some downwards transfer properties for n2 construction d un modele de zermelo fraenkel axiome du choix zfc hypothese du continu generalise ou le cardinal aleph 2 possede des proprietes de transfert descendant

nonlinear wave analysis of geochemical injection for multicomponent two phase flow in porous media abstract we study nonlinear waves in two phase geochemical flow in one spatial dimension in porous media we assume that each chemical species may flow in one or both phases with concentrations obeying thermodynamical equilibrium we present a new methodology for reducing a number of equations applicable to injection problems for general systems of conservation laws in geochemistry this reduction is achieved by solving a nonlinear inverse problem nevertheless we are able to perform a complete and explicit characteristic analysis obtaining rarefaction and shock waves that are used to solve the representative riemann problems besides the main bifurcations structures appearing in the phase space are derived with explicit expressions we illustrate the methodology by means of an example with four equations

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

the riemann problem near a hyperbolic singularity ii this paper is interested in classifying the solutions of riemann problems for the 2 times 2 conservation laws that have homogeneous quadratic flux functions such flux functions approximate an arbitrary 2 times 2 system in a neighborhood of an isolated point where strict hyperbolicity fails here the solution for the symmetric systems in region iii of the four region classification of schaeffer and shearer is given the solution is based on the qualitative shape of the integral curves described by schaeffer and shearer and a numerical calculation of the hugoniot loci and their shock types

a bound on the total variation of the conserved quantities for solutions of a general resonant nonlinear balance law we introduce a new potential interaction functional and use it to define a new glimm type functional that bounds the total variation of the conserved quantities at time t 0 by the total variation at time t 0 in glimm approximate solutions of a general resonant nonlinear balance law

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

a global formalism for nonlinear waves in conservation laws we introduce a unifying framework for treating all of the fundamental waves occurring in general systems ofn conservation laws fundamental waves are represented as pairs of states statisfying the rankine hugoniot conditions after trivial solutions have been eliminated by means of a blow up procedure these pairs form an n 1 dimensional manifold the fundamental wave manifold there is a distinguishedn dimensional submanifold of containing a single one dimensonal foliation that represents the rarefaction curves for all families similarly there is a foliation of itself that represent shock curves we identify othern dimensional submanifolds of that are naturally interpreted as boundaries of regions of admissible shock waves these submanifolds also have one dimensional foliations which represent curves of composite waves this geometric framework promises to simplify greatly the study of the stability and bifurcation properties

c algebras of toeplitz type associated with algebraic number fields we associate with the ring r of algebraic integers in a number field a c algebra mathfrak t r it is an extension of the ring c algebra mathfrak a r studied previously by the first named author in collaboration with x li in contrast to mathfrak a r it is functorial under homomorphisms of rings it can also be defined using the left regular representation of the ax b semigroup r rtimes r times on ell 2 r rtimes r times the algebra mathfrak t r carries a natural one parameter automorphism group sigma t t in mathbb r we determine its kms structure the technical difficulties that we encounter are due to the presence of the class group in the case where r is not a principal ideal domain in that case for a fixed large inverse temperature the simplex of kms states splits over the class group the partition functions are partial dedekind zeta functions we prove a result characterizing the asymptotic behavior of quotients of such partial zeta functions which we then use to show uniqueness of the beta kms state for each inverse temperature beta in 1 2

a class of n nonlinear hyperbolic conservation laws the riemann problem for a class of nonlinear systems of first order hyperbolic conservation laws is studied the class consists of systems where the derivative of the flux function is a lower triangular matrix there are no assumptions on genuine nonlinearity and strict hyperbolicity existence and uniqueness are proved except in a set with measure zero in the phase space and a set with measure zero in the flux function space where there is a one parameter family of solutions travelling waves are used as an entropy condition and examples show that the lax or liu entropy conditions are not sufficient an example shows that the solution does not necessarily depend continuously on the data the model may be used to describe three phase and tracer flow and flow in a neighborhood of a heterogeneity in porous media

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

vanishing viscosity solutions for conservation laws with regulated flux abstract in this paper we introduce a concept of regulated function v t x of two variables which reduces to the classical definition when v is independent of t we then consider a scalar conservation law of the form u t f v t x u x 0 where f is smooth and v is a regulated function possibly discontinuous w r t both t and x by adding a small viscosity one obtains a well posed parabolic equation as the viscous term goes to zero the uniqueness of the vanishing viscosity limit is proved relying on comparison estimates for solutions to the corresponding hamilton jacobi equation as an application we obtain the existence and uniqueness of solutions for a class of 2 2 triangular systems of conservation laws with hyperbolic degeneracy

nuclearity of semigroup c algebras and the connection to amenability abstract we study c algebras associated with subsemigroups of groups for a large class of such semigroups including positive cones in quasi lattice ordered groups and left ore semigroups we describe the corresponding semigroup c algebras as c algebras of inverse semigroups groupoid c algebras and full corners in associated group crossed products these descriptions allow us to characterize nuclearity of semigroup c algebras in terms of faithfulness of left regular representations and amenability of group actions moreover we also determine when boundary quotients of semigroup c algebras are uct kirchberg algebras this leads to a unified approach to cuntz algebras and ring c algebras

analysis of a singular hyperbolic system of conservation laws abstract we solve the riemann and cauchy problems globally for a singular system of n hyperbolic conservation laws the system which arises in the study of oil reservoir simulation has only two wave speeds and these coincide on a surface of codimension one in state space the analysis uses the random choice method of glimm comm pure appl math 18 1965 697 715

semigroup crossed products and the toeplitz algebras of nonabelian groups abstract we consider the quasi lattice ordered groups g p recently introduced by nica we realise their universal toeplitz algebra as a crossed product b p p by a semigroup of endomorphisms and show that the toeplitz representation is faithful precisely when an amenability condition is satisfied we then show directly that many interesting free products are amenable the final result contains theorems of cuntz and dinh on the uniqueness of toeplitz cuntz algebras as special cases

ring c algebras we associate reduced and full c algebras to arbitrary rings and study the inner structure of these ring c algebras as a result we obtain conditions for them to be purely infinite and simple we also discuss several examples originally our motivation comes from algebraic number theory

universally meager sets and principles of generic continuity and selection in banach spaces abstract in this paper we use large cardinals to address some problems about generic continuity and generic selection that occur in the study of fragmentability and differentiability in the context of banach spaces

semigroup c algebras of ax b semigroups we study semigroup c algebras of ax b semigroups over integral domains the goal is to generalize several results about c algebras of ax b semigroups over rings of algebraic integers we prove results concerning k theory and structural properties like the ideal structure or pure infiniteness our methods allow us to treat ax b semigroups over a large class of integral domains containing all noetherian integrally closed domains and coordinate rings of affine varieties over infinite fields

topological applications of generic huge embeddings in the foreman laver model obtained by huge cardinal collapse for many p ni implies ofi there are a variety of set theoretic and topo logical applications in particular to paracompactness the key tools are generic huge embeddings and preservation via k centred forcing we also formulate potent axioms a la foreman which enable us to transfer from n to all car dinals one such axiom implies that all n collectionwise normal moore spaces are metrizable it also implies as does martin s maximum that a first count able generalized ordered space is hereditarily paracompact iff every subspace of size ki is paracompact assuming the consistency of a huge cardinal foreman and laver fl con struct a model of zfc gch in which as they put it n2 possesses some downwards transfer properties for example in their model every graph of size and chromatic number n2 has a vertex induced subgraph of size and chro matic number ki and groups of size n2 are the union of ni free subgroups if all their subgroups of size ni are free we shall establish in their model a variety of such transfer propositions in a topological context actually we think of taking properties holding at ni and transferring them to n2 the method of proof is to use an elementary embedding with strong closure properties de finable in an ni centred k2 closed forcing extension of the model to perform the transfer and then argue that the forcing could not have accomplished the transfer unless it already held it is natural to generalize this idea to trans fer from n to other cardinals it is not known at present how to accomplish this from the usual large cardinals so we proceed axiomatically in the style of fo3 our results impinge on a number of areas of set theoretic topology but the most interesting ones deal with paracompactness e g the one mentioned in the abstract the techniques however are widely applicable as can be seen not only from our results on trees and ideals but from those of foreman and laver quoted above the paper is organized as follows we state the basic features of the foreman laver model and use them to go from ni to n2 next we present a variety of counterexamples to show our results do not hold in zfc we then introduce axioms that enable us to transfer from ni to arbitrary cardinals we finish with some results proved from supercompact cardinals and then a list of problems

type iii1 equilibrium states of the toeplitz algebra of the affine semigroup over the natural numbers abstract we complete the analysis of kms states of the toeplitz algebra t n n of the affine semigroup over the natural numbers recently studied by raeburn and the first author by showing that for every inverse temperature in the critical interval 1 2 the unique kms state is of type iii1 we prove this by reducing the type classification from t n n to that of the symmetric part of the bost connes system with a shift in inverse temperature to carry out this reduction we first obtain a parametrization of the nica spectrum of n n in terms of an adelic space combining a characterization of traces on crossed products due to the second author with an analysis of the action of n n on the nica spectrum we can also recover all the kms states of t n n originally computed by raeburn and the first author our computation sheds light on why there is a free transitive circle action on the extremal kms states for 2 that does not ostensibly come from an action of t on the c algebra

on the consistency strength of the proper forcing axiom abstract in recent work the second author extended combinatorial principles due to jech and magidor that characterize certain large cardinal properties so that they can also hold true for small cardinals for inaccessible cardinals these modifications have no effect and the resulting principles still give the same characterization of large cardinals we prove that the proper forcing axiom pfa implies these principles hold for 2 using this we argue to show that any of the known methods for forcing models of pfa from a large cardinal assumption requires a strongly compact cardinal if one forces pfa using a proper forcing then we get the optimal result that a supercompact cardinal is necessary

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

riemann solutions without an intermediate constant state for a system of two conservation laws abstract we present a class of systems consisting of two conservation laws in one spatial dimension that share an intriguing property they admit structurally stable riemann solutions without the standard constant state this striking phenomenon emerges in sharp contrast to what is known for strictly hyperbolic systems of conservation laws in which the existence of constant states is necessary for the structural stability of riemann solutions we prove that together coincidence of characteristic speeds and a certain amount of genuine nonlinearity are sufficient to trigger the aforementioned phenomenon the proof revolves about the presence of a singular point in the coincidence set that organizes the construction of our riemann solutions

thin stationary sets and disjoint club sequences we describe two opposing combinatorial properties related to adding clubs to 2 the existence of a thin stationary subset of p 1 2 and the existence of a disjoint club sequence on 2 a special aronszajn tree on 2 implies there exists a thin stationary set if there exists a disjoint club sequence then there is no thin stationary set and moreover there is a fat stationary subset of 2 which cannot acquire a club subset by any forcing poset which preserves 1 and 2 we prove that the existence of a disjoint club sequence follows from martin s maximum and is equiconsistent with a mahlo cardinal

stability of conservation laws with discontinuous coefficients we prove l1 contractivity of weak solutions to a conservation law with a flux function that may depend discontinuously on the space variable furthermore we show that the l1 difference between solutions to conservation laws with different flux functions is bounded by the total variation with respect to the space variable of the difference between the flux functions

stable numerical methods for conservation laws with discontinuous flux function abstract we develop numerical methods for solving nonlinear equations of conservation laws with flux function that depends on discontinuous coefficients using a relaxation approximation the nonlinear equation is transformed to a semilinear diagonalizable problem with linear characteristic variables eulerian and lagrangian methods are used for the advection stage while an implicit explicit scheme solves the relaxation stage the main advantages of this approach are neither riemann problem solvers nor linear iterations are required during the solution process moreover the characteristic based relaxation method is unconditionally stable such that no cfl conditions are imposed on the selection of time steps numerical results are shown for models on traffic flows and two phase flows

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

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

the ideal structure of the hecke c algebra of bost and connes we compute explicitly the primitive ideal space of the bost connes hecke c algebra by embedding it as a full corner in a transformation group c algebra and applying a general theorem of williams this requires the computation of the quasi orbit space for the action of q on the space mathcal a f of finite adeles we then carry out a similar computation for the action of q on the space mathcal a mathcal a f times mathbb r of full adeles

convergence of the 2 2 godunov method for a general resonant nonlinear balance law we introduce a generalized solution of the riemann problem for a general resonant nonlinear balance law and we prove the convergence of the 2 x 2 godunov numerical method based on these solutions in particular we obtain generic conditions that guarantee a canonical structure for the elementary waves in the solution of the riemann problem and an interesting multiplicity of time asymptotic wave patterns is observed and characterized

the 2 adic ring c algebra of the integers and its representations abstract we study the 2 adic version of the ring c algebra of the integers first we work out the precise relation between the cuntz algebra o 2 and our 2 adic ring c algebra in terms of representations secondly we prove a 2 adic duality theorem identifying the crossed product arising from 2 adic affine transformations on the 2 adic numbers with the analogous crossed product algebra over the real numbers and finally as an outcome of this duality result we construct an explicit imprimitivity bimodule and prove that it transports one canonical representation into the other

the riemann problem near a hyperbolic singularity the classification of solutions of quadratic riemann problems 1 the purpose of this paper is to classify the solutions of riemann problems near a hyperbolic singularity in a nonlinear system of conservation laws hyperbolic singularities play the role in the theory of riemann problems that rest points play in the theory of ordinary differential equations indeed generically only a finite number of structures can appear in a neighborhood of such a singularity in this the first of three papers the program of classification is discussed in general and the simplest structure that occurs is characterized

on the cauchy problems for polymer flooding with gravitation abstract we study two systems of conservation laws for polymer flooding in secondary oil recovery one with gravitation force and one without for each model we prove global existence of weak solutions for the cauchy problems under rather general assumptions on the flux functions approximate solutions are constructed through a front tracking algorithm and compactness is achieved through the bound on suitably defined wave strengths as the main technical novelty we introduce some new nonlinear functionals that yield a uniform bound on the total variation of the flux function

forcing axioms and the continuum hypothesis woodin has demonstrated that in the presence of large cardinals there is a single model of zfc which is maximal for 2 sentences over the structure h 2 ns 1 in the sense that its h 2 ns 1 satisfies every 2 sentence for which h 2 ns 1 can be forced by set forcing in this paper we answer a question of woodin by showing that there are two 2 sentences over the structure h 2 1 which can each be forced to hold along with the continuum hypothesis but whose conjunction implies 2 aleph 0 2 aleph 1 in the process we establish that there are two preservation theorems for not introducing new real numbers by a countable support iterated forcing which cannot be subsumed into a single preservation theorem

phase transition on the toeplitz algebra of the affine semigroup over the natural numbers abstract we show that the group q q of orientation preserving affine transformations of the rational numbers is quasi lattice ordered by its subsemigroup n n the associated toeplitz c algebra t n n is universal for isometric representations which are covariant in the sense of nica we give a presentation of t n n in terms of generators and relations and use this to show that the c algebra q n recently introduced by cuntz is the boundary quotient of q q n n in the sense of crisp and laca the toeplitz algebra t n n carries a natural dynamics which induces the one considered by cuntz on the quotient q n and our main result is the computation of the kms equilibrium states of the dynamical system t n n r for all values of the inverse temperature for 1 2 there is a unique kms state and the kms 1 state factors through the quotient map onto q n giving the unique kms state discovered by cuntz at 2 there is a phase transition and for 2 the kms states are indexed by probability measures on the circle there is a further phase transition at where the kms states are indexed by the probability measures on the circle but the ground states are indexed by the states on the classical toeplitz algebra t n

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

a random sampling method for a family of temple class systems of conservation laws the aw rascle zhang traffic model a model of sedimentation and other applications lead to nonlinear 2 times 2 systems of conservation laws that are governed by a single scalar system velocity such systems are of the temple class since rarefaction wave curves and hugoniot curves coincide moreover one characteristic field is genuinely nonlinear almost everywhere and the other is linearly degenerate two well known problems associated with these systems are handled via a random sampling approach firstly godunov s and related methods produce spurious oscillations near contact discontinuities since the numerical solution invariably leaves the invariant region of the exact solution it is shown that alternating between averaging av and remap steps similar to the approach by chalons and goatin commun math sci 5 533 551 2007 generates numerical solutions that do satisfy an invariant region property if the remap step is made by random sampling rs then combined techniques due to glimm commun pure appl math 18 697 715 1965 leveque and temple trans am math soc 288 115 123 1985 prove that the resulting av rs scheme converges to a weak solution numerical examples illustrate that the new scheme is superior to godunov s method in accuracy and resolution secondly the vacuum state which may form even from positive initial data causes potential problems of non uniqueness and instability this is resolved by introducing an alternative riemann solution concept

on k theoretic invariants of semigroup c algebras attached to number fields abstract we show that semigroup c algebras attached to a x b semigroups over rings of integers determine number fields up to arithmetic equivalence under the assumption that the number fields have the same number of roots of unity for finite galois extensions this means that the semigroup c algebras are isomorphic if and only if the number fields are isomorphic

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

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

semigroup c algebras and amenability of semigroups we construct reduced and full semigroup c algebras for left cancellative semigroups our new construction covers particular cases already considered by a nica and also toeplitz algebras attached to rings of integers in number fields due to j cuntz moreover we show how left amenability of semigroups can be expressed in terms of these semigroup c algebras in analogy to the group case