The Lambek-Grishin calculus is a symmetric version of categorial grammar obtained by augmenting the standard inventory of type-forming operations (product and residual left and right division) with a dual family: coproduct, left and right difference. Interaction between these two families is provided by distributivity laws. These distributivity laws have pleasant invariance properties: stabilit...

Dedekind symbols generalize the classical Dedekind sums (symbols). The symbols are determined uniquely by their reciprocity laws up to an additive constant. There is a natural isomorphism between the space of Dedekind symbols with polynomial (Laurent polynomial) reciprocity laws and the space of cusp (modular) forms. In this article we introduce Hecke operators on the space of weighted Dedekind...

Hyperbolic geometry is developed in a purely algebraic fashion from first principles, without a prior development of differential geometry. The natural connection with the geometry of Lorentz, Einstein and Minkowski comes from a projective point of view, with trigonometric laws that extend to ‘points at infinity’, here called ‘null points’, and beyond to ‘ideal points’ associated to a hyperbolo...

Human behavior is normally very hard to describe, as the relevant variables are difficult to identify, to quantify, and to measure. However, scientists have recently discovered several natural constants and laws in pedestrian crowds and vehicle traffic, starting from propagation velocities of traffic jams and Mexican waves, over human trail formation, scaling laws in traveller behavior, interac...

A new experimental market environment is developed. Continuously arriving incentives replace the traditional period structure. The issue posed is whether classical principles of market behavior apply when the environment is constantly changing. Three broad results emerge. (1) Natural " flow " generalizations of the laws of demand and supply exist and dictate much of the market behavior. (2) Two...

Legislation and regulations are expressed in natural language. Machine-readable forms of the texts may be represented as linked documents, semantically tagged text, or translation to a logic. The paper considers the latter form, which is key to testing consistency of laws, drawing inferences, and providing explanations relative to input. To translate laws to a machine-readable logic, sentences ...

We study the problem of feedback control for skew-symmetric and skewHamiltonian transfer functions using skew-symmetric controllers. This extends work of Helmke, et al., who studied static symmetric feedback control of symmetric and Hamiltonian linear systems. We identify spaces of linear systems with symmetry as natural subvarieties of the moduli space of rational curves in a Grassmannian, giv...

We introduce a tensor product for symmetric monoidal categories with the following properties. Let SMC denote the 2-category with objects small symmetric monoidal categories, arrows symmetric monoidal functors and 2-cells monoidal natural transformations. Our tensor product together with a suitable unit is part of a structure on SMC that is a 2-categorical version of the symmetric monoidal clos...

Our concepts of biology, evolution and complexity are constrained by having observed only a single instance of life, life on Earth. A truly comparative biology is needed to extend these concepts. Because we can not observe life on other planets, we are left with the alternative of creating artificial life forms on Earth. I will discuss the approach of inoculating evolution by natural selection ...