hamiltonian gravity and noncommutative geometry a version of foliated spacetime is constructed in which the spatial geometry is described as a time dependent noncommutative geometry the adm version of the gravitational action is expressed in terms of these variables it is shown that the vector constraint is obtained without the need for an extraneous shift vector in the action

hausdorff moment problem and fractional moments a simplified procedure abstract the purpose of this paper is the recovering of a probability density function with support 0 1 from the knowledge of its sequence of moments i e the classical hausdorff moment problem to avoid the well known ill conditioning firstly the moment curve is calculated from the assigned sequence of moments next the unknown density is approximated by maximum entropy maxent technique selecting some proper points on the moment curve exploiting convergence in entropy a simplified quick procedure is suggested to recover the approximate density an application to laplace transform inversion is illustrated

noncommutative residues and manifolds with conical singularities abstract for every singular point on a manifold with conical singularities a trace can be constructed on the cone algebra with asymptotics introduced by b w schulze each of them vanishes on operators supported in the interior and is therefore different from the noncommutative residue established by wodzicki on the ideal of operators with zero conormal symbol however we find another trace which coincides with the noncommutative residue in the interior all these traces are shown to be essentially unique on a slightly extended version of the cone algebra

zeta functions on a product of einstein manifolds and the multiplicative anomaly the global additive and multiplicative properties of laplace type operators acting on irreducible rank 1 symmetric spaces are considered the explicit form of the zeta function on product spaces and of the multiplicative anomaly is derived

analytic torsion for twisted de rham complexes we dene analytic torsion x e h 2 det h x e h for the twisted de rham complex consisting of the spaces of dieren tial forms on a compact oriented riemannian man ifold x valued in a at vector bundle e with a dieren tial given by r e h where r e is a at connection on e h is an odd degree closed dieren tial form on x and h x e h denotes the cohomology of this z2 graded complex the denition uses pseudo dieren tial operators and residue traces we show that when dim x is odd x e h is independent of the choice of metrics on x and e and of the representative h in the cohomology class h we explain the relation to generalized geometry when h is a 3 form we demonstrate some basic func torial properties when h is a top degree form we compute the torsion dene its simplicial counterpart and prove an analogue of the cheeger mtheorem we also establish the rela tionship of an invariant form of the analytic torsion for t dual circle bundles with 3 form uxes

inference in linear time series models with some unit roots this paper considers estimation and hypothesis testing in linear time series models when some or all of the variables have unit roots our motivating example is a vector autoregression with some unit roots in the companion matrix which might include polynomials in time as regressors in the general formulation the variable might be integrated or cointegrated of arbitrary orders and might have drifts as well we show that parameters that can be written as coefficients on mean zero nonintegrated regressors have jointly normal asymptotic distributions converging at the rate t 2 in general the other coefficients including the coefficients on polynomials in time will have nonnormal asymptotic distributions the results provide a formal characterization of which t or f tests such as granger causality tests will be asymptotically valid and which will have nonstandard limiting distributions

a note on noncausality in this note the relationship between alternative concepts of noncausality is analyzed using the tool of conditional independence among a fields for the reader who is unfamiliar with this technique the appendix sketches the proofs and the basic technical apparatus along with some basic motivations furthermore the relationship between the concepts of noncausality and transitivity is made explicit in order to facilitate in econometric modelling the use of results already obtained in sequential analysis

moment information for probability distributions without solving the moment problem ii main mass tails and shape approximation how much information does a small number of moments carry about the unknown distribution function is it possible to explicitly obtain from these moments some useful information e g about the support the modality the general shape or the tails of a distribution without going into a detailed numerical solution of the moment problem in this previous and subsequent papers clear and easy to implement answers will be given to some questions of this type first the question of how to distinguish between the main mass interval and the tail regions in the case we know only a number of moments of the target distribution function will be addressed the answer to this question is based on a version of the chebyshev stieltjes markov inequality which provides us with upper and lower moment based bounds for the target distribution then exploiting existing asymptotic results in the main mass region an explicit moment based approximation of the target probability density function is provided although the latter cannot be considered in general as a satisfactory solution it can always serve as an initial approximation in any iterative scheme for the numerical solution of the moment problem numerical results illustrating all the theoretical statements are also presented

subadditivity and additivity of the noncommutative yang mills action functional we formulate notions of subadditivity and additivity of the yang mills action functional in noncommutative geometry we identify a suitable hypothesis on spectral triples which proves that the yang mills functional is always subadditive the additivity property is stronger in the sense that it implies the subadditivity property under our hypothesis we also obtain a necessary and sufficient condition for the additivity of the yang mills functional an instance of additivity is shown for the case of noncommutative n tori we also investigate the behavior of critical points of the yang mills functional under additivity in the end we discuss examples involving compact spin manifolds matrix algebras noncommutative n torus and the quantum heisenberg manifolds to validate our hypothesis

schwinger terms and cohomology of pseudodifferential operators we study the cohomology of the schwinger term arising in second quantization of the class of observables belonging to the restricted general linear algebra we prove that for all pseudodifferential operators in 3 1 dimensions of this type the schwinger term is equivalent to the twisted radul cocycle a modified version of the radul cocycle arising in non commutative differential geometry in the process we also show how the ordinary radul cocycle for any pair of pseudodifferential operators in any dimension can be written as the phase space integral of the star commutator of their symbols projected to the appropriate asymptotic component

dixmier traces and extrapolation description of noncommutative lorentz spaces we study the connections between dixmier traces functions and traces of heat semigroups beyond the dual of the macaev ideal and in the general context of semifinite von neumann algebras we show that the correct framework for this investigation is that of operator lorentz spaces possessing an extrapolation description we demonstrate the applicability of our results to hormander weyl pseudo differential calculus on rn in this context we prove that the dixmier trace of a pseudo differential operator coincides with the dixmier integral of its symbol

measures of conditional linear dependence and feedback between time series abstract measures of linear dependence and feedback for two multiple time series conditional on a third are defined the measure of conditional linear dependence is the sum of linear feedback from the first to the second conditional on the third linear feedback from the second to the first conditional on the third and instantaneous linear feedback between the first and second series conditional on the third the measures are non negative and may be expressed in terms of measures of unconditional feedback between various combinations of the three series the measures of conditional linear feedback can be additively decomposed by frequency estimates of these measures are straightforward to compute and their distribution can be routinely approximated by bootstrap methods an empirical example involving real output money and interest rates is presented

residues of the eta function for an operator of dirac type abstract we compute the asymptotics of tr l 2 pe tp 2 where p is a first order operator of dirac type this is equivalent to evaluating the residues of the eta function

a new proof of weyl s formula on the asymptotic distribution of eigenvalues one of the more perplexing problems of finite particle quantum mechanics is to find the appropriate quantum mechanical description of a given classical mechanical system unfortunately from the mathematical point of view this problem is not terribly well posed suppose the classical system in question is represented by a symplectic manifold x and a classical observable p x r suppose also for simplicity that this system is a confined system in the sense that the sets p 1 are all compact let h be a hilbert space and let p be a self adjoint operator on h representing a quantum observable without specifying anything further about p for instance its relation to other observables there is practically no way of ascertaining whether p is an appropriate quantum counterpart of p at first glance the only requirements this appears to impose on p are that the spectrum of p be discrete and that the number of eigenvalues n a less than 1 be finite for all 1 this is the analogue of the condtion that p 1 be compact by examining the statistical properties of a large aggregate of classical systems each a copy of x p one can in fact say a little bit more about the relation p to p by comparing classical and quantum partition functions one can show that for iz large

fredholm modules on p c f self similar fractals and their conformal geometry the aim of the present work is to show how using the differential calculus associated to dirichlet forms it is possible to construct non trivial fredholm modules on post critically finite fractals by regular harmonic structures d r the modules are d s summable the summability exponent d s coinciding with the spectral dimension of the generalized laplacian operator associated with d r the characteristic tools of the noncommutative infinitesimal calculus allow to define a d s energy functional which is shown to be a self similar conformal invariant

structure of stieltjes classes of moment equivalent probability laws a stieltjes class is a one parameter family of moment equivalent distribution functions constructed by modulation of a given indeterminate distribution function f called the center of the class members of a stieltjes class are mutually absolutely continuous and conversely any pair of moment equivalent and mutually absolutely continuous distribution functions can be joined by a stieltjes class the center of a stieltjes class is an equally weighted mixture of its extreme members and this places restrictions on which distributions can belong to a stieltjes class with a given center the lognormal law provides interesting illustrations of the general ideas in particular it is possible for two moment equivalent infinitely divisible distributions to be joined by a stieltjes class and random scaling can be used to construct new stieltjes classes from a given stieltjes class

dixmier measurability in marcinkiewicz spaces and applications abstract we present new results concerning the description of dixmier measurable elements in the sense of a connes in marcinkiewicz spaces and demonstrate some applications of our results in particular we complement famous connes trace theorem by describing the asymptotics of the singular function of order n classical pseudo differential operators on an n dimensional compact riemannian manifold

a construction of critical gjms operators using wodzicki s residue for an even dimensional compact conformal manifold without boundary we construct a conformally invariant differential operator of order the dimension of the manifold in the conformally flat case this operator coincides with the critical gjms operator of graham jenne mason sparling we use the wodzicki residue of a pseudo differential operator of order 2 originally defined by a connes acting on middle dimension forms

noncommutative integration calculus a noncommutative integration calculus arising in the mathematical description of schwinger terms of fermion yang mills systems is discussed the differential complexes of forms u0 e u1 e un with e a grading operator on a hilbert space h and ui bounded operators on h which naturally contains the compactly supported de rham forms on rd i e e is the sign of the free dirac operator on rd and h is a l2 space on rd are considered an elementary proof is presented in which the integral of d forms rdtrn x0dx1 dxd for xi c0 rd gln is equal up to a constant to the conditional hilbert space trace of x0 e x1 e xd where 1 for d odd and d 1 5 matrix a spin matrix anticommuting with e for d even this result provides a natural generalization of integration of de rham forms to the setting of connes noncommutative geometry which involves the ordinary hilbert space trace rather than the dixmier trace

on the application of maximum entropy to the moments problem the maximum entropy approach to the solution of classical inverse problem of moments in which one seeks to reconstruct a function p x where x 0 from the values of a finite set n 1 of its moments is studied it is shown that for n 4 such a function always exists while for n 2 and n 3 the acceptable values of the moments are singled out analytically the paper extends to the general case where the results were previously bounded to the case n 2

topological calculation of the phase of the determinant of a non self adjoint elliptic operator we study the zeta regularized determinant of a non self adjoint elliptic operator on a closed odd dimensional manifold we show that if the spectrum of the operator is symmetric with respect to the imaginary axis then the determinant is real and its sign is determined by the parity of the number of the eigenvalues of the operator which lie on the positive part of the imaginary axis it follows that for many geometrically defined operators the phase of the determinant is a topological invariant in numerous examples coming from geometry and physics we calculate the phase of the determinants in purely topological terms some of those examples were known in physical literature but no mathematically rigorous proofs and no general theory were available until now

measures from dixmier traces and zeta functions for l functions on a closed compact riemannian manifold the noncommutative residue and the dixmier trace formulation of the noncommutative integral are shown to equate to a multiple of the lebesgue integral the identifications are shown to continue to and be sharp at l2 functions for functions strictly in lp 1 p 2 symmetrised noncommutative residue and dixmier trace formulas must be introduced for which the identification is shown to continue for the noncommutative residue however a failure is shown for the dixmier trace formulation at l1 functions it is shown the noncommutative residue remains finite and recovers the lebesgue integral for any integrable function while the dixmier trace expression can diverge the results show that a claim in the monograph j m gracia bondia j c varilly h figueroa elements of noncommutative geometry birkhauser adv texts birkhauser boston 2001 that the equality on c functions between the lebesgue integral and an operator theoretic expression involving a dixmier trace obtained from connes trace theorem can be extended to any integrable function is false the results of this paper include a general presentation for finitely generated von neumann algebras of commuting bounded operators including a bounded borel or l functional calculus version of c results in iv 2 of a connes noncommutative geometry academic press new york 1994

spectral boundary conditions for generalizations of laplace and dirac operators spectral boundary conditions for laplace type operators on a compact manifold x with boundary are partly dirichlet partly oblique neumann conditions where the partitioning is provided by a pseudodifferential projection they have an interest in string and brane theory relying on pseudodifferential methods we give sufficient conditions for the existence of the associated resolvent and heat operator and show asymptotic expansions of their traces in powers and power log terms allowing a smearing function the leading log coefficient is identified as a non commutative residue which vanishes when 1 the study has new consequences for well posed spectral boundary problems for first order dirac like elliptic operators generalizing the atiyah patodi singer problem it is found e g that the zeta function is always regular at zero in the selfadjoint case there is a stability of the zeta function value and the eta function regularity at zero under perturbations of the boundary projection of order dim x

the dixmier trace and asymptotics of zeta functions abstract we obtain general theorems which enable the calculation of the dixmier trace in terms of the asymptotics of the zeta function and of the trace of the heat semigroup we prove our results in a general semi finite von neumann algebra we find for p 1 that the asymptotics of the zeta function determines an ideal strictly larger than l p on which the dixmier trace may be defined we also establish stronger versions of other results on dixmier traces and zeta functions

noncommutative residue and sub dirac operators for foliations in this paper we define lower dimensional volumes associated with sub dirac operators for foliations in some cases we compute these lower dimensional volumes we also prove the kastler kalau walze type theorems for foliations with or without boundary as a corollary we give an explanation of the gravitational action for the robertson walker space a b f m3

a dixmier trace formula for the density of states a version of connes trace formula allows to associate a measure on the essential spectrum of a schrodinger operator with bounded potential in solid state physics there is another celebrated measure associated with such operators the density of states in this paper we demonstrate that these two measures coincide we show how this equality can be used to give explicit formulae for the density of states in some circumstances

hierarchical statistical models and a generalized likelihood ratio test this paper deals with statistical inference for statistical models which have a nested structure the emphasis here is on the use of a classical test of significance and on a confidence set construction approach for model identification and the paper proposes a generalized likelihood ratio test with equal marginal error rate as a basic tool the power performance of this test is characterized and some applications are illustrated

noncommutative residue for heisenberg manifolds applications in cr and contact geometry abstract this paper has four main parts in the first part we construct a noncommutative residue for the hypoelliptic calculus on heisenberg manifolds that is for the class of h do operators introduced by beals greiner and taylor this noncommutative residue appears as the residual trace on integer order h dos induced by the analytic extension of the usual trace to non integer order h dos moreover it agrees with the integral of the density defined by the logarithmic singularity of the schwartz kernel of the corresponding h do in addition we show that this noncommutative residue provides us with the unique trace up to constant multiple on the algebra of integer order h dos in the second part we give some analytic applications of this construction concerning zeta functions of hypoelliptic operators logarithmic metric estimates for green kernels of hypoelliptic operators and the extension of the dixmier trace to the whole algebra of integer order h dos in the third part we present examples of computations of noncommutative residues of some powers of the horizontal sublaplacian and the contact laplacian on contact manifolds in the fourth part we present two applications in cr geometry first we give some examples of geometric computations of noncommutative residues of some powers of the horizontal sublaplacian and of the kohn laplacian second we make use of the framework of noncommutative geometry and of our noncommutative residue to define lower dimensional volumes in pseudohermitian geometry e g we can give sense to the area of any 3 dimensional cr manifold endowed with a pseudohermitian structure on the way we obtain a spectral interpretation of the einstein hilbert action in pseudohermitian geometry

the spectral action for moyal planes extending a result of vassilevich we obtain the asymptotic expansion for the trace of a spatially regularized heat operator l f e t where is a generalized laplacian defined with moyal products and l f is moyal left multiplication the moyal planes corresponding to any skewsymmetric matrix being spectral triples the spectral action introduced in noncommutative geometry by chamseddine and connes is computed this result generalizes the connes lott action previously computed by gayral for symplectic

the action functional in non commutative geometry we establish the equality between the restriction of the adler manin wodzicki residue or non commutative residue to pseudodifferential operators of order n on ann dimensional compact manifoldm with the trace which j dixmier constructed on the macaev ideal we then use the latter trace to recover the yang mills interaction in the context of non commutative differential geometry

global and local aspects of spectral actions the principal object in noncommutative geometry is the spectral triple consisting of an algebra a hilbert space and a dirac operator field theories are incorporated in this approach by the spectral action principle which sets the field theory action to where f is a real function such that the trace exists and is a cutoff scale in the low energy weak field limit the spectral action reproduces reasonably well the known physics including the standard model however not much is known about the spectral action beyond the low energy approximation in this paper after an extensive introduction to spectral triples and spectral actions we study various expansions of the spectral actions exemplified by the heat kernel we derive the convergence criteria for a commutative spectral triple we compute the heat kernel on the torus up to the second order in gauge connection and consider limiting cases this article is part of a special issue of journal of physics a mathematical and theoretical in honour of stuart dowker s 75th birthday devoted to applications of zeta functions and other spectral functions in mathematics and physics

strong connections on quantum principal bundles a gauge invariant notion of a strong connection is presented and characterized it is then used to justify the way in which a global curvature form is defined strong connections are interpreted as those that are induced from the base space of a quantum bundle examples of both strong and non strong connections are provided in particular such connections are constructed on a quantum deformation of the two sphere fibrations2 rp2 a certain class of stronguq 2 connections on a trivial quantum principal bundle is shown to be equivalent to the class of connections on a free module that are compatible with theq dependent hermitian metric a particular form of the yang mills action on a trivialuq 2 bundle is investigated it is proved to coincide with the yang mills action constructed by a connes and m rieffel furthermore it is shown that the moduli space of critical points of this action functional is independent ofq

spectral triples and manifolds with boundary abstract we investigate manifolds with boundary in noncommutative geometry spectral triples associated to a symmetric differential operator and a local boundary condition are constructed we show that there is no tadpole for classical dirac operators with a chiral boundary condition on spin manifolds

traces of compact operators and the noncommutative residue abstract we extend the noncommutative residue of m wodzicki on compactly supported classical pseudo differential operators of order d and generalise a connes trace theorem which states that the residue can be calculated using a singular trace on compact operators contrary to the role of the noncommutative residue for the classical pseudo differential operators a corollary is that the pseudo differential operators of order d do not have a unique trace pseudo differential operators can be non measurable in connes sense other corollaries are given clarifying the role of dixmier traces in noncommutative geometry including the definitive statement of connes original theorem

moment information and entropy evaluation for probability densities abstract how much information does the sequence of integer moments carry about the corresponding unknown absolutely continuous distribution we prove that a reliable evaluation of the corresponding shannon entropy can be done by exploiting some known theoretical results on the entropy convergence uniquely involving exact moments without solving the underlying moment problem all the procedure essentially rests on the solution of linear systems with nearly singular matrices and hence it requires both calculations in high precision and a pre conditioning technique numerical examples are provided to support the theoretical results

uniqueness of multiplicative determinants on elliptic pseudodifferential operators we describe all multiplicative determinants on the pathwise connected component of identity in the group of invertible classical pseudodifferential operators on a closed manifold that are continuous along continuous paths and the restriction to zero order operators of which is of class c1 this boils down to a description of all traces on zero order classical pseudodifferential operators which turn out to be linear combinations of the wodzicki residue w and leading symbol traces introduced in pr1 both of which are continuous consequently multiplicative determinants are parametrized by the residue determinant w87 sc and a new leading symbol determinant both of which are expressed in terms of a homogeneous component of the symbol of the logarithm of the operator

measurable operators and the asymptotics of heat kernels and zeta functions abstract in this note we answer some questions inspired by the introduction in connes 1988 1994 6 7 of the notion of measurable operators using dixmier traces these questions concern the relationship of measurability to the asymptotics of functions and heat kernels the answers have remained elusive for some 15 years

noncommutative geometry and fundamental physical interactions the lagrangian level historical sketch and description of the present situation these notes comprise i a descriptive account of the history of the subject showing how physics and mathematics interwove to develop a mathematical concept of quantum manifold relevant to elementary particle theory ii a detailed technical description from scratch of the spectral action formalism and computation

dixmier traces and residues on weak operator ideals we develop the theory of modulated operators in general principal ideals of compact operators for laplacian modulated operators we establish connes trace formula in its local euclidean model and a global version thereof it expresses dixmier traces in terms of the vector valued wodzicki residue we demonstrate the applicability of our main results in the context of log classical pseudo differential operators studied by lesch and a class of operators naturally appearing in noncommutative geometry c 2020 elsevier inc all rights reserved

inverse two sided laplace transform for probability density functions we present a method for the numerical inversion of two sided laplace transform of a probability density function the method assumes the knowledge of the first m derivatives at the origin of the function to be antitransformed the approximate analytical form is obtained by resorting to maximum entropy principle both entropy and l1 norm convergence are proved some numerical examples are illustrated

measurement of linear dependence and feedback between multiple time series abstract measures of linear dependence and feedback for multiple time series are defined the measure of linear dependence is the sum of the measure of linear feedback from the first series to the second linear feedback from the second to the first and instantaneous linear feedback the measures are nonnegative and zero only when feedback causality of the relevant type is absent the measures of linear feedback from one series to another can be additively decomposed by frequency a readily usable theory of inference for all of these measures and their decompositions is described the computations involved are modest

dixmier traces on noncompact isospectral deformations abstract we extend the isospectral deformations of connes landi and dubois violette to the case of riemannian spin manifolds carrying a proper action of the noncompact abelian group r l under deformation by a torus action a standard formula relates dixmier traces of measurable operators to integrals of functions on the manifold we show that this relation persists for actions of r l under mild restrictions on the geometry of the manifold which guarantee the dixmier traceability of those operators

universal cycles and homological invariants of locally convex algebras abstract using an appropriate notion of locally convex kasparov modules we show how to induce isomorphisms under a large class of functors on the category of locally convex algebras examples are obtained from spectral triples our considerations are based on the action of algebraic k theory on these functors and involve compatibility properties of the induction process with this action and with kasparov type products this is based on an appropriate interpretation of the connes skandalis connection formalism as an application we prove bott periodicity and a thom isomorphism for algebras of schwartz functions as a special case this applies to the theories kk for locally convex algebras considered by cuntz

nonminimal operators and non commutative residue in this paper we get the kastler kalau walze type theorem associated to nonminimal operators by heat equation asymptotics on compact manifolds without boundary we compute the lower dimensional volume associated to nonminimal operators for five dimensional compact manifolds with boundary and we also compute lower dimensional volume associated to the de rham hodge operators for six dimensional compact manifolds with boundary

supersymmetric quantum theory and non commutative geometry classical differential geometry can be encoded in spectral data such as connes spectral triples involving supersymmetry algebras in this paper we formulate non commutative geometry in terms of supersymmetric spectral data this leads to generalizations of connes non commutative spin geometry encompassing non commutative riemannian symplectic complex hermitian and hyper kahler geometry a general framework for non commutative geometry is developed from the point of view of supersymmetry and illustrated in terms of examples in particular the non commutative torus and the non commutative 3 sphere are studied in some detail

type ii non commutative geometry i dixmier trace in von neumann algebras abstract we define the notion of connes von neumann spectral triple and consider the associated index problem we compute the analytic chern connes character of such a generalized spectral triple and prove the corresponding local formula for its hochschild class this formula involves the dixmier trace for ii von neumann algebras in the case of foliations we identify this dixmier trace with the corresponding measured wodzicki residue

dixmier trace for toeplitz operators on symmetric domains abstract for toeplitz operators on bounded symmetric domains of higher rank there is no obvious way to define the dixmier trace within the toeplitz c algebra since commutators are in general not compact in this paper we solve this problem by constructing a hilbert quotient module corresponding to partitions of length 1 which leads to commutators in the macaev class l n we also obtain an explicit formula for the dixmier trace of such commutators in terms of the underlying boundary geometry which involves a new type of flag manifold defined in jordan theoretic terms

the kastler kalau walze type theorem for six dimensional manifolds with boundary in this paper we define lower dimensional volumes of spin manifolds with boundary we compute the lower dimensional volume v ol6 1 3 for 6 dimensional spin manifolds with boundary and derive the gravity on boundary from the noncommutative residue associated with dirac operators for 6 dimensional manifolds with boundary we also get a kastler kalau walze type theorem for a general fourth order operator

generalized ray singer conjecture i a manifold with a smooth boundary this paper is devoted to a proof of a generalized ray singer conjecture for a manifold with boundary the dirichlet and the neumann boundary conditions are independently given on each connected component of the boundary and the transmission boundary condition is given on the interior boundary the ray singer conjecture rs claims that for a closed manifold the combinatorial and the analytic torsion norms on the determinant of the cohomology are equal for a manifold with boundary the ratio between the analytic torsion and the combinatorial torsion is computed some new general properties of the ray singer analytic torsion are found the proof does not use any computation of eigenvalues and their asymptotic expansions or explicit expressions for the analytic torsions of any special classes of manifolds

a c algebraic approach to the principal symbol ii we introduce an abstract theory of the principal symbol mapping for pseudodifferential operators extending the results of sukochev and zanin j oper theory 2019 and providing a simple algebraic approach to the theory of pseudodifferential operators in settings important in noncommutative geometry we provide a variant of connes trace theorem which applies to certain noncommutative settings with a minimum of technical preliminaries our approach allows us to consider operators with non smooth symbols and we demonstrate the power of our approach by extending connes trace theorem to operators with non smooth symbols in three examples the lie group mathrm su 2 noncommutative tori and moyal planes

self similarity of volume measures for laplacians on p c f self similar fractals our main goal in this paper is to obtain a precise analogue of weyl s asymptotic formula for the eigenvalue distribution of laplacians on a certain class of finitely ramified or p c f self similar fractals building in particular on the work of 7 9 22 24 our main result consists in precisely identifying for the class of decimable fractals the volume measures constructed by the second author in 24 for general p c f fractals and showing that they are self similar

testing for short and long run causality a frequency domain approach abstract the framework of geweke 1982 journal of the american statistical association 77 304 324 and hosoya 1991 probability theory and related fields 88 429 444 is adopted to construct a simple test for causality in the frequency domain this test can also be applied to cointegrated systems to study the large sample properties of the test we analyze the power against a sequence of local alternatives the finite sample properties are investigated by means of monte carlo simulations our methodology is applied to investigate the predictive content of the yield spread for future output growth using quarterly us data we observe reasonable leading indicator properties at frequencies around one year and typical business cycle frequencies

applications of connes geodesic flow to trace formulas in noncommutative geometry the trace formula of chazarain duistermaat and guillemin expresses that the singularities of the distribution trace of the wave group on a compact riemannian manifoldxis included in the set of periods of the geodesic flow restricted tos x most of the objects involved in this trace formula have analogues in connes noncommutative geometry this paper shows on several significant examples of noncommutative geometry that connes definition of geodesic flow leads to statements analogous to the classical trace formula of chazarain duistermaat and guillemin

classical mechanics and geometric quantization on an infinite dimensional disc and grassmannian we discuss the classical mechanics on the grassmannian and the disc modeled on the ideal l 2 we apply methods of geometric quantization to these systems their relation to a flat symplectic space is also discussed

inference on one way effect and evidence in japanese macroeconomic data this paper provides an approach to testing a variety of causal characteristics expressed in terms of the measures of one way effect for cointegrated vector time series for this purpose we propose wald tests and their computational algorithm by means of the measures of one way effect incorporating johansen s algorithm for the maximum likelihood estimates and the likelihood ratio tests using the wald test statistics the paper also provides a method of confidence set construction for the causal measures the approach proposed in the paper includes testing granger s non causality as an instance of its multiple applications as an illustration the paper presents a characterization of the causal structure of the recent japanese macroeconomy on the basis of the proposed method and the derived evidence

the noncommutative lorentzian cylinder as an isospectral deformation we present a new example of a finite dimensional noncommutative manifold namely the noncommutative cylinder it is obtained by isospectral deformation of the canonical triple associated with the euclidean cylinder we discuss connes character formula for the cylinder in the second part we discuss noncommutative lorentzian manifolds here the definition of spectral triples involves krein spaces and operators on krein spaces a central role is played by the admissible fundamental symmetries on the krein space of square integrable sections of a spin bundle over a lorentzian manifold finally we discuss isospectral deformation of the lorentzian cylinder and determine all admissible fundamental symmetries of the noncommutative cylinder

maximum entropy in the problem of moments the maximum entropy approach to the solution of underdetermined inverse problems is studied in detail in the context of the classical moment problem in important special cases such as the hausdorff moment problem we establish necessary and sufficient conditions for the existence of a maximum entropy solution and examine the convergence of the resulting sequence of approximations a number of explicit illustrations are presented in addition to some elementary examples we analyze the maximum entropy reconstruction of the density of states in harmonic solids and of dynamic correlation functions in quantum spin systems we also briefly indicate possible applications to the lee yang theory of ising models to the summation of divergent series and so on the general conclusion is that maximum entropy provides a valuable approximation scheme a serious competitor of traditional pade like procedures

the truncated stieltjes moment problem solved by using kernel density functions in this work the problem of the approximate numerical determination of a semi infinite supported continuous probability density function pdf from a finite number of its moments is addressed the target space is carefully defined and an approximation theorem is proved establishing that the set of all convex superpositions of appropriate kernel density functions kdfs is dense in this space a solution algorithm is provided based on the established approximate representation of the target pdf and the exploitation of some theoretical results concerning moment sequence asymptotics the solution algorithm also permits us to recover the tail behavior of the target pdf and incorporate this information in our solution a parsimonious formulation of the proposed solution procedure based on a novel sequentially adaptive scheme is developed enabling a very efficient moment data inversion the whole methodology is fully illustrated by numerical examples

does sunspot numbers cause global temperatures a reconsideration using non parametric causality tests abstract in a recent paper gupta et al 2015 analyzed whether sunspot numbers cause global temperatures based on monthly data covering the period 1880 1 2013 9 the authors find that standard time domain granger causality test fails to reject the null hypothesis that sunspot numbers do not cause global temperatures for both full and sub samples namely 1880 1 1936 2 1936 3 1986 11 and 1986 12 2013 9 identified based on tests of structural breaks however frequency domain causality test detects predictability for the full sample at short 2 2 6 months cycle lengths but not the sub samples but since full sample causality cannot be relied upon due to structural breaks gupta et al 2015 conclude that the evidence of causality running from sunspot numbers to global temperatures is weak and inconclusive given the importance of the issue of global warming our current paper aims to revisit this issue of whether sunspot numbers cause global temperatures using the same data set and sub samples used by gupta et al 2015 based on an nonparametric singular spectrum analysis ssa based causality test based on this test we however show that sunspot numbers have predictive ability for global temperatures for the three sub samples over and above the full sample thus generally speaking our non parametric ssa based causality test outperformed both time domain and frequency domain causality tests and highlighted that sunspot numbers have always been important in predicting global temperatures

sums of commutators in non commutative banach function spaces let m be a ii factor and denote by its normal faithful semi finite trace for any rearrangement invariant kothe function space x on 0 let x m be the associated non commutative banach function space this paper is concerned with ideals in m of the form ix m m x m that are contained in lp m for some p 0 it is proved that an element t in ix m is a finite sum of commutators of the form a b with a ix m and b m if and only if the function t 1t t t d t belongs to x where t is the brown spectral measure of t and t t t is the non increasing rearrangement of the function with respect to t this extends to general banach function spaces a result obtained by kalton for quasi banach ideals of compact operators and implies that the dixmier s trace of a quasi nilpotent element in l1 m is always zero

entropy and density approximation from laplace transforms how much information does the laplace transforms on the real line carry about an unknown absolutely continuous distribution if we measure that information by the boltzmann gibbs shannon entropy the original question becomes how to determine the information in a probability density from the given values of its laplace transform we prove that a reliable evaluation both of the entropy and density can be done by exploiting some theoretical results about entropy convergence that involve only finitely many real values of the laplace transform without having to invert the laplace transform we provide a bound for the approximation error of in terms of the kullback leibler distance and a method for calculating the density to arbitrary accuracy

the residue of the global η function at the origin where i dvol x l denotes the riemannian measure on m using the calculus of pseudo differential operators depending upon a complex parameter which was developed by seeley 9 one can show that both q s x p and r s p can be meromorphically extended to the complex plane with simple poles at s m n u for n 0 the residues of l s x p at the poles for n 0 m are given by local formulas in the jets of the total symbol of p we define

connes s trace theorem for curved noncommutative tori application to scalar curvature in this paper we prove a version of connes s trace theorem for noncommutative tori of any dimension n 2 this allows us to recover and improve earlier versions of this result in dimensions n 2 and n 4 by fathizadeh and khalkhali we also recover connes s integration formula for flat noncommutative tori of mcdonald sukochev and zanin as a further application we prove a curved version of this integration formula in terms of the laplace beltrami operator defined by an arbitrary riemannian metric for the class of the so called self compatible riemannian metrics including the conformally flat metrics of connes and tretkoff this shows that connes s noncommutative integral allows us to recover the riemannian density this exhibits a neat link between this notion of noncommutative integral and noncommutative measure theory in the sense of operator algebras as an application of these results we set up a natural notion of scalar curvature for curved noncommutative tori

generalized limits with additional invariance properties and their applications to noncommutative geometry abstract let n be the set of all natural numbers and l l n resp l l 0 be the banach space of all bounded sequences x x 1 x 2 resp classes of lebesgue measurable bounded functions on 0 with the uniform norm we work with subsets of the set of all normalized positive functionals on l and l which are used in noncommutative geometry to define various classes of dixmier traces singular positive traces on the ideal m 1 of compact operators on an infinite dimensional hilbert space h with logarithmic divergence of the partial sums of their singular values these classes of traces are of importance in noncommutative geometry and with each such a class say b of traces a subset m 1 b m 1 on which all traces from b coincide is linked the main objective of the present article is to give new characterizations of m 1 b for commonly occurring classes of dixmier traces we also answer a question from benameur and fack 2006 4 concerning conditions on a dixmier trace under which lidskii type formulae from that article hold

complete determination of the singularity structure of zeta functions series of extended epstein type provide examples of non trivial zeta functions with important physical applications the regular part of their analytic continuation is seen to be a convergent or an asymptotic series their singularity structure is completely determined in terms of the wodzicki residue in its generalized form which is proven to yield the residua of all the poles of the zeta function and not just that of the rightmost pole obtainable from the dixmier trace the calculation is a very down to earth application of these powerful functional analytical methods in physics

connes integration formula for the noncommutative plane our aim is to prove the integration formula on the noncommutative moyal plane in terms of singular traces a la connes

quantum differentiability of essentially bounded functions on euclidean space abstract we investigate the properties of the singular values of the quantised derivatives of essentially bounded functions on r d with d 1 the commutator i sgn d 1 m f of an essentially bounded function f on r d acting by pointwise multiplication on l 2 r d and the sign of the dirac operator d acting on c 2 d 2 l 2 r d is called the quantised derivative of f we prove the condition that the function x f x 2 d 1 f x 2 d f x 2 d 2 x r d being integrable is necessary and sufficient for the quantised derivative of f to belong to the weak schatten d class this problem has been previously studied by rochberg and semmes and is also explored in a paper of connes sullivan and telemann here we give new and complete proofs using the methods of double operator integrals furthermore we prove a formula for the dixmier trace of the d th power of the absolute value of the quantised derivative for real valued f when x f x 2 d is integrable there exists a constant c d 0 such that for every continuous normalised trace on the weak trace class l 1 we have sgn d 1 m f d c d r d f x 2 d d x

the local and global parts of the basic zeta coefficient for operators on manifolds with boundary for operators on a compact manifold x with boundary x the basic zeta coefficient c0 b p1 t is the regular value at s 0 of the zeta function tr b p 1 t s where b p g is a pseudodifferential boundary operator in the boutet de monvel calculus for example the solution operator of a classical elliptic problem and p1 t is a realization of an elliptic differential operator p1 having a ray free of eigenvalues relative formulas e g for the difference between the constants with two different choices of p1 t have been known for some time and are local we here determine c0 b p1 t itself with even order p1 showing how it is put together of local residue type integrals generalizing the noncommutative residues of wodzicki guillemin fedosov golse leichtnam schrohe and global canonical trace type integrals generalizing the canonical trace of kontsevich and vishik formed of hadamard finite parts our formula generalizes a formula shown recently by paycha and scott for manifolds without boundary it leads in particular to new definitions of noncommutative residues of expressions involving log p1 t since the complex powers of p1 t lie far outside the boutet de monvel calculus the standard consideration of holomorphic families is not really useful here instead we have developed a resolvent parametric method where results from our calculus of parameter dependent boundary operators can be used

anomalies and analytic torsion on hyperbolic manifolds the global additive and multiplicative properties of the laplacian on j forms and related zeta functions are analyzed the explicit form of zeta functions on a product of closed oriented hyperbolic manifolds hd and of the multiplicative anomaly are derived we also calculate in an explicit form the analytic torsion associated with a connected sum of such manifolds

banach limits and traces on l1 abstract we introduce a new approach to traces on the principal ideal l 1 generated by any positive compact operator whose singular value sequence is the harmonic sequence distinct from the well known construction of j dixmier the new approach provides the explicit construction of every trace of every operator in l 1 in terms of translation invariant functionals applied to a sequence of restricted sums of eigenvalues the approach is based on a remarkable bijection between the set of all traces on l 1 and the set of all translation invariant functionals on l this bijection allows us to identify all known and commonly used subsets of traces dixmier traces connes dixmier traces etc in terms of invariance properties of linear functionals on l and definitively classify the measurability of operators in l 1 in terms of qualified convergence of sums of eigenvalues this classification has led us to a resolution of several open problems for the class l 1 from 7 as an application we extend connes classical trace theorem to positive normalised traces

moyal planes are spectral triples axioms for nonunital spectral triples extending those introduced in the unital case by connes are proposed as a guide and for the sake of their importance in noncommutative quantum field theory the spaces r2n endowed with moyal products are intensively investigated some physical applications such as the construction of noncommutative wick monomials and the computation of the connes lott functional action are given for these noncommutative hyperplanes

the noncommutative residue for manifolds with boundary abstract we construct a trace on the algebra of classical elements in boutet de monvel s calculus on a compact manifold with boundary of dimensionn 2 this trace coincides with wodzicki s noncommutative residue if the boundary is reduced to the empty set moreover we show that it is the unique continuous trace on this algebra up to multiplication by a constant

non commutative residues for anisotropic pseudo differential operators in rn abstract in this paper we define a non commutative residue for the algebra of classical anisotropic pseudo differential operators on r n modulo regularizing operators furthermore by means of the weyl formula we show that this trace coincides with the dixmier trace on operators of order m

the decomposition and measurement of the interdependency between second order stationary processes for a given pair of multivariate stationary processes the process of one way effect is extracted from each of the processes each process is decomposed into two orthogonal processes namely into the process generated by the one way effect of the other process and the process orthogonal to it based on the decomposition three measures characterizing the interdependency of the pair of processes are introduced they are the measure of association the measure of one way effect and the measure of reciprocity each of the measures is defined as overall as well as frequencywise measure the paper shows that the measure of association is equal to the sum of the others it discusses the relationships of those measures to the ones proposed by gel fand yaglom and by geweke