the structure of generalized complexity cores structural properties of proper hard cores generalized complexity cores are studied the setting involves a countable class c of sets of strings that is closed under finite union and under finite variation for any set a let ca denote the collection of all c c such that c a any such a has a proper hard core h h a and for every c ca h c is finite the collection of all proper hard cores of a with respect to c modulo finite differences forms a lattice that is isomorphic to one of just three possible lattices a set b is c levelable if for any c ca there exists a c ca such that c c is infinite the structure of the lattice of hard cores of a with respect to c depends only on the relationship between a and c if a is finite or in c then the lattice is the trivial one point lattice if a is infinite not in c and not c levelable then the lattice is isomorphic to the lattice of all subsets of modulo finite subsets where lcub0 1rcub if a is c levelable then the lattice is isomorphic to a special lattice f f

a classification of complexity core lattices abstract lynch 1975 has shown that every recursive set a not in p contains an infinite polynomial complexity core a set of elements c a such that any algorithm deciding a needs more than polynomial time almost everywhere on c actually any a not in p contains infinitely many different cores the collection of which forms a lattice under inclusion we study the structure of this lattice proving that surprisingly there are only three possibilities assuming the lattice is not trivial which happens if a is in p its shape depends only on whether a is almost p immune or not it is known that the natural intractable sets usually do not have this property

matching rota baxter algebras matching dendriform algebras and matching pre lie algebras abstract we introduce the notion of a matching rota baxter algebra motivated by the recent work on multiple pre lie algebras arising from the study of algebraic renormalization of regularity structures 10 19 this notion is also related to iterated integrals with multiple kernels and solutions of the associative polarized yang baxter equation generalizing the natural connection of rota baxter algebras with dendriform algebras to matching rota baxter algebras we obtain the notion of matching dendriform algebras as in the classical case of one operation matching rota baxter algebras and matching dendriform algebras are related to matching pre lie algebras which coincide with the aforementioned multiple pre lie algebras more general notions and results on matching tridendriform algebras and matching postlie algebras are also obtained

on the rectangularity of nonlinear block codes we give simple sufficient conditions for a code to be rectangular and show that large families of well known nonlinear codes are rectangular these include hadamard 1893 levenshtein 1964 delsarte goethals 1975 kerdock 1972 and nordstrom robinson 1967 codes being rectangular each of these codes has a unique minimal trellis that can be used for soft decision maximum likelihood decoding

on circuit size complexity and the low hierarchy in np let a be a set having polynomial size circuits if a is also known to be in np then we may conclude that the graph of the polynomial size circuits for a is actually in pi 2 p using this observation we show that sets in np which have polynomial size ciruits are in l 3 p the third level of the low hierarchy in np by a similar technique we are able to show that some other intuitively low sets in np are in l 2 p and even in a certain refinement of l 2 p as a consequence sparse sets are not strong nondeterministic polynomial time turing complete in np unless the polynomial time hierarchy collapses to delta 2 p

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

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

sparse sets lowness and highness we develop the notions of generalized lowness for sets in ph the union of the polynomial time hierarchy and of generalized highness for arbitrary sets also we develop the notions of extended lowness and extended highness for arbitrary sets these notions extend the decomposition of np into low sets and high sets developed by schoning 15 and studied by ko and schoning 9 we show that either every sparse set in ph is generalized high or no sparse set in ph is generalized high further either every sparse set is extended high or no sparse set is extended high in both situations the former case corresponds to the polynomial time hierarchy having only finitely many levels while the latter case corresponds to the polynomial time hierarchy extending infinitely many levels

on small generators abstract yap 1983 shows that each set having small generators is in the class np poly which was introduced by karp and lipton 1980 we show here that the converse is also true i e each set in np poly has small generators this settles a question left open by yap 1983 further an argument by meyer stated in berman and hartmanis 1977 is generalized to show that a set has small generators if and only if for some sparse set s a np s

renormalization in quantum field theory and the riemann hilbert problem i the hopf algebra structure of graphs and the main theorem this paper gives a complete selfcontained proof of our result announced in 6 showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the riemann hilbert problem we shall first show that for any quantum field theory the combinatorics of feynman graphs gives rise to a hopf algebra which is commutative as an algebra it is the dual hopf algebra of the enveloping algebra of a lie algebra whose basis is labelled by the one particle irreducible feynman graphs the lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa the corresponding lie group g is the group of characters of we shall then show that using dimensional regularization the bare unrenormalized theory gives rise to a loop where c is a small circle of complex dimensions around the integer dimension d of space time our main result is that the renormalized theory is just the evaluation at z d of the holomorphic part of the birkhoff decomposition of we begin to analyse the group g and show that it is a semi direct product of an easily understood abelian group by a highly non trivial group closely tied up with groups of diffeomorphisms the analysis of this latter group as well as the interpretation of the renormalization group and of anomalous dimensions are the content of our second paper with the same overall title

hard core theorems for complexity classes nancy lynch proved that if a decision problem a is not solvable in polynomial time then there exists an infinite recursive subset x of its domain on which the decision is almost everywhere complex in this paper general theorems of this kind that can be applied to several well known automata based complexity classes including a common class of randomized algorithms are proved

minimal trellises for block codes basic concepts in the study of trellises of block codes are defined it is shown that minimal proper trellises exists for all block codes bounds on the size of such trellises are established these bounds are shown to be exact for maximum distance separable codes and nearly so for perfect codes

the z sub 4 linearity of kerdock preparata goethals and related codes certain notorious nonlinear binary codes contain more codewords than any known linear code these include the codes constructed by nordstrom robinson 1967 kerdock 1972 preparata 1968 goethals 1974 and delsarte goethals 1975 it is shown here that all these codes can be very simply constructed as binary images under the gray map of linear codes over z sub 4 the integers mod 4 although this requires a slight modification of the preparata and goethals codes the construction implies that all these binary codes are distance invariant duality in the z sub 4 domain implies that the binary images have dual weight distributions the kerdock and preparata codes are duals over z sub 4 and the nordstrom robinson code is self dual which explains why their weight distributions are dual to each other the kerdock and preparata codes are z sub 4 analogues of first order reed muller and extended hamming codes respectively all these codes are extended cyclic codes over z sub 4 which greatly simplifies encoding and decoding an algebraic hard decision decoding algorithm is given for the preparata code and a hadamard transform soft decision decoding algorithm for the i kerdock code binary first and second order reed muller codes are also linear over z sub 4 but extended hamming codes of length n spl ges 32 and the golay code are not using z sub 4 linearity a new family of distance regular graphs are constructed on the cosets of the preparata code

on the associative analog of lie bialgebras abstract an infinitesimal bialgebra is at the same time an associative algebra and coalgebra in such a way that the comultiplication is a derivation this paper continues the basic study of these objects with emphasis on the connections with the theory of lie bialgebras it is shown that non degenerate antisymmetric solutions of the associative yang baxter equation are in one to one correspondence with non degenerate cyclic 2 cocycles the associative and classical yang baxter equations are compared it is studied when a solution to the first is also a solution to the second necessary and sufficient conditions for obtaining a lie bialgebra from an infinitesimal one are found in terms of a canonical map that behaves simultaneously as a commutator and a cocommutator the class of balanced infinitesimal bialgebras is introduced they have an associated lie bialgebra several well known lie bialgebras are shown to arise in this way the construction of drinfeld s double from earlier work by the author in press in contemp math amer math soc providence for arbitrary infinitesimal bialgebras is complemented with the construction of the balanced double for balanced ones this construction commutes with the passage from balanced infinitesimal bialgebras to lie bialgebras

immunity relativizations and nondeterminism it is known 8 13 that there is a recursive set a such that text np a contains a set that is text p a immune that is there is an infinite set l in text np a such that no infinite subset of l is in text p a the first result generalizes this fact to situations where the running times of the machines specifying the larger class bound the size of the sets of strings queried in the computation trees of the machines specifying the smaller class the second result is of a different type for relativized complexity classes specified by bounds on the number of oracle queries and the number of nondeterministic steps allowed in computations it is known 14 that one can describe a set a such that there is an infinite hierarchy of classes relative to a where each new class in the hierarchy is obtained by increasing the amount of nondeterminism here it is shown that the conditions allowing infinite hierarchies to exist also allow for each i geqq 0 the existence of a set

compatible o operators on bimodules over associative algebras abstract we recall the definition of the o operators on bimodules over associative algebras also called generalized rota baxter operators and we introduce the notions of compatible o operators and of o n structures we show that an o n structure gives rise to a hierarchy of o operators and that a solution of the strong maurer cartan equation on the associative twilled algebra associated to an o operator gives rise to a pair of o n structures which are naturally in duality we introduce the notions of pn n and p structures on associative algebras and study their relations

completeness approximation and density polynomial time approximations to exponential time computable problems exp are considered from the point of view of structure infinitely often speedable and almost everywhere complex problems are studied using the notions of polynomial time productivity and immunity in particular the existence of a polynomial time immune set which is not polynomial time approximable at all but which is polynomial time tt complete in exp is proven the relationship between completeness and approximability is also studied it is shown that being polynomial time m complete in exp does not provide any control of the probability of erroneous approximations

the density and complexity of polynomial cores for intractable sets let a be a recursive problem not in p lynch has shown that a then has an infinite recursive polynomial complexity core this is a collection c of instances of a such that every algorithm deciding a needs more than polynomial time almost everywhere on c we investigate the complexity of recognizing the instances in such a core and show that every recursive problem a not in p has an infinite core recognizable in subexponential time we further study how dense the core sets for a can be under various assumptions about the structure of a our main results in this direction are that if p np then np complete problems have polynomially nonsparse cores recognizable in subexponential time and that exptime complete problems have cores of exponential density recognizable in exponential time

the existence and density of generalized complexity cores if c is a class of sets and a is not in c then an infinite set h is a proper hard core for a with respect to c if h a and for every c e c such that c a c c h is finite it is shown that if c is a countable class of sets of strings that is closed under finite union and finite variation then every infinite set not in c has a proper hard core with respect to c in addition the density of such generalized complexity cores is studied

completeness and weak completeness under polynomial size circuits this paper investigates the distribution and nonuniform complexity of problems that are complete or weakly complete for espace under nonuniform many one reductions that are computed by polynomial size circuits p poly many one reductions every weakly p poly many one complete problem is shown to have a dense exponential nonuniform complexity core an exponential lower bound on the space bounded kolmogorov complexities of weakly p poly turing complete problems is established more importantly the p poly many one complete problems are shown to be unusually simple elements of espace in the sense that they obey nontrivial upper bounds on nonuniform complexity size of nonuniform complexity cores and space bounded kolmogorov complexity that are violated by almost every element of espace more generally a small span theorem for p poly many one reducibility in espace is proven and used to show that every p poly many one degree including the complete degree has measure 0 in espace in contrast almost every element of espace is weakly p many one complete all upper and lower bounds are shown to be tight

on inefficient special cases of np complete problems abstract every intractable set a has a polynomial complexity core a set h such that for any p subset s of a or of s h is finite a complexity core h of a is proper if h a it is shown here that if p np then every currently known i e either invertibly paddable or k creative np complete set a and its complement have proper polynomial complexity cores that are nonsparse and are accepted by deterministic machines in time 2 cn for some constant c turning to the intractable class d ext c 0 d time 2 cn it is shown that every set that is p m complete for d ext has an infinite proper polynomial complexity core that is nonsparse and recursive

optimal approximations and polynomially levelable sets a set a not in p is polynomially levelable if any algorithm for a has speedup to a polynomial infinitely often in a precisely if given any algorithm m for a and polynomial p it is possible to find another algorithm m for a and polynomial p such that m runs in time p x on infinitely many inputs x in a on which the running time of m exceeds p x intuitively this condition states that among polynomial time computable approximations to a there is no optimal one or one giving correct answers on a maximally large subset of a it appears that most naturally occurring intractable sets are polynomially levelable we prove this for sets not in p that are either paddable self reducible or complete for a deterministic time class we also discuss levelability preserving reductions and give a simple reducibility characterization of nonlevelable sets

some consequences of non uniform conditions on uniform classes we obtain some results of the form if certain complexity classes satisfy a non uniform condition then some unlikely consequences follow more precisely 1 1 if the non uniform polynomial time hierarchy collapses at level i 0 i e ipoly ipoly then the meyer stockmeyer hierarchy collapses at level i 2 i e i 2 i 2 this strengthens a generalization of a result of karp and lipton 1980 2 2 if co np is conjunctively reducible to a sparse set then p np this generalizes a theorem of fortune 1979 3 3 if np is conjunctively and disjunctively reducible to a sparse np complete set then p np this is a partial generalization of a result of mahaney 1980 conjuctive and disjunctive reducibility were introduced by ladner lynch and selman 1975 4 4 if co np is reducible to a sparse set then np co np reducibility was introduced 5 by adleman and manders 1977

deformations of associative rota baxter operators abstract rota baxter operators and more generally o operators on associative algebras are important in probability combinatorics associative yang baxter equation and splitting of algebras using a method of uchino we construct an explicit graded lie algebra whose maurer cartan elements are given by o operators this allows us to construct cohomology for an o operator this cohomology can also be seen as the hochschild cohomology of a certain algebra with coefficients in a suitable bimodule next we study linear and formal deformations of an o operator which are governed by the above defined cohomology we introduce nijenhuis elements associated with an o operator which give rise to trivial deformations as applications we conclude deformations of weight zero rota baxter operators associative r matrices and averaging operators