An Algebra of Pieces of Space -- Hermann Grassmann to Gian Carlo Rota

We sketch the outlines of Gian Carlo Rota's interaction with the ideas that Hermann Grassmann developed in his Ausdehnungslehre of 1844 and 1862, as adapted and explained by Giuseppe Peano in 1888. This leads us past what Rota variously called 'Grassmann-Cayley algebra', or 'Peano spaces', to the Whitney algebra of a matroid, and finally to a resolution of the question "What, really, was Grassmann's regressive product?". This final question is the subject of ongoing joint work with Andrea Brini, Francesco Regonati, and William Schmitt. The present paper was presented at the conference "The Digital Footprint of Gian-Carlo Rota: Marbles, Boxes and Philosophy" in Milano on 17 Feb 2009. It will appear in proceedings of that conference, to be published by Springer Verlag.Comment: 28 page

The strength of Ramsey Theorem for coloring relatively large sets

We characterize the computational content and the proof-theoretic strength of a Ramsey-type theorem for bi-colorings of so-called {\em exactly large} sets. An {\it exactly large} set is a set X\subset\Nat such that \card(X)=\min(X)+1. The theorem we analyze is as follows. For every infinite subset $M$ of \Nat, for every coloring $C$ of the exactly large subsets of $M$ in two colors, there exists and infinite subset $L$ of $M$ such that $C$ is constant on all exactly large subsets of $L$. This theorem is essentially due to Pudl\`ak and R\"odl and independently to Farmaki. We prove that --- over Computable Mathematics --- this theorem is equivalent to closure under the $\omega$ Turing jump (i.e., under arithmetical truth). Natural combinatorial theorems at this level of complexity are rare. Our results give a complete characterization of the theorem from the point of view of Computable Mathematics and of the Proof Theory of Arithmetic. This nicely extends the current knowledge about the strength of Ramsey Theorem. We also show that analogous results hold for a related principle based on the Regressive Ramsey Theorem. In addition we give a further characterization in terms of truth predicates over Peano Arithmetic. We conjecture that analogous results hold for larger ordinals

An integrated sequence stratigraphic, palaeoenvironmental, and chronostratigraphic analysis of the Tangahoe Formation, southern Taranaki coast, with implications for mid-Pliocene (c. 3.4–3.0 Ma) glacio-eustatic sea-level changes

Sediments of the mid-Pliocene (c. 3.4–3.0 Ma) Tangahoe Formation exposed in cliffs along the South Taranaki coastline of New Zealand comprise a 270 m thick, cyclothemic shallow-marine succession that has been gently warped into a north to south trending, low angle anticline. This study examines the sedimentologic, faunal, and petrographic characteristics of 10 Milankovitch-scale (6th order), shallow-marine depositional sequences exposed on the western limb of the anticline. The sequences are recognised on the basis of the cyclic vertical stacking of their constituent lithofacies, which are bound by sharp wave cut surfaces produced during transgressive shoreface erosion. Each sequence comprises three parts: (1) a 0.2–2 m thick, deepening upwards, basal suite of reworked bioclastic lag deposits (onlap shellbed) and/or an overlying matrix supported, molluscan shellbed of offshore shelf affinity (backlap shellbed); (2) a 5–20 m thick, gradually shoaling, aggradational siltstone succession; and (3) a 5–10 m thick, strongly progradational, well sorted “forced regressive” shoreline sandstone. The three-fold subdivision corresponds to transgressive, highstand, and regressive systems tracts (TSTs, HSTs, and RSTs) respectively, and represents deposition during a glacio-eustatic sea-level cycle. Lowstand systems tract sediments are not recorded because the outcrop is situated c. 100 km east of the contemporary shelf edge and was subaerially exposed at that time. Well developed, sharp- and gradational-based forced regressive sandstones contain a variety of storm-emplaced sedimentary structures, and represent the rapid and abrupt basinward translation of the shoreline on to a storm dominated, shallow shelf during eustatic sea-level fall. Increased supply of sediment from north-west South Island during “forced regression” is indicated from petrographic analyses of the heavy mineralogy of the sandstones. A chronology based on biostratigraphy and the correlation of a new magnetostratigraphy to the magnetic polarity timescale allows: (1) identification of the Mammoth (C2An.2r) and Kaena (C2An.1r) subchrons; (2) correlation of the coastal section to the Waipipian Stage; and (3) estimation of the age of the coastal section as 3.36–3.06 Ma. Qualitative assessment of foraminiferal census data and molluscan palaeoecology reveals cyclic changes in water depth from shelf to shoreline environments during the deposition of each sequence. Seven major cycles in water depth of between 20 and 50m have been correlated to individual 40 ka glacio-eustatic sea-level cycles on the marine oxygen isotope timescale. The coastal Tangahoe Formation provides a shallow-marine record of global glacio-eustasy prior to the development of significant ice sheets on Northern Hemisphere continents, and supports evidence from marine δ18O archives that changes in Antarctic ice volume were occurring during the Pliocene

Corporate Tax Systems and the Location of Industry

This paper analyzes the effects of different corporate tax systems on the location of industry within an economic geography model with regional size asymmetries. Both the North and the South gain industry by adopting a tax regime that produces the lowest tax level. As the share of expenditures in the North increases, the Nash equilibrium has this region setting regressive taxes, while the South introduces progressive taxation. The unilateral welfare-maximizing tax structure in the North (South) is the regressive (progressive) system when expenditures in the North increase. Welfare in the North (South) is however maximized if both regions set regressive (progressive) taxes, while regressive (progressive) taxation in both regions represents a joint welfare maximizing outcome if the economic size of the North is higher (lower) than a certain threshold value. As trade is liberalized, the equilibrium tax regime adopted depends on how pro ts respond to lower trade costs. Proportional taxation is never an equilibrium, neither as regional spending changes, nor as trade is liberalized.Economic Geography; Tax Systems; Corporate Taxation

Contrasting carbonate depositional systems for Pliocene cool-water limestones cropping out in central Hawke's Bay, New Zealand

Pliocene limestone formations in central Hawke's Bay (eastern North Island, New Zealand) accumulated on and near the margins of a narrow forearc basin seaway within the convergent Australia/Pacific plate boundary zone. The active tectonic setting and varied paleogeographic features of the limestone units investigated, in association with probable glacioeustatic sea-level fluctuations, resulted in complex stratigraphic architectures and contrasting types of carbonate accumulation on either side of the seaway. Here, we recognise recurring patterns of sedimentary facies, and sequences and systems tracts bounded by key physical surfaces within the limestone sheets. The facies types range from Bioclastic (B) to Siliciclastic (S) end-members via Mixed (M) carbonate-siliciclastic deposits. Skeletal components are typical cool-water associations dominated by epifaunal calcitic bivalves, bryozoans, and especially barnacles. Siliciclastic contents vary from one formation to another, and highlight siliciclastic-rich limestone units in the western ranges versus siliciclastic-poor limestone units in the eastern coastal hills. Heterogeneities in facies types, stratal patterns, and also in diagenetic pathways between eastern and western limestone units are considered to originate in the coeval occurrence in different parts of the forearc basin of two main morphodynamic carbonate systems over time

Anatomy and Sequence Architecture of the early Post-Rift in the Neuquén Basin (Argentina): A Response to Physiography and Relative Sea-level Changes

A detailed architectural and sequence stratigraphic analysis was carried out in the early-post-rift succession of central Neuque´n Basin (Middle Jurassic Cuyo Group), integrating outcrop and subsurface information from a 3,000 km2 area. Sedimentary facies analysis allowed the definition of six marine facies associations, which are grouped in two main depositional systems. During the late Toarcian to early Bathonian a storm- and wave-dominated shoreface to offshore system was developed. This is overlain by a late Bathonian–early Callovian fluvio-deltaic depositional system. The sequence stratigraphic analysis of this succession identified parasequences limited by marine flooding surfaces with little evidence of erosion. Parasequences could be grouped into four parasequence sets that show the evolution of the depositional systems and can be interpreted in terms of shoreline trajectories. In this sense, a log-term transgressive event defined by a complicated retrogradational stacking pattern and spanning for almost 10 My is identified for the older deposits. This succession is eventually replaced by a progradational unit that represents highstand conditions. The Cuyo succession identified in this part of the basin is completed by the development of a regressive surface and the onset of a deltaic depositional system with no parasequence development but indicating an abrupt basinward facies shift. The long-term transgression that dominates the early post-rift succession in this part of the basin is interpreted to represent a period of sustained accommodation creation produced by the combination of thermal subsidence, differential compaction of syn-rift deposits, and eustatic rise under a relatively low sediment supply. Changes in the stacking pattern of retrogradational parasequence sets are thought to be produced by changes in the bathymetry of the topography being flooded, which is a relic of a complex array of footwalls and shoulders generated during the syn-rift stage. Regional changes in thickness and timing for the early post-rift succession could be therefore the result of the relatively passive infill of an inherited topography of the syn-rift, suggesting that most of the accommodation was already created before the onset of the long-term transgressive trend that characterizes the post-rift succession in this part of the basin. The Cuyo Group succession reported here reflects the complexity in terms of depositional styles, sequence stratigraphic patterns, and controls that can be expected during the early-post rift infill of syn-rift structural depressions. Therefore, the results of this study can provide useful lessons for post-rift systems having prolific hydrocarbon production worldwide (e.g., in the Northern North Sea, Norwegian Sea, Thailand, and Argentina and Chile).Fil: Veiga, Gonzalo Diego. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Centro de Investigaciones Geológicas. Universidad Nacional de La Plata. Facultad de Ciencias Naturales y Museo. Centro de Investigaciones Geológicas; ArgentinaFil: Schwarz, Ernesto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Centro de Investigaciones Geológicas. Universidad Nacional de La Plata. Facultad de Ciencias Naturales y Museo. Centro de Investigaciones Geológicas; ArgentinaFil: Spalletti, Luis Antonio. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Centro de Investigaciones Geológicas. Universidad Nacional de La Plata. Facultad de Ciencias Naturales y Museo. Centro de Investigaciones Geológicas; ArgentinaFil: Massaferro, Jose L. YPF S.A; Argentin

A Theory of Stationary Trees and the Balanced Baumgartner-Hajnal-Todorcevic Theorem for Trees

Building on early work by Stevo Todorcevic, we describe a theory of stationary subtrees of trees of successor-cardinal height. We define the diagonal union of subsets of a tree, as well as normal ideals on a tree, and we characterize arbitrary subsets of a non-special tree as being either stationary or non-stationary. We then use this theory to prove the following partition relation for trees: Main Theorem: Let $\kappa$ be any infinite regular cardinal, let $\xi$ be any ordinal such that $2^{\left|\xi\right|} < \kappa$, and let $k$ be any natural number. Then $$\text{non-$\left(2^{<\kappa}\right)$-special tree } \to \left(\kappa + \xi \right)^2_k.$$ This is a generalization to trees of the Balanced Baumgartner-Hajnal-Todorcevic Theorem, which we recover by applying the above to the cardinal $(2^{<\kappa})^+$, the simplest example of a non-$(2^{<\kappa})$-special tree. As a corollary, we obtain a general result for partially ordered sets: Theorem: Let $\kappa$ be any infinite regular cardinal, let $\xi$ be any ordinal such that $2^{\left|\xi\right|} < \kappa$, and let $k$ be any natural number. Let $P$ be a partially ordered set such that $P \to (2^{<\kappa})^1_{2^{<\kappa}}$. Then $$P \to \left(\kappa + \xi \right)^2_k.$$Comment: Submitted to Acta Mathematica Hungaric