The Downs-Thomson Paradox: Existence, Uniqueness and Stability of User Equilibria

Consider a network where two routes are available for users wishing to travel from a source to a destination. On one route (which could be viewed as private transport) service slows as traffic increases. On the other (which could be viewed as public transport) the service frequency increases with demand. The Downs-Thomson paradox occurs when improvements in service produce an overall decline in performance as user equilibria adjust. Using the model proposed by Calvert [10], with a ⋅|M|1 queue corresponding to the private transport route, and a bulk-service infinite server queue modelling the public transport route, we give a complete analysis of this system in the setting of probabilistic routing. We obtain the user equilibria (which are not always unique), and determine their stability.AMS subject classification: 60K30, 90B15, 90B20, 91A10, 91A13This revised version was published online in June 2005 with corrected coverdate     

Formalization of the “Twin Paradox” for Non-uniformly Accelerated Motions

The use of hyperbolic numbers for studying space-time geometry and trigonometry is extended for demonstrating the Frenet’s formulas in space-time. By means of this introduction the twin paradox for non-uniformly accelerated motions is formalized in a straightforward way.     

Boolos‐style proofs of limitative theorems

Boolos's proof of incompleteness is extended straightforwardly to yield simple “diagonalization‐free” proofs of some classical limitative theorems of logic. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Envy-free rights assignments and self-oriented preferences

Very recently a new solution to Sen's “Impossibility of a Paretian liberal” has been suggested where the focus is on the rights assignments per se (Austen-Smith, 1979). It was shown that the concept of fairness, when applied to rights, admits the existence of social decision functions which satisfy Sen's original conditions. Unfortunately this result collapses when individuals have rights over more than one pair of alternatives.In order to obtain possibility results for this more general case the present paper proposes to restrict individuals' preference orderings. It is proved that envy-free collective choice rules exist if individual preferences are self-oriented and if, in addition, people attach primary importance to their own private sphere alternatives. These restrictions are quite severe, but they may be justified if one values the absence of envy in rights allocations very highly.     

关于完全气体无粘可压流动的解析解

本文导出了一套不定常二、三维完全气体无粘可压流动的代数显式解析解，这些解除在理论上有重要意义外，还可以用来作为检验与发展各种数值解法的标准解．另外，文中还将提出和讨论二维可压等熵流的一个很有意思的疑难．     

A note on the ‘loser wins’ game

A simple explanation if provided for the ‘loser wins paradox’ and properties of the corresponding game are explored.     

Extensivity of entropy and modern form of Gibbs paradox

The extensivity property of entropy is clarified in the light of a critical examination of the entropy formula based on quantum statistics and the relevant thermodynamic requirement. The modern form of the Gibbs paradox, related to the discontinuous jump in entropy due to identity or non-identity of particles, is critically investigated. Qualitative framework of a new resolution of this paradox, which analyses the general effect of distinction mark on the Hamiltonian of a system of identical particles, is outlined.     

Sometimes binary is better

This note uses material involving perfect numbers and Zeno's paradoxes to show that although most students prefer to use base 10 when working with mathematical concepts there are times when the binary system is best.     

Bilattices for deductions in multi-valued logic

In this exploratory paper we propose a framework for the deduction apparatus of multi-valued logics based on the idea that a deduction apparatus has to be a tool to manage information on truth values and not directly truth values of the formulas. This is obtained by embedding the algebraic structure V defined by the set of truth values into a bilattice B. The intended interpretation is that the elements of B are pieces of information on the elements of V. The resulting formalisms are particularized in the framework of fuzzy logic programming. Since we see fuzzy control as a chapter of multi-valued logic programming, this suggests a new and powerful approach to fuzzy control based on positive and negative conditions.     

Parrondo’s paradox and complementary Parrondo processes

Parrondo’s Paradox has gained a fair amount of attention due to it being counter-intuitive. Given two stochastic processes, both of which are losing in nature, it is possible to have an overall net increase in capital by periodically or randomly alternating between the two processes. In this paper, we analyze the paradox with a different approach, in which we start with one process and seek to derive its complementary process. We will also state the conditions required for this to occur. Possible applications of our results include the development of future models based on the paradox.