Physics, Topology, Logic and Computation: A Rosetta Stone

In physics, Feynman diagrams are used to reason about quantum processes. In the 1980s, it became clear that underlying these diagrams is a powerful analogy between quantum physics and topology: namely, a linear operator behaves very much like a "cobordism". Similar diagrams can be used to reason about logic, where they represent proofs, and computation, where they represent programs. With the rise of interest in quantum cryptography and quantum computation, it became clear that there is extensive network of analogies between physics, topology, logic and computation. In this expository paper, we make some of these analogies precise using the concept of "closed symmetric monoidal category". We assume no prior knowledge of category theory, proof theory or computer science.Comment: 73 pages, 8 encapsulated postscript figure

Hamstring muscles: Architecture and innervation

Knowledge of the anatomical organization of the hamstring muscles is necessary to understand their functions, and to assist in the development of accurate clinical and biomechanical models. The hamstring muscles were examined by dissection in six embalmed human lower limbs with the purpose of clarifying their gross morphology. In addition to obtaining evidence for or against anatomical partitioning ( as based on muscle architecture and pattern of innervation), data pertaining to architectural parameters such as fascicular length, volume, physiological cross-sectional area, and tendon length were collected. For each muscle, relatively consistent patterns of innervation were identified between specimens, and each was unique with respect to anatomical organization. On the basis of muscle architecture, three regions were identified within semimembranosus. However, this was not completely congruent with the pattern of innervation, as a primary nerve branch supplied only two regions, with the third region receiving a secondary branch. Semitendinosus comprised two distinct partitions arranged in series that were divided by a tendinous inscription. A singular muscle nerve or a primary nerve branch innervated each partition. In the biceps femoris long head the two regions were supplied via a primary nerve branch which divided into two primary branches or split into a series of branches. Being the only muscle to cross a single joint, biceps femoris short head consisted of two distinct regions demarcated by fiber direction, with each innervated by a separate muscle nerve. Architecturally, each muscle differed with respect to parameters such as physiological cross-sectional area, fascicular length and volume, but generally all partitions within an individual muscle were similar in fascicular length. The long proximal and distal tendons of these muscles extended into the muscle bellies thereby forming elongated musculotendinous junctions. Copyright (C) 2005 S. Karger AG, Basel

Changes in the area and condition of samphire marshes with time. In: McComb, A.J., Kobryn, H.T. and Latchford, J.A. (eds) Samphire marshes of the Peel-Harvey estuarine system Western Australia.

Aerial photography is especially suited to the study of vegetation, water resources and shoreline mapping (A.S.O.P, 1968). Air photographs provide a perspective of the earth’s geographical features that are generally readily understood. Air photographs also provide a plan view that can be spatially compared to the knowledge an individual may have about a similar area. For those reasons aerial photographs frequently provide working documents for planners and managers. They are however, limited in spatial accuracy because the images suffer geometric distortions, particularly near photograph margins or when terrain varies in height. Also, a single aerial photograph rarely covers an entire study area. There are manual and computer assisted techniques of joining air photographs and also eliminating the geometric distortions within and between individual photographs. An assembly of aerial photographs is called a photo mosaic. Photo mosaics can be "controlled" or "uncontrolled", the former having the geometric distortions removed. The amount of geometric distortions in an aerial photograph depends on many factors, including the physical optics of the camera and the orientation of the camera at the instant of exposure. Where the optical axis of the camera is near vertical (70°) and super-wide angle (>100°) lenses. Gross distortions of scale occur on individual photographs when terrain slope changes suddenly, as when a scarp, cliff or portion of a mountain is included in the photograph (Maling, 1989). Some of the distortions in vertical air photography can be avoided by using only central portions of each photograph