Exchangeable measures for subshifts

Let \Om be a Borel subset of $S^\Bbb N$ where $S$ is countable. A measure is called exchangeable on \Om, if it is supported on \Om and is invariant under every Borel automorphism of \Om which permutes at most finitely many coordinates. De-Finetti's theorem characterizes these measures when \Om=S^\Bbb N. We apply the ergodic theory of equivalence relations to study the case \Om\neq S^\Bbb N, and obtain versions of this theorem when \Om is a countable state Markov shift, and when \Om is the collection of beta expansions of real numbers in $[0,1]$ (a non-Markovian constraint)

Tort Liability and Vaccine Manufacturers

This paper has been written with future vaccines in mind. It is true, of course, that most vaccines currently available are extremely safe and not prohibitively expensive. For the few injuries caused by these vaccines, an insurance system paid for by their manufacturers might be feasible and reasonable. The small increase in a manufacturer's cost of doing business could be offset by a similarly small increase in the price of the particular vaccine. Disincentive to create new vaccines would be minimal. But vaccines of the future may only become this safe if government insures them during their earliest stages of development. In this light, a government-paid system may be a necessary bridge between the riskier and safer periods of a vaccine's life

On Hausdorff dimension of the set of closed orbits for a cylindrical transformation

We deal with Besicovitch's problem of existence of discrete orbits for transitive cylindrical transformations $T_\varphi:(x,t)\mapsto(x+\alpha,t+\varphi(x))$ where $Tx=x+\alpha$ is an irrational rotation on the circle \T and \varphi:\T\to\R is continuous, i.e.\ we try to estimate how big can be the set D(\alpha,\varphi):=\{x\in\T:|\varphi^{(n)}(x)|\to+\infty\text{as}|n|\to+\infty\}. We show that for almost every $\alpha$ there exists $\varphi$ such that the Hausdorff dimension of $D(\alpha,\varphi)$ is at least $1/2$. We also provide a Diophantine condition on $\alpha$ that guarantees the existence of $\varphi$ such that the dimension of $D(\alpha,\varphi)$ is positive. Finally, for some multidimensional rotations $T$ on \T^d, $d\geq3$, we construct smooth $\varphi$ so that the Hausdorff dimension of $D(\alpha,\varphi)$ is positive.Comment: 32 pages, 1 figur

Can closed timelike curves or nonlinear quantum mechanics improve quantum state discrimination or help solve hard problems?

We study the power of closed timelike curves (CTCs) and other nonlinear extensions of quantum mechanics for distinguishing nonorthogonal states and speeding up hard computations. If a CTC-assisted computer is presented with a labeled mixture of states to be distinguished--the most natural formulation--we show that the CTC is of no use. The apparent contradiction with recent claims that CTC-assisted computers can perfectly distinguish nonorthogonal states is resolved by noting that CTC-assisted evolution is nonlinear, so the output of such a computer on a mixture of inputs is not a convex combination of its output on the mixture's pure components. Similarly, it is not clear that CTC assistance or nonlinear evolution help solve hard problems if computation is defined as we recommend, as correctly evaluating a function on a labeled mixture of orthogonal inputs.Comment: 4 pages, 3 figures. Final version. Added several references, updated discussion and introduction. Figure 1(b) very much enhance

Operator renewal theory and mixing rates for dynamical systems with infinite measure

We develop a theory of operator renewal sequences in the context of infinite ergodic theory. For large classes of dynamical systems preserving an infinite measure, we determine the asymptotic behaviour of iterates $L^n$ of the transfer operator. This was previously an intractable problem. Examples of systems covered by our results include (i) parabolic rational maps of the complex plane and (ii) (not necessarily Markovian) nonuniformly expanding interval maps with indifferent fixed points. In addition, we give a particularly simple proof of pointwise dual ergodicity (asymptotic behaviour of $\sum_{j=1}^nL^j$) for the class of systems under consideration. In certain situations, including Pomeau-Manneville intermittency maps, we obtain higher order expansions for $L^n$ and rates of mixing. Also, we obtain error estimates in the associated Dynkin-Lamperti arcsine laws.Comment: Preprint, August 2010. Revised August 2011. After publication, a minor error was pointed out by Kautzsch et al, arXiv:1404.5857. The updated version includes minor corrections in Sections 10 and 11, and corresponding modifications of certain statements in Section 1. All main results are unaffected. In particular, Sections 2-9 are unchanged from the published versio

Teaching Problem-Solving Lawyering: An Exchange of Ideas

In the last issue of the Clinical Law Review, StefanKrieger argues that clinical law teachers who emphasize problem-solving approaches to lawyering incorrectly downplay as a necessary prerequisite to learning effective legal practice the significance of domain knowledge, which he mainly identifies as knowledge about legal doctrine(FN1) Among the writings on clinical law teaching criticized by Krieger are those of Mark Aaronson, who has articulated as a teaching goal helping students learn how to improve their practical judgment in lawyering, which he describes as a process of deliberation whose most prominent features are a contextual tailoring of knowledge, a dialogic form of reasoning that accounts for plural perspectives, an ability to be empathetic and detached at the same time an intertwining of intellectual and moral concerns, an instrumental and equitable interest in human affairs, and a heavy reliance on learning from cumulative experience.(FN2) Krieger stresses the foundational importance for law students of acquiring substantive legal knowledge; Aaronson focuses on developing the ability of students to think critically and appropriately in role as a lawyer. In this brief exchange of ideas, Aaronson comments on Krieger\u27s critique of problem-solving teaching in law schools, to which Krieger then responds

Influence Of Continuous Precipitation Upon The Growth Kinetics Of The Cellular Reaction In An Al-Ag Alloy

The influence of the prior formation of a continuous precipitate upon the growth kinetics of the cellular reaction has been evaluated in an Al-17.9 wt. % Ag alloy. The continuous precipitate, in the form of intragranular plates of the γ′ transition phase, was shown to have reduced the upper bound of the driving force for the cellular reaction from the silver content of the untransformed alloy to that corresponding to the coherent solvus. When this reduction (≥ 98 %) is taken into account, the growth of cells is found to be controlled by cell boundary rather than by volume diffusion on the basis of both the Turnbull and the Cahn theories of the cellular reaction. Changing the mode of heat treatment from the usual quenching-and-aging to that of isothermal transformation reduces both the rate of growth of cells and the proportion of cellular structure formed by about an order of magnitude and increases the interlamellar spacing by 50-100%. These effects appear to result from a further decrease in the driving force. This decrease is attributed to a higher rate of introduction of misfit dislocations into the broad faces of the γ′ plates constituting the continuous precipitate, and thus to smaller values of the coherent solvus. © 1968

Quantum Commuting Circuits and Complexity of Ising Partition Functions

Instantaneous quantum polynomial-time (IQP) computation is a class of quantum computation consisting only of commuting two-qubit gates and is not universal in the sense of standard quantum computation. Nevertheless, it has been shown that if there is a classical algorithm that can simulate IQP efficiently, the polynomial hierarchy (PH) collapses at the third level, which is highly implausible. However, the origin of the classical intractability is still less understood. Here we establish a relationship between IQP and computational complexity of the partition functions of Ising models. We apply the established relationship in two opposite directions. One direction is to find subclasses of IQP that are classically efficiently simulatable in the strong sense, by using exact solvability of certain types of Ising models. Another direction is applying quantum computational complexity of IQP to investigate (im)possibility of efficient classical approximations of Ising models with imaginary coupling constants. Specifically, we show that there is no fully polynomial randomized approximation scheme (FPRAS) for Ising models with almost all imaginary coupling constants even on a planar graph of a bounded degree, unless the PH collapses at the third level. Furthermore, we also show a multiplicative approximation of such a class of Ising partition functions is at least as hard as a multiplicative approximation for the output distribution of an arbitrary quantum circuit.Comment: 36 pages, 5 figure