Stokes equations with penalised slip boundary conditions

We consider the finite-element approximation of Stokes equations with slip boundary conditions imposed with the penalty method. In the case of a smooth curved boundary, our numerical results suggest that curved finite elements, regularised normal vectors or reduced integration techniques can be used to avoid a Babuska’s-type paradox and ensure the convergence of finite-element approximations to the exact solution. Convergence orders with these remedies are also compared.     

Testing Bell inequalities in photonic crystals

We show how entangled atomic pairs can be prepared in order to test the Bell inequalities. The scheme is based on the interaction of the atoms with a highly localized field mode within a photonic crystal. The potential of using optically separated transitions and the stability of the entangled state to spontaneous emission could lead to the closure of the communication and the detection loopholes appearing in experiments so far. The robustness of the scheme against detector inefficiencies, the spread in the atomic velocities and the fact that the entangled pairs are not generated simultaneously is also studied. Received 31 July 2001 and Received in final form 30 November 2001     

The liberal paradox and the pareto set

We say that the liberal paradox occurs where the set of alternatives is a disjoint union of sets $\text{X}{}_{\mathrm{i}}$, one for each voter, if and only if the Pareto set contains no element $\text{u}$ of any $\text{X}{}_{\mathrm{i}}$ such that voter $\text{i}$ prefers $\text{u}$ to any other element of $\text{X}{}_{\mathrm{i}}$.For random profiles, we estimate the size of the Pareto set, and show that (1) if the number of voters is held fixed but the number of alternatives tends to infinity, the probability of the liberal paradox tends to 1; (2) if the number of alternatives is a fixed ratio to the number of voters and the number of voters tends to infinity, the probability of the liberal paradox tends to zero; (3) the rights can always be redistributed to the voters so that the liberal paradox does not occur; and (4) for single-peaked preferences if the sets of rights form intervals, the liberal paradox does not occur.     

Negative probabilities and information gain in weak measurements

We study the outcomes in a general measurement with postselection, and derive upper bounds for the pointer readings in weak measurement. The probabilities inferred from weak measurements change along with the coupling strength; and the true probabilities can be obtained when the coupling is strong enough. By calculating the information gain of the measuring device about which path the particles pass through, we show that the “negative probabilities” only emerge for cases when the information gain is little due to very weak coupling between the measuring device and the particles. When the coupling strength increases, we can unambiguously determine whether a particle passes through a given path every time, hence the average shifts always represent true probabilities, and the strange “negatives probabilities” disappear.     

Construction of novel stochastic matrices for analysis of Parrondo’s paradox

In Parrondo’s paradox, a winning strategy is formed either by playing two losing games randomly or alternating them periodically. The paradox is commonly analyzed using stochastic matrices. In this paper, we modify the stochastic matrices to allow a more systematic introduction of bias into fair processes, while retaining the use of simple matrix operations throughout the analysis.     

Spin Correlations in Nonrelativistic Quantum Mechanics

Einstein—Podolsky—Rosen spin correlations in the framework of nonrelativistic quantum mechanics for moving observers are calculated. The measurements are performed in bounded regions of space (detectors), not necessarily simultaneously. The resulting correlation function depends not only on the directions of spin measurements but also on the relative velocity of the observers.     

Relativistic Sagnac Effect and Ehrenfest Paradox

There seems to exist a dilemma in the literature as to the correct relativistic formula for the Sagnac phase-shift. The paper addresses this issue in the light of a novel, kinematically equivalent linear Sagnac-type thought experiment, which provides a vantage point from which the effect of rotation in the usual Sagnac effect can be analyzed. The question is shown to be related to the so-called rotating disc problem known as the Ehrenfest paradox. The relativistic formula for the Sagnac phase-shift seems to depend on the way the paradox is resolved. Kinematic resolution of the Ehrenfest paradox proposed by some authors predicts the usually quoted formula for the Sagnac delay but the resolution itself is shown to be based upon some implicit assumptions regarding the behaviour of solid bodies under acceleration. In order to have a greater insight into the problem, a second version of the thought experiment involving linear motion of a special type of a non-rigid frame of reference is discussed. It is shown by analogy that the usually quoted special relativistic formula for the Sagnac delay follows, provided the material of the disc matches the special type.     

Quantum Brain States

If conscious observers are to be included in the quantum mechanical universe, we need to find the rules that engage observers with quantum mechanical systems. The author has proposed five rules that are discovered by insisting on empirical completeness; that is, by requiring the rules to draw empirical information from Schrödinger's solutions that is more complete than is currently possible with the (Born) probability interpretation. I discard Born's interpretation, introducing probability solely through probability current. These rules tell us something about brains. They require the existence of observer brain states that are neither conscious nor unconscious. I call them ready brain states because they are on stand-by, ready to become conscious the moment they are stochastically chosen. Two of the rules are selection rules involving ready brain states. The place of these rules in a wider theoretical context is discussed.     

Quantum theory does not require action at a distance

The question of whether or not the correlations predicted by quantum theory and actually observed in EPR-type experiments can be understood without assuming action at a distance is still controversial. The author's point of view, expressed by the title of this paper, is substantiated by the discussion of a specific example, and is defended against some recent criticism by H. P. Stapp.1. These predictions arenot exhausted by Eq. (1); see [9].