on the weak reflection principle the weak reflection principle for 2 or wrp 2 is the statement that every stationary subset of p 1 2 reflects to an uncountable ordinal in 2 the reflection principle for 2 or rp 2 is the statement that every stationary subset of p 1 2 reflects to an ordinal in 2 with cofinality 1 let be a supercompact cardinal and assume 2 then there exists a forcing poset p which collapses to become 2 and p wrp 2 rp 2 in this paper we will be concerned with reflection of stationary subsets of p 1 2 recall that for an uncountable ordinal a set s p 1 x x 1 is stationary if for any function f there is a set b in s which is closed under f let s p 1 2 be a stationary set we say that s reflects to where is an uncountable ordinal in 2 if s p 1 is stationary in p 1 the weak reflection principle for 2 or wrp 2 is the statement that for every stationary set s p 1 2 there is an uncountable ordinal in 2 such that s reflects to this principle is a special case of the stronger weak reflection principle introduced in 1 the statement wrp 2 has a number of interesting combinatorial consequences including 2 2 4 6 2 7 and every stationary subset of 2 cof reflects to an ordinal in 2 with cofinality 1 a related principle is the reflection principle for 2 or rp 2 which asserts that every stationary subset of p 1 2 reflects to an ordinal in 2 with cofinality 1 the standard models for obtaining wrp 2 for example by levy collapsing a large cardinal to become 2 all satisfy rp 2 also it tends to be easier to work with rp 2 than with wrp 2 thus it is a natural question and one which has been open for some time whether wrp 2 implies rp 2 a standard argument shows that if a holds for every stationary set a 2 cof then for any stationary set s p 1 2 there is a stationary set t s which does not reflect to any uncountable ordinal in 2 with cofinality so assuming the existence of such diamonds wrp 2 implies rp 2 classically the existence of such diamonds were known to follow from gch more recently shelah 5 has proven they are a consequence of 21 2 thus assuming 21 2 wrp 2 implies rp 2 originally this result was proven in 2 by a different argument the weak reflection principle for 2 is equiconsistent with a weakly compact cardinal however sakai 3 has shown that a kind of local reflection is consistent from zfc specifically assuming gch and 1 there is a generic extension in which there exists a stationary set s p 1 2 such that every stationary subset of s reflects to an uncountable ordinal in 2 with cofinality date february 2010 2000 mathematics subject classification primary 03e35 secondary 03e05

biorthogonal systems and quotient spaces via baire category methods we show that every banach space x of density smaller that the baire category number open image in new window admits a quotient with a long schauder basis that can be taken of length 1 if x is not separable so assuming that the baire category number open image in new window does not take its minimal possible value a banach space x is separable if and only if all biorthogonal systems of x are countable

cosmicity of cometrizable spaces a space x is cometrizable if x has a coarser metric topology such that each point of x has a neighborhood base of metric closed sets most examples in the literature of spaces obtained by modifying the topology of the plane or some other metric space are cometrizable assuming the proper forcing axiom pfa we show that the following statements are equivalent for a cometrizable space x a x is the continuous image of a separable metric space b xo is hereditarily separable and hereditarily lindelof c x2 has no uncountable discrete subspaces d x is a lindelof semimetric space e x has the pointed ccc this result is a corollary to our main result which states that under pfa if x is a cometrizable space with no uncountable discrete subspaces then either x is the continuous image of a separable metric space or x contains a copy of an uncountable subspace of the sorgenfrey line

localization transition of stiff directed lines in random media dimensions which is strongly supported by identical free energy distributions this shows that pair interactionsin the replicated hamiltonian determine the nature of directed line localization transitions with consequences forthe critical behavior of the kardar parisi zhang kpz equation furthermore we quantify how the persistencelength of the stiff directed line is reduced by disorder doi 10 1103 physreve 86 060102 pacs number s 05

nonperturbative renormalization group for the stationary kardar parisi zhang equation scaling functions and amplitude ratios in 1 1 2 1 and 3 1 dimensions we investigate the strong coupling regime of the stationary kardar parisi zhang equation for interfaces growing on a substrate of dimension d 1 2 and 3 using a nonperturbative renormalization group nprg approach we compute critical exponents correlation and response functions extract the related scaling functions and calculate universal amplitude ratios we work with a simplified implementation of the second order in the response field approximation proposed in a previous work phys rev e 84 061150 2011 and phys rev e 86 019904 e 2012 which greatly simplifies the frequency sector of the nprg flow equations while keeping a nontrivial frequency dependence for the two point functions the one dimensional scaling function obtained within this approach compares very accurately with the scaling function obtained from the full second order nprg equations and with the exact scaling function furthermore the approach is easily applicable to higher dimensions and we provide scaling functions and amplitude ratios in d 2 and d 3 we argue that our ansatz is reliable up to d symbol see text 3 5

bounded sets in topological vector spaces a subset b of a tvs topological vector space e is called bounded if for every neighborhood u of the identity there is an g 0 such that eb c u the behavior of the family e of all bounded subsets of e has a strong influence on many properties of e for example if e is an f space a metrizable locally convex complete tvs then determines how close e is to being a banach space see k6 sect 29 also the bounded sets in a normed space have some very special properties for example if e is a normed space then e contains a discrete bounded subset of cardinality equal to the density of e and every bounded subset of the completion of e is contained in the closure of a bounded subset of e in d j dieudonn6 asked whether these properties hold in the more general context of a metrizable locally convex tvs consider the following statements a i f e is a metrizable locally convex tvs and every bounded subset of e is separable then e is separable a if e is an f space and every bounded subset of e is separable then e is separable b i f e is a metrizable locally convex tvs and j is its completion then every bounded subset of e is contained in the closure of a bounded subset of e a grothendieck g showed that b holds for separable spaces in d dieudonn6 provided counterexamptes to a and b assuming the continuum hypothesis ch he asked whether ch can be eliminated from these

on a problem of rolewicz about banach spaces that admit support sets we construct an example of a nonseparable banach space which does not admit a support set 2 it is a consistent and necessarily independent from the axioms of zfc example of a space c k of continuous functions on a compact hausdorff k with the supremum norm the construction depends on a construction of a boolean algebra with some combinatorial properties the space is also hereditarily lindelof in the weak topology but it doesn t have any nonseparable subspace nor any nonseparable quotient which is a c k space for k dispersed

upper critical dimension of the kardar parisi zhang equation numerical results for the directed polymer model in 1 4 dimensions in various types of disorder are presented the results are obtained for a system size that is considerably larger than considered previously for the extreme strong disorder case min max system associated with the directed percolation model the expected value of the meandering exponent 0 5 is clearly revealed with very weak finite size effects for the weak disorder case associated with the kardar parisi zhang equation finite size effects are stronger but the value of is clearly seen in the vicinity of 0 57 in systems with strong disorder it is expected that the system will cross over sharply from min max behavior at short chains to weak disorder behavior at long chains our numerical results agree with that expectation to complete the picture we obtain the energy fluctuation exponent for weak disorder and we find that the value of is in the vicinity of 0 14 thus the meandering exponent and the energy fluctuation exponent obey the strong coupling scaling relation 2 1 our results indicate that 1 4 is not the upper critical dimension in the weak disorder case and thus 4 1 does not seem to be the upper critical dimension for the kardar parisi zhang equation

on densely isomorphic normed spaces abstract in the first part of our note we prove that every weakly lindelof determined wld in particular every reflexive non separable banach x space contains two dense linear subspaces y and z that are not densely isomorphic this means that there are no further dense linear subspaces y 0 and z 0 of y and z which are linearly isomorphic our main result theorem b concerns the existence of biorthogonal systems in normed spaces in particular we prove under the continuum hypothesis ch that there exists a dense linear subspace of l 2 1 or more generally every wld space of density 1 which contains no uncountable biorthogonal system this result lies between two fundamental results concerning biorthogonal systems namely the construction of kunen under ch of a non separable banach space which contains no uncountable biorthogonal system and the construction of todorcevic under martin maximum of an uncountable biorthogonal system in every non separable banach space

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

normality in x2 for compact x in 1977 the second author announced the following consistent neg ative answer to a question of katetov assuming ma gh there is a compact nonmetric space x such that x1 is hereditarily normal we give the details of this example and construct another example assuming ch we show that both examples can be constructed so that x2 a is perfectly normal we also construct in zfc a compact nonperfectly normal x such that x2 a is nor mal in his classical paper k katetov showed that if x and y are infinite compact spaces and x x y is hereditarily normal then x and y are per fectly normal by sneider s theorem that a compact space with a gvdiagonal is metrizable s katetov concludes that if x is compact and x3 is heredi tarily normal then x is metrizable he asked if the same conclusion could be obtained assuming only that x2 is hereditarily normal in 1977 the second au thor obtained a counterexample assuming martin s axiom plus the negation of the continuum hypothesis ma ch this result was announced in ny and an outline of the proof appeared there although with many details omitted a complete proof appears in this paper for the first time we also construct a necessarily different as will be seen counterexample assuming ch since any counterexample must be perfectly normal it is probably not sur prising that our examples are related to alexandrov s double arrow space d 0 1 x 0 1 with the lexicographic order topology for it is in some sense the only known example in zfc of a compact perfectly normal nonmetrizable space see gi for a discussion of this the double arrow space has also been called the split interval because one can think of obtaining it by splitting each x 0 1 into two points x and x and putting the order topology on these points where x is declared to be less than x and otherwise the order is the natural one inherited by 0 1 now if a c 0 1 let d a be the same as above but with only the points of a split of course d a is just the quotient space of d obtained by identifying x with x for all x a if a is uncountable then d a is a compact perfectly normal nonmetrizable space

the graded witt ring and galois cohomology ii a primary problem in the theory of quadratic forms over a field f of characteristic different from two is to prove that the rings hq f and gwf are isomorphic here hqf h gal fq f z 2z where fq is the quadratic closure of f and gwf is the graded witt nrng associated to the fundamental ideal of even dimensional forms in the witt ring wf of f in this paper we assume we are given a field extension k of f such that wk is close to wf or hq k is close to hq f a method is developed to obtain information about these graded rings over f and its 2 extensions from information about the corresponding graded ring of k this relative theory extends and includes the previously developed absolute case where k fq applications are also given to show that hq f and gwf are isomorphic for a collection of fields anrsing naturally from the theory of abstract witt rings

on large indecomposable banach spaces abstract hereditarily indecomposable banach spaces may have density at most continuum plichko and yost 2000 19 argyros and tolias 2004 1 in this paper we show that this cannot be proved for indecomposable banach spaces we provide the first example of an indecomposable banach space of density 2 2 the space exists consistently is of the form c k and it has few operators in the sense that any bounded linear operator t c k c k satisfies t f g f s f for every f c k where g c k and s c k c k is weakly compact strictly singular

a solution to the l space problem received by the editors january 8 2005 2000 mathematics subject classification primary 54d20 54d65 03e02 03e75 secondary 54f15

smoke and mirrors combinatorial properties of small cardinals equiconsistent with huge cardinals abstract since the work of godel and cohen many questions in infinite combinatorics have been shown to be independent of the usual axioms for mathematics zermelo frankel set theory with the axiom of choice zfc attempts to strengthen the axioms to settle these problems have converged on a system of principles collectively known as large cardinal axioms these principles are linearly ordered in terms of consistency strength as far as is currently known all natural independent combinatorial statements are equiconsistent with some large cardinal axiom the standard techniques for showing this use forcing in one direction and inner model theory in the other direction the conspicuous open problems that remain are suspected to involve combinatorial principles much stronger than the large cardinals for which there is a current fine structural inner model theory for the main results in this paper show that many standard constructions give objects with combinatorial properties that are in turn strong enough to show the existence of models with large cardinals are larger than any cardinal for which there is a standard inner model theory

mazur intersection property for asplund spaces abstract the main result of the present note states that it is consistent with the zfc axioms of set theory relying on martin s maximum mm axiom that every asplund space of density character 1 has a renorming with the mazur intersection property combined with the previous result of jimenez and moreno based upon the work of kunen under the continuum hypothesis we obtain that the mip renormability of asplund spaces of density 1 is undecidable in zfc

unconditional basic sequences in spaces of large density abstract we study the problem of the existence of unconditional basic sequences in banach spaces of high density we show in particular the relative consistency of the statement that every banach space of density contains an unconditional basic sequence

a partition relation for successors of large cardinals we address partition problems of erdos and hajnal by showing that kappa to kappa 2 1 alpha for all alpha kappa if kappa kappa kappa and kappa carries a kappa dense ideal if kappa is measurable we show that kappa to alpha 2 n for n omega alpha omega where omega is a very large ordinal less than kappa that is closed under all primitive recursive ordinal operations

directed sets and cofinal types we show that 1 s 1 x x xl and l are the only cofinal types of directed sets of size sl but that there exist many cofinal types of directed sets of size continuum a partially ordered set d is directed if every two elements of d have an upper bound in d in this note we consider some basic problems concerning directed sets which have their origin in the theory of moore smith convergence in topology 12 3 19 9 one such problem is to determine all essential kind of directed sets needed for defining the closure operator in a given class of spaces 3 p 47 concerning this problem the following important notion was introduced by j tukey 19 two directed sets d and e are cofinally similar if there is a partially ordered set c in which both can be embedded as cofinal subsets he showed that this is an equivalence relation and that d and e are cofinally similar iff there is a convergent map from d into e and also a convergent map from e into d the equivalence classes of this relation are called cofinal types this concept has been extensively studied since then by various authors 4 13 7 8 already from the first introduction of this concept it has been known that 1 1 x x 1 and x1 represent different cofinal types of directed sets of size btl but no more than five such types were known the main result of this paper shows that 1 1 x x 1 and x1 are the only cofinal types of convergence in spaces of character 81 which can be constructed without additional set theoretic assumptions on the other hand we shall construct many different cofinal types of directed sets of size continuum this gives a solution to problem 1 of j isbell 7 the paper also contains several results about the structure of the class of all cofinal types as well as a result about decomposing arbitrary partially ordered sets into directed sets the results of this note were proved in february march 1982 and presented to the asl in january 1983 1 a decomposition theorem in this section we show that an arbitrary partially ordered set can be decomposed into a number of its directed subsets depending on the sizes of its antichains this result is connected with an unpublished problem of f galvin concerning the dilworth decomposition theorem 6 and it generalizes a similar result of e milner and k prikry 11 the transitivity condition of a partial ordering is not used in our proof so we state our result so as to apply to an arbitrary received by the editors december 3 1984 1980 mathpmaties subjeet classification primary 03e05 03e35 secondary 06a10 18b35 54a15 c1985 american mathematical society 0002 9947 85 1 00 25 per page

two consequences of determinacy consistent with choice we begin with a ground model satisfying zf ad acr and from it construct a generic extension satisfying zfc 6 c2 the nonstationary ideal on coi is co2 saturated 0 introduction the axiom of determinacy henceforth ad implies many interesting propositions about small cardinal numbers for example c and o2 it is natural to ask whether these propositions are true or more realistically can be true in the full universe v of sets our working hypothesis is that in v the axiom of choice is true and all definable games are determined so the question becomes whether these propositions are consistent with zfc together with definable determinacy we shall be vague about definable when we can afford to in this paper we shall consider two such consequences of ad the first being that 2i w2 and the second that the nonstationary ideal on xl is o2 saturated we shall show that the theory consisting of zfc adl r together with these two propositions is consistent adl r is the assertion that all games in l r a large class of definable games are deterniined in consonance with our working hypothesis we would like to prove our result assuming only the consistency of zfc adl r or equivalently that of zf ad dc cf 1 at present we cannot do this we instead assume the consistency of zf ad acr where acr is the axiom of choice for families indexed by the reals consistency wise zf ad acr is one of the strongest theories known to man we show in 1 that it proves the consistency of zf ad dc the only upper bound we know on the consistency strength of zf ad acr is that of zf adr 0 is regular we offer some partial justification for our use of such a strong hypothesis at the end of the paper our consistency proof is by forcing we start with a ground model m satisfying zf ac acr by a two step iteration we construct a generic extension m g onto so that 3h h 2 rm the forcing will add no new reals and preserve co2 i e rm rm g and m 1m g we then pass to the inner model n l rm h and show it has the desired properties the paper is organized as follows in 1 we give some preliminaries and background information in particular we set forth those consequences of ad whose received by the editors february 18 1980 1980 mathematics subject classification primary 03e35 secondary 03e60 1 982 american mathematical society 0002 9947 8 1 001000 05 75

generic banach spaces and generic simplexes abstract we give a systematic study of certain class of generic banach spaces we show that they distinguish between an array of different properties related to smoothness of equivalent norms such as for example the mazur intersection property or the existence of convex sets supported by all of their points we also examine the dual constructions of generic choquet simplexes with extra requirements such as for example those of poulsen and bauer asking that the set of extremal points is dense or closed respectively

forcing consequences of pfa together with the continuum large we develop a new method for building forcing iterations with symmetric systems of structures as side conditions using our method we prove that the forcing axiom for the class of all the finitely proper posets of small size is compatible with the continuum being larger than the second uncountable cardinal in particular this answers a question of justin moore by showing that does not follow from this arithmetical assumption

partition relations for successor cardinals this paper investigates the relations 22 and its variants for uncountable cardinals first of all the extensive literature in this area is reviewed then some possibilities afforded by large cardinal hypotheses are derived for example if is measurable then 1 22 for every finally the limitations imposed on provability in zfc by l and by relative consistency via forcing are considered primarily the consistency of if is not weakly compact then 2

growth exponents of the etching model in high dimensions in this work we generalize the etching model mello et al 2001 phys rev e 63 041113 t o d 1 dimensions the dynamic exponents of this model are compatible with those of the kardar parisi zhang universality class we investigate the roughness dynamics with surfaces up to d 6 we show that the data from all substrate lengths and for all dimensions can be collapsed into one common curve we determine the dynamic exponents as a function of the dimension moreover our results suggest that d 4 is not an upper critical dimension for the etching model and that it fulfills the galilean invariance

thin very tall compact scattered spaces which are hereditarily separable we strengthen the property of a function f 2 2 2 considered by baumgartner and shelah this allows us to consider new types of amalgamations in the forcing used by rabus juhasz and soukup to construct thin very tall compact scattered spaces we consistently obtain spaces k as above where k n is hereditarily separable for each n n this serves as a counterexample concerning cardinal functions on compact spaces as well as having some applications in banach spaces the banach space c k is an asplund space of density 2 which has no frechet smooth renorming nor an uncountable biorthogonal system

extracting long basic sequences from systems of dispersed vectors abstract we study banach spaces satisfying some geometric or structural properties involving tightness of transfinite sequences of nested linear subspaces these properties are much weaker than wcg and closely related to corson s property c given a transfinite sequence of normalized vectors which is dispersed or null in some sense we extract a subsequence which is a biorthogonal sequence or even a weakly null monotone basic sequence depending on the setting the separable complementation property is established for spaces with an m basis under rather weak geometric properties we also consider an analogy of the baire category theorem for the lattice of closed linear subspaces

irredundant sets in boolean algebras it is shown that every uncountable boolean algebra a contains an uncountable subset i such that no a of i is in the subalgebra generated by i a using an additional axiom of set theory it is also shown that a use of some such axiom is necessary

i o2 can be the nonstationary ideal on cof o1 we answer a question of shelah by showing that it is consistent that every member of i 2 n cof 1 is nonstationary if and only if it is consistent that there is a mahlo cardinal k

perfect set properties inl r u abstract we study perfect set properties in the modell r u whenl r is elementarily equivalent to a solovay model anduis a selective ultrafilter on the integers generic overl r

multisurface coding simulations of the restricted solid on solid model in four dimensions we study the restricted solid on solid model for surface growth in spatial dimension d 4 by means of a multisurface coding technique that allows us to analyze samples of size up to 256 4 in the steady state regime for such large systems we are able to achieve a controlled asymptotic regime where the typical scale of the fluctuations are larger than the lattice spacing used in the simulations a careful finite size scaling analysis of the critical exponents clearly indicate that d 4 is not the upper critical dimension of the model

a kpz cocktail shaken not stirred the stochastic partial differential equation proposed nearly three decades ago by kardar parisi and zhang kpz continues to inspire intrigue and confound its many admirers here we i pay debts to heroic predecessors ii highlight additional experimentally relevant aspects of the recently solved 1 1 kpz problem iii use an expanding substrates formalism to gain access to the 3d radial kpz equation and lastly iv examining extremal paths on disordered hierarchical lattices set our gaze upon the fate of d infty kpz clearly there remains ample unexplored territory within the realm of kpz and for the hearty much work to be done especially in higher dimensions where numerical and renormalization group methods are providing a deeper understanding of this iconic equation

banach spaces from a construction scheme abstract we construct a banach space x e with an uncountable e biorthogonal system but no uncountable biorthogonal system for e 1 e 1 in particular the space has no uncountable biorthogonal system we also construct a banach space x k with an uncountable k basic sequence but no uncountable k basic sequence for 1 k k a common feature of these examples is that they are both constructed by recursive amalgamations using a single construction scheme

forcing axioms and stationary sets is stationary in general if x is any class of posets the forcing axioms xfa and xfa are obtained by replacing semi proper ly by 9 e x in the above definitions sometimes we denote these axioms by fa x and fa x thus for example fa ccc is the familiar martin s axiom man the notion of semi proper as an extension of a previous notion of proper was defined and extensively studied by shelah in shl j these forcing notions generalize ccc and countably closed forcing and can be iterated without collapsing n spfa is implicit in shl as an obvious strengthening of the proper forcing axiom pfa previously formulated and proved consistent by baumgartner see bal del pfa had settled many combinatorial questions on the uncountable and spfa was not

unconditional families in banach spaces it is shown that for every separable banach space x with non separable dual the space x contains an unconditional family of size x the proof is based on ramsey theory for trees and finite products of perfect sets of reals among its consequences it is proved that every dual banach space has a separable quotient