The Newtonian limit of the relativistic Boltzmann equation

The relativistic Boltzmann equation for a constant differential cross section and with periodic boundary conditions is considered. The speed of light appears as a parameter $c>c_0$ for a properly large and positive $c_0$. A local existence and uniqueness theorem is proved in an interval of time independent of $c>c_0$ and conditions are given such that in the limit $c\to +\infty$ the solutions converge, in a suitable norm, to the solutions of the non-relativistic Boltzmann equation for hard spheres.Comment: 12 page

Coframe teleparallel models of gravity. Exact solutions

The superstring and superbrane theories which include gravity as a necessary and fundamental part renew an interest to alternative representations of general relativity as well as the alternative models of gravity. We study the coframe teleparallel theory of gravity with a most general quadratic Lagrangian. The coframe field on a differentiable manifold is a basic dynamical variable. A metric tensor as well as a metric compatible connection is generated by a coframe in a unique manner. The Lagrangian is a general linear combination of Weitzenb\"{o}ck's quadratic invariants with free dimensionless parameters \r_1,\r_2,\r_3. Every independent term of the Lagrangian is a global SO(1,3)-invariant 4-form. For a special choice of parameters which confirms with the local SO(1,3) invariance this theory gives an alternative description of Einsteinian gravity - teleparallel equivalent of GR. We prove that the sign of the scalar curvature of a metric generated by a static spherical-symmetric solution depends only on a relation between the free parameters. The scalar curvature vanishes only for a subclass of models with \r_1=0. This subclass includes the teleparallel equivalent of GR. We obtain the explicit form of all spherically symmetric static solutions of the ``diagonal'' type to the field equations for an arbitrary choice of free parameters. We prove that the unique asymptotic-flat solution with Newtonian limit is the Schwarzschild solution that holds for a subclass of teleparallel models with \r_1=0. Thus the Yang-Mills-type term of the general quadratic coframe Lagrangian should be rejected.Comment: 28 pages, Latex error is fixe

Gravity on a parallelizable manifold. Exact solutions

The wave type field equation \square \vt^a=\la \vt^a, where \vt^a is a coframe field on a space-time, was recently proposed to describe the gravity field. This equation has a unique static, spherical-symmetric, asymptotically-flat solution, which leads to the viable Yilmaz-Rosen metric. We show that the wave type field equation is satisfied by the pseudo-conformal frame if the conformal factor is determined by a scalar 3D-harmonic function. This function can be related to the Newtonian potential of classical gravity. So we obtain a direct relation between the non-relativistic gravity and the relativistic model: every classical exact solution leads to a solution of the field equation. With this result we obtain a wide class of exact, static metrics. We show that the theory of Yilmaz relates to the pseudo-conformal sector of our construction. We derive also a unique cosmological (time dependent) solution of the described type.Comment: Latex, 17 page

On a class of invariant coframe operators with application to gravity

Let a differential 4D-manifold with a smooth coframe field be given. Consider the operators on it that are linear in the second order derivatives or quadratic in the first order derivatives of the coframe, both with coefficients that depend on the coframe variables. The paper exhibits the class of operators that are invariant under a general change of coordinates, and, also, invariant under the global SO(1,3)-transformation of the coframe. A general class of field equations is constructed. We display two subclasses in it. The subclass of field equations that are derivable from action principles by free variations and the subclass of field equations for which spherical-symmetric solutions, Minkowskian at infinity exist. Then, for the spherical-symmetric solutions, the resulting metric is computed. Invoking the Geodesic Postulate, we find all the equations that are experimentally (by the 3 classical tests) indistinguishable from Einstein field equations. This family includes, of course, also Einstein equations. Moreover, it is shown, explicitly, how to exhibit it. The basic tool employed in the paper is an invariant formulation reminiscent of Cartan's structural equations. The article sheds light on the possibilities and limitations of the coframe gravity. It may also serve as a general procedure to derive covariant field equations

Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials with integrable angular cross section

This paper focuses on the study of existence and uniqueness of distributional and classical solutions to the Cauchy Boltzmann problem for the soft potential case assuming $S^{n-1}$ integrability of the angular part of the collision kernel (Grad cut-off assumption). For this purpose we revisit the Kaniel--Shinbrot iteration technique to present an elementary proof of existence and uniqueness results that includes large data near a local Maxwellian regime with possibly infinite initial mass. We study the propagation of regularity using a recent estimate for the positive collision operator given in [3], by E. Carneiro and the authors, that permits to study such propagation without additional conditions on the collision kernel. Finally, an $L^{p}$-stability result (with $1\leq p\leq\infty$) is presented assuming the aforementioned condition.Comment: 19 page

Increased Expression of System x_c^- in Glioblastoma Confers an Altered Metabolic State and Temozolomide Resistance

Glioblastoma multiforme is the most aggressive malignant primary brain tumor in adults. Several studies have shown that glioma cells upregulate the expression of xCT (SLC7A11), the catalytic subunit of system x_c^-, a transporter involved in cysteine import, that modulates glutathione production and glioma growth. However, the role of system x_c^- in regulating the sensitivity of glioma cells to chemotherapy is currently debated. Inhibiting system x_c^- with sulfasalazine decreased glioma growth and survival via redox modulation, and use of the chemotherapeutic agent temozolomide together with sulfasalazine had a synergistic effect on cell killing. To better understand the functional consequences of system x_c^- in glioma, stable SLC7A11-knockdown and -overexpressing U251 glioma cells were generated. Modulation of SLC7A11 did not alter cellar proliferation but overexpression did increase anchorage-independent cell growth. Knockdown of SLC7A11 increased basal reactive oxygen species (ROS) and decreased glutathione generation resulting in increased cell death under oxidative and genotoxic stress. Overexpression of SLC7A11 resulted in increased resistance to oxidative stress and decreased chemosensitivity to temozolomide. In addition, SLC7A11 overexpression was associated with altered cellular metabolism including increased mitochondrial biogenesis, oxidative phosphorylation, and ATP generation. These results suggest that expression of SLC7A11 in the context of glioma contributes to tumorigenesis, tumor progression, and resistance to standard chemotherapy

Global existence and full regularity of the Boltzmann equation without angular cutoff

We prove the global existence and uniqueness of classical solutions around an equilibrium to the Boltzmann equation without angular cutoff in some Sobolev spaces. In addition, the solutions thus obtained are shown to be non-negative and $C^\infty$ in all variables for any positive time. In this paper, we study the Maxwellian molecule type collision operator with mild singularity. One of the key observations is the introduction of a new important norm related to the singular behavior of the cross section in the collision operator. This norm captures the essential properties of the singularity and yields precisely the dissipation of the linearized collision operator through the celebrated H-theorem