Natural and artificially created light fields in three-dimensional space contain lines of zero intensity, known as optical vortices 1-3. Here, we describe a scheme to create optical beams with isolated optical vortex loops in the forms of knots and links using algebraic topology. The required complex fields with fibred knots and links 4 are constructed from abstract functions with braided zeros and the knot function is then embedded in a propagating light beam. We apply a numerical optimization algorithm to increase the contrast in light intensity, enabling us to observe several optical vortex knots. These knotted nodal lines, as singularities of the wave's phase, determine the topology of the wave field in space, and should have analogues in other three-dimensional wave systems such as superfluids 5 and Bose-Einstein condensates 6,7. In nature, one never finds the plane waves described in textbooks; real physical waves, such as light beams, are superpositions of many plane waves propagating in different directions. Generically, interference of three or more plane waves results in optical vortex lines, where the intensity is zero, and around which the phase increases by 2π (refs 1, 2). This contrasts with a single plane wave, for which the intensity is uniform, and the wavefronts-the surfaces of constant phase-are nested planes perpendicular to the wavevector k. The presence of vortices disrupts this regular arrangement of wavefronts 3 : a straight vortex line parallel to k has helicoidal wavefronts; a circular vortex loop perpendicular to k is the edge of a punctured wavefront. The situation is more complicated when a single, isolated vortex loop is knotted 8. Here, we establish theoretically and demonstrate experimentally that waves may contain a single isolated knotted vortex loop. The global topology of such a knotted vortex field is nontrivial. Wavefront surfaces with all phase labels must intersect on the vortex lines in any optical field, and fill all space. If the vortex line is knotted, all wavefront surfaces have a knotted boundary curve, and are thus multiply connected. Although the vortices occupy a finite volume of space, this topology affects the entire wave field. Such non-trivial topology of physical fields arises elsewhere in physics, such as flows in fluid dynamics 9 (including Lord Kelvin's vortex atom hypothesis 10), quantum condensates 6,7 and field theory 11,12. Time-dependent solutions of Maxwell's equations in which all electric field lines have the form of torus knots (knots that can be drawn, without crossing, on a torus) were found in ref. 13. This knotting of light complements our present approach, in which the topology resides in the optical complex amplitude, giving knots that are directly observable in the intensity distribution of the beam. Knotted vortex loops are hypothesized to be generic in turbulent and chaotic wave fields, including superfluids 5 , optical 14 and biological waves 15 , although only unknotted linked vortex rings have been identified in large-scale simulations ...
This paper presents the principles of cooperation and briefly describes the history and development of agricultural cooperatives in developed and less-developed countries, with particular emphasis on South Africa. A new Cooperatives Act, based on international principles of cooperation, was promulgated in South Africa in August 2005. The theory of cooperatives, and new institutional economics theory (NIE) (including transaction cost economics, agency theory and property rights theory) and its applicability to the cooperative organizational form, are also presented, as are the inherent problems of conventional cooperatives, namely free-rider, horizon, portfolio, control and influence cost problems caused by vaguely defined property rights. An analysis of the future of cooperatives in general, based on a NIE approach, suggests a life cycle for cooperatives (formation, growth, reorganization or exit) as they adapt to a changing economic environment characterized by technological change, industrialization of agriculture and growing individualism.
No. 92-44 WAITE MEMORIAL BOOK COLLECTION DIPf, OF AG.
Abstmct-In his original paper on the subject, Shannon found upper and lower bounds for the entropy of printed English based on the number of trials required for a subject to guess subsequent symbols in a given text. The guessing approach precludes asymptotic consistency of either the upper or lower bounds except for degenerate ergodic processes. Shannon's technique of guessing the next symbol is altered by having the subject place sequential bets on the next symbol of text. lf S,, denotes the subject's capital after n bets at 27 for 1 odds, and lf it is assumed that the subject hnows the underlying prpbabillty distribution for the process X, then the entropy estimate ls H,(X) =(l -(l/n) log,, S,) log, 27 bits/symbol. If the subject does npt hnow the true probabllty distribution for the stochastic process, then Z&(X! ls an asymptotic upper bound for the true entropy. ff X is stationary, EH,,(X)+H(X), H(X) bell the true entropy of the process. Moreovzr, lf X is ergodic, then by the SLOW McMilhm-Brebnan theorem H,,(X)+H(X) with probability one. Preliminary indications are that English text has au entropy of approximately 1.3 bits/symbol, which agrees well with Shannon's estimate.
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