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

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

Money and Credit With Limited Commitment and Theft

We study the interplay among imperfect memory, limited commitment, and theft, in an environment that can support monetary exchange and credit. Imperfect memory makes money useful, but it also permits theft to go undetected, and therefore provides lucrative opportunities for thieves. Limited commitment constrains credit arrangements, and the constraints tend to tighten with imperfect memory, as this mitigates punishment for bad behavior in the credit market. Theft matters for optimal monetary policy, but at the optimum theft will not be observed in the model. The Friedman rule is in general not optimal with theft, and the optimal money growth rate tends to rise as the cost of theft falls.

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

Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation

The modified Camassa-Holm (mCH) equation is a bi-Hamiltonian system possessing $N$-peakon weak solutions, for all $N\geq 1$, in the setting of an integral formulation which is used in analysis for studying local well-posedness, global existence, and wave breaking for non-peakon solutions. Unlike the original Camassa-Holm equation, the two Hamiltonians of the mCH equation do not reduce to conserved integrals (constants of motion) for $2$-peakon weak solutions. This perplexing situation is addressed here by finding an explicit conserved integral for $N$-peakon weak solutions for all $N\geq 2$. When $N$ is even, the conserved integral is shown to provide a Hamiltonian structure with the use of a natural Poisson bracket that arises from reduction of one of the Hamiltonian structures of the mCH equation. But when $N$ is odd, the Hamiltonian equations of motion arising from the conserved integral using this Poisson bracket are found to differ from the dynamical equations for the mCH $N$-peakon weak solutions. Moreover, the lack of conservation of the two Hamiltonians of the mCH equation when they are reduced to $2$-peakon weak solutions is shown to extend to $N$-peakon weak solutions for all $N\geq 2$. The connection between this loss of integrability structure and related work by Chang and Szmigielski on the Lax pair for the mCH equation is discussed.Comment: Minor errata in Eqns. (32) to (34) and Lemma 1 have been fixe

The costs of remoteness: evidence from German division and reunification

This paper exploits the division of Germany after the Second World War and the reunification of East and West Germany in 1990 as a natural experiment to provide evidence of the importance of market access for economic development. In line with a standard new economic geography model, we find that following division cities in West Germany that were close to the new border between East and West Germany experienced a substantial decline in population growth relative to other West German cities. We provide several pieces of evidence that the decline of the border cities can be entirely accounted for by their loss in market access and is neither driven by differences in industrial structure nor differences in the degree of warrelated destruction. Finally, we also find some first evidence of a recovery of the border cities after the re-unification of East and West Germany

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