an improved algorithm for the multidimensional moment constrained maximum entropy problem the maximum entropy principle is a versatile tool for evaluating smooth approximations of probability density functions with the least bias beyond specified constraints in the recent paper we introduced new computational framework for the moment constrained maximum entropy problem in a multidimensional domain and developed a simple numerical algorithm capable of computing maximum entropy problem in a two dimensional domain with moment constraints of order up to 4 here we design an improved numerical algorithm for computing the maximum entropy problem in a two and higher dimensional domain with higher order moment constraints the algorithm features multidimensional orthogonal polynomial basis in the dual space of lagrange multipliers to achieve numerical stability and rapid convergence of newton iterations the new algorithm is found to be capable of solving the maximum entropy problem in the two dimensional domain with moment constraints of order up to 8 in the three dimensional domain with moment constraints of order up to 6 and in the four dimensional domain with moment constraints of order up to 4 corresponding to the total number of moment constraints of 44 83 and 69 respectively the two and higher dimensional maximum entropy test problems in the current work are based upon long term statistics of numerical simulation of the real world geophysical model for wind stress driven oceanic currents such as the gulf stream and the kuroshio

stability of the gumowski mira equation with period two coefficient abstract we investigate the stability of solutions of the gumowski mira equation with a period two coefficient y n 1 y n b n y n 2 y n 1 n 0 1 with b n 0 for n 2 k 0 for n 2 k 1 k 0 1 and the initial values y 1 y 0 are real numbers

optimal filtering of complex turbulent systems with memory depth through consistency constraints in this article we develop a linear theory for optimal filtering of complex turbulent signals with model errors through linear autoregressive models we will show that when the autoregressive model parameters are chosen such that they satisfy absolute stability and consistency conditions of at least order 2 of the classical multistep method for solving initial value problems the filtered solutions with autoregressive models of order p 2 are optimal in the sense that they are comparable to the estimates obtained from the true filter with perfect model this result is reminiscent of the lax equivalence fundamental theorem in the analysis of finite difference discretization scheme for the numerical solutions of partial differential equations we will apply this linear theory to filter two nonlinear problems the slowest mode of the truncated burgers hopf and the lorenz 96 model on these nonlinear problems we will show that whenever these linear conditions are satisfied the filtered solutions accuracy is significantly improved finally we will also apply the recently developed offline test criteria to understand the robustness of the multistep filter on various turbulent nature including the stochastically forced linear advection diffusion equation and a toy model for barotropic turbulent rossby waves

kumjian pask algebras of finitely aligned higher rank graphs abstract we extend the definition of kumjian pask algebras to include algebras associated to finitely aligned higher rank graphs we show that these kumjian pask algebras are universally defined and have a graded uniqueness theorem we also prove the cuntz krieger uniqueness theorem to do this we use a groupoid approach as a consequence of the graded uniqueness theorem we show that every kumjian pask algebra is isomorphic to the steinberg algebra associated to its boundary path groupoid we then use steinberg algebra results to prove the cuntz krieger uniqueness theorem and also to characterize simplicity and basic simplicity

challenges in climate science and contemporary applied mathematics this article discusses the challenges in climate science from the emerging viewpoint of stochastic statistical properties of turbulent dynamical systems the mathematical topics discussed here include empirical information theory fluctuation dissipation theorems reduced order stochastic modeling and the development of mathematically unambiguous exactly solvable test models for climate science that capture crucial features of vastly more complex scientific problems the applied mathematics topics include the emerging development of multiscale algorithms for filtering data assimilation and superparametrization for climate science and other problems in science and engineering as well as suitable unambiguous mathematical test problems for their behavior interesting contemporary research directions and specific open problems are mentioned throughout the article the perspective here should also be useful for applications to other complex dynamical systems arising in neural science material science and environmental mechanical engineering 2011 wiley periodicals inc

integrability and algebraic entropy of k periodic non autonomous lyness recurrences abstract this work deals with non autonomous lyness type recurrences of the form x n 2 a n x n 1 x n where a n n is a k periodic sequence of complex numbers with minimal period k we treat such non autonomous recurrences via the autonomous dynamical system generated by the birational mapping f a k f a k 1 f a 1 where f a is defined by f a x y y a y x for the cases k 1 2 3 6 the corresponding mappings have a rational first integral by calculating the dynamical degree we show that for k 4 and for k 5 generically the dynamical system is no longer rationally integrable we also prove that the only values of k for which the corresponding dynamical system is rationally integrable for all the values of the involved parameters are k 1 2 3 6

limits and degeneracies of discrete painleve equations a sequel abstract we present the discrete systems which result from the discrete painleve equations q p vi and d p v associated to the affine weyl group e 7 1 two different procedures limits and degeneracies are used giving rise to a host of new discrete painleve equations but also to some equations which are integrable through linearisation

dynamics of bimeromorphic maps of surfaces we classify bimeromorphic self maps f x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of compact kahler surfaces x in terms of their actions f h 1 1 x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i on cohomology we observe that the growth rate of f n is invariant under bimeromorphic conjugacy and that by conjugating one can always arrange that f n f n we show that the sequence f n can be bounded grow linearly grow quadratically or grow exponentially in the first three cases we show that after conjugating f is an automorphism virtually isotopic to the identity f preserves a rational fibration or f preserves an elliptic fibration respectively in the last case we show that there is a unique up to scaling expanding eigenvector for f that is nef and that f is bimeromorphically conjugate to an automorphism if and only if 2 0 we go on in this case to construct a dynamically natural positive current representing and we study the growth rate of periodic orbits of f we conclude by illustrating our results with a particular family of examples

pseudogroups and their etale groupoids abstract a pseudogroup is a complete infinitely distributive inverse monoid such inverse monoids bear the same relationship to classical pseudogroups of transformations as frames do to topological spaces the goal of this paper is to develop the theory of pseudogroups motivated by applications to group theory c algebras and aperiodic tilings our starting point is an adjunction between a category of pseudogroups and a category of etale groupoids from which we are able to set up a duality between spatial pseudogroups and sober etale groupoids as a corollary to this duality we deduce a non commutative version of stone duality involving what we call boolean inverse semigroups and boolean etale groupoids as well as a generalization of this duality to distributive inverse semigroups non commutative stone duality has important applications in the theory of c algebras it is the basis for the construction of cuntz and cuntz krieger algebras and in the case of the cuntz algebras it can also be used to construct the thompson groups we then define coverages on inverse semigroups and the resulting presentations of pseudogroups as applications we show that paterson s universal groupoid is an example of a booleanization and reconcile exel s recent work on the theory of tight maps with the work of the second author

the interplay between steinberg algebras and skew rings abstract we study the interplay between steinberg algebras and skew rings for a partial action of a group in a hausdorff locally compact totally disconnected topological space we realize the associated partial skew group ring as a steinberg algebra over the transformation groupoid attached to the partial action we then apply this realization to characterize diagonal preserving isomorphisms of partial skew group rings over commutative algebras in terms of continuous orbit equivalence of the associated partial actions finally we show that any steinberg algebra associated to a hausdorff ample groupoid can be seen as a partial skew inverse semigroup ring

mathematical strategies for filtering complex systems regularly spaced sparse observations real time filtering of noisy turbulent signals through sparse observations on a regularly spaced mesh is a notoriously difficult and important prototype filtering problem simpler off line test criteria are proposed here as guidelines for filter performance for these stiff multi scale filtering problems in the context of linear stochastic partial differential equations with turbulent solutions filtering turbulent solutions of the stochastically forced dissipative advection equation through sparse observations is developed as a stringent test bed for filter performance with sparse regular observations the standard ensemble transform kalman filter etkf has poor skill on the test bed and even suffers from filter divergence surprisingly at observable times with resonant mean forcing and a decaying energy spectrum in the partially observed signal systematic alternative filtering strategies are developed here including the fourier domain kalman filter fdkf and various reduced filters called strongly damped approximate filter sdaf variance strongly damped approximate filter vsdaf and reduced fourier domain kalman filter rfdkf which operate only on the primary fourier modes associated with the sparse observation mesh while nevertheless incorporating into the approximate filter various features of the interaction with the remaining modes it is shown below that these much cheaper alternative filters have significant skill on the test bed of turbulent solutions which exceeds etkf and in various regimes often exceeds fdkf provided that the approximate filters are guided by the off line test criteria the skill of the various approximate filters depends on the energy spectrum of the turbulent signal and the observation time relative to the decorrelation time of the turbulence at a given spatial scale in a precise fashion elucidated here

a groupoid approach to discrete inverse semigroup algebras abstract let k be a commutative ring with unit and s an inverse semigroup we show that the semigroup algebra ks can be described as a convolution algebra of functions on the universal etale groupoid associated to s by paterson this result is a simultaneous generalization of the author s earlier work on finite inverse semigroups and paterson s theorem for the universal c algebra it provides a convenient topological framework for understanding the structure of ks including the center and when it has a unit in this theory the role of gelfand duality is replaced by stone duality using this approach we construct the finite dimensional irreducible representations of an inverse semigroup over an arbitrary field as induced representations from associated groups generalizing the case of an inverse semigroup with finitely many idempotents more generally we describe the irreducible representations of an inverse semigroup s that can be induced from associated groups as precisely those satisfying a certain finiteness condition this finiteness condition is satisfied for instance by all representations of an inverse semigroup whose image contains a primitive idempotent

the multidimensional moment constrained maximum entropy problem a bfgs algorithm with constraint scaling in a recent paper we developed a new algorithm for the moment constrained maximum entropy problem in a multidimensional setting using a multidimensional orthogonal polynomial basis in the dual space of lagrange multipliers to achieve numerical stability and rapid convergence of the newton iterations here we introduce two new improvements for the existing algorithm adding significant computational speedup in situations with many moment constraints where the original algorithm is known to converge slowly the first improvement is the use of the bfgs iterations to progress between successive polynomial reorthogonalizations rather than single newton steps typically reducing the total number of computationally expensive polynomial reorthogonalizations for the same maximum entropy problem the second improvement is a constraint rescaling aimed to reduce relative difference in the order of magnitude between different moment constraints improving numerical stability of iterations due to reduced sensitivity of different constraints to changes in lagrange multipliers we observe that these two improvements can yield an average wall clock time speedup of 5 6 times compared to the original algorithm

the tight groupoid of the inverse semigroups of left cancellative small categories the second named author was partially supported by pai iii grant fqm 298 of the junta de andalucia and by the dgi mineco and european regional development fund jointly through grant mtm2017 83487 p

a practical computational framework for the multidimensional moment constrained maximum entropy principle the maximum entropy principle is a versatile tool for evaluating smooth approximations of probability density functions with a least bias beyond given constraints in particular the moment based constraints are often a common prior information about a statistical state in various areas of science including that of a forecast ensemble or a climate in atmospheric science with that in mind here we present a unified computational framework for an arbitrary number of phase space dimensions and moment constraints for both shannon and relative entropies together with a practical usable convex optimization algorithm based on the newton method with an additional preconditioning and robust numerical integration routine this optimization algorithm has already been used in three studies of predictability and so far was found to be capable of producing reliable results in one and two dimensional phase spaces with moment constraints of up to order 4 the current work extensively references those earlier studies as practical examples of the applicability of the algorithm developed below

filtering nonlinear spatio temporal chaos with autoregressive linear stochastic models abstract fundamental barriers in practical filtering of nonlinear spatio temporal chaotic systems are model errors attributed to the stiffness in resolving multiscale features recently reduced stochastic filters based on linear stochastic models have been introduced to overcome such stiffness one of them is the mean stochastic model msm based on a diagonal ornstein uhlenbeck process in fourier space despite model errors the msm shows very encouraging filtering skill especially when the hidden signal of interest is strongly chaotic in this regime the dynamical system statistical properties resemble to those of the energy conserving equilibrium statistical mechanics with gaussian invariant measure therefore the ornstein uhlenbeck process with appropriate parameters is sufficient to produce reasonable statistical estimates for the filter model in this paper we consider a generalization of the msm with a diagonal autoregressive linear stochastic model in fourier space as a filter model for chaotic signals with long memory depth with this generalization the filter prior model becomes slightly more expensive than the msm but it is still less expensive relative to integrating the perfect model which is typically unknown in real problems furthermore the associated kalman filter on each fourier mode is computationally as cheap as inverting a matrix of size d where d is the number of observed variables on each fourier mode in our numerical example d 1 using the lorenz 96 l 96 model as a testbed we show that the non markovian nature of this autoregressive model is an important feature in capturing the highly oscillatory modes with long memory depth second we show that the filtering skill with autoregressive models supersedes that with msm in weakly chaotic regime where the memory depth is longer in strongly chaotic regime the performance of the ar p filter is still better or at least comparable to that of the msm most importantly we find that this reduced filtering strategy is not as sensitive as standard ensemble filtering strategies to additional intrinsic model errors that are often encountered when model parameters are incorrectly specified

hamiltonian structure and statistically relevant conserved quantities for the truncated burgers hopf equation in dynamical systems with intrinsic chaos many degrees of freedom and many conserved quantities a fundamental issue is the statistical relevance of suitable subsets of these conserved quantities in appropriate regimes the galerkin truncation of the burgers hopf equation has been introduced recently as a prototype model with solutions exhibiting intrinsic stochasticity and a wide range of correlation scaling behavior that can be predicted successfully by simple scaling arguments here it is established that the truncated burgers hopf model is a hamiltonian system with hamiltonian given by the integral of the third power this additional conserved quantity beyond the energy has been ignored in previous statistical mechanics studies of this equation thus the question arises of the statistical significance of the hamiltonian beyond that of the energy first an appropriate statistical theory is developed that includes both the energy and hamiltonian then a convergent monte carlo algorithm is developed for computing equilibrium statistical distributions the probability distribution of the hamiltonian on a microcanonical energy surface is studied through the monte carlo algorithm and leads to the concept of statistically relevant and irrelevant values for the hamiltonian empirical numerical estimates and simple analysis are combined to demonstrate that the statistically relevant values of the hamiltonian have vanishingly small measure as the number of degrees of freedom increases with fixed mean energy the predictions of the theory for relevant and irrelevant values for the hamiltonian are confirmed through systematic numerical simulations for statistically relevant values of the hamiltonian these simulations show a surprising spectral tilt rather than equipartition of energy this spectral tilt is predicted and confirmed independently by monte carlo simulations based on equilibrium statistical mechanics together with a heuristic formula for the tilt on the other hand the theoretically predicted correlation scaling law is satisfied both for statistically relevant and irrelevant values of the hamiltonian with excellent accuracy the results established here for the burgers hopf model are a prototype for similar issues with significant practical importance in much more complex geophysical applications several interesting mathematical problems suggested by this study are mentioned in the final section 2002 wiley periodicals inc

non integrability of measure preserving maps via lie symmetries we consider the problem of characterizing for certain natural number m the local c m non integrability near elliptic fixed points of smooth planar measure preserving maps our criterion relates this non integrability with the existence of some lie symmetries associated to the maps together with the study of the finiteness of its periodic points one of the steps in the proof uses the regularity of the period function on the whole period annulus for non degenerate centers question that we believe that is interesting by itself the obtained criterion can be applied to prove the local non integrability of the cohen map and of several rational maps coming from second order difference equations

a maximum entropy method for particle filtering standard ensemble or particle filtering schemes do not properly represent states of low priori probability when the number of available samples is too small as is often the case in practical applications we introduce here a set of parametric resampling methods to solve this problem motivated by a general h theorem for relative entropy we construct parametric models for the filter distributions as maximum entropy minimum information models consistent with moments of the particle ensemble when the prior distributions are modeled as mixtures of gaussians our method naturally generalizes the ensemble kalman filter to systems with highly non gaussian statistics we apply the new particle filters presented here to two simple test cases a one dimensional diffusion process in a double well potential and the three dimensional chaotic dynamical system of lorenz

convex subshifts separated bratteli diagrams and ideal structure of tame separated graph algebras abstract we introduce a new class of partial actions of free groups on totally disconnected compact hausdorff spaces which we call convex subshifts these serve as an abstract framework for the partial actions associated with finite separated graphs in much the same way as classical subshifts generalize the edge shift of a finite graph we define the notion of a finite type convex subshift and show that any such subshift is kakutani equivalent to the partial action associated with a finite bipartite separated graph we then study the ideal structure of both the full and the reduced tame graph c algebras o e c and o r e c of a separated graph e c and of the abelianized leavitt path algebra l k ab e c as well these algebras are the reduced crossed products with respect to the above mentioned partial actions and we prove that there is a lattice isomorphism between the lattice of induced ideals and the lattice of hereditary d saturated subsets of a certain infinite separated graph f d built from e c called the separated bratteli diagram of e c we finally use these tools to study simplicity and primeness of the tame separated graph algebras

mappings of hirota kimura yahagi type can have periodic coefficients too we present an extension of mappings of hirota kimura yahagi type where the coefficients instead of being constant are periodic functions of the independent variable several examples of mappings with periodic coefficients of various periods are produced we prove the integrability of all the examples by constructing the corresponding conserved quantities

strongly graded groupoids and strongly graded steinberg algebras abstract we study strongly graded groupoids which are topological groupoids g equipped with a continuous surjective functor g to a discrete group such that 1 1 1 for all we introduce the category of graded g sheaves and prove an analogue of dade s theorem g is strongly graded if and only if every graded g sheaf is induced by a g e sheaf the steinberg algebra of a graded ample groupoid is graded and we prove that the algebra is strongly graded if and only if the groupoid is applying this result we obtain a complete graphical characterisation of strongly graded leavitt path and kumjian pask algebras

studying discrete dynamical systems through differential equations abstract in this paper we consider dynamical systems generated by a diffeomorphism f defined on u an open subset of r n and give conditions over f which imply that their dynamics can be understood by studying the flow of an associated differential equation x x x also defined on u in particular the case where f has n 1 functionally independent first integrals is considered in this case x is constructed by imposing that it shares with f the same set of first integrals and that the functional equation f x det d f x x x u has some non zero solution several examples for n 2 3 are presented most of them coming from several well known difference equations

zero entropy for some birational maps of c2 abstract in this study we consider a special case of the family of birational maps f c 2 c 2 which were dynamically classified by 13 we identify the zero entropy subfamilies of f and explicitly give the associated invariant fibrations in particular we highlight all of the integrable and periodic mappings

mobius functions and semigroup representation theory ii character formulas and multiplicities abstract we generalize the character formulas for multiplicities of irreducible constituents from group theory to semigroup theory using rota s theory of mobius inversion the technique works for a large class of semigroups including inverse semigroups semigroups with commuting idempotents idempotent semigroups and semigroups with basic algebras using these tools we are able to give a complete description of the spectra of random walks on finite semigroups admitting a faithful representation by upper triangular matrices over the complex numbers these include the random walks on chambers of hyperplane arrangements studied by bidigare hanlon rockmore brown and diaconis applications are also given to decomposing tensor powers and exterior products of rook matrix representations of inverse semigroups generalizing and simplifying earlier results of solomon for the rook monoid

the ideal structure of algebraic partial crossed products given a partial action of a discrete group g on a hausdorff locally compact totally disconnected topological space x we consider the correponding partial action of g on the algebra lc x consisting of all locally constant compactly supported functions on x taking values in a given field k we then study the ideal structure of the algebraic partial crossed product lc x g after developing a theory of induced ideals we show that every ideal in lc x g may be obtained as the intersection of ideals induced from isotropy groups thus proving an algebraic version of the effros hahn conjecture

interpolating irregularly spaced observations for filtering turbulent complex systems we present a numerically fast reduced filtering strategy the fourier domain kalman filter with appropriate interpolations to account for irregularly spaced observations of complex turbulent signals the design of such a reduced filter involves i interpolating irregularly spaced observations to the model regularly spaced grid points ii understanding under which situation the small scale oscillatory artifact from such interpolation won t degrade the filtered solutions iii understanding when the interpolated covariance structure can be approximated by its diagonal terms when observations are corrupted by independent gaussian noise and iv applying a scalar kalman filter formula on each fourier component independently with an approximate diagonal interpolated covariance matrix from the practical point of view there is an emerging need to understand the effect of i toward the filtered solutions for example in utilizing the data produced from various interpolation techniques that merge multiple satellite measurements of atmospheric and ocean dynamical quantities to understand point iii above and to see how many of nondiagonal terms are effectively ignored in iv we compute a ratio lambda between the largest nondiagonal components and the smallest diagonal components of the interpolated covariance matrix we find that for piecewise linear interpolation this ratio lambda is always smaller than that of the alternative interpolation schemes such as trigonometric and cubic spline for any irregularly spaced observation networks when observations are not so sparse with small noise we find that the small scale oscillatory artifact in ii above is negligible when piecewise linear interpolation is used whereas for the other schemes such as the nearest neighbor trigonometric and cubic spline interpolation the oscillatory artifact degrades the filtered solutions significantly finally we also find that the reduced filtering strategy with piecewise linear interpolation produces more accurate filtered solutions than conventional approaches when observations are extremely irregularly spaced such that the ratio lambda is not so small and very sparse

etale groupoids and their quantales abstract we establish close and previously unknown relations between quantales and groupoids in particular to each etale groupoid either localic or topological there is associated a unital involutive quantale we obtain a bijective correspondence between localic etale groupoids and their quantales which are given a rather simple characterization and here are called inverse quantal frames we show that the category of inverse quantal frames is equivalent to the category of complete and infinitely distributive inverse monoids and as a consequence we obtain a non functorial correspondence between these and localic etale groupoids that generalizes more classical results concerning inverse semigroups and topological etale groupoids this generalization is entirely algebraic and it is valid in an arbitrary topos as a consequence of these results we see that a localic groupoid is etale if and only if its sublocale of units is open and its multiplication map is semiopen and an analogue of this holds for topological groupoids in practice we are provided with new tools for constructing localic and topological etale groupoids as well as inverse semigroups for instance via presentations of quantales by generators and relations the characterization of inverse quantal frames is to a large extent based on a new quantale operation here called a support whose properties are thoroughly investigated and which may be of independent interest

diagonal preserving isomorphisms of etale groupoid algebras abstract work of jean renault shows that for topologically principal etale groupoids a diagonal preserving isomorphism of reduced c algebras yields an isomorphism of groupoids several authors have proved analogues of this result for ample groupoid algebras over integral domains under suitable hypotheses in this paper we extend the known results by allowing more general coefficient rings and by weakening the hypotheses on the groupoids our approach has the additional feature that we only need to impose conditions on one of the two groupoids applications are given to leavitt path algebras

folding transformations of the painleve equations new symmetries of the painleve differential equations called folding transformations are determined these transformations are not birational but algebraic transformations of degree 2 3 or 4 these are associated with quotients of the spaces of initial conditions of each painleve equation we make the complete list of such transformations up to birational symmetries we also discuss correspondences of special solutions of painleve equations

notes on the non autonomous lyness equation abstract we prove that the solutions of the non autonomous lyness equation x n 1 x n b n x n 1 b n 0 x 0 0 x 1 0 are bounded away from zero and infinity if the sequence b n is monotonic we also give an example to show that the solutions need not be bounded even when the sequence b n attains only two values

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

on quispel roberts thompson extensions and integrable correspondences we present several novel examples of an extension of the quispel roberts thompson qrt mapping in which the coefficients are periodic functions of the independent variable and we prove that for period 2 this extension includes all so called asymmetric qrt mappings we establish the integrability of these new mappings by explicitly constructing rational invariants for them which can be parametrised in terms of elliptic functions moreover we show that the correspondences induced by these conserved quantities are all integrable i e exhibit polynomial growth of the number of images they produce and we present a conjecture pertaining to the degree of the growth the case of hirota kimura yahagi mappings and their associated correspondences is examined as well and we give examples of integrable correspondences associated with such mappings

the ideal structure of steinberg algebras abstract given an ample action of an inverse semigroup on a locally compact and hausdorff topological space we study the ideal structure of the crossed product algebra associated with it by developing a theory of induced ideals we manage to prove that every ideal in the crossed product algebra may be obtained as the intersection of ideals induced from isotropy group algebras this can be interpreted as an algebraic version of the effros hahn conjecture finally as an application of our result we study the ideal structure of a steinberg algebra associated with an ample groupoid by interpreting it as an inverse semigroup crossed product algebra

a perspective on non commutative frame theory abstract this paper extends the fundamental results of frame theory to a non commutative setting where the role of locales is taken over by etale localic categories this involves ideas from quantale theory and from semigroup theory specifically ehresmann semigroups restriction semigroups and inverse semigroups we prove several main results to start with we establish a duality between the category of complete restriction monoids and the category of etale localic categories the relationship between monoids and categories is mediated by a class of quantales called restriction quantal frames this result builds on the work of pedro resende on the connection between pseudogroups and etale localic groupoids but in the process we both generalize and simplify for example we do not require involutions and in addition we render his result functorial a wider class of quantales called multiplicative ehresmann quantal frames is put into a correspondence with those localic categories where the multiplication structure map is semiopen and all the other structure maps are open we also project down to topological spaces and as a result extend the classical adjunction between locales and topological spaces to an adjunction between etale localic categories and etale topological categories in fact varying morphisms we obtain several adjunctions just as in the commutative case we restrict these adjunctions to spatial sober and coherent spectral equivalences the classical equivalence between coherent frames and distributive lattices is extended to an equivalence between coherent complete restriction monoids and distributive restriction semigroups consequently we deduce several dualities between distributive restriction semigroups and spectral etale topological categories we also specialize these dualities for the setting where the topological categories are cancellative or are groupoids our approach thus links unifies and extends the approaches taken in the work by lawson and lenz and by resende

filtering skill for turbulent signals for a suite of nonlinear and linear extended kalman filters the filtering skill for turbulent signals from nature is often limited by errors due to utilizing an imperfect forecast model in particular real time filtering and prediction when very limited or no a posteriori analysis is possible e g spread of pollutants storm surges tsunami detection etc introduces a number of additional challenges to the problem here a suite of filters implementing stochastic parameter estimation for mitigating model error through additive and multiplicative bias correction is examined on a nonlinear exactly solvable stochastic test model mimicking turbulent signals in regimes ranging from configurations with strongly intermittent transient instabilities associated with positive finite time lyapunov exponents to laminar behavior stochastic parameterization extended kalman filter spekf used as a benchmark here involves exact formulas for propagating the mean and covariance of the augmented forecast model including the unresolved parameters the remaining filters use the same nonlinear forecast model but they introduce model error through different moment closure approximations and or linear tangent approximation used for computing the second order statistics of the augmented stochastic forecast model a comprehensive study of filter performance is carried out in the presence of various moment closure errors which are enhanced by additional model errors due to incorrect parameters inducing additive and multiplicative stochastic biases the estimation skill of the unresolved stochastic parameters is also discussed and it is shown that the linear tangent filter despite its popularity is completely unreliable in many turbulent regimes for both parameter estimation and filtering moreover regimes of filter divergence for the linear tangent filter are identified the results presented here provide useful guidelines for filtering turbulent high dimensional spatially extended systems with more general model errors as well as for designing more skillful methods for superparameterization of unresolved intermittent processes in complex multi scale models they also provide unambiguous benchmarks for the capabilities of linear and nonlinear extended kalman filters using incorrect statistics on an exactly solvable test bed with rich and realistic dynamics

how to find the conserved quantities of nonlinear discrete equations a method of finding conserved quantities of nonlinear ordinary difference equations is briefly discussed the method is based on the hypothesis that the conserved quantities are expressed by a ratio of two polynomials of dependent variables we obtain the conserved quantities of the discrete equation of motion of an anharmonic oscillator and a nonlinear difference equation of third order

z graded rings as cuntz pimsner rings abstract given a z graded ring a and a subring r a it is natural to ask whether a can be realised as the cuntz pimsner ring of some r system in this paper we derive sufficient conditions on a and r for this to be the case as an application we give conditions under which the steinberg algebra a k g associated to a z graded groupoid g n z g n can be realised as the cuntz pimsner ring of an a k g 0 system

self similar graphs a unified treatment of katsura and nekrashevych c algebras abstract given a graph e an action of a group g on e and a g valued cocycle on the edges of e we define a c algebra denoted o g e which is shown to be isomorphic to the tight c algebra associated to a certain inverse semigroup s g e built naturally from the triple g e as a tight c algebra o g e is also isomorphic to the full c algebra of a naturally occurring groupoid g tight s g e we then study the relationship between properties of the action of the groupoid and of the c algebra with an emphasis on situations in which o g e is a kirchberg algebra our main applications are to katsura algebras and to certain algebras constructed by nekrashevych from self similar groups these two classes of c algebras are shown to be special cases of our o g e and many of their known properties are shown to follow from our general theory

mathematical test criteria for filtering complex systems plentiful observations an important emerging scientific issue is the real time filtering through observations of noisy turbulent signals for complex systems as well as the statistical accuracy of spatio temporal discretizations for such systems these issues are addressed here in detail for the setting with plentiful observations for a scalar field through explicit mathematical test criteria utilizing a recent theory a j majda m j grote explicit off line criteria for stable accurate time filtering of strongly unstable spatially extended systems proceedings of the national academy of sciences 104 4 2007 1124 1129 for plentiful observations the number of observations equals the number of mesh points these test criteria involve much simpler decoupled complex scalar filtering test problems with explicit formulas and elementary numerical experiments which are developed here as guidelines for filter performance the theory includes information criteria to avoid filter divergence with large model errors asymptotic kalman gain filter stability and accurate filtering with small ensemble size as well as rigorous results delineating the role of various turbulent spectra for filtering under mesh refinement these guidelines are also applied to discrete approximations for filtering the stochastically forced dissipative advection equation with very turbulent and noisy signals with either an equipartition of energy or 5 3 turbulent spectrum with infrequent observations as severe test problems the theory and companion simulations demonstrate accurate statistical filtering in this context with implicit schemes with large time step with very small ensemble sizes and even with unstable explicit schemes under appropriate circumstances provided the filtering strategies are guided by the off line theoretical criteria the surprising failure of other strongly stable filtering strategies is also explained through these off line criteria

quantifying predictability through information theory small sample estimation in a non gaussian framework many situations in complex systems require quantitative estimates of the lack of information in one probability distribution relative to another in short term climate and weather prediction examples of these issues might involve the lack of information in the historical climate record compared with an ensemble prediction or the lack of information in a particular gaussian ensemble prediction strategy involving the first and second moments compared with the non gaussian ensemble itself the relative entropy is a natural way to quantify the predictive utility in this information and recently a systematic computationally feasible hierarchical framework has been developed in practical systems with many degrees of freedom computational overhead limits ensemble predictions to relatively small sample sizes here the notion of predictive utility in a relative entropy framework is extended to small random samples by the definition of a sample utility a measure of the unlikeliness that a random sample was produced by a given prediction strategy the sample utility is the minimum predictability with a statistical level of confidence which is implied by the data two practical algorithms for measuring such a sample utility are developed here the first technique is based on the statistical method of null hypothesis testing while the second is based upon a central limit theorem for the relative entropy of moment based probability densities these techniques are tested on known probability densities with parameterized bimodality and skewness and then applied to the lorenz 96 model a recently developed toy climate model with chaotic dynamics mimicking the atmosphere the results show a detection of non gaussian tendencies of prediction densities at small ensemble sizes with between 50 and 100 members with a 95 confidence level

simplicity of algebras associated to non hausdorff groupoids the first named author was partially supported by marsden grant 15 uoo 071 from the royal society of new zealand the second named author was partially supported by cnpq the third named author was partially supported by pai iii grant fqm 298 of the junta de andaluc ia and by the dgi mineco and european regional development fund jointly through grants mtm2014 53644 p and mtm2017 83487 p the fourth named author was partially supported by the australian research council grant dp150101595 the fifth named author was partially supported by a carleton university internal research grant