Concurrent π\pi-vector fields and energy beta-change

The present paper deals with an \emph{intrinsic} investigation of the notion of a concurrent $\pi$-vector field on the pullback bundle of a Finsler manifold $(M,L)$. The effect of the existence of a concurrent $\pi$-vector field on some important special Finsler spaces is studied. An intrinsic investigation of a particular $\beta$-change, namely the energy $\beta$-change ($\widetilde{L}^{2}(x,y)=L^{2}(x,y)+ B^{2}(x,y) with \ B:=g(\bar{\zeta},\bar{\eta})$;$\bar{\zeta} $being a concurrent$\pi$-vector field), is established. The relation between the two Barthel connections$\Gamma$and$\widetilde{\Gamma}$, corresponding to this change, is found. This relation, together with the fact that the Cartan and the Barthel connections have the same horizontal and vertical projectors, enable us to study the energy$\beta$-change of the fundamental linear connection in Finsler geometry: the Cartan connection, the Berwald connection, the Chern connection and the Hashiguchi connection. Moreover, the change of their curvature tensors is concluded. It should be pointed out that the present work is formulated in a prospective modern coordinate-free form.Comment: 27 pages, LaTex file, Some typographical errors corrected, Some formulas simpifie

A natural Finsler--Laplace operator

We give a new definition of a Laplace operator for Finsler metric as an average with regard to an angle measure of the second directional derivatives. This definition uses a dynamical approach due to Foulon that does not require the use of connections nor local coordinates. We show using 1-parameter families of Katok--Ziller metrics that this Finsler--Laplace operator admits explicit representations and computations of spectral data.Comment: 25 pages, v2: minor modifications, changed the introductio

Gravitational deflection of light in Rindler-type potential as a possible resolution to the observations of Bullet Cluster 1E0657-558

The surface density $\Sigma$-map and the convergence $\kappa$-map of Bullet Cluster 1E0657-558 show that the center of baryonic matters separates from the center of gravitational force, and the distribution of gravitational force do not possess spherical symmetry. This hints that a modified gravity with difference to Newtonian inverse-square law at large scale, and less symmetry is worth investigating. In this paper, we study the dynamics in Randers-Finsler spacetime. The Newtonian limit and gravitational deflection of light in a Rindler-type potential is focused in particular. It is shown that the convergence in Finsler spacetime could account for the observations of Bullet Cluster.Comment: 11 page

Why Pleiotropic Interventions are Needed for Alzheimer's Disease

Alzheimer's disease (AD) involves a complex pathological cascade thought to be initially triggered by the accumulation of β-amyloid (Aβ) peptide aggregates or aberrant amyloid precursor protein (APP) processing. Much is known of the factors initiating the disease process decades prior to the onset of cognitive deficits, but an unclear understanding of events immediately preceding and precipitating cognitive decline is a major factor limiting the rapid development of adequate prevention and treatment strategies. Multiple pathways are known to contribute to cognitive deficits by disruption of neuronal signal transduction pathways involved in memory. These pathways are altered by aberrant signaling, inflammation, oxidative damage, tau pathology, neuron loss, and synapse loss. We need to develop stage-specific interventions that not only block causal events in pathogenesis (aberrant tau phosphorylation, Aβ production and accumulation, and oxidative damage), but also address damage from these pathways that will not be reversed by targeting prodromal pathways. This approach would not only focus on blocking early events in pathogenesis, but also adequately correct for loss of synapses, substrates for neuroprotective pathways (e.g., docosahexaenoic acid), defects in energy metabolism, and adverse consequences of inappropriate compensatory responses (aberrant sprouting). Monotherapy targeting early single steps in this complicated cascade may explain disappointments in trials with agents inhibiting production, clearance, or aggregation of the initiating Aβ peptide or its aggregates. Both plaque and tangle pathogenesis have already reached AD levels in the more vulnerable brain regions during the “prodromal” period prior to conversion to “mild cognitive impairment (MCI).” Furthermore, many of the pathological events are no longer proceeding in series, but are going on in parallel. By the MCI stage, we stand a greater chance of success by considering pleiotropic drugs or cocktails that can independently limit the parallel steps of the AD cascade at all stages, but that do not completely inhibit the constitutive normal functions of these pathways. Based on this hypothesis, efforts in our laboratories have focused on the pleiotropic activities of omega-3 fatty acids and the anti-inflammatory, antioxidant, and anti-amyloid activity of curcumin in multiple models that cover many steps of the AD pathogenic cascade (Cole and Frautschy, Alzheimers Dement 2:284–286, 2006)