Berkeley on God's Knowledge of Pain

Since nothing about God is passive, and the perception of pain is inherently passive, then it seems that God does not know what it is like to experience pain. Nor would he be able to cause us to experience pain, for his experience would then be a sensation (which would require God to have senses, which he does not). My suggestion is that Berkeley avoids this situation by describing how God knows about pain “among other things” (i.e. as something whose identity is intelligible in terms of the integrated network of things). This avoids having to assume that God has ideas (including pain) apart from his willing that there be perceivers who have specific ideas that are in harmony or not in harmony with one another

Direct evaluation of dynamical large-deviation rate functions using a variational ansatz

We describe a simple form of importance sampling designed to bound and compute large-deviation rate functions for time-extensive dynamical observables in continuous-time Markov chains. We start with a model, defined by a set of rates, and a time-extensive dynamical observable. We construct a reference model, a variational ansatz for the behavior of the original model conditioned on atypical values of the observable. Direct simulation of the reference model provides an upper bound on the large-deviation rate function associated with the original model, an estimate of the tightness of the bound, and, if the ansatz is chosen well, the exact rate function. The exact rare behavior of the original model does not need to be known in advance. We use this method to calculate rate functions for currents and counting observables in a set of network- and lattice models taken from the literature. Straightforward ansatze yield bounds that are tighter than bounds obtained from Level 2.5 of large deviations via approximations that involve uniform scalings of rates. We show how to correct these bounds in order to recover the rate functions exactly. Our approach is complementary to more specialized methods, and offers a physically transparent framework for approximating and calculating the likelihood of dynamical large deviations

Determination of the biquaternion divisors of zero, including the idempotents and nilpotents

The biquaternion (complexified quaternion) algebra contains idempotents (elements whose square remains unchanged) and nilpotents (elements whose square vanishes). It also contains divisors of zero (elements with vanishing norm). The idempotents and nilpotents are subsets of the divisors of zero. These facts have been reported in the literature, but remain obscure through not being gathered together using modern notation and terminology. Explicit formulae for finding all the idempotents, nilpotents and divisors of zero appear not to be available in the literature, and we rectify this with the present paper. Using several different representations for biquaternions, we present simple formulae for the idempotents, nilpotents and divisors of zero, and we show that the complex components of a biquaternion divisor of zero must have a sum of squares that vanishes, and that this condition is equivalent to two conditions on the inner product of the real and imaginary parts of the biquaternion, and the equality of the norms of the real and imaginary parts. We give numerical examples of nilpotents, idempotents and other divisors of zero. Finally, we conclude with a statement about the composition of the set of biquaternion divisors of zero, and its subsets, the idempotents and the nilpotents.Comment: 7 page

Adverse Selection, Segmented Markets, and the Role of Monetary Policy

A model is constructed in which trading partners are asymmetrically informed about future trading opportunities and where spatial and informational frictions limit arbitrage between markets. These frictions create an inefficiency relative to a full information equilibrium, and the extent of this inefficiency is affected by monetary policy. A Friedman rule is optimal under a wide range of circumstances, including ones where segmented markets limit the extent of monetary policy intervention.Adverse Selection; Monetary Policy; Search