We present a minimum message length (MML) framework for trajectory partitioning by point selection, and use it to automatically select the tolerance parameter ε for Douglas-Peucker partitioning, adapting to local trajectory complexity. By examining a range of ε for synthetic and real trajectories, it is easy to see that the best ε does vary by trajectory, and that the MML encoding makes sensible choices and is robust against Gaussian noise. We use it to explore the identification of micro-activities within (...) a longer trajectory. This MML metric is comparable to the TRACLUS metric – and shares the constraint of abstracting only by omission of points – but is a true lossless encoding. Such encoding has several theoretical advantages – particularly with very small segments (high frame rates) – but actual performance interacts strongly with the search algorithm. Both differ from unconstrained piecewise linear approximations, including other MML formulations. (shrink)
We present a survey of some results and problems concerning constructions which require a diagonalization of length continuum to be carried out, particularly constructions of almost disjoint families of various sorts. We emphasize the role of cardinal invariants of the continuum and their combinatorial characterizations in such constructions.
We present an exact solution of the one-dimensional modified Klein Gordon and Duffin Kemmer Petiau equations with a step potential in the presence of minimal length in the uncertainty relation, where the expressions of the new transmission and reflection coefficients are determined for all cases. As an application, the Klein paradox in the presence of minimal length is discussed for all equations.
We argue that Bohrian complementarity is a framework for making new ontological sense of scientific findings. It provides a conceptual pattern for making sense of the results of an empirical investigation into new realms or fields of natural properties. The idea of “formation length” engenders this mutual attunement of evidence and reality. Physicists want to be able to ascribe ontological features to atomic constituents and atomic processes such as “emission”, “impact”, or “change of energy-state”. These expressions supposedly refer to (...) “local” forms of physical change that in sum constitute the possibility of there being a “global” (for example an atomic) system of possible states. We argue that it is only because we can act it out in the design of experiments that we canmake sense of the link between classical and quantum theoretical systems. We need the notion of formation length in order to express the principle that the atom is a causal unity. This not in the sense of being the ground for a particular kind of causality, but in the sense of unifying the grounds for the variety of causal manifestations that constitutes the atom. (shrink)
It is shown that the feasibly constructive arithmetic theory IPV does not prove LMIN, unless the polynomial hierarchy CPV-provably collapses. It is proved that PV plus LMIN intuitionistically proves PIND. It is observed that PV + PIND does not intuitionistically prove NPB, a scheme which states that the extended Frege systems are not polynomially bounded.