered finally these results are applied to fuzzy linear programming and fuzzy quadratic programming

non mitotic sets we study the question of the existence of non mitotic sets in np we show under various hypotheses that 1 tt mitoticity and m mitoticity differ on np t autoreducibility and t mitoticity differ on np this contrasts the situation in the recursion theoretic setting where ladner showed that autoreducibility and mitoticity coincide 2 tt autoreducibility does not imply weak 2 tt mitoticity from this it follows that autoreducibility and mitoticity are not equivalent for all reducibilities between 2 tt and t although the notions coincide for m and 1 tt reducibility

reductions on np and p selective sets abstract p selective sets are used to distinguish polynomial time bounded reducibilities on np in particular we consider different kinds of positive reductions these preserve membership in np and are not a priori closed under complements we show that the class of all sets which are both p selective and have positive reductions to their complements is p this is used to show that if dext next then there exists a set in np p that is not positive reducible to its complement various similar results are obtained we also show that p is the class of all sets which are both p selective and positive truth table self reducible from this result it follows that various naturally defined apparently intractible problems are not p selective unless p np

splittings robustness and structure of complete sets we investigate the structure of exp complete and hard sets under various kinds of reductions in particular we are interested in the way in which information that makes the set complete is stored in the set we study for various types of reductions the question of whether the set difference a s for a hard set a and a sparse set s is still hard we also address the question of which complete sets a can be split into sets a1 and a2 such that a equiv p r a 1 equiv p r a 2 for reduction type r i e which complete sets are mitotic we obtain both positive and negative answers to these questions depending on the reduction type and the structure of the sparse set

a fuzzy approach to hierarchical multiobjective programming problems and its application to an industrial pollution control problem in this paper we focus on hierarchical multiobjective programming problems where multiple decision makers in a hierarchical organization have their own multiple objective functions and propose an interactive algorithm based on the dual decomposition method to obtain the satisfactory solution which reflects not only the hierarchical relationships between multiple decision makers but also their own preferences for their objective functions in the proposed algorithm assuming that each of the decision makers has fuzzy goals for his her objective functions corresponding membership functions are elicited from the decision makers in their subjective manner in order to deal with hierarchical multiobjective programming problems a new kind of pareto optimality concept in membership spaces is defined and the concept of decision powers of multiple decision makers in a hierarchical decision structure is introduced the proposed algorithm is applied to the industrial pollution control problem in osaka city in japan in order to show the efficiency of the proposed method

diagonalizations over polynomial time computable sets abstract a formal notion of diagonalization is developed which allows to enforce properties that are related to the class of polynomial time computable sets the class of polynomial time computable functions respectively like e g p immunity it is shown that there are sets called p generic which have all properties enforceable by such diagonalizations we study the behaviour and the complexity of p generic sets in particular we show that the existence of p generic sets in np is oracle dependent even if we assume p np

a three level optimization method for fuzzy large scale multiobjective nonlinear programming problems abstract in general mathematical models to reflect real world complicated decision situations are often formulated as largescale programming problems which include a large number of variables and or constraints furthermore considering the fuzziness of human s subjective judgement in real world decision making it is necessary to deal with the large scale programming problems in the fuzzy environment from such a point of view in this paper under the assumption that the decision maker dm has fuzzy goal for each of the objective functions and the corresponding membership function may be nonlinear a 3 level optimization method is proposed for large scale multiobjective nonlinear programming problems ls monlps with the block angular structure to obtain the compromise solution of the dm based on the proposed algorithm the computer program is developed in fortran language and applied to the numerical example to clarify the efficiency of the proposed algorithm

exceptional and cosmetic surgeries on knots we show that the bridge distance of a knot determines a lower bound on the genera of essential surfaces and heegaard surfaces in the manifolds that result from non trivial dehn surgeries on the knot in particular knots with high bridge distance do not admit non trivial non hyperbolic surgeries or non trivial cosmetic surgeries we further show that if a knot has bridge distance at least 3 then its bridge number is bounded above by a function of seifert genus or indeed by the genus of almost any essential surface or heegaard surface in the surgered manifold

a fuzzy dual decomposition method for large scale multiobjective nonlinear programming problems abstract in this paper we propose a fuzzy dual decomposition method for large scale multiobjective nonlinear programming problems ls monlps with the block angular structure by considering the vague nature of human judgements we assume that the decision maker dm may have a fuzzy goal for each of the objective functions in the ls monlp after eliciting the corresponding membership function for each of the objective functions through the interaction with the dm an extended primal problem and the corresponding extended dual problem are formulated then a two level optimization algorithm for the extended dual problem is proposed for deriving the compromise solution for the dm to the ls monlp based on the proposed algorithm fortran programs are developed and an illustrative numerical example is demonstrated

duality theorems and saddle point optimality conditions in fuzzy nonlinear programming problems based on different solution concepts under a general setting of partial ordering defined on the set of all fuzzy numbers the duality theorems and saddle point optimality conditions in fuzzy nonlinear programming problems based on two solution concepts for primal problem and three solution concepts for dual problem are derived in this paper those solution concepts are inspired by the nondominated solution concept employed in multiobjective programming problems since the ordering among fuzzy numbers introduced in this paper is a partial ordering not a total ordering we also provide a concept of fuzzy scalar inner product for fuzzy numbers then the fuzzy valued lagrangian function and the fuzzy valued lagrangian dual function are proposed via the concept of fuzzy scalar product under these settings the dual problem is formulated and the objective function of this dual problem is a point to set fuzzy valued function in this case three solution concepts for dual problem are proposed the duality theorems and saddle point optimality conditions can be naturally elicited based on these three solution concepts

reductions between disjoint np pairs disjoint np pairs are pairs a b of nonempty disjoint sets in np we prove that all of the following assertions are equivalent there is a many one complete disjoint np pair there is a strongly many one complete disjoint np pair there is a turing complete disjoint np pair such that all reductions are smart reductions there is a complete disjoint np pair for one to one invertible reductions the class of all disjoint np pairs is uniformly enumerable let a b c and d be nonempty sets belonging to np a smart reduction between the disjoint np pairs a b and c d is a turing reduction with the additional property that if the input belongs to a b then all queries belong to c d we prove under the reasonable assumption that up co up has a p bi immune set that there exist disjoint np pairs a b and c d such that a b is truth table reducible to c d but there is no smart reduction between them this paper contains several additional separations of reductions between disjoint np pairs we exhibit an oracle relative to which distnp has a truth table complete disjoint np pair but has no many one complete disjoint np pair

duality theorems in fuzzy mathematical programming problems based on the concept of necessity the fuzzy valued lagrangian function for the fuzzy mathematical programming problem via the concept of the fuzzy scalar inner product is proposed a solution concept of fuzzy optimization problems which is essentially similar to the notion of pareto solution in multiobjective programming problems is also introduced by ranking the fuzzy numbers using the necessity indices under these settings the weak and strong duality theorems can be elicited we show that the primal and dual fuzzy mathematical programming problems have no duality gap under suitable convexity assumptions for fuzzy valued functions

dehn surgery on knots in d dehn considered the following method for constructing 3 manifolds remove a solid torus neighborhood n k of some knot x in the 3 sphere s and sew it back differently in particular he showed that taking x to be the trefoil one could obtain infinitely many non simply connected homology spheres o in this way let mx s n k then the different resewings are parametrized by the isotopy class r of the simple closed curve on the torus omk that bounds a meridional disk in the re attached solid torus we denote the resulting closed oriented 3 manifold by mk and say that it is obtained by r dehn surgery on x more generally one can consider the manifolds ml obtained by r dehn surgery on a fc component link l xi u u kk in s where r r it turns out that every closed oriented 3 manifold can be constructed in this way wal lie thus a good understanding of dehn surgery might lead to progress on general questions about the structure of 3 manifolds starting with the case of knots it is natural to extend the context a little and consider the manifolds m r obtained by attaching a solid torus v to an arbitrary compact oriented irreducible every 2 sphere bounds a 3 ball 3 manifold m with dm an incompressible torus where r is the isotopy class slope on dm of the boundary of a meridional disk of v we say that m r is the result of r dehn filling on m an observed feature of this construction is that

the solution and duality of imprecise network problems duality properties have been investigated by many researchers in the recent literature they are introduced in this paper for a fully fuzzified version of the minimal cost flow problem which is a basic model in network flow theory this model illustrates the least cost of the shipment of a commodity through a capacitated network in terms of the imprecisely known available supplies at certain nodes which should be transmitted to fulfil uncertain demands at other nodes first we review on the most valuable results on fuzzy duality concepts to facilitate the discussion of this paper by applying hukuhara s difference approximated and exact multiplication and wu s scalar production we exhibit the flow in network models then we use combinatorial algorithms on a reduced problem which is derived from fully fuzzified mcfp to acquire fuzzy optimal flows to give duality theorems we utilize a total order on fuzzy numbers due to the level of risk and realize optimality conditions for providing some efficient combinatorial algorithms finally we compare our results with the previous worthwhile works to demonstrate the efficiency and power of our scheme and the reasonability of our solutions in actual decision making problems

strong nondeterministic polynomial time reducibilities abstract building on the work of adleman and manders 1 we define a new class of reducibilities the strong nondeterministic polynomial time reducibilities we show that several reducibilities in this class differ over the recursive sets and assuming np conp that they each determine different classes of np hard sets comparisons in strength are also made between the strong nondeterministic polynomial time reducibilities and the classes of deterministic and non deterministic polynomial time reducibilities in ladner lynch and selman 6