Abstract:
The quantization of gravitational waves in the Milne universe is discussed. The relation between positive frequency functions of the gravitational waves in the Milne universe and those in the Minkowski universe is clarified. Implications to the one-bubble open inflation scenario are also discussed.

Abstract:
The cosmological concordance model is consistent with all available observational data, including the apparent distance and redshift relationship for distant supernovae, but it is curious how the Milne cosmological model is able to make predictions that are similar to this preferred General Relativistic model. Milne's cosmological model is based solely on Special Relativity and presumes a completely incompatible redshift mechanism; how then can the predictions be even remotely close to observational data? The puzzle is usually resolved by subsuming the Milne Cosmological model into General Relativistic cosmology as the special case of an empty Universe. This explanation may have to be reassessed with the finding that spacetime is approximately flat because of inflation, whereupon the projection of cosmological events onto the observer's Minkowski spacetime must always be kinematically consistent with Special Relativity, although the specific dynamics of the underlying General Relativistic model can give rise to virtual forces in order to maintain consistency between the observation and model frames.

Abstract:
We prove: Let be real numbers with . Then we have for all real numbers : with the best possible exponents and . The left-hand side of (0.1) with is a discrete version of an integral inequality due to E.A. Milne [1]. Moreover, we present a matrix analogue of (0.1).

Abstract:
Let k be an algebraically closed field of characteristic p>0. Let W(k) be the ring of Witt vectors with coefficients in k. We prove a motivic conjecture of Milne that relates, in the case of abelian schemes, the \'etale cohomology with $\dbZ_p$ coefficients to the crystalline cohomology with integral coefficients, in the more general context of p-divisible groups endowed with {\it arbitrary} families of crystalline tensors over a finite, discrete valuation ring extension of W(k). This extends a result of Faltings in [Fa2]. As a main new tool we construct global deformations of p-divisible groups endowed with crystalline tensors over certain regular, formally smooth schemes over W(k) whose special fibers over k have a Zariski dense set of k-valued points.

Abstract:
We prove the following let , and be real numbers, and let be positive real numbers with . The inequalities hold for all real numbers if and only if and . Furthermore, we provide a matrix version. The first inequality (with and ) is a discrete counterpart of an integral inequality published by E. A. Milne in 1925.

Abstract:
The spectrum of vacuum fluctuations in the Milne space (i.e. the tau-eta coordinate system) is an important ingredient in the thermalization studies in relativistic heavy ion collisions. In this paper, the Schrodinger functional for the gauge theory perturbative vacuum is derived for the Milne space. The Wigner-transform of the corresponding vacuum density functional is also found together with the propagators. We finally identify the fluctuation spectrum in vacuum, and show the equivalence between the present approach and the symplectic product based method.

Abstract:
We prove the following let α , β , a > 0 , and b > 0 be real numbers, and let w j ( j = 1 , … , n ; n ≥ 2 ) be positive real numbers with w 1 + … + w n = 1 . The inequalities α ∑ j = 1 n w j / ( 1 p j a ) ≤ ∑ j = 1 n w j / ( 1 p j ) ∑ j = 1 n w j / ( 1 + p j ) ≤ β ∑ j = 1 n w j / ( 1 p j b ) hold for all real numbers p j ∈ [ 0 , 1 ) ( j = 1 , … , n ) if and only if α ≤ min ( 1 , a / 2 ) and β ≥ max ( 1 , ( 1 min 1 ≤ j ≤ n w j / 2 ) b ) . Furthermore, we provide a matrix version. The first inequality (with α = 1 and a = 2 ) is a discrete counterpart of an integral inequality published by E. A. Milne in 1925.

Abstract:
The \Lambda CDM standard model, although an excellent parametrization of the present cosmological data, requires two as yet unobserved components, Dark Matter and Dark Energy, for more than 95% of the Universe. Faced to this unsatisfactory situation, we study an unconventional cosmology, the Dirac-Milne universe, a matter-antimatter symmetric cosmology, in which antimatter is supposed to present a negative active gravitational mass. The main feature of this cosmology is the linear evolution of the scale factor with time which directly solves the age and horizon problems of a matter-dominated universe. We study the concordance of this model to the cosmological test of Type Ia Supernov\ae\ distance measurements and calculate the theoretical primordial abundances of light elements for this cosmology. We also show that the acoustic scale of the Cosmic Microwave Background naturally emerges at the degree scale despite an open geometry.

Abstract:
We generalize the Milne quantization condition to non-Hermitian systems. In the general case the underlying nonlinear Ermakov-Milne-Pinney equation needs to be replaced by a nonlinear integral differential equation. However, when the system is PT-symmetric or/and quasi/pseudo-Hermitian the equations simplify and one may employ the original energy integral to determine its quantization. We illustrate the working of the general framework with the Swanson model and two explicit examples for pairs of supersymmetric Hamiltonians. In one case both partner Hamiltonians are Hermitian and in the other a Hermitian Hamiltonian is paired by a Darboux transformation to a non-Hermitian one.