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A Pythagorean triple is an ordered triple of integers (a,b,c) = (0,0,0) such that a² +b² = c². It is well known that the set P of all Pythagorean triples has an intrinsic structure of commutative monoid with respect to a suitable binary operation, (P,*). In this article, we will introduce the "commensurability" relation R among Pythagorean triples, and we will see that it induces a group quotient, P/R, which is isomorphic with the direct product of infinite (countable) copies of C∞, the infinite cyclic group, and a cyclic group of order 4. As an application, we will see that the acute angles of Pythagorean triangles are irrational when measured in degrees. [ABSTRACT FROM AUTHOR]

Simple Summary: There is some evidence that the presence of dolphins in fishing areas represents a concrete economic loss for fishermen due to their depredation activities on the entangled fish on the nets. Bycatch events are one of the major sources of anthropogenic mortality of species of conservation interest in the world. T. truncatus is a plastic species and the more frequently observed species in the Adriatic Sea owing to the natural tendency to interact with the fishing activities in the area. This case report describes the acoustic parameters detected in whistle spectral contours associated with low-frequency signals recorded with a passive acoustic monitoring device in an exceptional event of bycatch that involved three individuals during a midwater commercial trawling in the Adriatic Sea. Marine mammal vocal elements have been investigated for decades to assess whether they correlate with stress levels or stress indicators. Due to their acoustic plasticity, the interpretation of dolphins' acoustic signals of has been studied most extensively. This work describes the acoustic parameters detected in whistle spectral contours, collected using passive acoustic monitoring (PAM), in a bycatch event that involved three Bottlenose dolphins during midwater commercial trawling. The results indicate a total number of 23 upsweep whistles recorded during the bycatch event, that were analyzed based on the acoustic parameters as follows: (Median; 25th percentile; 75th percentile) Dr (second), total duration (1.09; 0.88; 1.24); fmin (HZ), minimum frequency (5836.4; 5635.3; 5967.1); fmax (HZ), maximum frequency, (11,610 ± 11,293; 11,810); fc (HZ), central frequency; (8665.2; 8492.9; 8982.8); BW (HZ), bandwidth (5836.4; 5635.3; 5967.1); Step, number of step (5; 4; 6). Furthermore, our data show that vocal production during the capture event was characterized by an undescribed to date combination of two signals, an ascending whistle (upsweep), and a pulsed signal that we called "low-frequency signal" in the frequency band between 4.5 and 7 kHz. This capture event reveals a novel aspect of T. truncatus acoustic communication, it confirms their acoustic plasticity, and suggests that states of discomfort are conveyed through their acoustic repertoire. [ABSTRACT FROM AUTHOR]