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We critically examine a model that attempts to explain the emergence of power laws (e.g., Zipf's law) in human language. The model is based on the principle of least effort in communications-specifically, the overall effort is balanced between the speaker effort and listener effort, with some trade-off. It has been shown that an information-theoretic interpretation of this principle is sufficiently rich to explain the emergence of Zipf's law in the vicinity of the transition between referentially useless systems (one signal for all referable objects) and indexical reference systems (one signal per object). The phase transition is defined in the space of communication accuracy (information content) expressed in terms of the trade-off parameter. Our study explicitly solves the continuous optimization problem, subsuming a recent, more specific result obtained within a discrete space. The obtained results contrast Zipf's law found by heuristic search (that attained only local minima) in the vicinity of the transition between referentially useless systems and indexical reference systems, with an inverse-factorial (sub-logarithmic) law found at the transition that corresponds to global minima. The inverse-factorial law is observed to be the most representative frequency distribution among optimal solutions.

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The fluence exiting a patient during beam delivery can be used as treatment delivery quality assurance, either by direct comparison with expected exit fluences or by backprojection to reconstruct the patient dose. Multiple possible sources of measured exit fluence deviations exist, including changes in the beam delivery and changes in the patient anatomy. The purpose of this work is to compare the deviations caused by these sources. Machine deliveryrelated variability is measured by acquiring multiple dosimetric portal images (DPIs) of several test fields without a patient/phantom in the field over a time period of 2 months. Patient anatomy-related sources of fluence variability are simulated by computing transmission DPIs for a prostate patient using the same incident fluence for 11 different computed tomography (CT) images of the patient anatomy. The standard deviation (SD) and maximum deviation of the exit fluence, averaged over 5 mm × 5 mm square areas, is calculated for each test set. Machine delivery fluence SDs as large as 1% are observed for a sample patient field and as large as 2.5% for a picket-fence dMLC test field. Simulations indicate that day-to-day patient anatomy variations induce exit fluence SDs as large as 3.5%. The largest observed machine delivery deviations are 4% for the sample patient field and 7% for the picket-fence field, while the largest difference for the patient anatomy-related source is 8.5%. Since daily changes in patient anatomy can result in substantial exit fluence deviations, care should be taken when applying fluence back-projection to ensure that such deviations are properly attributed to their source.