Note on Bessaga-Klee classification

We collect several variants of the proof of the third case of the Bessaga-Klee relative classification of closed convex bodies in topological vector spaces. We were motivated by the fact that we have not found anywhere in the literature a complete correct proof. In particular, we point out an error in the proof given in the book of C.~Bessaga and A.~Pe\l czy\'nski (1975). We further provide a simplified version of T.~Dobrowolski's proof of the smooth classification of smooth convex bodies in Banach spaces which works simultaneously in the topological case.Comment: 14 pages; we made few corrections, added one reference and precised the abstrac

Rich families and elementary submodels

We compare two methods of proving separable reduction theorems in functional analysis -- the method of rich families and the method of elementary submodels. We show that any result proved using rich families holds also when formulated with elementary submodels and the converse is true in spaces with fundamental minimal system an in spaces of density $\aleph_1$. We do not know whether the converse is true in general. We apply our results to show that a projectional skeleton may be without loss of generality indexed by ranges of its projections

Optical geometry analysis of the electromagnetic self-force

We present an analysis of the behaviour of the electromagnetic self-force for charged particles in a conformally static spacetime, interpreting the results with the help of optical geometry. Some conditions for the vanishing of the local terms in the self-force are derived and discussed.Comment: 18 pages; 2 figure

Neutrino-driven explosions twenty years after SN1987A

The neutrino-heating mechanism remains a viable possibility for the cause of the explosion in a wide mass range of supernova progenitors. This is demonstrated by recent two-dimensional hydrodynamic simulations with detailed, energy-dependent neutrino transport. Neutrino-driven explosions were not only found for stars in the range of 8-10 solar masses with ONeMg cores and in case of the iron core collapse of a progenitor with 11 solar masses, but also for a ``typical'' progenitor model of 15 solar masses. For such more massive stars, however, the explosion occurs significantly later than so far thought, and is crucially supported by large-amplitude bipolar oscillations due to the nonradial standing accretion shock instability (SASI), whose low (dipole and quadrupole) modes can develop large growth rates in conditions where convective instability is damped or even suppressed. The dominance of low-mode deformation at the time of shock revival has been recognized as a possible explanation of large pulsar kicks and of large-scale mixing phenomena observed in supernovae like SN 1987A.Comment: 11 pages, 6 figures; review proceeding for "Supernova 1987A: 20 Years After: Supernovae and Gamma-Ray Bursters" AIP, New York, eds. S. Immler, K.W. Weiler, and R. McCra

Extremal spacings between eigenphases of random unitary matrices and their tensor products

Extremal spacings between eigenvalues of random unitary matrices of size N pertaining to circular ensembles are investigated. Explicit probability distributions for the minimal spacing for various ensembles are derived for N = 4. We study ensembles of tensor product of k random unitary matrices of size n which describe independent evolution of a composite quantum system consisting of k subsystems. In the asymptotic case, as the total dimension N = n^k becomes large, the nearest neighbor distribution P(s) becomes Poissonian, but statistics of extreme spacings P(s_min) and P(s_max) reveal certain deviations from the Poissonian behavior

A Proof of Tarski’s Fixed Point Theorem by Application of Galois Connections

Two examples of Galois connections and their dual forms are considered. One of them is applied to formulate a criterion when a given subset of a complete lattice forms a complete lattice. The second, closely related to the first, is used to prove in a short way the Knaster-Tarski’s fixed point theore

Wigner-Eckart theorem for tensor operators of Hopf algebras

We prove Wigner-Eckart theorem for the irreducible tensor operators for arbitrary Hopf algebras, provided that tensor product of their irreducible representation is completely reducible. The proof is based on the properties of the irreducible representations of Hopf algebras, in particular on Schur lemma. Two classes of tensor operators for the Hopf algebra U$_{t}$(su(2)) are considered. The reduced matrix elements for the class of irreducible tensor operators are calculated. A construction of some elements of the center of U$_{t}$(su(2)) is given.Comment: 14 pages, late

Degree of entanglement as a physically ill-posed problem: The case of entanglement with vacuum

We analyze an example of a photon in superposition of different modes, and ask what is the degree of their entanglement with vacuum. The problem turns out to be ill-posed since we do not know which representation of the algebra of canonical commutation relations (CCR) to choose for field quantization. Once we make a choice, we can solve the question of entanglement unambiguously. So the difficulty is not with mathematics, but with physics of the problem. In order to make the discussion explicit we analyze from this perspective a popular argument based on a photon leaving a beam splitter and interacting with two two-level atoms. We first solve the problem algebraically in Heisenberg picture, without any assumption about the form of representation of CCR. Then we take the $\infty$-representation and show in two ways that in two-mode states the modes are maximally entangled with vacuum, but single-mode states are not entangled. Next we repeat the analysis in terms of the representation of CCR taken from Berezin's book and show that two-mode states do not involve the mode-vacuum entanglement. Finally, we switch to a family of reducible representations of CCR recently investigated in the context of field quantization, and show that the entanglement with vacuum is present even for single-mode states. Still, the degree of entanglement is here difficult to estimate, mainly because there are $N+2$ subsystems, with $N$ unspecified and large.Comment: This paper is basically a reply to quant-ph/0507189 by S. J. van Enk and to the remarks we got from L. Vaidman after our preliminary quant-ph/0507151. Version accepted in Phys. Rev.