quasi linear truth table reductions to p selective sets we show that if sat is quasi linear truth table reducible to a p selective set then np p as a consequence it follows that for a class k pp c p if every set in k is quasi linear truth table reducible to a p selective set then k p

infinitely often autoreducible sets a set a is autoreducible if one can compute for all x the value a x by querying a only at places y neq x furthermore a is infinitely often autoreducible if for infinitely many x the value a x can be computed by querying a only at places y neq x for all other x the computation outputs a special symbol to signal that the reduction is undefined it is shown that for polynomial time turing and truth table autoreducibility there are a b c in the class exp of all exponential time computable sets such that a is not infinitely often turing autoreducible b is turing autoreducible but not infinitely often truth table autoreducible and c is truth table autoreducible with g n 1 queries but not infinitely often turing autoreducible with g n queries here n is the length of the input g is nondecreasing and there exists a polynomial p such that p n bounds both the computation time and the value of g at input of length n furthermore connections between notions of infinitely often autoreducibility and notions of approximability are investigated the hausdorff dimension of the class of sets which are not infinitely often autoreducible is shown to be 1

a uniform approach to obtain diagonal sets in complexity classes abstract a uniform method for constructing sets which diagonalize over certain complexity classes while preserving other complexity properties is given we obtain some known results as well as some new ones as corollaries of our main theorem the new results concern the complexity classes p np co np pspace apt almost polynomial time r random polynomial time and the polynomial hierarchy

query complexity of membership comparable sets this paper investigates how many queries to k membership comparable sets are needed in order to decide all k 1 membership comparable sets for k 2 this query complexity is at least linear and at most cubic as a corollary we obtain that more languages are o logn membership comparable than truth table reducible to p selective sets

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

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

oracle dependent properties of the lattice of np sets abstract we consider under the assumption p np questions concerning the structure of the lattice of np sets together with the sublattice p we show that two questions which are slightly more complex than the known splitting properties of this lattice cannot be settled by arguments which relativize the two questions which we consider are whether every infinite np set contains an infinite p subset and whether there exists an np simple set we construct several oracles all of which make p np and which in addition make the above mentioned statements either true or false in particular we give a positive answer to the question raised by bennett and gill 1981 whether an oracle b exists making p b np b and such that every infinite set in np b has an infinite subset in p b the constructions of the oracles are finite injury priority arguments

nonuniform proof systems a new framework to describe nonuniform and probabilistic complexity classes the concept of non uniform proof systems is introduced this notion allows a uniform description of non uniform complexity classes 10 probabilistic classes e g bpp 8 15 27 32 am 2 3 and language classes defined by simultaneous non uniform and nondeterministic time bounds non uniform proof systems provide a better understanding of many results concerning these classes particularly their connections to uniform complexity measures we give an uniform approach to lowness results 19 20 for various complexity classes for instance we show that co np poly np is contained in the third level of the low hierarchy and that np c np co np poly implies that the polynomial time hierarchy collapses to its second level see also 1 finally some evidence is given that the low hierarchy cannot be extended beyond its third level by the current techniques

on the complexity of kings a king in a directed graph is a vertex from which each vertex in the graph can be reached through paths of length at most two there is a broad literature on tournaments completely oriented digraphs and it has been known for more than half a century that all tournaments have at least one king recently kings have proven useful in theoretical computer science in particular in the study of the complexity of reachability problems and semifeasible sets in this article we study the complexity of recognizing kings for each succinctly specified family of tournaments the king problem is already known to belong to p 2 p we prove that the complexity of kingship problems is a rich enough vocabulary to pinpoint every nontrivial many one degree in p 2 p that is we show that every set in p 2 p other than 0 and s is equivalent to a king problem under m p reductions indeed we show that the equivalence can even be realized by relatively simple padding and holds even if the notion of kings is redefined to refer to k kings for any fixed k 2 vertices from which all vertices can be reached through paths of length at most k in contrast we prove that for each succinctly specified family of tournaments the source problem the problem of deciding whether a given vertex v has the property that there exists a k such that v is a k king also falls within p 2 p yet cannot be p 2 p complete or even np hard unless p np using these and related techniques we obtain a broad range of additional results about the complexity of king problems diameter problems and radius problems it follows easily from our proof approach that the problem of testing kingship in succinctly specified graphs which need not be tournaments is p 2 p complete we show that the radius problem for arbitrary succinctly represented graphs is s 3 p complete but that in contrast the diameter problem for arbitrary succinctly represented graphs or even tournaments is p 2 p complete

autoreducibility and mitoticity of logspace complete sets for np and other classes abstract we study the autoreducibility and mitoticity of complete sets for np and other complexity classes where the main focus is on logspace reducibilities in particular we obtain for np and all other classes of the ph each m log complete set is t log autoreducible for p k p nexp each m log complete set is a disjoint union of two 2 tt log complete sets for pspace each dtt p complete set is a disjoint union of two dtt p complete sets

resource bounded immunity and simplicity revisiting the 30 years old notions of resource bounded immunity and simplicity we investigate the structural characteristics of various immunity notions strong immunity almost immunity and hyperimmunity as well as their corresponding simplicity notions we also study limited immunity and simplicity called k immunity and feasible k immunity and their simplicity notions finally we propose the k immune hypothesis as a working hypothesis that guarantees the existence of simple sets in np

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

vector bundles of rank 2 and linear systems on algebraic surfaces let s be a smooth complex algebraic surface and let l be a divisor on s it is well known that the linear system il ksi where ks is the canonical divisor plays an important role in understanding the geometry of s several authors see for example 8 11 studied various properties of this system our paper treats the same subject but the point of view is different from the works mentioned above the philosophy underlying our approach is almost classical points on s more generally effective 0 cycles in special position with respect to il ksi see definition below contain information about the geometry of s this point of view was recently revived in 5 also 10 in fact this paper contains the technique of going from the effective 0 cycle z in special position with respect to il ksi to the geometry of s for the sake of completeness we give a definition and state a theorem which can be found in 5 10

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

on symmetric differences of np hard sets with weakly p selective sets the symmetric differences of np hard sets with weakly p selective sets are investigated we show that if there exist a weakly p selective set a and an np pm hard set h such that h a pbtt sparse and a h pm sparse then p np so no np pm hard set has sparse symmetric difference with any weakly p selective set unless p np the proof of our main result is an interesting application of the tree prunning techniques fortune 1979 mahaney 1982 in addition we show that there exists a p selective set which has exponentially dense symmetric difference with every set in pbtt sparse

the stability of a cubic fixed point in three dimensions from the renormalization group the global structure of the renormalization group flows of a model with isotropic and cubic interactions is studied using the massive field theory directly in three dimensions the four loop expansions of the functions are calculated for arbitrary n the critical dimensionality nc 2 89 0 02 and the stability matrix eigenvalues estimates obtained on the basis of the generalized pade borel leroy resummation technique are shown to be in a good agreement with those found recently by exploiting the five loop e expansions

index sets and presentations of complexity classes this paper draws close connections between the ease of presenting a given complexity class b and the position of the index sets ib lcub i l mi b rcub and jb lcub i miis total l mi b rcub in the arithmetical hierarchy for virtually all classes b studied in the literature the lowest levels attainable are ib 03 and jb 02 the first holds iff b is 02 presentable and the second iff b is recursively presentable a general kind of priority argument is formulated and it is shown that every property enforcible by it is not recursively presentable it follows that the classes of p immune and p biimmune languages in exponential time are not recursively presentable it is shown that for all b with ib 03 many members of b do not provably in true 2 arithmetic belong to b a class h is exhibited such that ih 03 is open and ih 03 implies that the polynomial hierarchy is infinite

almost every set in exponential time is p bi immune a set a is p bi immune if neither a nor its complement has an infinite subset in p we investigate here the abundance of p bi immune languages in linear exponential time e using resource bounded measure we prove that the class of p bi immune sets has measure 1 in e this implies that almost every language in e is p bi immune a bit further we show that every p random pseudorandom language is e bi immune regarding the existence of p bi immune sets in np we show that if np does not have measure 0 in e then np contains a p bi immune set another consequence is that the class of 1 tt p complete languages for e has measure 0 in e in contrast using the approach of resource bounded category it is shown that in e the class of p bi immune languages lacks the property of baire the baire category analogue of lebesgue measurability

genericity randomness and polynomial time approximations polynomial time safe and unsafe approximations for intractable sets were introduced by meyer and paterson technical report tm 126 laboratory for computer science mit cambridge ma 1979 and yesha siam j comput 12 1983 pp 411 425 respectively the question of which sets have optimal safe and unsafe approximations has been investigated extensively duris and rolim lecture notes in comput sci 841 springer verlag berlin new york 1994 pp 38 51 and ambos spies proc 22nd icalp springer verlag berlin new york 1995 pp 384 392 showed that the existence of optimal polynomial time approximations for the safe and unsafe cases is independent using the law of the iterated logarithm for p random sequences which has been recently proven in proc 11th conf computational complexity ieee computer society press piscataway nj 1996 pp 180 189 we extend this observation by showing that both the class of polynomial time delta levelable sets and the class of sets which have optimal polynomial time unsafe approximations have p measure 0 hence typical sets in e in the sense of p measure do not have optimal polynomial time unsafe approximations we will also establish the relationship between resource bounded genericity concepts and the polynomial time safe and unsafe approximation concepts

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

minimal pairs and complete problems two sets are said to form a minimal pair for polynomial many one reductions if neither set is in p and the only sets which are polynomially reducible via polynomial many one reductions to both sets are in p we consider the question of which complete sets can be the top that is the supremum of a minimal pair our main result is that any elementary recursive set which is hard for deterministic exponential time cannot be the top of a minimal pair hence in particular any complete set for an elementary recursive class containing exponential time cannot be such a supremum for np complete sets the problem remains open we show here that it is oracle dependent this is the first example of an oracle dependent property which is completely degree theoretic

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

p selectivity intersections and indices the p selective sets selman 1979 are those sets for which there is a polynomial time algorithm that given any two strings determines which is more likely to belong to the set if either of the strings is in the set the algorithm chooses one that is in the set we prove that for each k the k ary boolean connectives under which the p selective sets are closed are exactly those that are either completely degenerate or almost completely degenerate we determine the complexity of the index set of the r e p selective sets 30 complete

critical exponents of the n vector model recently the series for two renormalization group functions corresponding to the anomalous dimensions of the fields and of the three dimensional field theory have been extended to next order seven loops by murray and nickel we examine the influence of these additional terms on the estimates of critical exponents of the n vector model using some new ideas in the context of the borel summation techniques the estimates have slightly changed but remain within the errors of the previous evaluation exponents such as related to the field anomalous dimension which were poorly determined in the previous evaluation of le guillou zinn justin have seen their apparent errors significantly decrease more importantly perhaps summation errors are better determined the change in exponents affects the recently determined ratios of amplitudes and we report the corresponding new values finally because an error has been discovered in the last order of the published expansions order we have also re analysed the determination of exponents from the expansion the conclusion is that the general agreement between expansion and three dimensional series has improved with respect to le guillou zinn justin

genericity and randomness over feasible probability measures this paper investigates the notion of resource bounded genericity developed by ambos spies fleischhack and huwig ambos spies neis and terwijn have recently shown that every language that is t n random over the uniform probability measure is t n generic it is shown here that in fact every language that is t n random over any strongly positive t n computable probability measure is t n generic roughly speaking this implies that when genericity is used to prove a resource bounded measure result the result is not specific to the underlying probability measure

the stability of the o invariant fixed point in three dimensions we study the stability of the o n fixed point in three dimensions under perturbations of the cubic type we address this problem in the three cases n 2 3 4 by using finite size scaling techniques and high precision monte carlo simulations it is well known that there is a critical value below which the o n fixed point is stable and above which the cubic fixed point becomes the stable one whilst we cannot exclude that as recently claimed our analysis strongly suggests that coincides with 3

promise problems complete for complexity classes abstract a general framework is given to obtain hardness results for promise problems that derive from self reducible decision problems the principal theorem is that if a set a is d p equivalent to a disjunctive self reducible set in np then the natural promise problem associated with a is as hard to solve as it is to recognize a np hardness of the satisfiability promise problem follows and graph isomorphism hardness of a promise problem that derives from the graph isomorphism problem is proved

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

on 4 generated bundles and directly linked subcanonical curves abstract given a subcanonical curve in p 3 we study conditions under which is directly linked to another subcanonical curve we show that this happens when the rank 2 bundle e associated to admits a surjection 4op3 ai e ai z we give an explicit construction of such a starting with surjection u and we prove that the numbers ai appearing in u are uniquely determined by e this implies that we have at most 3 possibilities for the numerical characters of a subcanonical curve directly linked to

relative to a random oracle a bf p a ne bf np a ne text co bf np a with probability 1 let a be a language chosen randomly by tossing a fair coin for each string x to determine whether x belongs to a with probability 1 each of the relativized classes textbf logspace a bf p a bf np a bf pp a and textbf pspace a is properly contained in the next also bf np a ne text co bf np a with probability 1 by contrast with probability 1 the class bf p a coincides with the class bf bpp a of languages recognized by probabilistic oracle machines with error probability uniformly bounded below tfrac 1 2 bf np a is shown with probability 1 to contain a bf p a immune set i e a set having no infinite subset in bf p a the relationship of bf p a immunity to p sparseness and bf np a completeness is briefly discussed bf p a immune sets in bf np a can be sparse or moderately dense but not co sparse relativization with respect to a random length preserving permutation pi instead of a rand

geometry of families of nodal curves on the blown up projective plane let p2r be the projective plane blown up at r generic points denote by e0 e1 er the strict transform of a generic straight line on p2 and the exceptional divisors of the blown up points on p2r respectively we consider the variety virr of all irreducible curves c with k nodes as the only singularities and give asymptotically nearly optimal sufficient conditions for its smoothness irreducibility and non emptiness moreover we extend our conditions for the smoothness and the irreducibility on families of reducible curves for r 9 we give the complete answer concerning the existence of nodal curves in virr

on polynomial time bounded truth table reducibility of np sets to sparse sets it is proved that if text p ne text np then there exists a set in text np that is not polynomial time bounded truth table reducible in short leqq text btt text p reducible to any sparse set in other words it is proved that no sparse leqq text btt text p hard set exists for text np unless text p text np by using the technique proving this result the intractability of several number theoretic decision problems i e decision problems defined naturally from number theoretic problems is investigated it is shown that for these number theoretic decision problems if it is not in text p then it is not leqq text btt text p reducible to any sparse set

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

multicritical behavior in models with two competing order parameters we employ the nonperturbative functional renormalization group to study models with an o n 1 o n 2 symmetry here different fixed points exist in three dimensions corresponding to bicritical and tetracritical behavior induced by the competition of two order parameters we discuss the critical behavior of the symmetry enhanced isotropic the decoupled and the biconical fixed point and analyze their stability in the n 1 n 2 plane we study the fate of nontrivial fixed points during the transition from three to four dimensions finding evidence for a triviality problem for coupled two scalar models in high energy physics we also point out the possibility of noncanonical critical exponents at semi gaussian fixed points and show the emergence of goldstone modes from discrete symmetries

reducibility classes of p selective sets a set is p selective if there is a polynomial time semi decision algorithm for the set an algorithm that given any two strings decides which is more likely to be in the set this paper establishes a strict hierarchy among the various reductions and equivalences to p selective sets

on the existence of certain families of curves introduction 25 w 1 generalities on families of curves in ip 28 w the map 0 d 33 w 3 the number of moduli of a family 40 w 4 plane curves existence of families with the expected number of moduli 43 w 5 smoothing certain reducible curves in ip r r 3 47 w 6 curves in ip r 3 existence of families with the expected number of moduli 52 w curves in ip3 a more general existence theorem 55

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

algebraic properties for selector functions the nondeterministic advice complexity of the p selective sets is known to be exactly linear regarding the deterministic advice complexity of the p selective sets i e the amount of karp lipton advice needed for polynomial time machines to recognize them in general the best current upper bound is quadratic k ko j comput system sci 26 1983 pp 209 221 and the best current lower bound is linear l hemaspaandra and l torenvliet theoret comput sci 154 1996 pp 367 377 we prove that every associatively p selective set is commutatively associatively p selective using this we establish an algebraic sufficient condition for the p selective sets to have a linear upper bound which thus would match the existing lower bound on their deterministic advice complexity if all p selective sets are associatively p selective then the deterministic advice complexity of the p selective sets is linear the weakest previously known sufficient condition was p np we also establish related results for algebraic properties of and advice complexity of the nondeterministically selective sets

resource bounded randomness and weakly complete problems we introduce and study resource bounded random sets based on lutz s concept of resource bounded measure 5 6 we concentrate on nc randomness c 2 which corresponds to the polynomial time bounded p measure of lutz and which is adequate for studying the internal and quantative structure of e dtime 2lin first we show that the class of nc random sets has p measure 1 this provides a new simplified approach to p measure 1 results next we compare randomness with genericity in the sense of 1 2 and we show that nc 1 random sets are nc generic whereas the converse fails from the former we conclude thatnc random sets are not p btt complete for e our technical main results describe the distribution of the nc random sets under p m reducibility we show that every nc random set in e has nk random predecessors in e for any k 1 whereas the amount of randomness of the successors is bounded we apply this result to answer a question raised by lutz 8 we show that the class of weakly complete sets has measure 1 in e and that there are weakly complete problems which are not p btt complete for e

bi immunity separates strong np completeness notions we prove that if for some e 0 np contains a set that is dtime 2ne bi immune then np contains a set that is 2 turing complete for np hence 3 truth table complete but not 1 truth table complete for np thus this hypothesis implies a strong separation of completeness notions for np lutz and mayordomo theor comput sci 164 1996 141 163 and ambos spies and bentzien j comput syst sci 61 3 2000 335 361 previously obtained the same consequence using strong hypotheses involving resource bounded measure and or category theory our hypothesis is weaker and involves no assumptions about stochastic properties of np

nodal curves on surfaces of general type plane irreducible curves of degree d having exactly nodes and no other singularities is non empty and everywhere smooth of codimension in the linear system o d if c vd severi uses the non speciality of the normal line bundle to the composition c c p where c is the normalization of c to prove that vd is smooth of the asserted codimension at the point c this proof can be extended to rational ruled and k3 surfaces with little changes we discuss this point in sect 1 see also t for the case of rational surfaces in the case of a surface of general type s the approach of severi fails and in fact it is easy to see that the analogous of severi s theorem does not hold in general if we impose too many nodes to the curves of a complete linear system d one may nevertheless look for an upper bound on ensuring that the family vd of irreducible curves in d with nodes is smooth of codimension in sect 2 we give the following partial answer to this problem

families of nodal curves on projective threefolds and their regularity via postulation of nodes the main purpose of this paper is to introduce a new approach to study families of nodal curves on projective threefolds precisely given x a smooth projective threefold e a rank two vector bundle on x l a very ample line bundle on x and k 0 0 integers and denoted by v e l k the subscheme of p h 0 e l k parametrizing global sections of e l k whose zero loci are irreducible and nodal curves on x we present a new cohomological description of the tangent space t s v e l k at a point s 2 v e l k this description enable us to determine effective and uniform upper bounds for which are linear polynomials in k such that the family v e lk is smooth and of the expected dimension regular for short the almost sharpness of our bounds is shown by some interesting examples furthermore when x is assumed to be a fano or a calaby yau threefold we study in detail the regularity property of a point s 2 v e lk related to the postulation of the nodes of its zero locus c v s x roughly speaking when the nodes of c are assumed to be in general position either on x or on an irreducible divisor of x having at worst log terminal singularities or to lie on a l c i and subcanonical curve in x we find upper bounds onwhich are respectively cubic quadratic and linear polynomials in k ensuring the regularity of v e l k at s finally when x p 3 we also discuss some interesting geometric properties of the curves given by sections parametrized by v e ox k

cook versus karp levin separating completeness notions if np is not small under the hypothesis that np does not have p measure 0 roughly that np contains more than a negligible subset of exponential time it is shown that there is a language that is t p complete cook complete but not m p complete karp levin complete for np this conclusion widely believed to be true is not known to follow from p np or other traditional complexity theoretic hypotheses