what is good mathematics i good mathematical problem solving e g a major breakthrough on an important mathematical problem ii good mathematical technique e g a masterful use of existing methods or the development of new tools iii good mathematical theory e g a conceptual framework or choice of notation which systematically unifies and generalises an existing body of results iv good mathematical insight e g a major conceptual simplification or the realisation of a unifying principle heuristic analogy or theme v good mathematical discovery e g the revelation of an unexpected and intriguing new mathematical phenomenon connection or counterexample vi good mathematical application e g to important problems in physics engineering computer science statistics etc or from one field of mathematics to another vii good mathematical exposition e g a detailed and informative survey on a timely mathematical topic or a clear and well motivated argument viii good mathematical pedagogy e g a lecture or writing style which enables others to learn and do mathematics more effectively ix good mathematical vision e g a long range and fruitful program or set of conjectures x good mathematical taste e g a research goal which is inherently interesting and impacts important topics themes or questions xi good mathematical public relations e g an effective showcasing of a mathematical achievement to non mathematicians or from one field of mathematics to another xii good meta mathematics e g advances in the foundations philosophy history or practice of mathematics xiii rigorous mathematics with all details correctly and carefully given in full etc xiv beautiful mathematics e g the amazing identities of ramanujan results which are easy and pretty to state but not to prove xv elegant mathematics e g paul erd s concept of proofs from the book achieving a difficult result with a minimum of effort

isoperimetry and integrability of the sum of independent banach space valued random variables we develop a new method to study the tails of a sum of independent mean zero banach space valued random variables x i i leq n it relies on a new isoperimetric inequality for subsets of a product of probability spaces in particular we prove that for p geq 1 bigg sum i leq n x i bigg p leq frac kp 1 log p bigg bigg sum i leq n x i bigg 1 max i leq n x i p bigg where k is a universal constant other optimal inequalities for exponential moments are obtained

some new results in multiplicative and additive ramsey theory there are several notions of largeness that make sense in any semigroup and others such as the various kinds of density that make sense in sucien tly well behaved semigroups including n and n it is known that sets with positive multiplicative density must contain arbitrarily large geoarithmetic progressions that is sets of the form fr j a id i j 2 f0 1 kgg we establish some combined additive and multiplicative ramsey theoretic consequences of known algebraic results in the semigroups n and n derive some new algebraic results and derive consequences of them involving geoarithmetic progressions for example we show that in any nite partition of n there must be for each k sets of the form fb a id j i j 2 f0 1 kgg together with d the arithmetic progression fa id i 2 f0 1 kgg and the geometric progression fbd j j 2 f0 1 kgg in one cell of the partition

universal coalgebra a theory of systems in the semantics of programming finite data types such as finite lists have traditionally been modelled by initial algebras later final coalgebras were used in order to deal with infinite data types coalgebras which are the dual of algebras turned out to be suited moreover as models for certain types of automata and more generally for transition and dynamical systems an important property of initial algebras is that they satisfy the familiar principle of induction such a principle was missing for coalgebras until the work of aczel 1988 on a theory of non wellfounded sets in which he introduced a proof principle nowadays called coinduction it was formulated in terms of bisimulation a notion originally stemming from the world of concurrent programming languages milner 1980 park 1981 using the notion of coalgebra homomorphism the definition of bisimulation on coalgebras can be shown to be formally dual to that of congruence on algebras aczel and mendler 1989 thus the three basic notions of universal algebra algebra homomorphism of algebras and congruence turn out to correspond to coalgebra homomorphism of coalgebras and bisimulation respectively in this paper the latter are taken as the basic ingredients of a theory called universal coalgebra some standard results from universal algebra are reformulated using the afore mentioned correspondence and proved for a large class of coalgebras leading to a series of results on e g the lattices of subcoalgebras and bisimulations simple coalgebras and coinduction and a covariety theorem for coalgebras similar to birkhoff s variety theorem

coalgebraic constructions of canonical nondeterministic automata for each regular language l we describe a family of canonical nondeterministic acceptors nfas their construction follows a uniform recipe build the minimal dfa for l in a locally finite variety v and apply an equivalence between the category of finite v algebras and a suitable category of finite structured sets and relations by instantiating this to different varieties we recover three well studied canonical nfas v boolean algebras yields the atomaton of brzozowski and tamm v semilattices yields the jiromaton of denis lemay and terlutte and v z 2 vector spaces yields the minimal xor automaton of vuillemin and gama moreover we obtain a new canonical nfa called the distromaton by taking v distributive lattices each of these nfas is shown to be minimal relative to a suitable measure and we derive sufficient conditions for their state minimality our approach is coalgebraic exhibiting additional structure and universal properties of the canonical nfas

a non singular dynamical system without maximal ergodic inequality abstract it is shown that there is a non singular dynamical system for which the maximal ergodic inequality does not hold we further discuss the connection between a non singular dynamical systems and the pointwise convergence of the furstenberg ergodic averages

two applications of analytic functors we apply the theory of analytic functors to two topics related to theoretical computer science one is a mathematical foundation of certain syntactic well quasi orders and well orders appearing in graph theory the theory of term rewriting systems and proof theory the other is a new verification of the lagrange good inversion formula using several ideas appearing in semantics of lambda calculi especially the relation between categorical traces and fixpoint operators copyright 2002 elsevier science b v

on final coalgebras of continuous functors continuous endofunctors f of locally finitely presentable categories carry a natural metric on their final coalgebra whenever f 0 has an element this metric is proved to be a cauchy completion of the initial algebra of f this is illustrated on the poset of real numbers represented as a final coalgebra of an endofunctor of pos by pavlovic and pratt under additional assumptions on the locally finitely presentable category all finitary endofunctors are proved to have a final coalgebra constructed in steps of the natural iteration construction

the ergodic theoretical proof of szemeredi s theorem partial results were obtained previously by k f roth 1952 who established the existence of arithmetic progressions of length three in subsets of z of positive upper density and by e szemeredi 1969 who proved the existence of progressions of length four in 1976 furstenberg noticed that the statement of theorem i is equivalent to a statement about multiple recurrence of measure preserving transformations namely

a new proof of the density hales jewett theorem the hales jewett theorem asserts that for every r and every k there exists n such that every r colouring of the n dimensional grid 1 k contains a monochromatic combinatorial line this result is a generalization of van der waerden s theorem and it is one of the fundamental results of ramsey theory the theorem of van der waerden has a famous density version conjectured by erd s and turan in 1936 proved by szemeredi in 1975 and given a different proof by furstenberg in 1977 the hales jewett theorem has a density version as well proved by furstenberg and katznelson in 1991 by means of a significant extension of the ergodic techniques that had been pioneered by furstenberg in his proof of szemeredi s theorem in this paper we give the first elementary proof of the theorem of furstenberg and katznelson and the first to provide a quantitative bound on how large n needs to be in particular we show that a subset of 1 2 3 of density contains a combinatorial line if n is at least as big as a tower of 2s of height o 1 our proof is surprisingly simple indeed it gives arguably the simplest known proof of szemeredi s theorem

coalgebraic semantics for timed processes we give a coalgebraic formulation of timed processes and their operational semantics we model time by a monoid called a time domain and we model processes by timed transition systems which amount to partial monoid actions of the time domain or equivalently coalgebras for an evolution comonad generated by the time domain all our examples of time domains satisfy a partial closure property yielding a distributive law of a monad for total monoid actions over the evolution comonad and hence a distributive law of the evolution comonad over a dual comonad for total monoid actions we show that the induced coalgebras are exactly timed transition systems with delay operators we then integrate our coalgebraic formulation of time qua timed transition systems into turi and plotkin s formulation of structural operational semantics in terms of distributive laws we combine timing with action via the more general study of the combination of two arbitrary sorts of behaviour whose operational semantics may interact we give a modular account of the operational semantics for a combination induced by that of each of its components our study necessitates the investigation of products of comonads in particular we characterise when a monad lifts to the category of coalgebras for a product comonad providing constructions with which one can readily calculate

the burnside algebra of a quasigroup abstract the paper extends the concept of a burnside algebra from finite groups to finite quasigroups based on an earlier specification of quasigroup permutation actions as members of a certain covariety of coalgebras for quasigroups the burnside algebra carries two distinct product operations denoted respectively as direct product and reduced tensor product the direct product arises from categorical products in the covariety while the reduced tensor product arises from bisimulations in the associative case the direct product and reduced tensor product agree with respect to the direct product the burnside algebra of a quasigroup is semisimple and its structure may be described in terms of marks exactly generalizing the associative case with respect to the reduced tensor product nilpotent elements may exist the problem of finding a primitive set of idempotents for the semisimple part of the reduced tensor product reduct of the burnside algebra is raised as a first step towards the solution of this problem the primitive reduced tensor idempotent multiple of the regular representation is identified in terms of the orbitals of the action of the right multiplication group there is a characterization of those quasigroups for which this multiple is the same as in the group case

equational and implicational classes of coalgebras if f set set is a functor which is bounded and preserves weak generalized pullbacks then a class of f coalgebras is a covariety i e closed under h homomorphic images s sub coalgebras and sums if and only if it can be de0ned by a set of coequations similarly quasi covarieties i e classes closed under h and can be characterized by implications of coequations these results are analogous to the theorems of birkho4 and of mal cev in classical universal algebra c 2001 elsevier science b v all rights reserved

distributivity of categories of coalgebras for any set endofunctor f the category setf of f coalgebras has preimages i e pullbacks along an injective map if f preserves preimages then setf is distributive and the converse holds whenever setf has finite products

a polynomial bound in freiman s theorem earlier bounds involved exponential dependence in in the second estimate ourargument combines i ruzsa s method which we improve in several places as well asy bilu s proof of freiman s theorem a fundamental result in the theory of set addition is freiman s theorem let a z be a nite set of integers with small sumset thus assume a a a 0 1 wherea a x y x y a 0 2 and denotes the cardinality the factor should be thought of as a possiblylarge constant then freiman s theorem states that a is contained in a d dimensionalprogression p whered d 0 3 and p a c 0 4 precise de nitions are given in section 1 although this statement is very intuitive there is no simple proof so far and it is one of the deep results in additive numbertheory g freiman s book fr on the subject is not easy to read which perhaps explainswhy in earlier years the result did not get its deserved publicity more recently two

on the structure of categories of coalgebras abstract consideration of categories of transition systems and related constructions leads to the study of categories of f coalgebras where f is an endofunctor of the category of sets or of some more general set like category it is fairly well known that if e is a topos and f e e preserves pullbacks and generates a cofree comonad then the category of f coalgebras is a topos unfortunately in most of the examples of interest in computer science the endofunctor f does not preserve pullbacks though it comes close to doing so in this paper we investigate what can be said about the category of coalgebras under various weakenings of the hypothesis that f preserves pullbacks it turns out that almost all the elementary properties of a topos except for effectiveness of equivalence relations are still inherited by the category of coalgebras and the latter can be recovered by embedding the category in its effective completion however we also show that in the particular cases of greatest interest the category of coalgebras is not itself a topos

the primes contain arbitrarily long arithmetic progressions we prove that there are arbitrarily long arithmetic progressions of primes there are three major ingredients the first is szemeredi s theorem which asserts that any subset of the integers of positive density contains progressions of arbitrary length the second which is the main new ingredient of this paper is a certain transference principle this allows us to deduce from szemeredi s theorem that any subset of a suficiently pseudorandom set or measure of positive relative density contains progressions of arbitrary length the third ingredient is a recent result of goldston and yildirim which we reproduce here using this one may place a large fraction of the primes inside a pseudorandom set of almost primes or more precisely a pseudorandom measure concentrated on almost primes with positive relative density

a formal abstract framework for modelling and testing complex software systems the contribution of this paper is twofold first it defines a unified framework for modelling abstract components as well as a formalization of integration rules to combine their behaviour this is based on a coalgebraic definition of components which is a categorical representation allowing the unification of a large family of formalisms for specifying state based systems second it studies compositional conformance testing i e checking whether an implementation made of correct interacting components combined with integration operators conforms to its specification

category theory for operational semantics we use the concept of a distributive law of a monad over a copointed endofunctor to define and develop a reformulation and mild generalisation of turi and plotkin s notion of an abstract operational rule we make our abstract definition and give a precise analysis of the relationship between it and turi and plotkin s definition following tuff and plotkin our definition suitably restricted agrees with the notion of a set of gsos rules allowing one to construct both an operational model and a canonical internally fully abstract denotational model going beyond turi and plotkin we construct what might be seen as large step operational semantics from small step operational semantics and we show how our definition allows one to combine distributive laws in particular accounting for the combination of operational semantics with congruences

recursively defined metric spaces without contraction in this paper we use the theory of accessible categories to find fixed points of endofunctors on the category of 1 bounded complete metric spaces and nonexpansive functions in contrast to previous approaches we do not assume that the endofunctors are locally contractive and our results do not depend on banach s fixed point theorem our approach is particularly suitable for constructing models of systems that feature quantitative data for instance using the kantorovich metric on probability measures we construct a quantitative model for probabilistic transition systems the metric in our model can reasonably be seen as measuring the behavioural distance between states of the system it depends exclusively on the transition probabilities and not on an arbitrary discount factor

on coalgebra based on classes the category class of classes and functions is proved to have a number of properties suitable for algebra and coalgebra every endofunctor is set based it has an initial algebra and a terminal coalgebra the categories of algebras and coalgebras are complete and cocomplete and every endofunctor generates a free completely iterative monad a description of a terminal coalgebra for the power set functor is provided

the category theoretic solution of recursive domain equations recursive specifications of domains plays a crucial role in denotational semantics as developed by scott and strachey and their followers the purpose of the present paper is to set up a categorical framework in which the known techniques for solving these equations find a natural place the idea is to follow the well known analogy between partial orders and categories generalizing from least fixed points of continuous functions over cpos to initial ones of continuous functors over omega categories to apply these general ideas we introduce wand s bf o categories where the morphism sets have a partial order structure and which include almost all the categories occurring in semantics the idea is to find solutions in a derived category of embeddings and we give order theoretic conditions which are easy to verify and which imply the needed categorical ones the main tool is a very general form of the limit colimit coincidence remarked by scott in the concluding section we outline how compatibilit

terminal coalgebras in well founded set theory abstract this paper shows that in order to obtain the theorem of aczel and mendler on the existence of terminal coalgebras for an endofunctor on the category of sets it is entirely unnecessary to delve into such exotica as non well founded set theory in addition we discuss the canonical map from the initial algebra for an endofunctor on sets to the terminal coalgebra and show that in many cases it embeds the former as a dense subset of the latter in a certain natural topology by way of example we calculate the terminal coalgebra for various simple endofunctors

trees and semantics abstract several types of trees are proposed in the literature to model the behaviour of nondeterministic processes the purpose of this paper is to clarify the distinction between the various types of trees and to discuss their use as semantic domains trees in the sense of milner s derivation trees and bisimulation equivalence classes of trees are considered in three different settings in the category of sets in the metric setting and in the partial order setting each type of tree is characterized by a domain equation special attention is given to the use of trees as models for processes which allow for unbounded nondeterminism as for instance milners ccs processes in this case one has to deal with infinitely branching trees which cause problems to give denotational semantics that are fully abstract w r t a transition system based operational semantics

multiplicity of holomorphic functions the main result of this paper concerns an analytic version of birkhoff s ergodic theorem it allows us to define asymptotic multiplicities given any algebraically stable rational map f mathbb p 2 to mathbb p 2 this is the key for our study of the singularities of the green current t associated to f we characterize the points where the lelong number of t is strictly positive

remarks on strong exponential integrability of vector valued random series and triangular arrays some optimal refinements of the results on strong exponential integrability of sums of independent zero mean uniformly bounded random vectors are given

processes as formal power series a coinductive approach to denotational semantics we characterize must testing equivalence on csp in terms of the unique homomorphism from the moore automaton of csp processes to the final moore automaton of partial formal power series over a certain semiring the final automaton is then turned into a csp algebra operators and fixpoints are defined respectively via behavioural differential equations and simulation relations this structure is then shown to be preserved by the final homomorphism as a result we obtain a fully abstract compositional model of csp phrased in purely set theoretical terms

polynomial extensions of van der waerden s and szemeredi s theorems an extension of the classical van der waerden and szemeredi the orems is proved for commuting operators whose exponents are polynomials as a consequence for example one obtains the following result let s zl be a set of positive upper banach density let p1 n pk n be polynomials with rational coefficients taking integer values on the integers and satisfying pi 0 0 i 1 k then for any v1 vk zl there exist an integer n and a vector u zl such that u pi n vi s for each i k department of mathematics ohio state university columbus ohio 43210 e mail address vitaly math ohio state edu department of mathematics technion haifa 23000 israel e mail address sashal techunix technion ac il current address department of mathematics stanford university stanford california 94305 e mail address leibman math stanford edu license or copyright restrictions may apply to redistribution see http www ams org journal terms of use