Gender Equality and Human Rights

The achievement of substantive equality is understood as having four dimensions: redressing disadvantage; countering stigma, prejudice, humiliation and violence; transforming social and institutional structures; and facilitating political participation and social inclusion. The paper shows that, although not articulated in this way, these dimensions are clearly visible in the application by the various interpretive bodies of the principles of equality to the enjoyment of treaty rights. At the same time, it shows that there are important ways in which these bodies could go further, both in articulating the goals of substantive equality and in applying them when assessing compliance by States with international obligations of equality. The substantive equality approach, in its four-dimensional form, provides an evaluative tool with which to assess policy in relation to the right to gender equality. The paper elaborates on the four-dimensional approach to equality and how it can be used to evaluate the impact of social and economic policies on women to determine how to make the economy 'work for women' and advance gender equality. The paper suggests that there is a growing consensus at the international level on an understanding of substantive equality that reflects the four dimensional framework. This paper was produced for UN Women's flagship report "Progress of the World's Women 2015-2016" and is released as part of the UN Women discussion paper series

Canonical extensions and ultraproducts of polarities

J{\'o}nsson and Tarski's notion of the perfect extension of a Boolean algebra with operators has evolved into an extensive theory of canonical extensions of lattice-based algebras. After reviewing this evolution we make two contributions. First it is shown that the failure of a variety of algebras to be closed under canonical extensions is witnessed by a particular one of its free algebras. The size of the set of generators of this algebra can be made a function of a collection of varieties and is a kind of Hanf number for canonical closure. Secondly we study the complete lattice of stable subsets of a polarity structure, and show that if a class of polarities is closed under ultraproducts, then its stable set lattices generate a variety that is closed under canonical extensions. This generalises an earlier result of the author about generation of canonically closed varieties of Boolean algebras with operators, which was in turn an abstraction of the result that a first-order definable class of Kripke frames determines a modal logic that is valid in its so-called canonical frames

A secular increase in continental crust nitrogen during the Precambrian

Recent work indicates the presence of substantial geologic nitrogen reservoirs in the mantle and continental crust. Importantly, this geologic nitrogen has exchanged between the atmosphere and the solid Earth over time. Changes in atmospheric nitrogen (i.e. atmospheric mass) have direct effects on climate and biological productivity. It is difficult to constrain, however, the evolution of the major nitrogen reservoirs through time. Here we show a secular increase in continental crust nitrogen through Earth history recorded in glacial tills (2.9 Ga to modern), which act as a proxy for average upper continental crust composition. Archean and earliest Palaeoproterozoic tills contain 66 $\pm$ 100 ppm nitrogen, whereas Neoproterozoic and Phanerozoic tills contain 290 $\pm$ 165 ppm nitrogen, whilst the isotopic composition has remained constant at ~4\permil. Nitrogen has accumulated in the continental crust through time, likely sequestered from the atmosphere via biological fixation. Our findings support dynamic, non-steady state behaviour of nitrogen through time, and are consistent with net transfer of atmospheric N to geologic reservoirs over time.Comment: 14 pages, 2 figures, 2 tables, supplemental informatio

Modal Logics that Bound the Circumference of Transitive Frames

For each natural number $n$ we study the modal logic determined by the class of transitive Kripke frames in which there are no cycles of length greater than $n$ and no strictly ascending chains. The case $n=0$ is the G\"odel-L\"ob provability logic. Each logic is axiomatised by adding a single axiom to K4, and is shown to have the finite model property and be decidable. We then consider a number of extensions of these logics, including restricting to reflexive frames to obtain a corresponding sequence of extensions of S4. When $n=1$, this gives the famous logic of Grzegorczyk, known as S4Grz, which is the strongest modal companion to intuitionistic propositional logic. A topological semantic analysis shows that the $n$-th member of the sequence of extensions of S4 is the logic of hereditarily $n+1$-irresolvable spaces when the modality $\Diamond$ is interpreted as the topological closure operation. We also study the definability of this class of spaces under the interpretation of $\Diamond$ as the derived set (of limit points) operation. The variety of modal algebras validating the $n$-th logic is shown to be generated by the powerset algebras of the finite frames with cycle length bounded by $n$. Moreover each algebra in the variety is a model of the universal theory of the finite ones, and so is embeddable into an ultraproduct of them

Morphisms and Duality for Polarities and Lattices with Operators

Structures based on polarities have been used to provide relational semantics for propositional logics that are modelled algebraically by non-distributive lattices with additional operators. This article develops a first order notion of morphism between polarity-based structures that generalises the theory of bounded morphisms for Boolean modal logics. It defines a category of such structures that is contravariantly dual to a given category of lattice-based algebras whose additional operations preserve either finite joins or finite meets. Two different versions of the Goldblatt-Thomason theorem are derived in this setting

British fair play: sport across diasporas at the BBC World Service

This chapter uses archive material to explore the role of sports broadcasting on the BBC World Service in the twentieth century.Whereas the service's early audiences were expatriate British listeners, the BBC WS recruited different diasporic audiences later into the twentieth century.This chapter looks at specific sports and at the accommodations of the tensions between the WS's links with both Empire and with discourses of impartiaility, which have led to both endurances in the postcolonialist aspects of sport as well as new opportunities for reconfigurations of diaspora

Habitability of waterworlds: runaway greenhouses, atmospheric expansion and multiple climate states of pure water atmospheres

There are four different stable climate states for pure water atmospheres, as might exist on so-called "waterworlds". I map these as a function of solar constant for planets ranging in size from Mars size to 10 Earth-mass. The states are: globally ice covered (Ts< 245K), cold and damp (270 < Ts< 290K), hot and moist (350< Ts< 550K) and very hot and dry (Ts< 900K). No stable climate exists for 290< Ts < 350K or 550 < Ts < 900K. The union of hot moist and cold damp climates describe the liquid water habitable zone, the width and location of which depends on planet mass. At each solar constant, two or three different climate states are stable. This is a consequence of strong non-linearities in both thermal emission and the net absorption of sunlight. Across the range of planet sizes, I account for the atmospheres expanding to high altitudes as they warm. The emitting and absorbing surfaces (optical depth of unity) move to high altitude, making their area larger than the planet surface, so more thermal radiation is emitted and more sunlight absorbed (the former dominates). The atmospheres of small planets expand more due to weaker gravity: the effective runaway greenhouse threshold is about 35Wm-2 higher for Mars, 10Wm-2 higher for Earth or Venus but only a few Wm-2 higher for a 10 Earth-mass planet. There is an underlying (expansion neglected) trend of increasing runaway greenhouse threshold with planetary size (40Wm-2 higher for a 10 Earth-mass planet than for Mars). Summing these opposing trends means that Venus-size (or slightly smaller) planets are most susceptible to a runaway greenhouse. The habitable zone for pure water atmospheres is very narrow, with an insolation range of 0.07 times the solar constant. A wider habitable zone requires background gas and greenhouse gas; N2 and CO2 on Earth, which are biologically controlled. Thus, habitability depends on inhabitance.Comment: Accepted for publication in Astrobiolog