关于典范体积形式的一个公式

本文给出了一个典范体积（Canonical volume)形式公式的代数证明。     

Triangularizing matrices over GF(2) by congruence

Necessary and sufficient conditions are presented for a square matrix over GF(2) to be triangulable by congruence.     

The physical role of gravitational and gauge degrees of freedom in general relativity – II: Dirac versus Bergmann observables and the objectivity of space-time

This is the second of a couple of papers in which we aim to show the peculiar capability of the Hamiltonian ADM formulation of metric gravity to grasp a series of conceptual and technical problems that appear to have not been directly discussed so far. In this paper we also propose new viewpoints about issues that, being deeply rooted into the foundational level of Einstein theory, seem particularly worth of clarification in connection with the alternative programs of string theory and loop quantum gravity. The achievements of the present work include: (1) the analysis of the so-called Hole phenomenology in strict connection with the Hamiltonian treatment of the initial value problem. The work is carried through in metric gravity for the class of spatially non-compact Christoudoulou-Klainermann space-times, in which the temporal evolution is ruled by the weak ADM energy. It is crucial to our analysis the re-interpretation of active diffeomorphisms as passive and metric-dependent dynamical symmetries of Einstein's equations, a re-interpretation which enables to disclose their (nearly unknown) connection to gauge transformations on-shell; this is expounded in the first paper (gr-qc/0403081); (2) the utilization of the Bergmann-Komar intrinsic pseudo-coordinates, defined as suitable functionals of the Weyl curvature scalars, as tools for a specific gauge-fixing to the super-hamiltonian and super-momentum constraints; (3) the consequent construction of a physical atlas of 4-coordinate systems for the 4-dimensional mathematical manifold, in terms of the highly non-local degrees of freedom of the gravitational field (its four independent Dirac observables). Such construction embodies the physical individuation of the points of space-time as point-events, both in absence and presence of matter, and associates a non-commutative structure to each gauge fixing or 4-dimensional coordinate system; (4) a clarification of the multiple definition given by Peter Bergmann of the concept of (Bergmann) observable in general relativity. This clarification leads to the proposal of a main conjecture asserting the existence of: i) special Dirac's observables which are also Bergmann's observables, ii) gauge variables that are coordinate independent (namely they behave like the tetradic scalar fields of the Newman-Penrose formalism). A by-product of this achievements is the falsification of a recently advanced argument asserting the absence of (any kind of) change in the observable quantities of general relativity; (5) a proposal showing how the physical individuation of point-events could in principle be implemented as an experimental setup and protocol leading to a standard of space-time more or less like atomic clocks define standards of time. In the end, against the well-known Einstein's assertion according to which general covariance takes away from space and time the last remnant of physical objectivity, we conclude that point-events maintain a peculiar sort of objectivity. Also, besides being operationally essential for building measuring apparatuses for the gravitational field, the role of matter in the non-vacuum gravitational case is also that of participating directly in the individuation process, being involved in the determination of the Dirac observables. Finally, some hints following from our approach for the quantum gravity programme are suggested.     

On the Distribution of the Wave Function for Systems in Thermal Equilibrium

For a quantum system, a density matrix ρ that is not pure can arise, via averaging, from a distribution μ of its wave function, a normalized vector belonging to its Hilbert space ?. While ρ itself does not determine a unique μ, additional facts, such as that the system has come to thermal equilibrium, might. It is thus not unreasonable to ask, which μ, if any, corresponds to a given thermodynamic ensemble? To answer this question we construct, for any given density matrix ρ, a natural measure on the unit sphere in ?, denoted GAP(ρ). We do this using a suitable projection of the Gaussian measure on ? with covariance ρ. We establish some nice properties of GAP(ρ) and show that this measure arises naturally when considering macroscopic systems. In particular, we argue that it is the most appropriate choice for systems in thermal equilibrium, described by the canonical ensemble density matrix ρ_β = (1/Z) exp (?β H). GAP(ρ) may also be relevant to quantum chaos and to the stochastic evolution of open quantum systems, where distributions on ? are often used.     

可扩映射

设f为不带边的紧致流形M上的自覆盖映射,本义证明了,若f∈intE（M）,则f满足弱公理A.     

Anodic Polarographic Determination of Nilvadipine in Dosage Forms and Spiked Human Urine

 The anodic polarographic behaviour of nilvadipine was studied by using direct-current (DC), polarography and differential-pulse polarography (DPP). Nilvadipine, being a dihydropyridine derivative, exhibits well-defined anodic waves over the whole pH range of the Britton-Robinson buffers (BRb). In BRb of pH = 4, the diffusion-current constant was 6.45 ± 0.07 μA · mM⁻¹. The current-concentration plots are rectilinear over the ranges 1.6–12.8 and 0.2–12.8 μg/ml for DC polarography and DPP, respectively, with minimum detectability of 0.05 μg/ml (1.3 × 10⁻⁷ M) in case of the latter technique. The proposed method was successfully applied for determination of the drug in commercial capsules containing the drug, the percentage recoveries being in good agreement with the label claim. Furthermore, the method was applied for the determination of the drug in spiked human urine, the percentage recovery being 96.84 ± 2.83. Received January 25, 2001 Revision June 18, 2001     

Small Eigenvalues of the Laplacians on Vector Bundles over Towers of Covers

Under the conditions that a compact Riemannian manifold is of sufficiently pinched negative sectional curvature and that a smooth Hermitian vector bundle over the manifold is also of sufficiently small curvature, we prove some pinching results on the asymptotic behavior of the numbers of small eigenvalues of the Laplacians on the induced Hermitian vector bundles over a tower of covers of the manifold. In the process we also obtain interesting results on the non-existence of square integrable 'almost harmonic' vector bundle-valued forms omitting the middle degree(s) on the universal cover. This revised version was published online in June 2006 with corrections to the Cover Date.