Did Lobachevsky Have A Model Of His "imaginary Geometry"?

The invention of non-Euclidean geometries is often seen through the optics of Hilbertian formal axiomatic method developed later in the 19th century. However such an anachronistic approach fails to provide a sound reading of Lobachevsky's geometrical works. Although the modern notion of model of a given theory has a counterpart in Lobachevsky's writings its role in Lobachevsky's geometrical theory turns to be very unusual. Lobachevsky doesn't consider various models of Hyperbolic geometry, as the modern reader would expect, but uses a non-standard model of Euclidean plane (as a particular surface in the Hyperbolic 3-space). In this paper I consider this Lobachevsky's construction, and show how it can be better analyzed within an alternative non-Hilbertian foundational framework, which relates the history of geometry of the 19th century to some recent developments in the field.Comment: 31 pages, 8 figure

Electromagnetic transitions between giant resonances within a continuum-RPA approach

A general continuum-RPA approach is developed to describe electromagnetic transitions between giant resonances. Using a diagrammatic representation for the three-point Green's function, an expression for the transition amplitude is derived which allows one to incorporate effects of mixing of single and double giant resonances as well as to take the entire basis of particle-hole states into consideration. The radiative widths for E1 transition between the charge-exchange spin-dipole giant resonance and Gamow-Teller states are calculated for ^{90}Nb and ^{208}Bi nuclei. The importance of the mixing is stressed.Comment: 10 pages, 2 figures, uses elsart.st

Categories without structures

The popular view according to which Category theory provides a support for Mathematical Structuralism is erroneous. Category-theoretic foundations of mathematics require a different philosophy of mathematics. While structural mathematics studies invariant forms (Awodey) categorical mathematics studies covariant transformations which, generally, don t have any invariants. In this paper I develop a non-structuralist interpretation of categorical mathematics and show its consequences for history of mathematics and mathematics education.Comment: 28 page

First application of the continuum-QRPA to description of the double beta decay

A continuum-QRPA approach to calculation of the $2\nu\beta\beta$- and $0\nu\beta\beta$-amplitudes has been formulated. For $^{130}$Te a regular suppression (about 20%) of the high-multipole contributions to the $0\nu\beta\beta$-amplitude has been found which can be associated with additional ground state correlations appearing from the transitions to collective states in the continuum. At the same time the total calculated $0\nu\beta\beta$-amplitude for $^{130}$Te gets suppressed by about 20% as compared to the result of the usual, discretized, QRPA.Comment: 8 pages, 2 figures. Proceedings of the 27th Int. School on Nuclear Physics "Neutrinos in Cosmology, in Astro, Particle and Nuclear Physics", Erice, Italy, Sept. 16-24, 2005. To appear in Prog.Part.Nucl.Phy