On a theory of the bb-function in positive characteristic

We present a theory of the $b$-function (or Bernstein-Sato polynomial) in positive characteristic. Let $f$ be a non-constant polynomial with coefficients in a perfect field $k$ of characteristic $p>0.$ Its $b$-function $b_f$ is defined to be an ideal of the algebra of continuous $k$-valued functions on $\mathbb{Z}_p.$ The zero-locus of the $b$-function is thus naturally interpreted as a subset of $\mathbb{Z}_p,$ which we call the set of roots of $b_f.$ We prove that $b_f$ has finitely many roots and that they are negative rational numbers. Our construction builds on an earlier work of Musta\c{t}\u{a} and is in terms of $D$-modules, where $D$ is the ring of Grothendieck differential operators. We use the Frobenius to obtain finiteness properties of $b_f$ and relate it to the test ideals of $f.$Comment: Final versio

Scalable quantum computation with fast gates in two-dimensional microtrap arrays of trapped ions

We theoretically investigate the use of fast pulsed two-qubit gates for trapped ion quantum computing in a two-dimensional microtrap architecture. In one dimension, such fast gates are optimal when employed between nearest neighbours, and we examine the generalisation to a two-dimensional geometry. We demonstrate that fast pulsed gates are capable of implementing high-fidelity entangling operations between ions in neighbouring traps faster than the trapping period, with experimentally demonstrated laser repetition rates. Notably, we find that without increasing the gate duration, high-fidelity gates are achievable even in large arrays with hundreds of ions. To demonstrate the usefulness of this proposal, we investigate the application of these gates to the digital simulation of a 40-mode Fermi-Hubbard model. This also demonstrates why shorter chains of gates required to connect arbitrary pairs of ions makes this geometry well suited for large-scale computation

The Market Price of Aggregate Risk and the Wealth Distribution

We introduce limited liability in a model with a continuum of ex ante identical agents who face aggregate and idiosyncratic income risk. These agents can trade a complete menu of contingent claims, but they cannot commit and shares in a Lucas tree serve as collateral to back up their state-contingent promises. The limited liability option gives rise to a second risk factor, in addition to aggregate consumption growth risk. This liquidity risk is created by binding solvency constraints, and it is measured by the growth rate of one moment of the wealth distribution. The economy is said to experience a negative liquidity shock when this growth rate is high and a large fraction of agents faces severely binding solvency constraints. The adjustment to the Breeden-Lucas stochastic discount factor induces substantial time variation in equity risk premia that is consistent with the data at business cycle frequencies.

Toward Sustainability: Technology Transition and Endogenous Population Growth

In order to reach the state of economic sustainability, the problem of technology transition emphasizes the possibility of substituting for the exhaustible resource with an everlasting source of energy input. This paper aims at providing an analysis of this problem in an overlapping-generation model where the population is not a datum, but endogenous in the sense that it results from fertility decisions made by economic agents. First, we provide a new proof of the existence of competitive equilibrium under infinite time horizon. Here the difficulty lies in the fact that the market size is itself endogenous, because fertility - hence the population - is an individual decision at every point in time. Second, and perhaps most interestingly, the oil stock might not be entirely depleted, and the unused part in situ may serve the role of storing value for wealth transmission over time, just as money. But in contrast with paper money, which has no intrinsic value, leaving productive oil in situ as a bubble certainly adds another dimension to the inefficiency of overlapping-generation model. In this case, there are infinitely many equilibria as well as many steady states, depending on the data that characterize the initial state of the economy. Moreover, the convergence to some steady state, far from being simply monotone, might exhibit cyclical behavior, such as damped oscillation, limit cycles, etc.Endogenous Population, Overlapping Generations, Oil Bubble, Dynamic Equilibria, Complex Dynamics

Monetary Policy with Incomplete Markets

We consider an extension of a general equilibrium model with incomplete markets that considers cash-in-advance constraints. The total amount of money is supplied by an authority, which produces at no cost and lends money to agents at short term nominal rates of interest, meeting the demand. Agents have initial nominal claims, which in the aggregate, are the counterpart of an initial public debt. The authority covers its expenditures, including initial debt, through public revenues which consists of taxes and seignorage, and distributes its eventual budget surpluses through transfers to individuals, while no further instruments are available to correct eventual budget deficits. We define a concept of equilibrium in this extended model, and prove that there exists a monetary equilibrium with no transfers. Moreover, we show that if the price level is high enough, a monetary equilibrium with positive transfers exists.Cash-in-advance constraints, incomplete markets, nominal assets, monetary equilibrium, money, nominal interest rate, transfers, price levels

Incomplete markets and monetary policy

We consider an extension of a general equilibrium model with incomplete markets that considers cash-in-advance constraints. The total amount of money is supplied by an authority, which produces at no cost and lends money to agents at short term nominal rates of interest, meeting the demand. Agents have initial nominal claims, which in the aggregate, are the counterpart of an initial public debt. The authority covers its expenditures, including initial debt, through public revenues which consists of taxes and seignorage, and distributes its eventual budget surpluses through transfers to individuals, while no further instruments are available to correct eventual budget deficits. We define a concept of equilibrium in this extended model, and prove that there exists a monetary equilibrium with no transfers. Moreover, we show that if the price level is high enough, a monetary equilibrium with transfers exists.Cash-in-advance constraints, incomplete markets, nominal assets, monetary equilibrium, money, nominal interest rate.

Parameterizing Quasiperiodicity: Generalized Poisson Summation and Its Application to Modified-Fibonacci Antenna Arrays

The fairly recent discovery of "quasicrystals", whose X-ray diffraction patterns reveal certain peculiar features which do not conform with spatial periodicity, has motivated studies of the wave-dynamical implications of "aperiodic order". Within the context of the radiation properties of antenna arrays, an instructive novel (canonical) example of wave interactions with quasiperiodic order is illustrated here for one-dimensional (1-D) array configurations based on the "modified-Fibonacci" sequence, with utilization of a two-scale generalization of the standard Poisson summation formula for periodic arrays. This allows for a "quasi-Floquet" analytic parameterization of the radiated field, which provides instructive insights into some of the basic wave mechanisms associated with quasiperiodic order, highlighting similarities and differences with the periodic case. Examples are shown for quasiperiodic infinite and spatially-truncated arrays, with brief discussion of computational issues and potential applications.Comment: 29 pages, 10 figures. To be published in IEEE Trans. Antennas Propagat., vol. 53, No. 6, June 200

Spatiotemporal dynamics in 2D Kolmogorov flow over large domains

Kolmogorov flow in two dimensions - the two-dimensional Navier-Stokes equations with a sinusoidal body force - is considered over extended periodic domains to reveal localised spatiotemporal complexity. The flow response mimicks the forcing at small forcing amplitudes but beyond a critical value develops a long wavelength instability. The ensuing state is described by a Cahn-Hilliard-type equation and as a result coarsening dynamics are observed for random initial data. After further bifurcations, this regime gives way to multiple attractors, some of which possess spatially-localised time dependence. Co-existence of such attractors in a large domain gives rise to interesting collisional dynamics which is captured by a system of 5 (1-space and 1-time) PDEs based on a long wavelength limit. The coarsening regime reinstates itself at yet higher forcing amplitudes in the sense that only longest-wavelength solutions remain attractors. Eventually, there is one global longest-wavelength attractor which possesses two localised chaotic regions - a kink and antikink - which connect two steady one-dimensional flow regions of essentially half the domain width each. The wealth of spatiotemporal complexity uncovered presents a bountiful arena in which to study the existence of simple invariant localised solutions which presumably underpin all of the observed behaviour

Endogenous debt constraints in collateralized economies with default penalties

In infinite horizon financial markets economies, competitive equilibria fail to exist if one does not impose restrictions on agents' trades that rule out Ponzi schemes. When there is limited commitment and collateral repossession is the unique default punishment, Araujo, Páscoa and Torres-Martínez (2002) proved that Ponzi schemes are ruled out without imposing any exogenous/endogenous debt constraints on agents' trades. Recently Páscoa and Seghir (2009) have shown that this positive result is not robust to the presence of additional default punishments. They provide several examples showing that, in the absence of debt constraints, harsh default penalties may induce agents to run Ponzi schemes that jeopardize equilibrium existence.The objective of this paper is to close a theoretical gap in the literature by identifying endogenous borrowing constraints that rule out Ponzi schemes and ensure existence of equilibria in a model with limitedcommitment and (possible) default. We appropriately modify the definition of finitely effective debt constraints, introduced by Levine and Zame (1996) (see also Levine and Zame (2002)), to encompass models with limited commitment, default penalties and collateral. Along this line, we introduce in the setting of Araujo, Páscoa and Torres-Martínez (2002), Kubler and Schmedders (2003) and Páscoa and Seghir (2009) the concept of actions with finite equivalent payoffs. We show that, independently of the level of default penalties, restricting plans to have finite equivalent payoffs rules out Ponzi schemes and guarantees the existence of an equilibrium that is compatible with the minimal ability to borrow and lend that we expect in our model.An interesting feature of our debt constraints is that they give rise to budget sets that coincide with the standard budget sets of economies having a collateral structure but no penalties (as defined in Araujo,Páscoa and Torres-Martínez (2002)). This illustrates the hidden relation between finitely effective debt constraints and collateral requirements.