the atiyah singer index theorem as passage to the classical limit in quantum mechanics a new approach to the atiyah singer index theorem is described using the technique of continuous fields ofc algebras the proof is given in the case of elliptic pseudodifferential operators on n

equivariant deformations of homogeneous spaces abstract c algebraic deformations of homogeneous spaces g are constructed by completing dense subspaces of c 0 g in a different multiplication and c norm these deformations are equivariant in the sense that they still carry a natural action of g by left translation the motivating examples are the noncommutative heisenberg manifolds of rieffel but the construction given here is based on the authors twisted dual group algebras which are equivariant deformations of c 0 g the procedure is general enough to give other recently studied families of noncommutative spaces and some interesting new examples of simple c algebras

validity of the third law of thermodynamics for the tsallis entropy bento et al phys rev e 91 022105 2015 recently stated that the tsallis entropy violates the third law of thermodynamics for 0 1 is already inherent in the fundamental incomplete structure of the deformed logarithms and exponentials underlying the tsallis entropy then we provide the complete deformed functions and show that the tsallis entropy conforms to the third law of thermodynamics for both superadditive q 1 regimes finally we remark that the tsallis entropy does not require the use of an escort averaging scheme once it is expressed in terms of the complete deformed functions

when shannon and khinchin meet shore and johnson equivalence of information theory and statistical inference axiomatics we propose a unified framework for both shannon khinchin and shore johnson axiomatic systems we do it by rephrasing shannon khinchine axioms in terms of generalized arithmetics of kolmogorov and nagumo we prove that the two axiomatic schemes yield identical classes of entropic functionals the uffink class of entropies this allows to re establish the entropic parallelism between information theory and statistical inference that has seemed to be broken by the use of non shannonian entropies

renyi entropy yields artificial biases not in the data and incorrect updating due to the finite size data we show that the renyi entropy implies artificial biases not warranted by the data and incorrect updating information due to the finite size of the data despite being additive it is demonstrated that this is so because it does not conform to the system and subset independence axioms of shore and johnson j ieee trans inf theory 26 26 1980 iettaw0018 944810 1109 tit 1980 1056144 we finally show that the escort averaged constraints do not remedy the situation

an improved method for finding multiple roots and it s multiplicity of nonlinear equations in r abstract the aim of this paper is to develop an improved method for finding the multiple roots of nonlinear equations f x 0 in r along with an approximation to it s multiplicity when it is not known explicitly this is done by first describing a derivative free transformation g x of f x which reduces the multiple roots of f x 0 to simple roots of g x 0 then a second order newton type method is used to compute the simple roots of g x 0 and the approximation to the multiplicity of the roots of f x 0 the method is tested on two numerical examples considered by king r f king a secant method for multiple roots bit 17 1977 321 328 and results obtained are compared it is found that our method is more efficient than that given by king and obtain multiple roots as well as it s multiplicity much faster

a non iterative method for solving non linear equations abstract in this paper using a hyperbolic tangent function tanh x 0 we develop a non iterative method to estimate a root of an equation f x 0 the problem of finding root is transformed to evaluating an integral and thus we need not take account of choosing initial guess the larger the value of the better the approximation to the root alternatively we employ the signum function sgn x instead of the hyperbolic tangent function which results in an exact formula for the root availability of the present method is shown by some numerical examples for which the traditional newton s method is not appropriate

letter to the editor new higher order methods for solving nonlinear equations with multiple roots for a nonlinear equation f x 0 having a multiple root we consider steffensen s transformation t using the transformation say f q x t qf x for integer q 2 repeatedly we develop higher order iterative methods which require neither derivatives of f x nor the multiplicity of the root it is proved that the convergence order of the proposed iterative method is 1 2 q 2 for any equation having a multiple root of multiplicity m 2 the efficiency of the new method is shown by the results for some numerical examples

systemes dynamiques non commutatifs et moyennabilite on introduit la notion de fonction de type positif relativement a un systeme dynamique c dont les proprietes sont analogues a celles des fonctions de type positif classiques on montre que dans le cas d un groupe discret g agissant sur une algebre de von neumann m la moyennabilite de l action est equivalente a l existence d une suite bornee de fonctions de type positif sur g a support fini a valeurs dans le centre z m de m qui converge vers 1 pour la topologie de la convergence simple ultrafaible

harmonic analysis approach to gromov hausdorff convergence for noncommutative tori we show that the rotation algebras are limits of matrix algebras in a very strong sense of convergence for algebras with additional lipschitz structure our results generalize to higher dimensional noncommutative tori and operator valued coefficients in contrast to previous results by rieffel li kerr and latremoliere we use lipschitz norms induced by the carre du champ of certain natural dynamical systems including the heat semigroup

on the k theory of twisted higher rank graph c algebras abstract we investigate the k theory of twisted higher rank graph algebras by adapting parts of elliott s computation of the k theory of the rotation algebras we show that each 2 cocycle on a higher rank graph taking values in an abelian group determines a continuous bundle of twisted higher rank graph algebras over the dual group we use this to show that for a circle valued 2 cocycle on a higher rank graph obtained by exponentiating a real valued cocycle the k theory of the twisted higher rank graph algebra coincides with that of the untwisted one

nonadditive entropy maximization is inconsistent with bayesian updating the maximum entropy method char22 used to infer probabilistic models from data char22 is a special case of bayes s model inference prescription which in turn is grounded in basic propositional logic by contrast to the maximum entropy method the compatibility of nonadditive entropy maximization with bayes s model inference prescription has never been established here we demonstrate that nonadditive entropy maximization is incompatible with bayesian updating and discuss the immediate implications of this finding we focus our attention on special cases as illustrations

thermodynamics from first principles correlations and nonextensivity the standard formulation of thermostatistics being based on the boltzmann gibbs distribution and logarithmic shannon entropy describes idealized uncorrelated systems with extensive energies and short range interactions in this rapid communication we use the fundamental principles of ergodicity via liouville s theorem the self similarity of correlations and the existence of the thermodynamic limit to derive generalized forms of the equilibrium distribution for long range interacting systems significantly our formalism provides a justification for the well studied nonextensive thermostatistics characterized by the tsallis distribution which it includes as a special case we also give the complementary maximum entropy derivation of the same distributions by constrained maximization of the gibbs shannon entropy the consistency between the ergodic and maximum entropy approaches clarifies the use of the latter in the study of correlations and nonextensive thermodynamics

lie groupoid c algebras and weyl quantization a strict quantization of a poisson manifold p on a subset containing 0 as an accumulation point is defined as a continuous field of with a dense subalgebra on which the poisson bracket is defined and a set of continuous cross sections for which here for all whereas for one requires that in norm for any lie groupoid the vector bundle dual to the associated lie algebroid is canonically a poisson manifold let and for let be the c algebra of the family of c algebras forms a continuous field and we construct a dense subalgebra and an associated family of continuous cross sections of this field generalizing weyl quantization which define a strict quantization of many known strict quantizations are a special case of this procedure on the maps reduce to standard weyl quantization for where q is a riemannian manifold one recovers connes tangent groupoid as well as a recent generalization of weyl s prescription when is the gauge groupoid of a principal bundle one is led to the weyl quantization of a particle moving in an external yang mills field in case that is a lie group with lie algebra one recovers rieffel s quantization of the lie poisson structure on a transformation group c algebra defined by a smooth action of a lie group on a manifold q turns out to be the quantization of the poisson manifold defined by this action

nonadditive entropies yield probability distributions with biases not warranted by the data different quantities that go by the name of entropy are used in variational principles to infer probability distributions from limited data shore and johnson showed that maximizing the boltzmann gibbs form of the entropy ensures that probability distributions inferred satisfy the multiplication rule of probability for independent events in the absence of data coupling such events other types of entropies that violate the shore and johnson axioms including nonadditive entropies such as the tsallis entropy violate this basic consistency requirement here we use the axiomatic framework of shore and johnson to show how such nonadditive entropy functions generate biases in probability distributions that are not warranted by the underlying data

metric aspects of noncommutative homogeneous spaces for a closed cocompact subgroup of a locally compact group g given a compact abelian subgroup k of g and a homomorphism k g satisfying certain conditions landstad and raeburn constructed equivariant noncommutative deformations c g of the homogeneous space g generalizing rieffel s construction of quantum heisenberg manifolds we show that when g is a lie group and g is connected given any norm on the lie algebra of g the seminorm on c g induced by the derivation map of the canonical g action defines a compact quantum metric furthermore it is shown that this compact quantum metric space depends on continuously with respect to quantum gromov hausdorff distances

continuous fields of c algebras coming from group cocycles and actions recently i have been attempting to formulate a suitable c algebraic framework for the subject of deformation quantization 3 19 continuous fields of c algebras provide one of the key elements for this framework the main examples of deformation quantizations which i have constructed up to now in this c algebra framework come from letting either cocycles on groups or actions of groups vary it has thus become necessary to show that one obtains in this way fields of c aigebras that are indeed continuous since this material is of a general nature and can be useful in other situations 4 6 11 13 it has seemed appropriate to give a separate exposition of it in the present article section 1 of this article contains a review of the published results on continuous fields which we will need as well as a discussion of the fact that the approach which we will take involves treating upper and lower semi continuity separately in sect 2 we discuss the continuity of fields of c algebras which arise from varying cocycles on groups while in sect 3 we do the same for actions which vary

dynamical behaviors of some iterative methods for multiple roots of nonlinear equations recently more and more higher order iterative methods for finding the multiple roots of nonlinear equations have been presented most of them require the information of the multiplicity of roots in this paper the conjugacy maps and the extraneous fixed points of some iterative methods for finding the multiple roots are discussed basins of attractions of them are also given to demonstrate their dynamical behaviors around the multiple roots for several polynomials

order unit quantum gromov hausdorff distance we introduce a new distance distoq between compact quantum metric spaces we show that distoq is lipschitz equivalent to rieel s distance distq and give criteria for when a parameterized family of compact quantum metric spaces is continuous with respect to distoq as applications we show that the continuity of a parameterized family of quantum metric spaces induced by ergodic actions of a fixed compact group is determined by the multiplicities of the actions generalizing rieel s work on noncommutative tori and integral coadjoint orbits of semisimple compact connected lie groups we also show that the deformations of connes and landi are continuous in the parameter

generalized nonadditive entropies and quantum entanglement we examine the inference of quantum density operators from incomplete information by means of the maximization of general nonadditive entropic forms extended thermodynamic relations are given when applied to a bipartite spin 1 2 system the formalism allows one to avoid fake entanglement for data based on the bell clauser horne shimony holt observable and in general on any set of bell constraints particular results obtained with the tsallis entropy and with an introduced exponential entropic form are also discussed

strongly asymptotic morphisms on separable metrisable algebras abstract asymptotic morphisms on c algebras and their compositions were introduced by connes and higson this paper considers definitions of asymptotic morphisms on separable metrisable algebras and a compatibility condition is given which allows the composition of such morphisms a class of algebras is defined with the property that every bounded set has compact closure where the compatibility conditions are automatically satisfied three examples are given in detail the first involving a non normable algebra due to elliott natsume and nest the second is integration with respect to a certain quasi commutative spectral measure on the algebra of paths on a c algebra and the third the equivalence between the suspensions of the mapping cone and the ideal for a short exact sequence of c algebras

on q non extensive statistics with non tsallisian entropy we combine an axiomatics of renyi with the qq deformed version of khinchin axioms to obtain a measure of information i e entropy which accounts both for systems with embedded self similarity and non extensivity we show that the entropy thus obtained is uniquely solved in terms of a one parameter family of information measures the ensuing maximal entropy distribution is phrased in terms of a special function known as the lambert w function we analyze the corresponding high and low temperature asymptotics and reveal a non trivial structure of the parameter space salient issues such as concavity and schur concavity of the new entropy are also discussed

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

expansion of solution in terms of generalized eigenfunctions for a hyperbolic system with static boundary condition this paper studies a linear hyperbolic system with static boundary condition that was first studied in neves et al j funct anal 67 1986 320 344 it is shown that the spectrum of the system consists of zeros of a sine type function and the generalized eigenfunctions of the system constitute a riesz basis with parentheses for the root subspace the state space thereby decomposes into topological direct sum of root subspace and another invariant subspace in which the associated semigroup is superstable that is to say the semigroup is identical to zero after a finite time period

deformation quantization and the baum connes conjecture alternative titles of this paper would have been index theory without index or the baum connes conjecture without baum

a class of three point root solvers of optimal order of convergence the construction of a class of three point methods for solving nonlinear equations of the eighth order is presented these methods are developed by combining fourth order methods from the class of optimal two point methods and a modified newton s method in the third step obtained by a suitable approximation of the first derivative based on interpolation by a nonlinear fraction it is proved that the new three step methods reach the eighth order of convergence using only four function evaluations which supports the kung traub conjecture on the optimal order of convergence numerical examples for the selected special cases of two step methods are given to demonstrate very fast convergence and a high computational efficiency of the proposed multipoint methods some computational aspects and the comparison with existing methods are also included

an equivariant index for proper actions i abstract equivariant indices taking values in group theoretic objects have previously been defined in cases where either the group acting or the orbit space of the action is compact in this paper we define an equivariant index without assuming the group or the orbit space to be compact this allows us to generalise an index of deformed dirac operators defined for compact groups by braverman in parts ii and iii of this series we explore some properties and applications of this index

geometric invariants of the quantum hall effect we study the two dimensional hall effect with a random potential the hall conductivity is identified as a geometric invariant associated with an algebra of observables using the pairing betweenk theory and cyclic cohomology theory we identify this geometric invariant with a topological index thereby giving the hall conductivity a new interpretation

approximation of quantum tori by finite quantum tori for the quantum gromov hausdorff distance we establish that given a compact abelian group g endowed with a continuous length function l and a sequence hn n n of closed subgroups of g converging to g for the hausdorff distance induced by l then c g is the quantum gromov hausdorff limit of any sequence c hn nn n for the natural quantum metric structures and when the lifts of n to g converge pointwise to this allows us in particular to approximate the quantum tori by finite dimensional c algebras for the quantum gromov hausdorff distance moreover we also establish that if the length function l is allowed to vary we can collapse quantum metric spaces to various quotient quantum metric spaces

spectral continuity for aperiodic quantum systems i general theory abstract how does the spectrum of a schrodinger operator vary if the corresponding geometry and dynamics change is it possible to define approximations of the spectrum of such operators by defining approximations of the underlying structures in this work a positive answer is provided using the rather general setting of groupoid c algebras a characterization of the convergence of the spectra by the convergence of the underlying structures is proved in order to do so the concept of continuous field of groupoids is slightly extended by adding continuous fields of cocycles with this at hand magnetic schrodinger operators on dynamical systems or delone systems fall into this unified setting various approximations used in computational physics like the periodic or the finite cluster approximations are expressed through the tautological groupoid which provides a universal model for fields of groupoids the use of the hausdorff topology turns out to be fundamental in understanding why and how these approximations work

high order nonlinear solver for multiple roots a method of order four for finding multiple zeros of nonlinear functions is developed the method is based on jarratt s fifth order method for simple roots and it requires one evaluation of the function and three evaluations of the derivative the informational efficiency of the method is the same as previously developed schemes of lower order for the special case of double root we found a family of fourth order methods requiring one less derivative thus this family is more efficient than all others all these methods require the knowledge of the multiplicity

troublesome aspects of the renyi maxent treatment we study in great detail the possible existence of a renyi associated thermodynamics with negative results in particular we uncover a hidden relation in renyi s variational problem maxent this relation connects the two associated lagrange multipliers canonical ensemble with the mean energy u and the renyi parameter as a consequence of such relation we obtain anomalous renyi maxent thermodynamic results

cyclic cocycles on deformation quantizations and higher index theorems abstract we construct a nontrivial cyclic cocycle on the weyl algebra of a symplectic vector space using this cyclic cocycle we construct an explicit local quasi isomorphism from the complex of differential forms on a symplectic manifold to the complex of cyclic cochains of any formal deformation quantization thereof we give a new proof of nest tsygan s algebraic higher index theorem by computing the pairing between such cyclic cocycles and the k theory of the formal deformation quantization furthermore we extend this approach to derive an algebraic higher index theorem on a symplectic orbifold as an application we obtain the analytic higher index theorem of connes moscovici and its extension to orbifolds

route from discreteness to the continuum for the tsallis q entropy the existence and exact form of the continuum expression of the discrete nonlogarithmic q entropy is an important open problem in generalized thermostatistics since its possible lack implies that nonlogarithmic q entropy is irrelevant for the continuous classical systems in this work we show how the discrete nonlogarithmic q entropy in fact converges in the continuous limit and the negative of the q entropy with continuous variables is demonstrated to lead to the csiszar type q relative entropy just as the relation between the continuous boltzmann gibbs expression and the kullback leibler relative entropy as a result we conclude that there is no obstacle for the applicability of the q entropy to the continuous classical physical systems

index theory for boundary value problems via continuous fields of c algebras we prove an index theorem for boundary value problems in boutet de monvel s calculus on a compact manifold x with boundary the basic tool is the tangent semi groupoid t x generalizing the tangent groupoid defined by connes in the boundaryless case and an associated continuous field cr t x of c algebras over 0 1 its fiber in 0 cr t x can be identified with the symbol algebra for boutet de monvel s calculus for 0 the fibers are isomorphic to the algebra k of compact operators we therefore obtain a natural map k0 cr t x k0 c0 t x k0 k z using deformation theory we show that this is the analytic index map on the other hand using ideas from noncommutative geometry we construct the topological index map and prove that it coincides with the analytic index map

on the structure of twisted group c algebras we first give general structural results for the twisted group algebras c g of a locally compact group g with large abelian subgroups in particular we use a theorem of williams to realise c g as the sections of a c bundle whose fibres are twisted group algebras of smaller groups and then give criteria for the simplicity of these algebras next we use a device of rosenberg to show that when is a discrete subgroup of a solvable lie group g the k groups k c are isomorphic to certain twisted k groups k g of the homogeneous space g and we discuss how the twisting class h 3 g z depends on the cocycle

iterative methods for solving nonlinear equations with finitely many roots in an interval in this paper we consider a nonlinear equation f x 0 having finitely many roots in a bounded interval based on the so called numerical integration method b i yun a non iterative method for solving non linear equations appl math comput 198 2008 691 699 without any initial guess we propose iterative methods to obtain all the roots of the nonlinear equation in the result an algorithm to find all of the simple roots and multiple ones as well as the extrema of f x is developed moreover criteria for distinguishing zeros and extrema are included in the algorithm availability of the proposed method is demonstrated by some numerical examples

convergence radius of halley s method for multiple roots under center holder continuous condition recently a new treatment based on taylor s expansion to give the estimate of the convergence radius of iterative method for multiple roots has been presented it has been successfully applied to enlarge the convergence radius of the modified newton s method and osada s method for multiple roots this paper re investigates the convergence radius of halley s method under the condition that the derivative f m 1 of function f satisfies the center holder continuous condition we show that our result can be obtained under much weaker condition and has a wider range of application than that given by bi et al 2011 21

comment on renyi entropy yields artificial biases not in the data and incorrect updating due to the finite size data in their recent paper phys rev e 99 032134 2019 2470 004510 1103 physreve 99 032134 oikonomou and bagci have argued that renyi entropy is ill suited for inference purposes because it is not consistent with the shore johnson axioms of statistical estimation theory in this comment we seek to clarify the latter statement by showing that there are several issues in oikonomou s and bagci s reasonings which lead to erroneous conclusions when all these issues are properly accounted for no violation of shore johnson axioms is found

three step iterative methods with eighth order convergence for solving nonlinear equations a family of eighth order iterative methods for the solution of nonlinear equations is presented the new family of eighth order methods is based on king s fourth order methods and the family of sixth order iteration methods developed by chun et al per iteration the new methods require three evaluations of the function and one evaluation of its first derivative therefore this family of methods has the efficiency index which equals 1 682 kung and traub conjectured that a multipoint iteration without memory based on n evaluations could achieve optimal convergence order 2 n 1 thus we provide a new example which agrees with the conjecture of kung traub for n 4 numerical comparisons are made to show the performance of the presented methods

eighth order methods with high efficiency index for solving nonlinear equations in this paper we derive a new family of eighth order methods for solving simple roots of nonlinear equations by using weight function methods per iteration these methods require three evaluations of the function and one evaluation of its first derivative which implies that the efficiency indexes are 1 682 numerical comparisons are made to show the performance of the derived methods as shown in the illustration examples

deformation quantization of heisenberg manifolds form a smooth manifold equipped with a poisson bracket we formulate ac algebra framework for deformation quantization including the possibility of invariance under a lie group of diffeomorphisms preserving the poisson bracket we then show that the much studied non commutative tori give examples of such deformation quantizations invariant under the usual action of ordinary tori going beyond this the main results of the paper provide a construction of invariant deformation quantizations for those poisson brackets on heisenberg manifolds which are invariant under the action of the heisenberg lie group and for various generalizations suggested by this class of examples interesting examples are obtained of simplec algebras on which the heisenberg group acts ergodically

reply to comment on renyi entropy yields artificial biases not in the data and incorrect updating due to the finite size data we reply to the preceding comment by jizba and korbel jizba and korbel phys rev e 100 026101 2019 10 1103 physreve 100 026101 by first pointing out that the schur concavity proposed by them falls short of identifying the correct intervals of normalization for the optimum probability distribution even though normalization is a necessary ingredient in the entropy maximization procedure second their treatment of the subset independence axiom requires a modification of the lagrange multipliers one begins with thereby rendering the optimization less trustworthy we also explicitly demonstrate that the renyi entropy violates the subset independence axiom and compare it with the shannon entropy third the composition rule offered by jizba and korbel is shown to yield probability distributions even without a need for the entropy maximization procedure at the expense of creating artificial bias in the data

convergence radius of osada s method under center holder continuous condition recently a new treatment based on taylor s expansion to give the estimate of the convergence radius of iterative method for multiple roots has been presented it has been successfully applied to enlarge the estimate of the convergence radius of the modified newton s method for multiple roots using the similarly treatment this paper investigates the convergence radius of the osada s method under the condition that the derivative f m 1 of function f satisfies the center holder continuous condition by some examples we show the treatment is simpler and efficient once again the uniqueness ball of solution is also discussed