magic identities for the conformal four point integrals the minkowski metric case abstract the original magic identities are due to j m drummond j henn v a smirnov and e sokatchev they assert that all n loop box integrals for four scalar massless particles are equal to each other 3 the authors give a proof of the magic identities for the euclidean metric case only and claim that the result is also true in the minkowski metric however the minkowski case is much more subtle and requires specification of the relative positions of cycles of integration to make these identities correct in this article we prove the magic identities in the minkowski metric case and in particular specify the cycles of integration our proof of magic identities relies on previous results from 7 8 where we give a mathematical interpretation of the n loop box integrals in the context of representations of a lie group u 2 2 and quaternionic analysis the main result of 7 8 is a weaker operator version of the magic identities no prior knowledge of physics or feynman diagrams is assumed from the reader we provide a summary of all relevant results from quaternionic analysis to make the article self contained

fundamental group of uniquely ergodic cantor minimal systems abstract we introduce the fundamental group f r g of a uniquely ergodic cantor minimal g system r g where g is a countable discrete group we compute fundamental groups of several uniquely ergodic cantor minimal g systems we show that if r g arises from a free action of a finitely generated abelian group then there exists a unital countable subring r of r such that f r g r we also consider the relation between fundamental groups of uniquely ergodic cantor minimal z n systems and fundamental groups of crossed product c algebras c x z n

probability inequalities for empirical processes and a law of the iterated logarithm given a class f of functions on x we can view vn as a stochastic process indexed by f and consider limit theorems for this process to prove such theorems it is often helpful to have bounds on the tail of the r v sup w i n f i see for example dudley 1978 kuelbs and dudley 1980 dudley and phillip 1983 or gine and zinn 1983 our main question of interest is for what classes f can best possible bounds be obtained so first we must ask what are these best possible bounds of course

a geometrical angle on feynman integrals a direct link between a one loop n point feynman diagram and a geometrical representation based on the n dimensional simplex is established by relating the feynman parametric representations to the integrals over contents of n 1 dimensional simplices in non euclidean geometry of constant curvature in particular the four point function in four dimensions is proportional to the volume of a three dimensional spherical or hyperbolic tetrahedron which can be calculated by splitting into birectangular ones it is also shown that the known formula of reduction of the n point function in n 1 dimensions corresponds to splitting the related n dimensional simplex into n rectangular ones

some classifiable groupoid c algebras with prescribed k theory given a simple acyclic dimension group g 0 and countable torsion free abelian group g 1 we construct a minimal amenable etale equivalence relation r on a cantor set whose associated groupoid c algebra c r is tracially af and hence classifiable in the elliott classification scheme for simple amenable separable c algebras and with k c r cong g 0 g 1

the rotation number for almost periodic potentials we define and analyze the rotation number for the almost periodic schrodinger operatorl d2 dx2 q x we use the rotation number to discuss i the spectrum ofl ii its relation to the korteweg de vries equation

speeding up mcmc by efficient data subsampling we propose subsampling markov chain monte carlo mcmc an mcmc framework where the likelihood function for n observations is estimated from a random subset of m observations we introduce a highly

cohomology of canonical projection tilings we define the cohomology of a tiling as the cocycle cohomology of its associated groupoid and consider this cohomology for the class of tilings which are obtained from a higher dimensional lattice by the canonical projection method in schlottmann s formulation we prove the cohomology to be equivalent to a certain cohomology of the lattice we discuss one of its qualitative features namely that it provides a topological obstruction for a generic tiling to be substitutional we develop and demonstrate techniques for the computation of cohomology for tilings of codimension smaller than or equal to 2 presenting explicit formulae these in turn give computations for the k theory of certain associated non commutative c algebras

transverse laplacians for substitution tilings pearson and bellissard recently built a spectral triple the data of riemannian noncommutative geometry for ultrametric cantor sets they derived a family of laplace beltrami like operators on those sets motivated by the applications to specific examples we revisit their work for the transversals of tiling spaces which are particular self similar cantor sets we use bratteli diagrams to encode the self similarity and cuntz krieger algebras to implement it we show that the abscissa of convergence of the function of the spectral triple gives indications on the exponent of complexity of the tiling we determine completely the spectrum of the laplace beltrami operators give an explicit method of calculation for their eigenvalues compute their weyl asymptotics and a seeley equivalent for their heat kernels

rates of convergence for the empirical quantization error for n k n and r 0 let e n r p k r inf 1 k k i 1 x i f x i r where the infimum is taken over all measurable maps f r d r d with f r d n and x 1 x k are i i d r d valued random variables we analyse the asymptotic a s behaviour of the nth empirical quantization error e n r p k

on kahler einstein metrics on certain kahler manifolds with c1 m 0 on demontre qu il existe une metrique de kahler einstein sur une hypersurface de fermat a m dimensions de degre superieur a m 1

spaces of tilings finite telescopic approximations and gap labeling the continuous hull of a repetitive tiling t in d with the finite pattern condition fpc inherits a minimal d lamination structure with flat leaves and a transversal open image in new window which is a cantor set this class of tiling includes the penrose the amman benkker ones in 2d as well as the icosahedral tilings in 3d we show that the continuous hull with its canonical d action can be seen as the projective limit of a suitable sequence of branched oriented and flat compact d manifolds as a consequence the longitudinal cohomology and the k theory of the corresponding c algebra open image in new window are obtained as direct limits of cohomology and k theory of ordinary manifolds moreover the space of invariant finite positive measures can be identified with a cone in the d th homology group canonically associated with the orientation of d at last the gap labeling theorem holds given an invariant ergodic probability measure on the hull the corresponding integrated density of states ids of any selfadjoint operators affiliated to open image in new window takes on values on spectral gaps in the module generated by the occurrence probabilities of finite patches in the tiling

homology and topological full groups of etale groupoids on totally disconnected spaces for almost finite groupoids we study how their homology groups reflect dynamical properties of their topological full groups it is shown that two clopen subsets of the unit space has the same class in h0 if and only if there exists an element in the topological full group which maps one to the other it is also shown that a natural homomorphism called the index map from the topological full group to h1 is surjective and any element of the kernel can be written as a product of four elements of finite order in particular the index map induces a homomorphism from h1 to k1 of the groupoid c algebra explicit computations of homology groups of af groupoids and etale groupoids arising from subshifts of finite type are also given

proof of the magnetic gap labelling conjecture for principal solenoidal tori abstract in this note we prove the magnetic spectral gap labelling conjecture as stated in 8 in all dimensions for principal solenoidal tori

rokhlin dimension for flows we introduce a notion of rokhlin dimension for one parameter automorphism groups of c algebras this generalizes kishimoto s rokhlin property for flows and is analogous to the notion of rokhlin dimension for actions of the integers and other discrete groups introduced by the authors and zacharias in previous papers we show that finite nuclear dimension and absorption of a strongly self absorbing c algebra are preserved under forming crossed products by flows with finite rokhlin dimension and that these crossed products are stable furthermore we show that a flow on a commutative c algebra arising from a free topological flow has finite rokhlin dimension whenever the spectrum is a locally compact metrizable space with finite covering dimension for flows that are both free and minimal this has strong consequences for the associated crossed product c algebras those containing a non zero projection are classified by the elliott invariant for compact manifolds this consists of topological k theory together with the space of invariant probability measures and a natural pairing given by the ruelle sullivan map

minimal configurations for the frenkel kontorova model on a quasicrystal in this paper we consider the frenkel kontorova model of a one dimensional chain of atoms submitted to a potential this potential splits into an interaction potential and a potential induced by an underlying substrate which is a quasicrystal under standard hypotheses we show that every minimal configuration has a rotation number that the rotation number varies continuously with the minimal configuration and that every non negative real number is the rotation number of a minimal configuration this generalizes well known results obtained by s aubry and p y le daeron in the case of a crystalline substrate

the pinwheel tilings of the plane prototiles in euclidean n space en for n 2 the prototiles are usually required to be rather nice topologically at least homeomorphs of the closed unit ball one then makes arbitrarily many congruent copies called tiles of these prototiles and considers all ways called tilings that such tiles may provide a simultaneous covering and packing of en a tiling is thus an unordered collection of tiles for which the union is all of en but such that the interiors of each pair of tiles do not intersect wang s original problem was to determine if it was possible to have a finite set of prototiles with associated tilings of space for which all the tilings were nonperiodic a tiling is nonperiodic if it is not invariant under any single simultaneous translation of its tiles this was settled in the affirmative in the thesis of wang s student robert berger 1

exact number of ergodic invariant measures for bratteli diagrams abstract for a bratteli diagram b we study the simplex m 1 b of probability measures on the path space of b which are invariant with respect to the tail equivalence relation equivalently m 1 b is formed by probability measures invariant with respect to a homeomorphism of a cantor set we study relations between the number of ergodic measures from m 1 b and the structure and properties of the diagram b we prove a criterion and find sufficient conditions of unique ergodicity of a bratteli diagram in which case the simplex m 1 b is a singleton for a finite rank k bratteli diagram b having exactly l k ergodic invariant measures we explicitly describe the structure of the diagram and find the subdiagrams which support these measures we find sufficient conditions under which i a bratteli diagram has a prescribed number finite or infinite of ergodic invariant measures and ii the extension of a measure from a uniquely ergodic subdiagram gives a finite ergodic invariant measure several examples including stationary bratteli diagrams pascal bratteli diagrams and toeplitz flows are considered

orbit equivalence for cantor minimal ℤd systems we show that every minimal action of any finitely generated abelian group on the cantor set is topologically orbit equivalent to an af relation as a consequence this extends the classification up to orbit equivalence of minimal dynamical systems on the cantor set to include af relations and d actions

continuous cocycle superrigidity for shifts and groups with one end we prove that if a finitely generated group g is not torsion then a necessary and sufficient condition for every full shift over g has continuous cocycle superrigidity is that g has one end it is a topological version of the well known popa s measurable cocycle superrigidity theorem cocycle and orbit equivalence superrigidity for malleable actions of w rigid groups invent math 170 2 243 295 2007 for the proof of sufficiency we introduce a new specification property for shifts over general groups which plays a similar role as malleable property in the measurable setting this new specification property is good enough for us to extend the method of using homoclinic equivalence relation that was introduced by schmidt pac j math 170 1 237 269 1995 to study cocycle rigidity for mathbb z d shifts indeed in this direction we prove this superrigidity result for certain more general systems and for the necessary condition we apply specker s characterization for ends of groups via the associated first cohomology groups to get the result finally combining our results with li ergod theory dyn syst arxiv 1503 01704 theorem 1 6 we have an application in continuous orbit equivalence rigidity

split quaternionic analysis and separation of the series for sl 2 r and sl 2 c sl 2 r abstract we extend our previous study of quaternionic analysis based on representation theory to the case of split quaternions h r the special role of the unit sphere in the classical quaternions h identified with the group su 2 is now played by the group sl 2 r realized by the unit quaternions in h r as in the previous work we use an analogue of the cayley transform to relate the analysis on sl 2 r to the analysis on the imaginary lobachevski space sl 2 c sl 2 r identified with the one sheeted hyperboloid in the minkowski space m we study the counterparts of cauchy fueter and poisson formulas on h r and m and show that they solve the problem of separation of the discrete and continuous series the continuous series component on h r gives rise to the minimal representation of the conformal group sl 4 r while the discrete series on m provides its k types realized in a natural polynomial basis we also obtain a surprising formula for the plancherel measure of sl 2 r in terms of the poisson type integral on the split quaternions h r finally we show that the massless singular functions of four dimensional quantum field theory are nothing but the kernels of projectors onto the discrete and continuous series on the imaginary lobachevski space sl 2 c sl 2 r our results once again reveal the central role of the minkowski space in quaternionic and split quaternionic analysis as well as a deep connection between split quaternionic analysis and the four dimensional quantum field theory

an analogue of the thom isomorphism for crossed products of a c algebra by an action of r abstract in this paper we show the existence and uniqueness of a natural isomorphism oj of kj a with kj 1 a r j z 2 where a r is a c dynamical r system k is the functor of topological k theory and a r is the crossed product of a by the action of r the pimsner voiculescu exact sequence is obtained as a corollary we show that given an invariant trace on a with dual trace gt one has gto 1 u 1 2 i u u for any unitary u in the domain of the derivation of a associated to the action finally we show that the crossed product of c s3 continuous functions on the 3 sphere by a minimal diffeomorphism is a simple c algebra with no nontrivial idempotent

orbit equivalence for cantor minimal mathbb z 2 systems we show that every minimal action of any finitely generated abelian group on the cantor set is topologically orbit equivalent to an af relation as a consequence this extends the classification up to orbit equivalence of minimal dynamical systems on the cantor set to include af relations and d actions

quaternionic analysis representation theory and physics we develop quaternionic analysis using as a guiding principle representation theory of various real forms of the conformal group we first review the cauchy fueter and poisson formulas and explain their representation theoretic meaning the requirement of unitarity of representations leads us to the extensions of these formulas in the minkowski space which can be viewed as another real form of quaternions representation theory also suggests a quaternionic version of the cauchy formula for the second order pole remarkably the derivative appearing in the complex case is replaced by the maxwell equations in the quaternionic counterpart we also uncover the connection between quaternionic analysis and various structures in quantum mechanics and quantum field theory such as the spectrum of the hydrogen atom polarization of vacuum one loop feynman integrals we also make some further conjectures the main goal of this and our subsequent paper is to revive quaternionic analysis and to show profound relations between quaternionic analysis representation theory and four dimensional physics

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

silver exudation a silver is a tetrahedon whose four vertices lie close to a plane and whose orthogonal projection to that plane is a convex quadrilateral with no short edge silvers are notoriously common in 3 dimensional delaunay triangulations even for well spaced point sets we show that if the delaunay triangulation has the ratio property introduced in miller et al 1995 then there is an assignment of weights so the weighted delaunay traingulation contains no silvers we also give an algorithm to compute such a weight assignment

the conformal four point integrals magic identities and representations of u 2 2 abstract in 3 5 we found mathematical interpretations of the one loop conformal four point feynman integral as well as the vacuum polarization feynman integral in the context of representations of a lie group u 2 2 and quaternionic analysis then we raised a natural question of finding mathematical interpretation of other feynman diagrams in the same setting in this article we describe this interpretation for all conformal four point integrals using this interpretation we give a representation theoretic proof of an operator version of the magic identities for the conformal four point integrals described by the box diagrams the original magic identities are due to j m drummond j henn v a smirnov and e sokatchev they assert that all n loop box integrals for four scalar massless particles are equal to each other 2 the authors give a proof of the magic identities for the euclidean metric case only and claim that the result is also true for the minkowski metric case however the minkowski case is much more subtle in this article we prove an operator version of the magic identities in the minkowski metric case and in particular specify the relative positions of cycles of integration that make these identities correct no prior knowledge of physics or feynman diagrams is assumed from the reader we provide a summary of all relevant results from quaternionic analysis to make the article self contained

orbit equivalence of cantor minimal systems and their continuous spectra to any continuous eigenvalue of a cantor minimal system x t we associate an element of the dimension group k 0 x t associated to x t we introduce and study the concept of irrational miscibility of a dimension group the main property of these dimension groups is the absence of irrational values in the additive group of continuous spectrum of their realizations by cantor minimal systems the strong orbit equivalence respectively orbit equivalence class of a cantor minimal system associated to an irrationally miscible dimension group g u resp with trivial infinitesimal subgroup with trivial rational subgroup have no non trivial continuous eigenvalues

classifying crossed product c algebras i combine recent results in the structure theory of nuclear rm c algebras and in topological dynamics to classify certain types of crossed products in terms of their elliott invariants in particular transformation group rm c algebras associated to free minimal bbb z d actions on the cantor set with compact space of ergodic measures are classified by their ordered rm k theory in fact the respective statement holds for finite dimensional compact metrizable spaces provided that projections of the crossed products separate tracial states moreover rm c algebras associated to certain minimal homeomorphisms of spheres s 2n 1 are only determined by their spaces of invariant borel probability measures without a condition on the space of ergodic measures finally i show that for a large collection of classifiable rm c algebras crossed products by bbb z d actions are generically again classifiable

topological full groups of one sided shifts of finite type we explore the topological full group g of an essentially principal etale groupoid g on a cantor set when g is minimal we show that g and its certain normal subgroup is a complete invariant for the isomorphism class of the etale groupoid g furthermore when g is either almost finite or purely infinite the commutator subgroup d g is shown to be simple the etale groupoid g arising from a onesided irreducible shift of finite type is a typical example of a purely infinite minimal groupoid for such g g is thought of as a generalization of the higman thompson group we prove that g is of type f and so in particular it is finitely presented this gives us a new infinite family of finitely presented infinite simple groups also the abelianization of g is calculated and described in terms of the homology groups of g