Borel circle squaring

We give a completely constructive solution to Tarski's circle squaring problem. More generally, we prove a Borel version of an equidecomposition theorem due to Laczkovich. If $k \geq 1$ and $A, B \subseteq \mathbb{R}^k$ are bounded Borel sets with the same positive Lebesgue measure whose boundaries have upper Minkowski dimension less than $k$, then $A$ and $B$ are equidecomposable by translations using Borel pieces. This answers a question of Wagon. Our proof uses ideas from the study of flows in graphs, and a recent result of Gao, Jackson, Krohne, and Seward on special types of witnesses to the hyperfiniteness of free Borel actions of $\mathbb{Z}^d$.Comment: Minor typos correcte

Measurable realizations of abstract systems of congruences

An abstract system of congruences describes a way of partitioning a space into finitely many pieces satisfying certain congruence relations. Examples of abstract systems of congruences include paradoxical decompositions and $n$-divisibility of actions. We consider the general question of when there are realizations of abstract systems of congruences satisfying various measurability constraints. We completely characterize which abstract systems of congruences can be realized by nonmeager Baire measurable pieces of the sphere under the action of rotations on the $2$-sphere. This answers a question of Wagon. We also construct Borel realizations of abstract systems of congruences for the action of $\mathsf{PSL}_2(\mathbb{Z})$ on $\mathsf{P}^1(\mathbb{R})$. The combinatorial underpinnings of our proof are certain types of decomposition of Borel graphs into paths. We also use these decompositions to obtain some results about measurable unfriendly colorings.Comment: minor correction

Inflation Targeting and the Anchoring of Inflation Expectations in The Western Hemisphere

We investigate the extent to which long-run inflation expectations are well anchored in three western hemisphere countries—Canada, Chile, and the United States—using a high-frequency event-study analysis. Specifically, we use daily data on far-ahead forward inflation compensation—the difference between forward rates on nominal and inflation-indexed bonds—as an indicator of financial market perceptions of inflation risk and the expected level of inflation at long horizons. For the United States, we find that far-ahead forward inflation compensation reacts significantly to macroeconomic data releases, suggesting that long-run inflation expectations are not completely anchored. In contrast, the Canadian inflation compensation data do not exhibit significant sensitivity to either Canadian or U.S. macroeconomic news, consistent with the view that inflation targeting in Canada has been successful in anchoring long-run inflation expectations. Finally, while the requisite data for Chile is only available for a limited sample period (2002-2005), our results are consistent with the hypothesis that inflation targeting in Chile has also succeeded in anchoring long-run inflation expectations.

Does inflation targeting anchor long-run inflation expectations? evidence from long-term bond yields in the U.S., U.K., and Sweden

We investigate the extent to which inflation targeting helps anchor long-run inflation expectations by comparing the behavior of daily bond yield data in the United Kingdom and Sweden--both inflation targeters--to that in the United States, a non-inflation-targeter. Using the difference between far-ahead forward rates on nominal and inflation-indexed bonds as a measure of compensation for expected inflation and inflation risk at long horizons, we examine how much, if at all, far-ahead forward inflation compensation moves in response to macroeconomic data releases and monetary policy announcements. In the U.S., we find that forward inflation compensation exhibits highly significant responses to economic news. In the U.K., we find a level of sensitivity similar to that in the U.S. prior to the Bank of England gaining independence in 1997, but a striking absence of such sensitivity since the central bank became independent. In Sweden, we find that forward inflation compensation has been insensitive to economic news over the whole period for which we have data. Our findings support the view that a well-known and credible inflation target helps to anchor the private sector's perceptions of the distribution of long-run inflation outcomes.Inflation (Finance) ; Prices ; Monetary policy

The mid-domain effect: It’s not just about space

Ecologists and biogeographers have long sought to understand how and why diversity varies across space. Up until the late 20th century, the dominant role of environmental gradients and historical processes in driving geographical species richness patterns went largely undisputed. However, almost 20 years ago, Colwell & Hurtt (1994) proposed a radical reappraisal of ecological gradient theory that called into question decades of empirical and theoretical research. That controversial idea was later termed the ‘the mid-domain effect’: the simple proposition that in the absence of environmental gradients, the random placement of species ranges within a bounded domain will give rise to greatest range overlap, and thus richness, at the center of the domain (Colwell & Lees, 2000) (Fig. 1a). The implication of this line of reasoning is that the conventional null model of equal species richness regardless of latitude, elevation or depth should be replaced by one where richness peaks at some midpoint in geographical space. Our intention here is to draw attention to a neglected, yet important manifestation of the mid-domain effect, namely the application of mid-domain models (also referred to as geometric constraint models) to non-spatial domains. If individual species have ranges that exist not just in geographical space but also in environmental factors, such as temperature, rainfall, pH, productivity or disturbance, shouldn’t we also expect mid-domain richness peaks along non-spatial gradients? A mid-domain model applied to non-spatial gradients predicts the maximum potential richness for every value of an environmental factor. As with spatial mid-domain models, realized richness would probably be less, but the limits to richness are still predicted to be hump-shaped. Indeed, hump-shaped relationships emerge with remarkably high frequency across various non-spatial gradients. For instance, two of ecology’s most fundamental, albeit controversial theories – the productivity–diversity relationship and the intermediate disturbance hypothesis – predict mid-domain peaks in species richness. However, the potential of non-spatial mid-domain models has gone largely ignored

A Combined Preconditioning Strategy for Nonsymmetric Systems

We present and analyze a class of nonsymmetric preconditioners within a normal (weighted least-squares) matrix form for use in GMRES to solve nonsymmetric matrix problems that typically arise in finite element discretizations. An example of the additive Schwarz method applied to nonsymmetric but definite matrices is presented for which the abstract assumptions are verified. A variable preconditioner, combining the original nonsymmetric one and a weighted least-squares version of it, is shown to be convergent and provides a viable strategy for using nonsymmetric preconditioners in practice. Numerical results are included to assess the theory and the performance of the proposed preconditioners.Comment: 26 pages, 3 figure