Hahn-Jordan decomposition for gleason measures

The possibility of the Hahn-Jordan decomposition forn-finite signed measures, wheren is a cardinal, defined on a quantum logic of all closed subspaces of a Hilbert space whose dimension is a nonmeasurable cardinal 2, is investigated.     

Converse of the Eilers-Horst theorem

We generalize the theorem of Eilers and Horst, showing that any finite as well as any-finite measure on a quantum logic of all closed subspaces of a Hilbert spaceH of dimension 2 is a Gleason one iff the dimension ofH is a nonmeasurable cardinal.     

A theorem on the first heteroclinic tangency in two-dimensional maps. Orientation-preserving cases

Using the properties of the Jordan curve, the following theorem on the heteroclinic tangency in orientation-preserving two-dimensional maps is proved: LetT :R ² R ² be a one-parameter family ofC ¹ diffeomorphisms andJ=DetDT be such that 0<J1 or 1J<. LetW _u ⁿ be the unstable manifold of a hyperbolicn-cycle andW _s ^m the stable manifold of a hyperbolicm-cycle. Suppose that for< _c ,W _u ⁿ andW _s ^m have no common points, and that for> _c ,W _u ⁿ andW s/m have a transversal heteroclinic point. Then at= _c ,W _u ⁿ andW _s ^m are in the first asymptotic heteroclinic tangency except for the following three cases: (1)n=m; both cycles are without reflection. (2)m=2n; then- andm-cycles are with and without reflection, respectively; (3)n=2m; then- andm-cycles are without and with reflection, respectively.     

Gleason's theorem and completeness of inner product spaces

We generalize the result of Hamhalter and Pták showing that an inner product space whose dimension is either a nonmeasurable cardinal or an arbitrary cardinal is complete iff its lattice of strongly closed subspaces possesses at least one state or one completely additive state, respectively. Moreover, we show that this lattice of any separable space possesses many-finite measures, and we give the Gleason analogue for them.     

Different Orthocomplementations on the Subspace Lattice of a Finite-Dimensional Complex Vector Space

In this paper we prove, by using real closed fields and model theory, the followingresult: for any integer n 3, there exist, on the lattice of all subspaces on thevector space C ⁿ , 2(^2K0)orthocomplementations leading to nonisomorphic structuresof orthomodular lattices.     

Completeness of inner product spaces and quantum logic of splitting subspaces

We show that an inner product space S is complete whenever its system E(S) of all splitting subspaces, i.e., of all subspaces E for which EE =S holds, is a quantum logic, that is, an orthocomplemented orthomodular -orthoposet. It is well known that the quantum logic is an important axiomatic model of quantum mechanics. This generalizes the result of G. Cattaneo and G. Marino (Lett. Math. Phys. 11, 15–20 (1986)) who required that E(S) be a lattice. Moreover, the conditions are weakened to show that S is complete whenever E(S) contains the join of any sequence of one-dimentional orthogonal subspaces.     

SL(n+1, C) strata and orbits in the solution space of Euclidean CP n models

We further study the action of SL(n+1, C) on the space of finite action solutions of the bidimensional Euclidean CP ⁿ models. We decompose the space of k-instantons into strata. Each stratum in characterized by an integer m with 0mmin(k, n) which can be calculated from the instanton by purely algebraic means. The k-instantons with m=n are called generic. Their stratum is shown to be dense in the space of k-instantons when kn. The isotropy subgroups for each stratum are identified.

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Résumé Nous poursuivons l'étude de l'action de SL(n+1, C) sur l'espace des solutions à action finie du modèle CP ⁿ sur l'espace euclidien bi-dimensionnel. L'espace des k-instantons est décomposé en strates. Chaque strate est caractérisée par un entier m tel que 0 mmin(k, n) et qui peut être calculé à partir de l'instanton par des méthodes purement algébriques. Les k-instantons avec m=n sont dits génériques. Leur strate est dense dans l'espace des k-instantons (lorsque k n). Les sous-groupes d'isotropie de chacune des strates sont identifiés.

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Supported in part by the Natural Sciences and Engineering Research Council of Canada and by the Fonds FCAR pour l'aide et le soutien à la recherche.     

Crossover scaling function for the free energy

A cubic field, coupling tos|s|², inn-component spin models induces a bicritical crossover fromn-isotropic to Ising like (m=1) critical behaviour for 1<n<, but to classical behaviour in the limitn. By following the analysis of Nelson and Domany, the bicritical scaling function for the free energy ind dimensions is obtained correct to order =4–d and for general (m,n). The mechanism responsible for the breakdown of hyperscaling in the classical behaviour is discussed.     

Surface tension,step free energy,and facets in the equilibrium crystal

Some aspects of the microscopic theory of interfaces in classical lattice systems are developed. The problem of the appearance of facets in the (Wulff) equilibrium crystal shape is discussed, together with its relation to the discontinuities of the derivatives of the surface tension (n) (with respect to the components of the surface normaln) and the role of the step free energy ^step(m) (associated with a step orthogonal tom on a rigid interface). Among the results are, in the case of the Ising model at low enough temperatures, the existence of ^step(m) in the thermodynamic limit, the expression of this quantity by means of a convergent cluster expansion, and the fact that 2^step(m) is equal to the value of the jump of the derivative / (when varies) at the point =0 [withn=(m ₁ sin ,m ₂ sin , cos )]. Finally, using this fact, it is shown that the facet shape is determined by the function ^step(m).     

Phase transitions on fractal lattices with long-range interactions

A fractal latticeF is defined here to comprise all points of the forma + ma+ m² a+ ... +m^qa^(q), whereq is a nonnegative integer anda, a,..., a^(q)A, whereA is a finite set of points in some Euclidean space. Providedm is not too small (in particular,m must be at least 2), the dimension ofF is shown to beD = log n/logm, wheren is the number of points inA. It is shown further that an Ising model onF, with a ferromagnetic pair interaction r^– between spins separated by a distancer, has a phase transition ifD < < 2D. On the other hand, for > 2D, provided a certain condition which rules out periodic lattices is satisfied, there can be no finite-temperature transition leading to spontaneous magnetization.     

Co-symplectic geometry and co-Lagrangian subspaces

A new natural structure on the tangent spaces of a co-tangent bundle is introduced and some of its properties are investigated. This structure is based on a symmetric bilinear form and leads to a geometry that is, in many respects, analogous to the symplectic geometry. The new structure can thus justifiably be called co-symplectic geometry. The null structure of co-symplectic vector spaces is investigated in detail. It is found that the manifold of all maximally isotropic subspaces of a co-symplectic vector space is a homogeneous compact manifold of dimension 1/2n(n–1) consisting of two diffeomorphic components and having fundamental groupZ ₂Z ₂. A representation of the fundamental group of this manifold is explicitly constructed in terms of quadrupoles of co-Lagrangian subspaces.     

Completeness of inner product spaces with respect to splitting subspaces

The set E(S) of all splitting subspaces, i.e., of all subspaces M of an inner product space S for which MM =S holds, is an orthocomplemented orthomodular orthoposet and it has already been shown that the ordering property on E(S) of being a complete lattice characterizes the completeness of inner product spaces. In this work this last result is generalized proving that S is a Hilbert space under the weaker request that E(S) is a -lattice. As a marginal result, we also prove that an inner product space is complete if and only if the complete lattice (S) of all subspaces is orthomodular.     

Range of Charges on Orthogonally Closed Subspaces of an Inner Product Space

It is still an open question whether the complete lattice F(S) of all orthogonally closed subspaces of an incomplete inner product space S admits a nonzero charge. A negative answer would result in a new way of completeness characterization of inner product spaces. Many partial results have been established regarding what has now turned to be a highly nontrivial problem. Recently, in Dvureenskij and Ptak (Letters in Mathematical Physics, 62, 63–70, 2002) the range of a finitely additive state s on F(S), dim S = , was shown to include the whole interval [0, 1]. This was then generalized in Dvureenskij (International Journal of Theoretical Physics, 2003) for general inner product spaces satisfying the Gleason property. Motivated by these results, we give a thorough investigation of the possible ranges of charges on F(S), dim S q3. We show that if the nonzero charge m is bounded, then for infinite dimensional inner product spaces, Range(m) is always convex. We also show that this need not be the case with unbounded charges. Finally, in the last section, we investigate the range of charges on F(S), dim S = , satisfying the sign-preserving and Jauch-Piron properties. We show that for such measures the range is again always convex.     

Constitutive relations in multidimensional isotropic elasticity and their restrictions to subspaces of lower dimensions

The mechanical meaning and the relationships among material constants in an n-dimensional isotropic elastic medium are discussed. The restrictions of the constitutive relations (Hooke’s law) to subspaces of lower dimension caused by the conditions that an m-dimensional strain state or an m-dimensional stress state (1 ≤ m < n) is realized in the medium. Both the terminology and the general idea of the mathematical construction are chosen by analogy with the case n = 3 and m = 2, which is well known in the classical plane problem of elasticity theory. The quintuples of elastic constants of the same medium that enter both the n-dimensional relations and the relations written out for any m-dimensional restriction are expressed in terms of one another. These expressions in terms of the known constants, for example, of a three-dimensional medium, i.e., the classical elastic constants, enable us to judge the material properties of this medium immersed in a space of larger dimension.     

The extended phase space of the BRS approach

The origin of the classical BRS symmetry is discussed for the case of a first class constrained system consisting of a 2n-dimensional phase spaceS with free action of a Lie gauge groupG of dimensionm. The extended phase spaceS _ext of the Fradkin-Vilkovisky approach is identified with a globally trivial vector bundle overS with fibreL*(G)L(G), whereL(G) is the Lie algebra ofG andL*(G) its dual. It is shown that the structure group of the frame bundle of the supermanifoldS _ext is the orthosymplectic group OSp(m,m; 2n), which is the natural invariance group of the super Poisson bracket structure on the function spaceC (S _ext). The action of the BRS operator is analyzed for the caseS=R ²ⁿ with constraints given by pure momenta. The breaking of the osp(m,m; 2n)-invariance down to sp(2n–2m) occurs via an intermediate osp(m; 2n–m). Starting from a (2n+2m)-dimensional system with orthosymplectic invariance, different choices for the BRS operator correspond to choosing different 2n-dimensional constraint supermanifolds inS _ext, which in general characterize different constrained systems. There is a whole family of physically equivalent BRS operators which can be used to describe a particular constrained system.     

Random walk with persistence

The exact analytic result is obtained for the Fourier transform of the generating functionF(R,s)= _n=0 s ⁿ P(R,n), whereP(R,n) is the probability density for the end-to-end distanceR inn steps of a random walk with persistence. The moments R ²(n), R ⁴(n), and R ⁶(n) are calculated and approximate results forP(R,n) and R ^–1(n) are given.     

The properties of some derivative autocorrelation functions computed with the atom-atom potential

High-resolution quasi-elastic neutron-scattering measurements have been made on two nematogens: DMBCA with a nematic range 108 to 119°C, and 5CB and a tail-deuteriated sample (D5CB), having a nematic range 22·6 to 35·1°C.

Results on 5CB in the crystal phase at ～18°C showed no significant quasielastic broadening, which means that any random motions of the alkyl chain are slower than about 5 × 10⁹ rad s^-1. Measurements were made at a single temperature in the nematic phases on specimens aligned in a magnetic field of 0·25T; for DMBCA with scattering vector Q⊥n (n is the nematic director) and for 5CB and D5CB with Q⊥n and Q∥n and also on the isotropic liquid phase of D5CB at 45°C. Analysis of the coherent scattering from nematic D5CB at Q = 1·2 Å^-1 and 25°C gave an order parameter <P ₂>=0·55, close to the simple mean field value for this temperature. The coherent scattering from DMBCA is too weak to allow this experiment to be performed.

The most remarkable qualitative feature of the results is the close similarity of the scattering law S(Q, ω) for D5CB (and 5CB) with Q⊥n and Q∥n. Analysis of the results in all cases was made using values for the translational diffusion constants measured previously. Corrections for multiple scattering are shown to be important and a single simple model has been devised which fits the line shapes of all the results for D5CB in nematic (Q⊥n and Q∥n) and isotropic liquid phases and DMBCA. The model involves uniaxial rotational diffusion about the long molecular axis m coupled to a displacement along the rotation axis giving a net rotation in a plane whose normal makes an angle ∝ relative to the direction m. Values for the rotational diffusion constant D _rd ns^-1 are as follows: D5CB, 25°C, 6 (∝ ～ 50°); 45°C, 10. DMBCA, 112°C, 16, (all ±10–15 per cent).

The results for D5CB and 5CB are so similar that no additional detailed model fitting was attempted for the fully hydrogenous sample and it is concluded that while the motion of the alkyl tails is freer, the time scale of the motions is not more than about a factor of 2 faster than that of the molecular cores.     

F-Information, a Unitless Variant of Fisher Information

A new information matrix [F] with elements F _mn = (y _m - a _m )(y _n - a n) ( ln p(y | a)/a _m ) ( ln p(y | a)/a _n ) is analyzed. The PDF p(y | a) is the usual likelihood law. [F] differs from the Fisher information matrix by the presence of the first two factors in the given expectation. These factors make F _mn unitless, in contrast with the Fisher information. This lack of units allows F _mn values from entirely different phenomena to be compared as, for example, Shannon information values can be compared. Each element F _mn defines an error inequality analogous to the Cramer-Rao inequality. In the scalar case F _mn F, for a normal p(y|a) law F = 3, while for an exponential law F = 9. A variational principle F = min (called FMIN) allows an unknown PDF p(x) to be estimated in the presence of weak information. Under certain conditions F obeys a Boltzmann F-theorem F/t 0, indicating that F is a physical entropy. Finally, the trace of [F] may be used as the scalar information quantity in an information-based principle for deriving distribution laws p of physics.     

On States on Orthogonally Closed Subspaces of an Inner Product Space

We show that every finitely additive state on the system F(S) of all orthogonally closed subspaces of an infinite-dimensional inner product space S attains all values from the real interval [0,1]. In particular, we show that there is no finitely additive countably-valued state on F(S) whenever dim S=. The main technique we use is an embedding of L(H _n ) into F(S).     

A signed measure completeness criterion

We show that an inner product space S is complete whenever its system E(S) of all splitting subspaces, i.e. of all subspaces M for which M+M =S holds, possesses at least one nonzero completely additive signed measure or, equivalently, iff S possesses at least one nonzero frame function. Moreover, we show a new and simple proof that S is complete iff E(S) contains the join of any sequence of orthogonal one-dimensional subspaces.