On the properties of level spacings for decomposable systems

In this paper we show that the quantum theory of chaos, based on the statistical theory of energy spectra, presents inconsistencies difficult to overcome. In classical mechanics a system described by an hamiltonian $H = H_1 + H_2$ (decomposable) cannot be ergodic, because there are always two dependent integrals of motion besides the constant of energy. In quantum mechanics we prove the existence of decomposable systems \linebreak $H^q = H^q_1 + H^q_2$ whose spacing distribution agrees with the Wigner law and we show that in general the spacing distribution of $H^q$ is not the Poisson law, even if it has often the same qualitative behaviour. We have found that the spacings of $H^q$ are among the solutions of a well defined class of homogeneous linear systems. We have obtained an explicit formula for the bases of the kernels of these systems, and a chain of inequalities which the coefficients of a generic linear combination of the basis vectors must satisfy so that the elements of a particular solution will be all positive, i.e. can be considered a set of spacings.Comment: LateX, 13 page

Non-linear Quantization of Integrable Classical Systems

It is demonstrated that the so-called "unavoidable quantum anomalies" can be avoided in the farmework of a special non-linear quantization scheme. A simple example is discussed in detail.Comment: LaTeX, 14 p

E7 groups from octonionic magic square

In this paper we continue our program, started in [2], of building up explicit generalized Euler angle parameterizations for all exceptional compact Lie groups. Here we solve the problem for E7, by first providing explicit matrix realizations of the Tits construction of a Magic Square product between the exceptional octonionic algebra J and the quaternionic algebra H, both in the adjoint and the 56 dimensional representations. Then, we provide the Euler parametrization of E7 starting from its maximal subgroup U=(E6 x U(1))/Z3. Next, we give the constructions for all the other maximal compact subgroups.Comment: 23 pages, added sections with new construction

Mapping the geometry of the F4 group

In this paper we present a construction of the compact form of the exceptional Lie group F4 by exponentiating the corresponding Lie algebra f4. We realize F4 as the automorphisms group of the exceptional Jordan algebra, whose elements are 3 x 3 hermitian matrices with octonionic entries. We use a parametrization which generalizes the Euler angles for SU(2) and is based on the fibration of F4 via a Spin(9) subgroup as a fiber. This technique allows us to determine an explicit expression for the Haar invariant measure on the F4 group manifold. Apart from shedding light on the structure of F4 and its coset manifold OP2=F4/Spin(9), the octonionic projective plane, these results are a prerequisite for the study of E6, of which F4 is a (maximal) subgroup.Comment: 50 pages; some typos correcte

Mapping the geometry of the E6 group

In this paper we present a construction for the compact form of the exceptional Lie group E6 by exponentiating the corresponding Lie algebra e6, which we realize as the the sum of f4, the derivations of the exceptional Jordan algebra J3 of dimension 3 with octonionic entries, and the right multiplication by the elements of J3 with vanishing trace. Our parametrization is a generalization of the Euler angles for SU(2) and it is based on the fibration of E6 via a F4 subgroup as the fiber. It makes use of a similar construction we have performed in a previous article for F4. An interesting first application of these results lies in the fact that we are able to determine an explicit expression for the Haar invariant measure on the E6 group manifold.Comment: 30 page

Flow network indices signal a directional change in ecosystems: Evidence from a small mountain lake (Lake Santo, northern Italy)

Empirical evidence of the theoretically expected trends of ecosystem development is scarce so far. In this research, we used long-term empirical data about the plankton community of a small mountain lake (Lake Santo, northern Apennines, Italy) to reconstruct its developmental trajectory during a period comprised between early 1970 s and 2010 s. We exploited these data to build yearly ecological networks and from their configuration of energy flows we computed network information indices. The trends of these indices enlighten about the developmental trajectory of this ecosystem during the period covered by the data set. In particular, they indicate that Lake Santo evolved in the direction of increasing stability at the expense of efficiency in energy transfer. We compared these results with current hypotheses about the directionality of ecosystem development, which are rooted in ecosystem theory, and discussed the possibility that, counter to some theoretical models of ecosystem development, Lake Santo followed an unimpeded direction of development rather than a trajectory typical of an ecosystem under stress. Finally, the long-term trends of flow network indices provided insights about the health status of the ecosystem

Understanding Social–Ecological Systems using Loop Analysis

The sustainable management of social–ecological systems (SESs) requires that we understand the complex structure of relationships and feedbacks among ecosystem components and socioeconomic entities. Therefore, the construction and analysis of models integrating ecological and human actors is crucial for describing the functioning of SESs, and qualitative modeling represents an ideal tool since it allows studying dependencies among variables of diverse types. In particular, the qualitative technique of loop analysis yields predictions about how a system’s variables respond to stress factors. Different interaction types, scarce information about functional relationships among variables, and uncertainties in the values of the parameters are the rule rather than exceptions when studying SESs. Accordingly, loop analysis seems to be perfectly suitable to investigate them. Here, we introduce the key aspects of loop analysis, discuss its applications to SESs, and suggest it enables making the first steps toward the integration of the three dimensions of sustainability

Euler angles for G2

We provide a simple parametrization for the group G2, which is analogous to the Euler parametrization for SU(2). We show how to obtain the general element of the group in a form emphasizing the structure of the fibration of G2 with fiber SO(4) and base H, the variety of quaternionic subalgebras of octonions. In particular this allows us to obtain a simple expression for the Haar measure on G2. Moreover, as a by-product it yields a concrete realization and an Einstein metric for H.Comment: 21 pages, 2 figures, some misprints correcte

Comparison between high-field 3 Tesla MRI and computed tomography with and without arthrography for visualization of canine carpal ligaments: a cadaveric study

OBJECTIVE:To compare the quality of visualization of canine carpal ligaments by using computed tomography (CT), MRI, CT arthrography (CTA), and magnetic resonance arthrography (MRA). STUDY DESIGN: Prospective descriptive study. STUDY POPULATION: Cadavers from dogs weighing more than 20 kg. METHODS: A 16-slice CT scanner and a 3 Tesla MRI were used for the investigation. A dilute contrast medium was injected into the middle carpal and radiocarpal joints under fluoroscopic control, and CTA and MRA images were acquired. To evaluate the difference between imaging modalities, 3 observers graded carpal ligaments of clinical interest using a scale from 0 to 4 for their quality of visualization. Data were analyzed by using a random-effect ordinal logistic regression with Bonferroni adjustment. The interobserver agreement was calculated by using the weighted Cohen's κ. RESULTS: Normal carpal joints (n = 9) were investigated. Magnetic resonance arthrography improved visualization of the majority of carpal ligaments compared with MRI (P < .05) and offered the best visualization overall. Magnetic resonance imaging and MRA offered better visualization compared with both CT and CTA (P < .05). There was no difference between CT and CTA. Interobserver agreement was discrete (0.2 < κ ≤ 0.4) for all observers. CONCLUSION: Arthrography improved the capabilities of MRI but not of CT for visualization of the canine carpal ligaments. Magnetic resonance arthrography was particularly useful for evaluation of the stabilizers of the antebrachiocarpal joint. CLINICAL SIGNIFICANCE: 3 Tesla MRA and MRI allow excellent visualization of the ligamentous morphology and may be helpful in the diagnostic process of carpal sprains in dogs