Thermodynamic stability conditions for nonadditive composable entropies

The thermodynamic stability conditions (TSC) of nonadditive and composable entropies are discussed. Generally the concavity of a nonadditive entropy with respect to internal energy is not necessarily equivalent to the corresponding TSC. It is shown that both the TSC of Tsallis' entropy and that of the $\kappa$-generalized Boltzmann entropy are equivalent to the positivity of the standard heat capacity.Comment: 6pages; Contribution to a topical issue of Continuum Mechanics and Thermodynamics (CMT), edited by M. Sugiyam

Resource Bounded Unprovability of Computational Lower Bounds

This paper introduces new notions of asymptotic proofs, PT(polynomial-time)-extensions, PTM(polynomial-time Turing machine)-omega-consistency, etc. on formal theories of arithmetic including PA (Peano Arithmetic). This paper shows that P not= NP (more generally, any super-polynomial-time lower bound in PSPACE) is unprovable in a PTM-omega-consistent theory T, where T is a consistent PT-extension of PA. This result gives a unified view to the existing two major negative results on proving P not= NP, Natural Proofs and relativizable proofs, through the two manners of characterization of PTM-omega-consistency. We also show that the PTM-omega-consistency of T cannot be proven in any PTM-omega-consistent theory S, where S is a consistent PT-extension of T.Comment: 78 page

Railways competition in a park-and-ride model

It is well understood that the way of use of automobiles within urban areas should be reconsidered for environmental improvements, such as reduction of CO2 or energy itself. In this paper, scale effect of city size and cost advantage of railway over automobiles is examined in a simple setting with so-called Park-and-Ride commuter system. Our results are as follows: The fare per mile charged by a monopoly railway firm may not be relevant to the city size; the fare in a symmetric equilibrium under monopolistic competition is decreasing with the city size; the fare in a symmetric zero-profit kinked equilibrium is increasing with the city size; the fare of social optimum is decreasing with the city size.