Lattice tensor products. III - Congruences

G. Grätzer and F. Wehrung introduced the lattice tensor product, A?B, of the lattices Aand B. In Part I of this paper, we showed that for any finite lattice A, we can "coordinatize" A?B, that is, represent A?,B as a subset A of B ^A, and provide an effective criteria to recognize the A-tuples of elements of B that occur in this representation. To show the utility of this coordinatization, we prove, for a finite lattice A and a bounded lattice B, the isomorphism Con A ≌ (Con A)B>, which is a special case of a recent result of G. Grätzer and F. Wehrung and a generalization of a 1981 result of G. Grätzer, H. Lakser, and R.W. Quackenbush.     

Induced product representations of extended Cuntz algebras

Let _H be the extended Cuntz algebra over the Hilbert space H. Since its zero grade part _H⁰ is the C^*-inductive limit of B(H^r), we look for some family of representations on an inductive limit of H^r as r. When such construction is shaped according to the structure of _H⁰, von Neumanns notion of a reference sequence of unit vectors for Hilbert infinite tensor products emerges; after a further Rieffel induction step, a class IPR[H] of representations of _H arises. For any two such representations, we describe explicitly their associated intertwiners. Any two representations in IPR[H] are either disjoint or unitarily equivalent. Actions of the group by translation on sequences of unit vectors are involved, as well as the ideals of .     

Conformal metrics and Ricci tensors in the pseudo-Euclidean space

We consider constant symmetric tensors on , , and we study the problem of finding metrics conformal to the pseudo-Euclidean metric such that . We show that such tensors are determined by the diagonal elements and we obtain explicitly the metrics . As a consequence of these results we get solutions globally defined on for the equation Moreover, we show that for certain unbounded functions defined on , there are metrics conformal to the pseudo-Euclidean metric with scalar curvature .

     

High-field 127I NMR of solid sheelite structures: periodates revisited

We report new solid-state ¹²⁷I NMR results for sheelite periodates, MIO₄ (M = Na⁺ , K⁺ , Rb⁺ , and NH⁺ ₄), and for pseudo-scheelite CsIO₄ and HIO₄. The observed ¹²⁷I quadrupole coupling constants were between 1.0 and 43.0 MHz in agreement with previous NQR data. In contrast to an early ¹²⁷I NMR study (S. L. Segel and H. M. Vyas, 1980, J. Chem. Phys.72, 1406), we found that the ¹²⁷I chemical shift anisotropy is negligibly small in sheelite periodates. A small but definite ¹²⁷I chemical shift tensor was observed for pseudo-scheelite CsIO₄.     

A Homotopy 2-Groupoid of a Hausdorff Space

If X is a Hausdorff space we construct a 2-groupoid G ₂ X with the following properties. The underlying category of G ₂ X is the `path groupoid" of X whose objects are the points of X and whose morphisms are equivalence classes f, g of paths f, g in X under a relation of thin relative homotopy. The groupoid of 2-morphisms of G ₂ X is a quotient groupoid X / N X, where X is the groupoid whose objects are paths and whose morphisms are relative homotopy classes of homotopies between paths. N X is a normal subgroupoid of X determined by the thin relative homotopies. There is an isomorphism G ₂ X(f,f) ₂(X, f(0)) between the 2-endomorphism group of f and the second homotopy group of X based at the initial point of the path f. The 2-groupoids of function spaces yield a 2-groupoid enrichment of a (convenient) category of pointed spaces.We show how the 2-morphisms may be regarded as 2-tracks. We make precise how cubical diagrams inhabited by 2-tracks can be pasted.     

Condition number as a measure of noise performance of diffusion tensor data acquisition schemes with MRI

Diffusion tensor mapping with MRI can noninvasively track neural connectivity and has great potential for neural scientific research and clinical applications. For each diffusion tensor imaging (DTI) data acquisition scheme, the diffusion tensor is related to the measured apparent diffusion coefficients (ADC) by a transformation matrix. With theoretical analysis we demonstrate that the noise performance of a DTI scheme is dependent on the condition number of the transformation matrix. To test the theoretical framework, we compared the noise performances of different DTI schemes using Monte-Carlo computer simulations and experimental DTI measurements. Both the simulation and the experimental results confirmed that the noise performances of different DTI schemes are significantly correlated with the condition number of the associated transformation matrices. We therefore applied numerical algorithms to optimize a DTI scheme by minimizing the condition number, hence improving the robustness to experimental noise. In the determination of anisotropic diffusion tensors with different orientations, MRI data acquisitions using a single optimum b value based on the mean diffusivity can produce ADC maps with regional differences in noise level. This will give rise to rotational variances of eigenvalues and anisotropy when diffusion tensor mapping is performed using a DTI scheme with a limited number of diffusion-weighting gradient directions. To reduce this type of artifact, a DTI scheme with not only a small condition number but also a large number of evenly distributed diffusion-weighting gradients in 3D is preferable.     

Horizon-less Spherically Symmetric Vacuum-Solutions in a Higgs Scalar-Tensor Theory of Gravity

The exact static and spherically symmetric solutions of the vacuum field equations for a Higgs Scalar-Tensor theory (HSTT) are derived in Schwarzschild coordinates. It is shown that in general there exists no Schwarzschild horizon and that the fields are only singular (as naked singularity) at the center (i.e. for the case of a point-particle). However, the Schwarzschild solution as in usual general relativity (GR) is obtained for the vanishing limit of Higgs field excitations.     

The conformal status of o = ‐3/2 Brans‐Dicke cosmology

Following recent fit of supernovae data to Brans‐Dicke theory which favours the model with o = ‐ 3/2 [1] we discuss the status of this special case of Brans‐Dicke cosmology in both isotropic and anisotropic framework. It emerges that the limit o = ‐3/2 is consistent only with the vacuum field equations and it makes such a Brans‐Dicke theory conformally invariant. Then it is an example of the conformal relativity theory which allows the invariance with respect to conformal transformations of the metric. Besides, Brans‐Dicke theory with o = ‐3/2 gives a border between a standard scalar field model and a ghost/phantom model. In this paper we show that in o = ‐3/2 Brans‐Dicke theory, i.e., in the conformal relativity there are no isotropic Friedmann solutions of non‐zero spatial curvature except for k=‐1 case. Further we show that this k=‐1 case, after the conformal transformation into the Einstein frame, is just the Milne universe and, as such, it is equivalent to Minkowski spacetime. It generally means that only flat models are fully consistent with the field equations. On the other hand, it is shown explicitly that the anisotropic non‐zero spatial curvature models of Kantowski‐Sachs type are admissible in o = ‐3/2 Brans‐Dicke theory. It then seems that an additional scale factor which appears in anisotropic models gives an extra deegre of freedom and makes it less restrictive than in an isotropic Friedmann case.     

Symmetries of Energy-Momentum Tensor： Some Basic Facts

It has been pointed out by Hall et al. [Gen. Rel. Gray. 28 （1996） 299.] that matter collineations can be defined by using three different methods. But there arises the question whether one studies matter collineations by using LεTab=0, or LεT^ab = 0 or LεT^b a=0. These alternative conditions are, of. course, not generally equivalent. This problem has been explored by applying these three definitions to general static spherically symmetric spacetimes. We compare the results with each definition.     

Heat flow disturbed by periodic array of insulated interface cracks

The thermal effects of an interface crack between two dissimilar half-spaces is considered. The interface cracks are partially or fully insulated, and spaced in a periodic array. Using the complex variable technique, the temperature and fluxes are found in closed form, and the interactions between heat flows due to nearby cracks are determined.