the regular spaces with countably based models the regular spaces which may be realized as the set of maximal elements in an continuous dcpo are the polish spaces in addition we give a new and conceptually simple model for complete metric spaces these results enable us to prove that the probabilistic powerdomain of a countably based model of a metric space always contains a copy of the normalized borel measures in their weak topology and to establish the hierarchy for countably based models

classification problems in continuum theory we study several natural classes and relations occurring in continuum theory from the viewpoint of descriptive set theory and infinite combinatorics we provide useful characterizations for the relation of likeness among dendrites and show that it is a bqo with countably many equivalence classes for dendrites with finitely many branch points the homeomorphism and quasi homeomorphism classes coincide and the minimal quasi homeomorphism classes among dendrites with infinitely many branch points are identified in contrast we prove that the homeomorphism relation between dendrites is s infty universal it is shown that the classes of trees and graphs are both d 2 boldsymbol sigma 0 3 complete the class of dendrites is boldsymbol pi 0 3 complete and the class of all continua homeomorphic to a graph or dendrite with finitely many branch points is boldsymbol pi 0 3 complete we also show that if g is a nondegenerate finitely triangulable continuum then the class of g like continua is bolbsymbol pi 0 2 complete

rational recursion operators for integrable differential difference equations in this paper we introduce the concept of prehamiltonian pairs of difference operators demonstrate their connections with nijenhuis operators and give a criteria for the existence of weakly nonlocal inverse recursion operators for differential difference equations we begin with a rigorous setup of the problem in terms of the skew field of rational pseudo difference operators over a difference field with a zero characteristic subfield of constants and the principal ideal ring of matrix rational pseudo difference operators in particular we give a criteria for a rational operator to be weakly nonlocal a difference operator is called prehamiltonian if its image is a lie subalgebra with respect to the lie bracket on the difference field two prehamiltonian operators form a prehamiltonian pair if any linear combination of them is prehamiltonian then we show that a prehamiltonian pair naturally leads to a nijenhuis operator and a nijenhuis operator can be represented in terms of a prehamiltonian pair this provides a systematic method to check whether a rational operator is nijenhuis as an application we construct a prehamiltonian pair and thus a nijenhuis recursion operator for the differential difference equation recently discovered by adler and postnikov the nijenhuis operator obtained is not weakly nonlocal we prove that it generates an infinite hierarchy of local commuting symmetries we also illustrate our theory on the well known examples including the toda the ablowitz ladik and the kaup newell differential difference equations

a note on well posedness for camassa holm equation in this note we investigate the problem of well posedness for a shallow water equation with data having critical regularity our results are based on the use of besov spaces b2 rs which generalize the sobolev spaces hs with critical index s 3 2

polish group actions dichotomies and generalized elementary embeddings the results in this paper involve two different topics in the descriptive theory of polish group actions the book becker kechris 6 is an introduction to that theory our two topics and two collections of theorems are rather unrelated but the proofs for both topics are essentially the same locally compact polish groups i e second countable locally compact groups are the traditional objects of study in the field known as topological groups more recently there has been interest in nonlocally compact polish groups such as s the symmetric group on n topologized as a subspace of nn much of this paper is concerned with a proper subclass of the class of polish groups namely those polish groups which admit a complete left invariant metric we call these cli groups the class of cli groups includes all locally compact groups proposition 3 c 2 d but also much more for example all solvable groups are cli hjorth solecki see proposition 3 c 2 f and a closed subgroup g of s is cli iff g is closed in nn see example 3 c 3 a we study continuous actions by polish groups on polish spaces and the associated orbit equivalence relation for locally compact groups all of our results are known or trivial or both but some of our results can be viewed as generalizations of known theorems about locally compact groups to the larger class of cli groups one of the two topics considered in this paper is dichotomy theorems for equivalence relations these are theorems which assert that either the quotient space is small or else it contains a copy of a specific large set we prove some dichotomy theorems for the orbit space of actions by cli groups there are two types of dichotomies which we call the silver vaught dichotomy and the glimm effros dichotomy the silver vaught dichotomy asserts that either there are only countably many equivalence classes or else there is a perfect set of pairwise inequivalent elements assuming the negation of the continuum hypothesis the silver vaught dichotomy is equivalent to the proposition that there are either countably many equivalence classes or 2 0 equivalence classes the topological vaught conjecture which is open is for any continuous action by a polish group g on a polish space x the orbit equivalence relation satisfies the silver vaught dichotomy this conjecture implies and was motivated by the vaught conjecture in model theory indeed

on the number of permutatively inequivalent basic sequences in a banach space let x be a banach space with a schauder basis en n n the relation e0 is borel reducible to permutative equivalence between normalized block sequences of en n n or x is c0 or lp saturated for some 1 p if en n n is shrinking unconditional then either it is equivalent to the canonical basis of c0 or lp 1 p or the relation e0 is borel reducible to permutative equivalence between sequences of normalized disjoint blocks of x or of x if en n n is unconditional then either x is isomorphic to l2 or x contains 2 subspaces or 2 quotients which are spanned by pairwise permutatively inequivalent normalized unconditional bases

a glimm effros dichotomy for borel equivalence relations a basic dichotomy concerning the structure of the orbit space of a transformation group has been discovered by glimm g12 in the locally compact group action case and extended by effros e 1 e2 in the polish group action case when additionally the induced equivalence relation is f it is the purpose of this paper to extend the glimm effros dichotomy to the very general context of an arbitrary borel equivalence relation not even necessarily induced by a group action despite the totally classical descriptive set theoretic nature of our result our proof requires the employment of methods of effective descriptive set theory and thus ultimately makes crucial use of computability or recursion theory on the integers

on the algebraic bethe ansatz for the xxx spin chain creation operators beyond the equator considering the xxx spin 1 2 chain in the framework of the algebraic bethe ansatz we make the following short comment the product of the creation operators corresponding to the recently found solution of the bethe equations on the wrong side of the equator 1 is just zero not only its action on the pseudovacuum

ideal models of spaces ideal domains have an elementary order theoretic structure every element is either compact or maximal despite this we establish that 1 they can model any space currently known to possess a countably based model and 2 the metric spaces with ideal models are exactly the completely metrizable spaces

unified approach to action angle representation of real and complex multisolitons abstract using a method of complexification a unified approach to action angle representation of complex and real multisolitons is given within a purely algebraic framework the action angle variables the interacting solitons as well as the eigenstates of the recursion operator are explicitly expressed in terms of the physical field variable several plots of these quantities are given

the smooth ideal we give a classical proof of the generalization of the characterization of smoothness to quotients of polish spaces by borel equivalence relations as an application we describe the extent to which any given borel equivalence relation on a polish space is encoded by the corresponding ideal generated by the family of borel sets on which it is smooth

lipschitz metric for the novikov equation we consider the lipschitz continuous dependence of solutions for the novikov equation with respect to the initial data in particular we construct a finsler type optimal transport metric which renders the solution map lipschitz continuous on bounded sets of h 1 mathbb r cap w 1 4 mathbb r although it is not lipschitz continuous under the natural sobolev metric from an energy law due to the finite time gradient blowup by an application of thom s transversality theorem we also prove that when the initial data is in an open dense subset of h 1 mathbb r cap w 1 4 mathbb r the solution is piecewise smooth this generic regularity result helps us extend the lipschitz continuous metric to the general weak solutions our method of constructing the metric can be used to treat other kinds of quasi linear equations provided a good knowledge about the energy concentration

essential countability of treeable equivalence relations we establish a dichotomy theorem characterizing the circumstances under which a treeable borel equivalence relation e is essentially countable under additional topological assumptions on the treeing we in fact show that e is essentially countable if and only if there is no continuous embedding of e1 into e our techniques also yield the first classical proof of the analogous result for hypersmooth equivalence relations and allow us to show that up to continuous kakutani embeddability there is a minimum borel function which is not essentially countable to one

the interacton equation given the lie algebra lfp of vector fields of a free particle then in a joint field of interacting particles an interacton is defined as a direct summand isomorphic to lfp for several soliton equations it is shown how interactons can be obtained from various methods of symmetry analysis and how their dynamics can be expressed in terms of self interaction alone this procedure leads to new integrable systems furthermore in case of nonlinear schrodinger equations it is shown how the dynamics of interactons can be formulated in terms of self consistent potentials with antlinear parts

complete analytic equivalence relations we prove that various concrete analytic equivalence relations arising in model theory or analysis are complete i e maximum in the borel reducibility ordering the proofs use some general results concerning the wider class of analytic quasi orders

depth zero boolean algebras we study the class of depth zero boolean algebras both from a classical viewpoint and an effective viewpoint in particular we provide an algebraic characterization constructing an explicit measure for each depth zero boolean algebra and demonstrating there are no others and an effective characterization providing a necessary and sufficient condition for a depth zero boolean algebra of rank at most to have a computable presentation

the classification problem for torsion free abelian groups of finite rank in 1937 baer 5 introduced the notion of the type of an element in a torsion free abelian group and showed that this notion provided a complete invariant for the classification problem for torsion free abelian groups of rank 1 since then despite the efforts of such mathematicians as kurosh 23 and malcev 25 no satisfactory system of complete invariants has been found for the torsion free abelian groups of finite rank n 2 so it is natural to ask whether the classification problem is genuinely more difficult for the groups of rank n 2 of course if we wish to show that the classification problem for the groups of rank n 2 is intractible it is not enough merely to prove that there are 2 such groups up to isomorphism for there are 2 pairwise nonisomorphic groups of rank 1 and we have already pointed out that baer has given a satisfactory classification for this class of groups in this paper following friedman stanley 11 and hjorth kechris 15 we shall use the more sensitive notions of descriptive set theory to measure the complexity of the classification problem for the groups of rank n 2 recall that up to isomorphism the torsion free abelian groups of rank n are exactly the additive subgroups of the n dimensional vector space qf which contain n linearly independent elements thus the collection of torsion free abelian groups of rank 1 r n can be naturally identified with the set s qn of all nontrivial additive subgroups of qn notice that s qn is a borel subset of the polish space p qn of all subsets of qn and hence s qn can be regarded as a standard borel space i e a polish space equipped with its associated o algebra of borel subsets here we are identifying p qn with the space 2 qn of all functions h qn 0 1 equipped with the product topology furthermore the natural action of gln q on the vector space qn induces a corresponding borel action on s qn and it is easily checked that if a b c s qn then a v b iff there exists an element f e gln q such that p a b it follows that the isomorphism relation on s qn is a countable borel equivalence relation if x is a standard borel space then a borel equivalence relation on x is an equivalence relation e c x2 which is a borel subset of x2 the borel equivalence relation e is said to be countable iff every e equivalence class is countable

direct and inverse scattering problems of the nonlinear intermediate long wave equation the inverse scattering transformation method associated with a nonlinear singular integrodifferential equation is discussed the equation describes long internal gravity waves in a stratified fluid of finite depth and reduces to the korteweg de vries equation as shallow water limit and the benjamin ono equation as deep water limit both limits of the method and novel aspects of the theory are also discussed

turbulence representations and trace preserving actions we establish criteria for turbulence in certain spaces of c algebra repre sentations and apply this to the problem of nonclassiability by countable structures for group actions on a standard atomless probability space x and on the hypernite ii1 factor r we also prove that the conjugacy action on the space of free actions of a countably innite amenable group on r is turbulent and that the conjugacy action on the space of ergodic measure preserving ows on x is generically turbulent

global weak solutions for a new periodic integrable equation with peakon solutions abstract we prove the existence of global weak solutions for a new periodic integrable equation with peakon solutions

a dichotomy characterizing analytic digraphs of uncountable borel chromatic number in any dimension we study the extension of the kechris solecki todorcevic dichotomy on analytic graphs to dimensions higher than 2 we prove that the extension is possible in any dimension finite or infinite the original proof works in the case of the finite dimension we first prove that the natural extension does not work in the case of the infinite dimension for the notion of continuous homomorphism used in the original theorem then we solve the problem in the case of the infinite dimension finally we prove that the natural extension works in the case of the infinite dimension but for the notion of baire measurable homomorphism

the classification of hypersmooth borel equivalence relations this paper is a contribution to the study of borel equivalence relations in standard borel spaces i e polish spaces equipped with their borel structure a class of such equivalence relations which has received particular attention is the class of hyperfinite borel equivalence relations these can be defined as the increasing unions of sequences of borel equivalence relations all of whose equivalence classes are finite or as it turns out equivalently those induced by the orbits of a single borel auto morphism hyperfinite equivalence relations have been classified in djk under two notions of equivalence borel bi reducibility and borel isomorphism

some new integrable systems constructed from the bi hamiltonian systems with pure differential hamiltonian operators abstract when both hamiltonian operators of a bi hamiltonian system are pure differential operators we show that the generalized kupershmidt deformation gkd developed from the kupershmidt deformation in 10 offers an useful way to construct new integrable system starting from the bi hamiltonian system we construct some new integrable systems by means of the generalized kupershmidt deformation in the cases of harry dym hierarchy classical boussinesq hierarchy and coupled kdv hierarchy we show that the gkd of harry dym equation gkd of classical boussinesq equation and gkd of coupled kdv equation are equivalent to the new integrable rosochatius deformations of these soliton equations with self consistent sources we present the lax pair for these new systems therefore the generalized kupershmidt deformation provides a new way to construct new integrable systems from bi hamiltonian systems and also offers a new approach to obtain the rosochatius deformation of soliton equation with self consistent sources

non classification of cartan subalgebras for a class of von neumann algebras abstract we study the complexity of the classification problem for cartan subalgebras in von neumann algebras we construct a large family of ii1 factors whose cartan subalgebras up to unitary conjugacy are not classifiable by countable structures providing the first such examples additionally we construct examples of ii1 factors whose cartan subalgebras up to conjugacy by an automorphism are not classifiable by countable structures finally we show directly that the cartan subalgebras of the hyperfinite ii1 factor up to unitary conjugacy are not classifiable by countable structures and deduce that the same holds for any mcduff ii1 factor with at least one cartan subalgebra

bethe equations on the wrong side of the equator we analyse baxter s famous t q equations for the xxx xxz spin chain and show that apart from its usual polynomial trigonometric solution which provides the solution of bethe ansatz equations there also exists a second solution which should correspond to the bethe ansatz beyond this second solution of baxter s equation plays an essential role and together with the first one gives rise to all fusion relations

vaught s conjecture and the glimm effros property for polish transformation groups we extend the original glimm effros theorem for locally compact groups to a class of polish groups including the nilpotent ones and those with an invariant metric for this class we thereby obtain the topological vaught conjecture

well posedness and analyticity for the cauchy problem for the generalized camassa holm equation abstract in this paper we are concerned with the cauchy problem for the generalized camassa holm equation which was proposed by hakkaev and kirchev s hakkaev k kirchev local well posedness and orbital stability of solitary wave solutions for the generalized camassa holm equation comm partial differential equations 30 2005 761 781 we establish the local well posedness for a range of besov spaces moreover with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time which extends some results of danchin r danchin a few remarks on the camassa holm equation differential integral equations 14 2001 953 988 and himonas and misiolek a himonas g misiolek analyticity of the cauchy problem for an integrable evolution equation math ann 327 2003 575 584 to more general equations

borel complexity and automorphisms of c algebras abstract we show that if a is z o 2 o a uhf algebra of infinite type or the tensor product of a uhf algebra of infinite type and o then the conjugation action aut a aut a is generically turbulent for the point norm topology we moreover prove that if a is either i a separable c algebra which is stable under tensoring with z or k or ii a separable ii 1 factor which is mcduff or a free product of ii 1 factors then the automorphisms of a are not classifiable up to conjugacy by countable structures

descriptive complexity of some isomorphism classes of banach spaces abstract we present a topological frame in which it is possible to estimate the complexity of some borel families of separable banach spaces we use this frame for evaluating the complexity of the isomorphism class of the hilbert space l 2 of certain asymptotically hilbertian spaces and of the l p spaces under the condition 1 p the complexity appears to increase with the distance to the hilbert space commented open problems conclude the article

on the classification of positions and complex structures in banach spaces abstract a topological setting is defined to study the complexities of the relation of equivalence of embeddings or position of a banach space into another and of the relation of isomorphism of complex structures on a real banach space the following results are obtained a if x is not uniformly finitely extensible then there exists a space y for which the relation of position of y inside x reduces the relation e 0 and therefore is not smooth b the relation of position of l p inside l p or inside l p p 2 reduces the relation e 1 and therefore is not reducible to an orbit relation induced by the action of a polish group c the relation of position of a space inside another can attain the maximum complexity e max d there exists a subspace of l p 1 p 2 on which isomorphism between complex structures reduces e 1 and therefore is not reducible to an orbit relation induced by the action of a polish group

the space of composants of an indecomposable continuum abstract the family of all composants of an indecomposable continuum is studied we investigate the equivalence relation induced on an indecomposable continuum by its partition into composants we show that up to borel bireducibility such an equivalence relation can be of only two types e 0 and e 1 the e 0 type being simple and the e 1 type being complicated as a consequence of this we show that each indecomposable continuum carries a borel probability measure which assigns 0 to each composant and 0 or 1 to each borel set which is the union of a family of composants in particular it follows that there is no borel transversal for the family of all composants this solves an old problem in the theory of continua we prove that all hereditarily indecomposable continua are of the complicated type that is they fall into the e 1 type we analyze the properties of being of type e 1 or of type e 0 we show using effective descriptive set theory that the first of these properties is analytic and so the second one is coanalytic we construct examples of continua of both types in fact we produce a family of indecomposable continua and use it to prove that these properties are complete analytic and complete coanalytic respectively hence non borel so they do not admit simple topological characterizations we also use continua from this family to show that an indecomposable continuum may be of type e 1 only because of the behavior of composants on a small subset of the continuum this in particular shows that certain natural approaches to solving kuratowski s problem on generic ergodicity of the composant equivalence relation will not work we finish with some open problems

on isometry and isometric embeddability between ultrametric polish spaces abstract we study the complexity with respect to borel reducibility of the relations of isometry and isometric embeddability between ultrametric polish spaces for which a set d of possible distances is fixed in advance these are respectively an analytic equivalence relation and an analytic quasi order and we show that their complexity depends only on the order type of d when d contains a decreasing sequence isometry is borel bireducible with countable graph isomorphism and isometric embeddability has maximal complexity among analytic quasi orders if d is well ordered the situation is more complex for isometry we have an increasing sequence of borel equivalence relations of length 1 which are cofinal among borel equivalence relations classifiable by countable structures while for isometric embeddability we have an increasing sequence of analytic quasi orders of length at least 3 we then apply our results to solve various open problems in the literature for instance we answer a long standing question of gao and kechris by showing that the relation of isometry on locally compact ultrametric polish spaces is borel bireducible with countable graph isomorphism

uniformisations partielles et criteres a la hurewicz dans le plan we give characterizations of the borel sets potentially in some wadge class among the borel sets with countable vertical sections of a product of two polish spaces to do this we use some partial uniformization results

smooth sets for a borel equivalence relation we study some properties of smooth borel sets with respect to a borel equivalence relation showing some analogies with the collection of countable sets from a descriptive set theoretic point of view we found what can be seen as an analog of the hyperarithmetic points in the context of smooth sets we generalize a theorem of weiss from z actions to actions by arbitrary countable groups we show that the ci ideal of closed smooth sets is ill non borel

decomposing borel sets and functions and the structure of baire class 1 functions all spaces considered are metric separable and are denoted usually by the letters x y or z w stands for the set of all natural numbers if a metric separable space is additionally complete we call it polish if it is a continuous image of ww or equivalently of a polish space it is called souslin the main subject of the present paper is the structure of baire class 1 functions recent developments in banach space theory in particular discoveries of rosenthal and bourgain fremlin and talagrand see r for a survey of these developments stimulated investigations into the structure of baire class 1 functions the interested reader may consult papers by haydon odel rosenthal hor kechris louveau kl rosenthal r2 and references quoted there these investigations have already provided new applications in banach space theory see for example ri in our study of baire class 1 functions we will be interested in two decomposition properties one of them defined by lusin the other one by jayne and rogers first however we want to consider a more general problem of determining how difficult it is to represent a borel set as a union of simpler borel sets or the graph of a borel function as a union of the graphs of simpler borel functions using effective descriptive set theory in particular louveau s theorem we show that if a c x is borel x polish then a e e or there is a continuous injection q ww a such that q 1 b is meager for any b c a which is e2 this gives a new proof of j stern s result that if a borel set a is the union of cov m sets in e then a is itself e cov m is the smallest cardinality of a family of meager sets covering ill we prove similar results for functions put for f x y and a family of functions g

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

an integrable semi discretization of the camassa holm equation and its determinant solution an integrable semi discretization of the camassa holm ch equation is presented the keys of its construction are bilinear forms and determinant structure of solutions of the ch equation determinant formulas of n soliton solutions of the continuous and semi discrete camassa holm equations are presented based on determinant formulas we can generate multi soliton multi cuspon and multi soliton cuspon solutions numerical computations using the integrable semi discrete camassa holm equation are performed it is shown that the integrable semi discrete camassa holm equation gives very accurate numerical results even in the cases of cuspon cuspon and soliton cuspon interactions the numerical computation for an initial value condition which is not an exact solution is also presented

modulating properties of harmonic breather solutions of kdv we show that a class of solutions of kdv can be used to modulate miscellaneous wavepackets similarly to how the trigonometric functions are used in linear theory to perform the same task

borel actions of polish groups we show that a borel action of a polish group on a standard borel space is borel isomorphic to a continuous action of the group on a polish space and we apply this result to three aspects of the theory of borel actions of polish groups universal actions invariant probability measures and the topological vaught conjecture we establish the existence of universal actions for any given polish group extending a result of mackey and varadarajan for the locally compact case we prove an analogue of tarski s theorem on paradoxical decompositions by showing that the existence of an invariant borel probability measure is equivalent to the nonexistence of paradoxical decompositions with countably many borel pieces we show that various natural versions of the topological vaught conjecture are equivalent to each other and in the case of the group of permutations of n with the model theoretic vaught conjecture for infinitary logic this depends on our identification of the universal action for that group

linear algebraic groups and countable borel equivalence relations this paper is a contribution to the study of borel equivalence relations on standard borel spaces i e polish spaces equipped with their borel structure in mathematics one often deals with problems of classification of objects up to some notion of equivalence by invariants frequently these objects can be viewed as elements of a standard borel space x and the equivalence turns out to be a borel equivalence relation e on x a complete classification of x up to e consists of finding a set of invariants i and a map c x i such that xey c x c y for this to be of any interest both i and c must be explicit or definable and as simple and concrete as possible the theory of borel equivalence relations studies the set theoretic nature of possible invariants and develops a mathematical framework for measuring the complexity of such classification problems

on a camassa holm type equation with two dependent variables we consider a generalization of the camassa holm ch equation with two dependent variables called ch2 introduced in a paper by liu and zhang liu s q and zhang y 2005 j geom phys 54 427 53 we briefly provide an alternative derivation of it based on the theory of hamiltonian structures on the dual of a lie algebra the lie algebra involved here is the same algebra as underlies the nls hierarchy we study the structural properties of the hierarchy defined by the ch2 equation within the bi hamiltonian theory of integrable pdes and provide its lax representation then we explicitly discuss how to construct classes of solutions both of peakon and of algebro geometrical type finally we sketch the construction of a class of singular solutions defined by setting to zero one of the two dependent variables

on the complexity of the uniform homeomorphism relation between separable banach spaces we investigate the uniform homeomorphism relation between separable banach spaces and the related relation of local equivalence we completely characterize the descriptive complexity of local equivalence in the borel reducibility hierarchy this also provides a lower bound for the complexity of the uniform homeomorphism

completeness of the isomorphism problem for separable c algebras this paper studies the descriptive set theoretical complexity of the isomorphism problem for separable c algebras we prove that the isomorphism problem for separable simple ai c algebras is complete in the class of orbit equivalence relations this means that any isomorphism problem arising from a continuous action of a separable completely metrizable group can be reduced to the isomorphism of simple separable ai c algebras

unitary equivalence of automorphisms of separable c algebras abstract we prove that the automorphisms of any separable c algebra that does not have continuous trace are not classifiable by countable structures up to unitary equivalence this implies a dichotomy for the borel complexity of the relation of unitary equivalence of automorphisms of a separable unital c algebra such relation is either smooth or not even classifiable by countable structures

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

the structure of hyperfinite borel equivalence relations we study the structure of the equivalence relations induced by the orbits of a single borel automorphism on a standard borel space we show that any two such equivalence relations which are not smooth i e do not admit borel selectors are borel embeddable into each other this utilizes among other things work of effros and weiss using this and also results of dye varadarajan and recent work of nadkarni we show that the cardinality of the set of ergodic invariant measures is a complete invariant for borel isomorphism of aperiodic nonsmooth such equivalence relations in particular since the only possible such cardinalities are the finite ones countable infinity and the cardinality of the continuum there are exactly countably infinitely many isomorphism types canonical examples of each type are also discussed

dichotomy theorems for families of non cofinal essential complexity we prove that for every borel equivalence relation e either e is borel reducible to mathbb e 0 or the family of borel equivalence relations incompatible with e has cofinal essential complexity it follows that if f is a borel equivalence relation and cal f is a family of borel equivalence relations of non cofinal essential complexity which together satisfy the dichotomy that for every borel equivalence relation e either e in cal f or f is borel reducible to e then cal f consists solely of smooth equivalence relations thus the dichotomy is equivalent to a known theorem

on the isomorphism problem for measures on boolean algebras abstract the paper investigates possible generalizations of maharam s theorem to a classification of boolean algebras that support a finitely additive measure we prove that boolean algebras that support a finitely additive non atomic uniformly regular measure are metrically isomorphic to a subalgebras of the jordan algebra with the lebesgue measure we give some partial analogues to be used for a classification of algebras that support a finitely additive non atomic measure with a higher uniform regularity number we show that some naturally induced equivalence relations connected to metric isomorphism are quite complex even on the cantor algebra and therefore probably we cannot hope for a nice general classification theorem for finitely additive measures we present an example of a boolean algebra which supports only separable measures but no uniformly regular one

finite propagation speed for the camassa holm equation we prove that any classical solution of the camassa holm equation will have compact support if its initial data has this property

completeness of the bethe ansatz for the six and eight vertex models we discuss some of the difficulties that have been mentioned in the literature in connection with the bethe ansatz for the six vertex model and xxz chain and for the eight vertex model in particular we discuss the beyond the equator infinite momenta and exact complete string problems we show how they can be overcome and conclude that the coordinate bethe ansatz does indeed give a complete set of states as expected

well posedness of a class of solutions to an integrable two component camassa holm system abstract in this paper we study a class of solutions to the cauchy problem for an integrable two component camassa holm system with the initial data u 0 0 1 h 1 r w 1 r l 2 r l r based on characteristics we study a corresponding ode and obtain a unique local solution by applying the contraction mapping principle we prove local existence and uniqueness of the solution to the camassa holm system by constructing a solution obtained from the ode and studying the regularity of the solution finally we show continuous dependence of the solution on the initial data in some weak sense

borel chromatic numbers we study in this paper graph coloring problems in the context of descriptive set theory we consider graphs g x r where the vertex set x is a standard borel space i e a complete separable metrizable space equipped with its algebra of borel sets and the edge relation r x 2 is definable i e borel analytic co analytic etc a borel n coloring of such a graph where 1 n n 0 is a borel map c x y with card y n such that xry c x c y if such a borel coloring exists we define the borel chromatic number of g in symbols x b g to be the smallest such n otherwise we say that g has uncountable borel chromatic number in symbols x b g n 0