tensor product of quantum logics a quantum logic is the couple l m where l is an orthomodular lattice and m is a strong set of states on l the jauch piron property in the form is also supposed for any state of m a tensor product of quantum logics is defined this definition is compared with the definition of a free orthodistributive product of orthomodular lattices the existence and uniqueness of the tensor product in special cases of hilbert space quantum logics and one quantum and one classical logic are studied

extensions of convexity models the notion of convexity model is introduced to provide a general frame for statistical theories of physical interest this frame encompasses in particular the classical and the quantum cases in a convexity model the states of the physical system and the convex structure they form play a basic role observables and related quantities are then naturally defined the notion of extensions of a convexity model is studied it appears physically relevant to cope with several needs paradigmatically with the one of viewing the physical system as a part of a compound system we focus attention on quantum like extensions of both the usual classical and quantum convexity models as well as on classical like extensions of the quantum model the behavior of state overlapping and state superposition under model extension is briefly examined

nilpotency almost nonnegative curvature and the gradient flow on alexandrov spaces we show that almost nonnegatively curved m manifolds are up to finite cover nilpotent spaces in the sense of homotopy theory and have c m nilpotent fundamental groups we also show that up to a finite cover almost nonnegatively curved manifolds are fiber bundles with simply connected fibers over nilmanifolds

a connection between distributivity and locality in compound p lattices a p lattice is a complete orthomodular atomic lattice l which is formed by the set of propositions of a physical system a composition of physical systems is considered and some concept of locality in a compound physical system is represented in terms of p lattices we give a remark toward necessary and sufficient conditions for it to hold which have been provided implicitly in antecedent studies and we show that it can be provided under weaker conditions

the long time behavior of 3 dimensional ricci flow on certain topologies de gruyter 2017 in this paper we analyze the long time behavior of 3 dimensional ricci flow with surgery we prove that under the topological condition that the initial manifold only has non aspherical or hyperbolic components in its geometric decomposition there are only finitely many surgeries and the curvature is bounded by ct 1 for large t this proves a conjecture of perelman for this class of initial topologies the proof of this fact illustrates the fundamental ideas that are used in the subsequent papers of the author

cohomogeneity one topological manifolds revisited we prove a structure theorem for closed topological manifolds of cohomogeneity one this result corrects an oversight in the literature we complete the equivariant classification of closed simply connected cohomogeneity one topological manifolds in dimensions 5 6 and 7 and obtain topological characterizations of these spaces in these dimensions these manifolds are homeomorphic to smooth manifolds

geometric structures gromov norm and kodaira dimensions abstract we define the kodaira dimension for 3 dimensional manifolds through thurston s eight geometries along with a classification in terms of this kodaira dimension we show this is compatible with other existing kodaira dimensions and the partial order defined by non zero degree maps for higher dimensions we explore the relations of geometric structures and mapping orders with various kodaira dimensions and other invariants especially we show that a closed geometric 4 manifold has nonvanishing gromov norm if and only if it has geometry h 2 h 2 h 2 c or h 4

hard and soft packing radius theorems singular spaces are playing an increasingly important role in riemannian geometry in particular this is true of the so called alexandrov spaces i e finite hausdorff dimensional complete inner metric spaces with a lower curvature bound in the local triangle comparison sense interest in alexandrov spaces is largely explained by the fact that the natural process of taking gromov hausdorff limits is closed in alexandrov geometry but not in riemannian geometry this simple observation has become a powerful tool in riemannian geometry see for example fy gp2 or gpw and pe in this paper we use a new means to obtain results in riemannian geometry by studying alexandrov spaces the operation of forming spherical suspensions of positively curved spaces an operation which is also closed in alexandrov geometry but not in riemannian geometry will be applied to prove a differentiable sphere theorem which is unlike all previous solutions to the following basic problem

collapsing manifolds obtained by kummer type constructions we construct f structures on a bott manifold and on some other manifolds obtained by kummer type constructions we also prove that if m e x where e is a fiber bundle with structure group g and a fiber admitting a g invariant metric of non negative sectional curvature and x admits an f structure with one trivial covering then one can construct on m a sequence of metrics with sectional curvature uniformly bounded from below and volume tending to zero i e vol k m 0 as a corollary we prove that all the elements in the spin cobordism ring can be represented by manifolds m with vol k m 0

collapsing with a lower bound on the curvature operator cheeger and gromov showed that f structures are related to collapse with a double sided curvature bound we define fibered f structures and extend some of the cheeger gromov results to the setting of collapse with a lower bound on the curvature operator

small bound for birational automorphism groups of algebraic varieties with an appendix by yujiro kawamata we give an effective upper bound of bir x for the birational automorphism group of an irregular n fold with n 3 of general type in terms of the volume v v x under an albanese smoothness and simplicity condition to be precise rm bir x leq d 3 v 10 an optimum linear bound rm bir x leq frac 1 3 times 42 3 v is obtained for those threefolds with non maximal albanese dimension for all n 3 a bound rm bir x leq d n v 10 is obtained when albx is generically finite alb x is smooth and alb x is simple

ml parameter estimation of a multiscale stochastic process using the em algorithm an algorithm for estimation of the parameters of a multiscale stochastic process based on scale recursive dynamics on trees is presented the expectation maximization algorithm is used to provide maximum likelihood estimates for the general case of a nonhomogeneous tree with no fixed structure for the process dynamics experimental results are presented using synthetic data

on inferring quantum energies and expectation values abstract a maximum entropy like inference approach is proposed that yields ground and excited states energies and expectation values without solving schrodinger s equation the method is tested with reference to the quartic anharmonic oscillator and other model potentials and yields good results without undue numerical effort

vanishing quantum discord is necessary and sufficient for completely positive maps two long standing open problems in quantum theory are to characterize the class of initial system bath states for which quantum dynamics is equivalent to i a map between the initial and final system states and ii a completely positive cp map the cp map problem is especially important due to the widespread use of such maps in quantum information processing and open quantum systems theory here we settle both these questions by showing that the answer to the first is all with the resulting map being hermitian and that the answer to the second is that cp maps arise exclusively from the class of separable states with vanishing quantum discord

sharp holder continuity of tangent cones for spaces with a lower ricci curvature bound and applications we prove a new estimate on manifolds with a lower ricci bound which asserts that the geometry of balls centered on a minimizing geodesic can change in at most a holder continuous way along the geodesic we give examples that show that the holder exponent along with essentially all the other consequences that follow from this estimate are sharp among the applications is that the regular set is convex for any non collapsed limit of einstein metrics in the general case of a potentially collapsed limit of manifolds with just a lower ricci curvature bound we show that the regular set is weakly convex and a e convex we also show two conjectures of cheeger colding one of these asserts that the isometry group of any even collapsed limit of manifolds with a uniform lower ricci curvature bound is a lie group the other asserts that the dimension of any limit space is the same everywhere

a boundary of the set of the riemannian manifolds with bounded curvatures and diameters on etudie la structure de la fermeture de l ensemble constitue par des classes d isometrie de varietes de riemann a courbures et diametres bornes

finite subgroups of algebraic groups generalizing a classical theorem of jordan to arbitrary characteristic we prove that every finite subgroup of gln over a field of any characteristic p possesses a subgroup of bounded index which is composed of finite simple groups of lie type in characteristic p a commutative group of order prime to p and a p group while this statement can be deduced from the classification of finite simple groups our proof is self contained and uses methods only from algebraic geometry and the theory of linear algebraic groups we believe that our results can serve as a viable substitute for classification in a range of applications in various areas of mathematics

morse novikov cohomology of almost nonnegatively curved manifolds abstract let m n be a closed manifold of almost nonnegative sectional curvature and nonzero first de rham cohomology group using a topological argument we show that the morse novikov cohomology group h p m n vanishes for any p and h d r 1 m n 0 based on a new integral formula we also show that a similar result holds for a closed manifold of almost nonnegative ricci curvature under the additional assumption that its curvature operator is uniformly bounded from below

geometric diffeomorphism finiteness in low dimensions and homotopy group finiteness the main results of this note consist in the following two geometric finiteness theorems for diffeomorphism types and homotopy groups of closed simply connected manifolds

locality and orthomodular structure of compound systems a p lattice is defined as a complete orthomodular atomic lattice l which is formed by the set of propositions of a physical system a composition of physical systems in the framework of p lattices is considered and some notions of locality are given it is shown that the following statements about compound systems are equivalent a all atoms of a compound system are reducible to those of its subsystems b all pure states of a compound system are separable into those of its subsystems c a compound system has statistical property independence d at least one of the subsystems is classical e bell type inequalities hold

post classification version of jordan s theorem on finite linear groups abstract using classification of finite simple groups i show that a finite subgroup g of gln c where c the complex numbers contains a commutative normal subgroup m of index at most n 1 nalogn b moreover if g is primitive and does not contain normal subgroups that are direct products of large alternating groups then the factor n 1 can be dropped i further show that similar statements hold also in characteristics p 2 if one takes m to be an extension of a group of lie type of characteristic p by a solvable group that has a normal p subgroup with commutative p quotient these results improve the celebrated theorems of jordan and of brauer and feit

ricci curvature and volume convergence the purpose of this paper is to give a new integral estimate of distances and angles on manifolds with a given lower ricci curvature bound this will provide us with an integral version of the toponogov comparison triangle theorem for ricci curvature and almost extreme triangles see the earlier works cl and c2 for an analog of this when the manifold has positive ricci curvature using this we prove the anderson cheeger conjecture saying that the volume is a continuous function on the space of all closed n manifolds with ricci curvature greater or equal to n 1 equipped with the gromovhausdorff metric we also prove gromov s conjecture for n 57 3 saying that an almost nonnegatively ricci curved n manifold with first betti number equal to n is a torus further we prove a conjecture of anderson cheeger saying that an open n manifold with nonnegative ricci curvature whose tangent cone at infinity is in is itself in finally we prove a conjecture of fukaya yamaguchi we will now describe these results in more detail let dgh denote the gromov hausdorff distance glp first we have the following result which was conjectured by anderson cheeger

restrictions on collapsing with a lower sectional curvature bound we obtain new topological information about the local structure of collapsing under a lower sectional curvature bound as an application we prove a new sphere theorem and obtain a partial result towards the conjecture that not every alexandrov space can be obtained as a limit of a sequence of riemannian manifolds with sectional curvature bounded from below

some remarks on assignment maps we study the properties of general linear assignment maps showing that positivity axiom can be suitably relaxed and propose a new class of dynamical maps generalized dynamics a puzzling result arising in such a context in quantum information theory is also discussed

a quantum logical and geometrical approach to the study of improper mixtures we study improper mixtures from a quantum logical and geometrical point of view taking into account the fact that improper mixtures do not admit an ignorance interpretation and must be considered as states in their own right we do not follow the standard approach which considers improper mixtures as measures over the algebra of projections instead of it we use the convex set of states in order to construct a new lattice whose atoms are all physical states pure states and improper mixtures this is done in order to overcome one of the problems which appear in the standard quantum logical formalism namely that for a subsystem of a larger system in an entangled state the conjunction of all actual properties of the subsystem does not yield its actual state in fact its state is an improper mixture and cannot be represented in the von neumann lattice as a minimal property which determines all other properties as is the case for pure states or classical systems the new lattice also contains all propos

filtering via simulation auxiliary particle filters this article analyses the recently suggested particle approach to filtering time series we suggest that the algorithm is not robust to outliers for two reasons the design of the simulators and the use of the discrete support to represent the sequentially updating prior distribution here we tackle the first of these problems

volume collapsed three manifolds with a lower curvature bound in this paper we determine the topology of three dimensional complete orientable riemannian manifolds with a uniform lower bound of sectional curvature whose volume is sufficiently small

counting primes groups and manifolds let lambda sl 2 z be the modular group and let c n lambda be the number of congruence subgroups of lambda of index at most n we prove that lim n infinity log c n lambda log n 2 log log n 3 2 sqrt 2 4 the proof is based on the bombieri vinogradov riemann hypothesis on the average and on the solution of a new type of extremal problem in combinatorial number theory similar surprisingly sharp estimates are obtained for the subgroup growth of lattices in higher rank semisimple lie groups if g is such a lie group and gamma is an irreducible lattice of g it turns out that the subgroup growth of gamma is independent of the lattice and depends only on the lie type of the direct factors of g it can be calculated easily from the root system the most general case of this result relies on the generalized riemann hypothesis but many special cases are unconditional the proofs use techniques from number theory algebraic groups finite group theory and combinatorics

margulis lemma for compact lie groups we improve margulis lemma for a compact connected lie group g there is a neighborhood u of the identity such that for any finite subgroup gamma subset g u cap gamma generates an abelian group we show that for each n there exists an integer w n 0 such that if h is a closed subgroup of a compact connected lie group g of dimension n then the quotient group h h0 has an abelian subgroup of index le w n where h0 is the identity component of h as an application we show that the fundamental group of the homogeneous space g h has an abelian subgroup of index le w n we show this same property for the fundamental groups of almost non negatively curved n manifolds whose universal coverings are not collapsed

multiscale systems kalman filters and riccati equations an algorithm analogous to the rauch tung striebel algorithm spl minus consisting of a fine to coarse kalman filter like sweep followed by a coarse to fine smoothing step spl minus was developed previously by the authors ibid vol 39 no 3 p 464 78 1994 in this paper they present a detailed system theoretic analysis of this filter and of the new scale recursive riccati equation associated with it while this analysis is similar in spirit to that for standard kalman filters the structure of the dyadic tree leads to several significant differences in particular the structure of the kalman filter error dynamics leads to the formulation of an ml version of the filtering equation and to a corresponding smoothing algorithm based on triangularizing the hamiltonian for the smoothing problem in addition the notion of stability for dynamics requires some care as do the concepts of reachability and observability using these system theoretic constructs the stability and steady state behavior of the fine to coarse kalman filter and its riccati equation are analysed

bounds for finite primitive complex linear groups abstract in 1878 jordan showed that a finite complex linear group must possess a normal abelian subgroup whose index is bounded by a function of the degree n alone in this paper we study primitive groups when n 12 the optimal bound is n 1 achieved by the symmetric group of degree n 1 we obtain the optimal bounds in smaller degree also our proof uses known lower bounds for the degrees of the faithful representations of each quasisimple group for which the classification of finite simple groups is required in a subsequent paper m j collins on jordan s theorem for complex linear groups j group theory 10 2007 411 423 we will show that n 1 is the optimal bound in general for jordan s theorem when n 71

construction of the tensor product for the lattices of properties of physical entities we construct the tensor product for the property lattices of two entities we give a physical interpretation for this tensor product we show that the tensor product is never an orthocomplemented lattice if both entities are nontrivial it also never satisfies the covering law this is the reason why this tensor product does not exist in quantum mechanics and does not exist in quantum logic we analyze the relation of the tensor product with the usual description of two entities in quantum mechanics at the end we give a mathematical way of constructing this tensor product

quantal effects and maxent convex operational models coms are considered as great extrapolations to larger settings of any statistical theory in this article we generalize the maximum entropy principle maxent of jaynes to any com after expressing maxent in a geometrical and lattice theoretical setting we are able to cast it for any com this scope amplification opens the door to a new systematization of the principle and sheds light into its geometrical structure

metrics of positive ricci curvature on quotient spaces one of the classical problems in differential geometry is the investigation of closed manifolds which admit riemannian metrics with given lower bounds for the sectional or the ricci curvature and the study of relations between the existence of such metrics and the topology and geometry of the underlying manifold despite many efforts during the past decades this problem is still far from being understood for example so far the only obstructions to the existence of a metric with positive ricci curvature come from the obstructions to the existence of metrics with positive scalar curvature li hi ro schy ta and the bonnet myers theorem which implies that the fundamental group of a closed manifold with positive ricci curvature must be finite fruitful constructions of metrics with positive ricci curvature on closed manifolds have so far been established by techniques that include deformation of metrics au eh we kahler geometry yau1 yau2 bundles and warping po na bb gpt special kinds of surgery sy wr metrical glueing gz2 and sasakian geometry bgn particularly large classes of examples of manifolds with positive ricci curvature are given by all compact homogeneous spaces with finite fundamental group na and all closed cohomogeneity one manifolds with finite fundamental group gz2 in this article we present several new classes of manifolds which admit metrics of positive or nonnegative ricci curvature the idea is to consider quotients of manifolds m g of positive or nonnegative ricci curvature by a free isometric action while taking such a quotient non decreases the sectional curvature it may well happen in general that the ricci curvature of the quotient is inferior to the one of m our first results however state that the quotient of m does admit metrics of positive ricci curvature if m belongs to one of the aforementioned classes of homogeneous spaces or spaces with a cohomogeneity one action more precisely we prove

ignorance is a bliss mathematical structure of many box models we show that the propositional system of a many box model is always a set representable effect algebra in particular cases of 2 box and 1 box models it is an orthomodular poset and an orthomodular lattice respectively we discuss the relation of the obtained results with the so called local orthogonality principle we argue that non classical properties of box models are the result of a dual enrichment of the set of states caused by the impoverishment of the set of propositions on the other hand quantum mechanical models always have more propositions as well as more states than the classical ones consequently we show that the box models cannot be considered as generalizations of quantum mechanical models and seeking additional principles that could allow us to recover quantum correlations in box models are at least from the fundamental point of view pointless

collapsing and pinching under a lower curvature bound in this paper we are concerned with collapsing phenomena and pinching problems of riemannian manifolds whose sectional curvatures are uniformly bounded from below for a positive integer n and for d 0 let x be the set of compact riemannian n manifolds m with sectional curvatures km 1 and diameters diam m d by the weak compactness theorem of gromov glp x is relatively compact in the set of compact metric spaces with respect to the hausdorff distance furthermore gromov proved in g2 that the sum of betti numbers of any element in x is uniformly bounded in terms of the given constants these results suggest that x is an object of study on which we could develop some geometry and topology in fact when volumes are uniformly bounded away from zero the following results are known let v be the subset of x with volume vol m v grove and petersen gp1 proved that the set of homotopy types of elements in 4 v is finite related results are in y2 for pinching problems otsu shiohama and the author osy y3 obtained some differential sphere theorems in the class x v according to a recent paper gpw if n 3 4 then v contains at most finitely many diffeomorphism types let mi i 1 2 be a convergent sequence in a and x be the limit of them we say that mi collapses to x if the hausdorff dimension of x is less than n by the volume condition no collapsing occurs in 4 v from this point of view it is quite natural to ask what collapsing phenomena occur and what types of pinching theorems hold in the class x in this situation it seems difficult to determine the singularities of x some examples are given in section 1 in this paper we study collapsing phenomena in the case when x is a riemannian manifold and establish a pinching theorem for a riemannian manifold m we denote by inj m the injectivity radius of m when n is written in place of x and the metric of n is normalized our

lower bounds on ricci curvature and the almost rigidity of warped products the basic rigidity theorems for manifolds of nonnegative or positive ricci curvature are the volume cone implies metric cone theorem the maximal diameter theorem cg and the splitting theorem cg each asserts that if a certain geometric quantity volume or diameter is as large as possible relative to the pertinent lower bound on ricci curvature then the metric on the manifold in question is a warped product metric of a particular type in this paper we provide quantitative generalizations of the above mentioned results among the applications are the splitting theorem for gromovhausdorff limit spaces x where mn x ricmn i see fy other applications include the assertion that for complete manifolds m with ricmf 0 and euclidean volume growth all tangent cones at infinity are metric cones compare bkn ct p1 via resealing arguments there are also strong consequences for the local structure of manifolds whose ricci curvature satisfies a fixed lower bound and for their gromov hausdorff limits some of these are announced in ccol for a more detailed discussion see cco2 cco3 cco4 our work further develops and significantly extends techniques which were introduced in col co2 and significantly extended in co3 in order to prove certain stability conjectures of anderson cheeger gromov and perelman the results of col co3 were announced in co4 we briefly review some of those results let dgh denote the gromov hausdorff distance between metric spaces see glp let s denote the unit sphere and recall that s is the unique complete

a parameter estimation framework for multiscale kalman smoother algorithm in precipitation data fusion a new effective parameter estimation approach is presented for the multiscale kalman smoother mks algorithm as demonstrated it shows promising potentials in deriving better data products involving sources from different spatial scales and precisions the proposed approach employs a multiobjective parameter estimation framework which includes three multiobjective estimation schemes mo schemes rather than using the conventional maximum likelihood scheme ml scheme to estimate the mks parameters unlike the ml scheme the mo schemes are not built on strict statistical assumptions related to prediction errors and observation errors rather they directly associate the fused data of multiple scales with multiple objective functions in the mo schemes objective functions are defined to facilitate consistency among the fused data at multiple scales and the input data at their original scales as well in terms of spatial patterns and magnitudes merits of the new approach are evaluated through a monte carlo experiment and a series of comparison analyses using synthetic precipitation data that contain noises which follow either the multiplicative error model or the additive error model our results show that the mks fused precipitation performs better using the mo framework improvements are particularly significant for the fused precipitation associated with fine spatial resolutions this is due mainly to the adoption of more criteria and constraints in the mo framework the weakness of the original ml scheme arising from its blindly putting more weights into the data associated with finer resolutions is circumvented in the proposed new mo framework