Theme #9 is about exploiting dynamics already present in a situation to advance one’s interests. Many Sun Tzu ideas find a place here, reflecting Sun Tzu’s keen appreciation of war’s larger context (Passage #1.1) conjoined with the inherently dynamic quality of Sun Tzu’s core concept of shi.

A ‘manifold with corners’ is a space locally modelled on [0,infinity)^k x R^{n-k}, just as manifolds are locally modelled on R^n. Triangles, squares and cubes are examples. A manifold with corners X has a boundary dX, a manifold with corners of dimension one less. The boundary of a triangle is the three edges. If f : X -> Y is a smooth map of manifolds with corners then f need not map dX -> dY, i.e. boundaries are not ‘functorial’. But we can extend the boundary to the ‘corners’ C(X), a manifold with corners of mixed dimension, which is functorial, i.e. f extends to C(f) : X -> Y.

There are several notions of smooth map of manifolds with corners. We choose the ‘b-maps’ of Richard Melrose. There are two notions of tangent bundle, the ordinary tangent bundle TX, functorial under smooth maps, and the ‘b-tangent bundle’ bTX.

‘Manifolds with g-corners’ are a generalization with exotic corner structure. They have some nicer properties, e.g. existence of b-transverse fibre products.

Manifolds with corners occur in many places. e.g. in TQFTs, and analysis of partial differential equations. Some moduli spaces in Morse theory, Floer theories, and Symplectic Geometry are manifolds with corners.

If X is a manifold with corners of dimension n, the boundary dX is a manifold with corners of dimension n-1, and the k-fold boundary d^kX a manifold with corners of dimension n-k. The ‘k-corners’ C_k(X) is d^kX/S_k, also a manifold with corners of dimension n-k. The ‘corners’ C(X) is the disjoint union of all C_k(X), a manifold with corners of mixed dimension.

A smooth map f : X -> Y of manifolds with corners need not map dX -> dY, that is, boundaries are not functorial. But there is a natural map C(f) : C(X) -> C(Y), which need not map C_k(X) -> C_k(Y). This is the ‘corner functor’ for manifolds with corners.

We extend the corner functor to $C^\infty$-schemes with corners. It is right adjoint to the inclusion functor from interior $C^\infty$-schemes with corners to all $C^\infty$-schemes with corners, and so is canonically determined by the notion ‘interior’. Using the corner functor we define boundaries and corners of ‘firm’ $C^\infty$-schemes with corners.

We use the corner functor to study fibre products of $C^\infty$-schemes with corners, and show that b-transverse fibre products of manifolds with (g-)corners map to fibre products of $C^\infty$-schemes with corners.

The chapter unites anthropological accounts of blood. It introduces refrains that unify themes of the entire book. It argues that blood marks the bounds of religious and social bodies, using Durkheim, Douglas, and Bildhauer; Irenaeus, Maximus, and Aquinas. Iron compounds make blood red, but societies draft its color and stickiness for their own purposes. Inside, blood carries life. Outside, blood marks the body fertile or at risk. But that’s a social fiction. Skin makes a membrane to pass when a body breathes, eats, perspires, eliminates, menstruates, ejaculates, conceives, or bleeds. Only blood evokes so swift and social a response: It brings parent to child, bystander to victim, ambulance to patient, soldier to comrade, midwife to mother, defender to border. The New Testament names the blood of Christ three times as often as his cross – five times as often as his death. The blood of Jesus is the blood of Christ; the wine of communion is the blood of Christ; the means of atonement is the blood of Christ; the kinship of believers is the blood of Christ; the cup of salvation is the blood of Christ; icons ooze with the blood of Christ; and the blood of Christ is the blood of God.

We start with an odd mutation in flies that causes their legs to be double-jointed, but what is even stranger is that the extra joints are upside-down. This leads to a discussion of cell polarity not only in flies but also in the inner ear of humans. Two intercellular signaling pathways are involved:PCP and Notch.

The first section of the chapter sets out the methodology for understanding two key dimensions of the spatial patterns of Etruria: hierarchy and boundaries. These are addressed by rank size and XTENT respectively. The second section of the chapter brings Etruria into the analysis by tackling issues of chronology, post-depostional distortions, sampling, site definition, prior use of rank size and causal mechanisms.

Chapter 8 concerns a group of WEC units that may be realised in a more distant future, namely groups or arrays of individual WEC units and two-dimensional WEC units, which needs to be rather big structures. Firstly, a group of WEC bodies is analysed. Next a group consisting of WEC bodies as well as OWCs is analysed. Then the previous real radiation resistance needs to be replaced by a complex radiation damping matrix which is complex, but Hermitian, which means that its eigenvalues are real.

Chapter 4 introduces basic differential equations and boundary conditions for gravity waves propagating along a water surface. Assuming low wave amplitudes, equations are linearised. Then a quantitative discussion is given for harmonical (sinusoidal) waves propagating either on deep water, or otherwise on water of constant depth. Phase and group velocities are introduced, and then formulas are derived for the potential energy and the kinetic energy associated with a water wave. A closely related result is an important formula for the wave-power level, which equals the wave’s group velocity multiplied by the wave’s stored – kinetic + potential – energy per unit of sea surface. An additional subject is the wave’s momentum density. A section concerns real sea waves. Further, circular waves are mathematically described. Two sections of the chapter concern mathematical tools to be applied in Chapters 5–8 of the book. A final section considers water waves analysed in the time domain.

Signed in July 1893 and ratified four years later, the Spenser–Mariscal Treaty between Great Britain and Mexico signaled the end of British involvement in the Caste War. The divergence between imperial and colonial interests latent in the governance of Belize since the beginning of the Caste War period becomes particularly salient in the dissents surrounding the Spenser–Mariscal Treaty. The belief that complicated struggles over land, labor and people in Belize could be resolved by a simple line on the map reflected British imperial hubris and the lack of understanding of ground-level reality. Colonial officials, on the other hand, entrenched in the local sphere, found their interests aligned more with the Creole and Hispanic Belizeans and the Maya at the borders than with the British Crown that they purported to represent. Examination of the correspondences surrounding the Spenser–Mariscal Treaty reveals this contradiction at the heart of the imperial project in the Belizean northern frontier at the end of the Caste War.

Chapter One explores ancestors of the idea that the physical sciences were relevant and significant to the study of obscure powers associated with the human body and mind.In the late eighteenth and early nineteenth centuries, practitioners of animal magnetism and mesmerism linked the study of a supposed new imponderable ‘magnetic’ fluid affecting health to better-known physical imponderables.In the mid-nineteenth century the German chemist Karl von Reichenbach and his followers stimulated much debate for their alleged discovery of new imponderable ‘od’ that they believed extended the domain of physics into the realm of physiology.From the 1840s onwards ’Modern Spiritualism’ prompted many natural philosophers to intervene on controversies over its startling physical effects.The final section of the chapter contextualises these attempts to link physical and psychical realms in terms of the fluid state of the physical sciences in the early and mid-nineteenth century.