Lens spaces, rational balls and the ribbon conjecture

We apply Donaldson's theorem on the intersection forms of definite 4--manifolds to characterize the lens spaces which smoothly bound rational homology 4--dimensional balls. Our result implies, in particular, that every smoothly slice 2--bridge knot is ribbon, proving the ribbon conjecture for 2--bridge knots.Comment: 45 pages, 8 figures; accepted for publication in Geometry and Topolog

Evolution or revolution? a study of price and wage volatility in England, 1200-1900

Using annual data 1209-1914, this paper examines whether there are structural breaks in the movements of prices and wages that correspond to the major ‘revolutions’ identified in historical narratives. Econometric modelling of trend and volatility in prices and wages confirms the importance of the Commercial Revolution and the Glorious Revolution, but suggests that the Industrial Revolution may be better described in evolutionary terms. The evidence also points to a late medieval revolution at the time of the Good Parliament, shortly after the Black Death and just before the Peasant’s Revolt. This supports Britnell and Campbell’s commercialisation hypothesis - that the institutional pre-conditions for the Industrial Revolution began to develop at a very early date.Economic evolution; Economic revolution; Historical economics;

The theory of international business: the role of economic models

This paper reviews the scope for economic modelling in international business studies. It argues for multi-level theory based on classic internalisation theory. It present a systems approach that encompasses both firm-level and industry-level analysis

On the slice genus of links

We define Casson-Gordon sigma-invariants for links and give a lower bound of the slice genus of a link in terms of these invariants. We study as an example a family of two component links of genus h and show that their slice genus is h, whereas the Murasugi-Tristram inequality does not obstruct this link from bounding an annulus in the 4-ball.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-30.abs.htm

Derivatives of Knots and Second-order Signatures

We define a set of "second-order" L^(2)-signature invariants for any algebraically slice knot. These obstruct a knot's being a slice knot and generalize Casson-Gordon invariants, which we consider to be "first-order signatures". As one application we prove: If K is a genus one slice knot then, on any genus one Seifert surface, there exists a homologically essential simple closed curve of self-linking zero, which has vanishing zero-th order signature and a vanishing first-order signature. This extends theorems of Cooper and Gilmer. We introduce a geometric notion, that of a derivative of a knot with respect to a metabolizer. We also introduce a new equivalence relation, generalizing homology cobordism, called null-bordism.Comment: 40 pages, 22 figures, typographical corrections, to appear in Alg. Geom. Topolog

Modelling the medieval economy: money, prices and income in England, 1263-1520

This chapter presents a simple econometric model of the medieval English economy, focusing on the relationship between money, prices and incomes. The model is estimated using annual data for the period 1263-1520 obtained from various sources. The start date is determined by the availability of continuous runs of annual data, while the finishing date immediately precedes the take-off of Tudor price inflation. Accounts from the ecclesiastical and monastic estates have survived in great numbers for this period, thereby ensuring that crop yields can be estimated from a regionally representative set of estates

Filtering smooth concordance classes of topologically slice knots

We propose and analyze a structure with which to organize the difference between a knot in the 3-sphere bounding a topologically embedded 2-disk in the 4-ball and it bounding a smoothly embedded disk. The n-solvable filtration of the topological knot concordance group, due to Cochran-Orr-Teichner, may be complete in the sense that any knot in the intersection of its terms may well be topologically slice. However, the natural extension of this filtration to what is called the n-solvable filtration of the smooth knot concordance group, is unsatisfactory because any topologically slice knot lies in every term of the filtration. To ameliorate this we investigate a new filtration, {B_n}, that is simultaneously a refinement of the n-solvable filtration and a generalization of notions of positivity studied by Gompf and Cochran. We show that each B_n/B_{n+1} has infinite rank. But our primary interest is in the induced filtration, {T_n}, on the subgroup, T, of knots that are topologically slice. We prove that T/T_0 is large, detected by gauge-theoretic invariants and the tau, s, and epsilon-invariants; while the non-triviliality of T_0/T_1 can be detected by certain d-invariants. All of these concordance obstructions vanish for knots in T_1. Nonetheless, going beyond this, our main result is that T_1/T_2 has positive rank. Moreover under a "weak homotopy-ribbon" condition, we show that each T_n/T_{n+1} has positive rank. These results suggest that, even among topologically slice knots, the fundamental group is responsible for a wide range of complexity.Comment: 41 pages, slightly revised introduction, minor corrections and up-dated references, this is the final version to appear in Geometry and Topolog

Revolutionary change and structural breaks: A time series analysis of wages and commodity prices in Britain 1264-1913

In this paper we empirically test the hypothesis that economic revolutions are associated with structural breaks in historical economic data. A simple test for structural breaks in economic time series is applied to British wage and price data from the medieval to the modern period. Evidence for structural change is found in nearly half of the series studied -- suggesting that structural breaks are an intrinsic feature of such historic data. Structural changes are most closely linked to the Commercial Revolution followed by the Agricultural Revolution and the Industrial Revolution, with changes linked to an underlying process of price stabilisation as measured by a decrease in the long-term level of volatility.historical economics; economic revolutions; structural breaks; price stabilisation

Performance metrics for characterization of a seizure detection algorithm for offline and online use

Purpose: To select appropriate previously reported performance metrics to evaluate a new seizure detection algorithm for offline and online analysis, and thus quantify any performance variation between these metrics. Methods: Traditional offline algorithms mark out any EEG section (epoch) of a seizure (event), so that neurologists only analyze the detected and adjacent sections. Thus, offline algorithms could be evaluated using number of correctly detected events, or event-based sensitivity (SEVENT), and epoch-based specificity (percentage of incorrectly detected background epochs). In contrast, online seizure detection (especially, data selection) algorithms select for transmission only the detected EEG sections and hence need to detect the entire duration of a seizure. Thus, online algorithms could be evaluated using percentage of correctly detected seizure duration, or epoch-based sensitivity (SEPOCH), and epoch-based specificity. Here, a new seizure detection algorithm is evaluated using the selected performance metrics for epoch duration ranging from 1s to 60s. Results: For 1s epochs, the area under the event-based sensitivity-specificity curve was 0.95 whilst SEPOCH achieves 0.81. This difference is not surprising, as intuitively, detecting any epoch within a seizure is easier than detecting every epoch - especially as seizures evolve over time. For longer epochs of 30s or 60s, SEVENT falls to 0.84 and 0.82 respectively and SEPOCH reduces to 0.76. Here, decreased SEVENT shows that fewer seizures are detected, possibly due to easy-to-detect short seizure sections being masked by surrounding EEG. However, detecting one long epoch constitutes a larger percentage of a seizure than a shorter one and thus SEPOCH does not decrease proportionately. Conclusions: Traditional offline and online seizure detection algorithms require different metrics to effectively evaluate their performance for their respective applications. Using such metrics, it has been shown that a decrease in performance may be expected when an offline seizure detection algorithm (especially with short epoch duration) is used for online analysis.Accepted versio

Knot Concordance and Higher-Order Blanchfield Duality

In 1997, T. Cochran, K. Orr, and P. Teichner defined a filtration {F_n} of the classical knot concordance group C. The filtration is important because of its strong connection to the classification of topological 4-manifolds. Here we introduce new techniques for studying C and use them to prove that, for each natural number n, the abelian group F_n/F_{n.5} has infinite rank. We establish the same result for the corresponding filtration of the smooth concordance group. We also resolve a long-standing question as to whether certain natural families of knots, first considered by Casson-Gordon and Gilmer, contain slice knots.Comment: Corrected Figure in Example 8.4, Added Remark 5.11 pointing out an important strengthening of Theorem 5.9 that is needed in a subsequent pape