prinjective modules reflection functors quadratic forms and auslander reiten sequences let a b be artinian rings and let a m b be an a b bimodule which is a finitely generated left a module and a finitely generated right b module a right a m b prinjective module is a finitely genrated module x r x a x b x a a m b x b over the triangular matrix ring r a o a m b b such that x a is a projective a module x b is an injective b module and is a b homomorphism we study the category prin r b a of right a m b prinjective modules it is an additive krull schmidt subcategory of mod r closed under extensions

almost split sequences for complexes of fixed size abstract let a be an additive k category k a commutative artinian ring and n 1 we denote by c n a the category of complexes x x i d x i i z in a with x i 0 if i 1 n we see that c n a is endowed with a natural exact structure and its global dimension is at most n 1 in case a is a dualizing category we prove that c n a has almost split sequences in the sense of p draxler i reiten s o smalo o solberg exact categories and vector space categories with an appendix by b keller trans amer math soc 351 2 1999 647 682 or r bautista the category of morphisms between projective modules comm algebra 32 11 2004 4303 4331 if a is the category of finitely generated projective modules an artin algebra we prove that the ends of an almost split sequence are related by an auslander reiten translation functor which is defined in the most general category c n proj

non autonomous second order hamiltonian systems abstract we study the existence of periodic solutions for a second order non autonomous dynamical system containing variable kinetic energy terms our assumptions balance the interaction between the kinetic energy and the potential energy with neither one dominating the other we study sublinear problems and the existence of non constant solutions

existence of periodic solutions for a class of second order p laplacian systems abstract in this paper the existence of periodic solutions are obtained for a class of non autonomous second order p laplacian systems by the least action principle

classifying exact categories via wakamatsu tilting abstract using the morita type embedding we show that any exact category with enough projectives has a realization as a pre resolving subcategory of a module category when the exact category has enough injectives the image of the embedding can be described in terms of wakamatsu tilting semi dualizing subcategories if moreover the exact category has higher kernels then its image coincides with the category naturally associated with a cotilting subcategory up to summands we apply these results to the representation theory of artin algebras in particular we show that the ideal quotient of a module category by a functorially finite subcategory closed under submodules is a torsionfree class of some module category

homogeneity for open partitions of pairs of reals we prove a partition theorem for analytic sets of reals namely if a r is analytic and a 2 k 0 k 1 with k 0 relatively open then either there is a perfect 0 homogeneous subset or a is a countable union of 1 homogeneous subsets we also show that such a partition property for coanalytic sets is the same as that each uncountable coanalytic set contains a perfect subset a two person game for this partition property is also studied there are some applications of such partition properties

regular variation without limits karamata theory n h bingham et al 1987 8 ch 1 explores functions f for which the limit function g f x f x exists as x and for which g subject to mild regularity assumptions on f further karamata theory n h bingham et al 1987 8 ch 2 explores functions f for which the upper limit as x remains bounded here the usual regularity assumptions invoke boundedness of f on a baire non meagre measurable non null set with f baire measurable and the conclusions assert uniformity over compact sets implying upper bounds of the form f x f x k for all large x we give unifying combinatorial conditions which include the two classical cases deriving them from a combinatorial semigroup theorem we examine character degradation in the passage from f to f using some standard descriptive set theory and thus identify natural classes in which the theory may be established

on the n back and forth types of boolean algebras the objective of this paper is to uncover the structure of the back and forth equivalence classes at the nite levels for the class of boolean algebras as an application we obtain bounds on the computational complexity of determining the back and forth equivalence classes of a boolean algebra for nite levels this result has implications for characterizing the relatively intrinsically 0 relations of boolean algebras as existential formulas over a nite set of relations

coherent sequences and threads abstract the combinatorial principle says that there is a coherent sequence of length that cannot be threaded if then the related principle implies let 2 and x assume both and fail then there is an inner model n with a proper class of strong cardinals such that x n if in addition 2 0 and n then there is an inner model m n x with n woodin cardinals such that x m n x in particular by martin and steel projective determinacy holds as a corollary to this and results of todorcevic and velickovic the proper forcing axiom for posets of cardinality 2 0 implies projective determinacy

scott ranks of models of a theory abstract the scott rank of a countable structure is a measure coming from the proof of scott s isomorphism theorem of the complexity of that structure the scott spectrum of a theory by which we mean a sentence of l 1 is the set of scott ranks of countable models of that theory in z f c p d we give a descriptive set theoretic classification of the sets of ordinals which are the scott spectrum of a theory they are particular 1 1 classes of ordinals our investigation of scott spectra leads to the resolution in zfc of a number of open problems about scott ranks we answer a question of montalban by showing for each 1 that there is a 2 in theory with no models of scott rank less than we also answer a question of knight and calvert by showing that there are computable models of high scott rank which are not computably approximable by models of low scott rank finally we answer a question of sacks and marker by showing that 2 1 is the least ordinal such that if the models of a computable theory t have scott rank bounded below 1 then their scott ranks are bounded below

periodic second order superlinear hamiltonian systems abstract we consider periodic solutions for superlinear second order non autonomous dynamical systems including both kinetic and potential terms we study the existence of nontrivial and ground state solutions

phantom covering ideals in categories without enough projective morphisms abstract we give sufficient conditions to ensure that the ideal e of e phantom maps in a locally presentable exact category a e is a special pre covering ideal where e is an exact substructure of a e as a byproduct we infer the existence of various covering ideals in categories of sheaves which have a meaningful geometrical motivation in particular we deal with a zariski local notion of phantom maps in categories of sheaves we would like to point out that our approach is necessarily different from 18 as the categories involved in most of the examples we are interested in do not have enough projective morphisms

martin s maximum saturated ideals and non regular ultrafilters part ii we prove assuming the existence of a huge cardinal the consistency of fully non regular ultrafilters on the successor of any regular cardinal we also construct ultrafilters with ultraproducts of small cardinality part ii is logically independent of part i

the left distributive law and the freeness of an algebra of elementary embeddings the left distributive law for a single binary operation is the law a a it has been studied in universal algebra stein s kepka and nemec kn kepka kl k2 see also jezek et al jkn for a bibliography on the two sided distributive law and it has been studied more recently by set theorists because of its connection with elementary embeddings for e a limit ordinal let 6 be the collection of all j i i j an elementary embedding of v e into itself j not the identity then the existence of a 3 such that c is a large cardinal axiom see gaifman g and kanamori et ul krs for jo let k cr j the critical point of j and ti i j then 1 must equal sup ii w kunen ku martin ml woodin m2 martin and steel msl ms2 and woodin w in deep work showed that slight strengthenings of this axiom imply determinacy properties of the real line and subsequently that considerable weakenings suffice making the axiom unnecessary for those determinacy properties there is a natural operation on 8 write ul for u t in this and similar contexts below for jo gj j extends to a map j v i by defining for a s v j a u j a n i then j may or may not be an elementary embedding of vi e into itself but at least j is elementary from vi e a into v e ja in the special case that a as a set of ordered pairs is a k e we have that j k e j s let j k j k then the operation on j is nonassociative noncommutative and left distributive another operation on 4 is composition if k ie then k ie let z be the set of laws a b c u b c ucb c u bc u b c ub uc a b ub 0 a then j satisfies z and zimplies the left distributive law u bc u b c uboa c ub uc for jo c let 4 be the closure of j under let q the set of polynomials in j be the closure of jj under and 3 results in which 4 9j and their governing equations are involved appear in ml m2 l dougherty

auslander gorenstein algebras and precluster tilting abstract we generalize the notions of n cluster tilting subcategories and selfinjective algebras into n precluster tilting subcategories and n selfinjective algebras where we show that a subcategory naturally associated to n precluster tilting subcategories has a higher auslander reiten theory furthermore we give a bijection between n precluster tilting subcategories and n minimal auslander gorenstein algebras which is a higher dimensional analog of auslander solberg correspondence 8 as well as a gorenstein analog of n auslander correspondence 22 the auslander reiten theory associated to an n precluster tilting subcategory is used to classify the n minimal auslander gorenstein algebras into four disjoint classes our method is based on relative homological algebra due to auslander solberg

almost split sequences in subcategories let r be a commutative artin ring and let i be an r algebra which is a finitely generated r module in 5 auslander and reiten introduced the notion of an almost split sequence in mod a the category of finitely generated modules over a and the existence and uniqueness of such short exact sequences were established there the theory of almost split sequences developed further and similar results for certain subcategories of mod i were proved by bautista and martinez 8 and roggenkamp 9 the main purpose of this paper is to develop a more general theory for subcategories of mod i having almost split sequences which has the previous examples as special cases notions closely related to that of almost split sequences are those of minimal left and right almost split morphisms these notions were introduced in 61 for mod i and further developed in 7 for subcategories of mod ii this paper is based upon 7 and we therefore use the same notations and conventions as used there we now give some definitions and then proceed to describe the content of the paper section by section let c be a subcategory of mod ii then a morphism g b c in c is said to be a right almost split morphism in c if i g is not a splittable epimorphism and ii whenever there is a nonsplittable epimorphism h c c in c there exists an h c b such that gh h dually a morphismfi a f b in c is said to be left almost split if i f is not a splittable monomorphism and ii whenever there is a nonsplittable monomorphism h a a there exists an h b a such that h f h we say that c has right almost split morphisms if for all indecomposable objects

boolean algebra approximations knight and stob proved that every low4 boolean algebra is 0 6 isomorphic to a computable one furthermore for n 1 2 3 4 every lown boolean algebra is 0 n 2 isomorphic to a computable one we show that this is not true for n 5 there is a low5 boolean algebra that is not 0 7 isomorphic to any computable boolean algebra it is worth remarking that because of the machinery developed the proof uses at most a 0 00 priority argument the technique used to construct this boolean algebra is new and might be useful in other applications such as to solve the lown boolean algebra problem either positively or negatively

supercompact cardinals sets of reals and weakly homogeneous trees it is shown that if there exists a supercompact cardinal then every set of reals which is an element of l r is the projection of a weakly homogeneous tree as a consequence of this theorem and recent work of martin and steel martin d a steel j r 1988 proc natl acad sci usa 85 6582 6586 it follows that if there is a supercompact cardinal every set of reals in l r is determined

periodic solutions of non autonomous second order systems we obtain an existence theorem of periodic solutions of non autonomous second order systems with classical theorems of variational calculus

the maximality of the core model our main results are 1 every countably certified extender that coheres with the core model k is on the extender sequence of k 2 k computes successors of weakly compact cardinals correctly 3 every model on the maximal 1 small construction is an iterate of k 4 joint with w j mitchell k is universal for mice of height whenever 2 5 if there is a such that is either a singular countably closed cardinal or a weakly compact cardinal and fails then there are inner models with woodin cardinals and 6 an erdos cardinal suffices to develop the basic theory of k

stratifying endomorphism algebras using exact categories abstract this paper constructs enlargements of hecke algebras over z t t 1 to certain standardly stratified algebras the latter are obtained as endomorphism algebras of modules with dual left cell module filtrations in the sense of kazhdan lusztig a novel feature of the proofs is the use of suitably chosen exact categories to avoid difficult ext 1 vanishing conditions

ranked structures and arithmetic transfinite recursion atr0 is the natural subsystem of second order arithmetic in which one can develop a decent theory of ordinals sim99 we investigate classes of structures which are in a sense the well founded part of a larger sim pler class for example superatomic boolean algebras within the class of all boolean algebras the other classes we study are well founded trees reduced abelian p groups and countable compact topological spaces using com putable reductions between these classes we show that arithmetic transfinite recursion is the natural system for working with them natural statements such as comparability of structures in the class are equivalent to atr0 the reductions themselves are also objects of interest

periodic solutions of a class of non autonomous second order systems some existence theorems are obtained by the least action principle for periodic solutions of nonautonomous second order systems with a potential which is the sum of a subconvex function and a subquadratic function

on the c0 extension property for compact lines abstract we present a characterization of the continuous increasing surjections k l between compact lines k and l for which the corresponding subalgebra c l has the c 0 extension property in c k a natural question arising in connection with this characterization is shown to be independent of the axioms of zfc

closed subbifunctors of the extension functor abstract given a subbifunctor f of ext 1 one can ask if one can generalize the construction of the derived category to obtain a relative derived category where one localizes with respect to f acyclic sequences we show that this is possible if and only if f is closed we also show that for artin algebras the closed subbifunctors correspond to serre subcategories of a category of finitely presented functors that vanish on projectives and we use this to find new examples of closed subbifunctors using relatively derived categories we give a relative version of happel s result on derived equivalences induced by tilting and we show in an example how this can be used to find ordinary derived equivalences

a computability theoretic equivalent to vaught s conjecture we prove that for every theory t which is given by an l 1 sentence t has less than 2 0 many countable models if and only if we have that for every x 2 on a cone of turing degrees every x hyperarithmetic model of t has an x computable copy we also find a concrete description relative to some oracle of the turing degree spectra of all the models of a counterexample to vaught s conjecture

some existence results on periodic solutions of ordinary p laplacian systems some existence theorems are obtained for periodic solutions of ordinary p laplacian systems by the minimax methods in critical point theory

periodic solutions of non autonomous second order hamiltonian systems the purpose of this paper is to study the existence of periodic solutions for a class of non autonomous second order hamiltonian system some new existence theorems are obtained by the least action principle

on faithfully balanced modules f cotilting and f auslander algebras abstract the aim of this paper is to give a unification of the theory of co tilting modules and higher auslander correspondence by using faithfully balanced modules i e faithful modules with the double centralizer property in the language of relative homological algebra following auslander solberg a relative version of faithfully balancedness is given which will be called 1 f faithful the relative versions of higher auslander correspondence and brenner butler theorem are also developed

ideal approximation theory abstract let a e be an exact category and f ext a subfunctor a morphism in a is an f phantom if the pullback of an e conflation along is a conflation in f if the exact category a e has enough injective objects and projective morphisms it is proved that an ideal i of a is special precovering if and only if there is a subfunctor f ext with enough injective morphisms such that i is the ideal of f phantom morphisms a crucial step in the proof is a generalization of salce s lemma for ideal cotorsion pairs if i is a special precovering ideal then the ideal cotorsion pair i i generated by i in a e is complete this theorem is used to verify 1 that the ideal cotorsion pair cogenerated by the pure injective modules of r mod is complete 2 that the ideal cotorsion pair cogenerated by the contractible complexes in the category of complexes ch r mod is complete and using auslander and reiten s theory of almost split sequences 3 that the ideal cotorsion pair cogenerated by the jacobson radical jac mod of the category mod of finitely generated representations of an artin algebra is complete

periodic solutions of non autonomous second order systems with γ quasisubadditive potential abstract the existence of periodic solutions to non autonomous second order systems with quasisubadditive potential is proved with classical theorems of variational calculus