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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]

Using the basic properties of the base-b representation of rational numbers, we will give an elementary proof of Gauss's lemma: Every real root of a monic polynomial with integer coefficients is either an integer or irrational. The paper offers a new perspective in understanding the meaning of ‘irrational numbers’ from a deeper perspective, and a good occasion to improve the student's skills with point of view on number systems. [ABSTRACT FROM AUTHOR]

In this paper, we will show a characterization for pairs of commensurable diagonals of regular n-gons. This result gives a positive answer to a recent question. [ABSTRACT FROM AUTHOR]

In this paper, we propose an elementary proof of Niven's Theorem in which the tangent function will have a primary role. [ABSTRACT FROM AUTHOR]

In this article, we will give a geometric interpretation of certain finite arithmetic progressions. For this purpose, we will introduce the concept of the n-regular partition { P n } of a quadrilateral. [ABSTRACT FROM AUTHOR]

The Padovan sequence and the Plastic number are mathematical objects of central interest among architects, and from this point of view, their importance is comparable to that of the Fibonacci sequence and the Golden mean. Padovan-like sequences, such as Perrin sequence or the Cordonnier sequence, also have mathematical applications. In this article, we will characterize a large family of Padovan-like sequences in terms of 'Padovan diagonals' of generalized Pascal's triangles. [ABSTRACT FROM AUTHOR]

In this paper, inspired by recent articles of A. Szynal-Liana & I. Wloch and F. T. Aydin & S. Yüce (see [26] and [2]), we will introduce the ̄hJacobsthal quaternions and the ̄hJacobsthal-Lucas sequences and their associated quaternions. The new results that we have obtained extend most of those obtained in [26]. [ABSTRACT FROM AUTHOR]

Until the seventeenth century, rational numbers were represented as fractions. It was starting from that century, thanks to Simon Stevin, that for all practical purposes the use of decimal notation became widespread. Although decimal numbers are widely used, their organic development is lacking, while a vast literature propagates misconceptions about them, among both students and teachers. In particular, the case of the period 9 is especially interesting. In this paper, changing the point of view, we propose substituting the usual deffinitions of the periodic decimal representation of a rational num- ber - the one obtained through long division introduced in the secondary school and the one obtained through the series concept for undergraduate level - with one which, instead of infinite progressions, uses a known property of fractions deducible from Euler's Totient Theorem. The long division becomes a convenient algorithm for obtaining the decimal expansion of a rational number. The new deffinition overcomes the difficulties noted in the literature. An elementary proof of Euler's Totient Theorem will be given depending only on Euclidean Division Theorem. [ABSTRACT FROM AUTHOR]

In this article, we introduce the notion of continue quadrilateral, that is a quadrilateral whose sides are in geometric progression. We obtain an extension of the principal result referring to the growth of continue triangles. Precisely, we will see that the growth of a continue quadrilateral belongs to the interval (1/Φ2, Φ2), where Φ2 is the Silver mean. The main result is that in any circle a continue quadrilateral of growth µ can be inscribed for every µ belonging to the interval (1/Φ2, Φ2). Our investigation is supported by dynamical software. [ABSTRACT FROM AUTHOR]

In this survey we highlight the relations between some subgroup embedding properties that characterise groups in which normality is a transitive relation in certain universes of groups with some finiteness properties. [ABSTRACT FROM AUTHOR]

In this study the description of underwater vocal repertoire of bearded seal in Svalbard (Norway) was extended. Two autonomous passive acoustic recorders were deployed for one year (August 2014-July 2015) in the inner and outer parts of the Kongsfjorden, and 1728 h were recorded and 17 220 vocalizations were found. Nine different vocalization classes were identified and characterized using ten acoustic parameters. The calls showed heterogeneous spectral features, but share the descending trend of frequency modulation. The different classes emerged were discriminated primarily by bandwidth and duration, and then by minimum frequency, central frequency, and maximum frequency in this order. This study represents a step forward to improve the understanding of the acoustic behaviour and the social function of these calls, and identified long passive acoustic monitoring as an effective method to assess vocal complexity and the ecology of marine species producing sounds. [ABSTRACT FROM AUTHOR]

A subgroup X of a group G is said to be an H-subgroup if NG(X) ∩ Xg ⩽ X for each element g belonging to G. In [M. Bianchi and e.a., On finite soluble groups in which normality is a transitive relation, J. Group Theory, 3 (2000) 147-156.] the authors showed that finite groups in which every subgroup has the H-property are exactly soluble groups in which normality is a transitive relation. Here we extend this characterization to groups without simple sections. [ABSTRACT FROM AUTHOR]

In this article we will use some special triangles, to construct polygonal chains that describe the families of logarithmic spirals, among which are the celebrated Golden Spiral, Spira solaris and Pheidia Spiral. [ABSTRACT FROM AUTHOR]

We study groups in which normality is a weakly transitive relation, giving an extension of in Theorem A [On finite T-groups, J. Aust. Math. Soc. 75 (2003) 181-191] due to Ballester-Bolinches and Esteban-Romero pointing out the relations between these groups and those in which all subgroups are almost pronormal. Moreover, we extend a well-known theorem of Peng [Finite groups with pro-normal subgroups, Proc. Amer. Math. Soc. 20 (1969) 232-234] proving that for a large class of generalized FC-groups the weak transitivity of normality is equivalent to having finitely many maximal pronormalizers of subgroups. [ABSTRACT FROM AUTHOR]

A well-known result of Feinberg and Shannon states that the tribonacci sequence can be detected by the so-called Pascal's pyramid. Here we will show that any tribonacci-like sequence can be obtained by the diagonals of the Feinberg's triangle associated to a suitable generalized Pascal's pyramid. The results also extend similar properties of Fibonacci-like sequences. [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]

Simple Summary: To date there is not much information regard the role that sounds may play in the life of elasmobranchs. This gains particular importance if we consider the current understanding about noise pollution at sea. In fact, in the past few years, the effects of anthropogenic noise on marine fauna have received increasing attentions considering the plethora of repercussions deriving from the expansion of this type of pollution. Here, we exposed small-spotted catshark specimens kept in an aquarium, to different acoustic conditions to analyse the possible changes in swimming behaviour. Four different acoustic conditions consisted of biological sounds and anthropogenic noises. Moreover, the amplitude levels were differentiated among them, to analyse the effects caused by different signal-to-noise ratios. The results highlighted both a tendency of the animals to increase the overall time spent swimming and to avoid the noisiest section of the aquarium when subjected to higher amplitude levels of noise. Despite the growing interest in human-made noise effects on marine wildlife, few studies have investigated the potential role of underwater noise on elasmobranch species. In this study, twelve specimens of small-spotted catshark (Scyliorhinus canicula) were exposed to biological and anthropogenic sounds in order to assess their behavioural changes in response to prey acoustic stimuli and to different amplitude levels of shipping noise. The sharks, individually held in aquariums, were exposed to four experimental acoustic conditions characterized by different spectral (Hz) components and amplitude (dB re 1 µPa) levels. The swimming behaviour and spatial distribution of sharks were observed. The results highlighted significant differences in swimming time and in the spatial use of the aquarium among the experimental conditions. When the amplitude levels of biological sources were higher than those of anthropogenic sources, the sharks' swimming behaviour was concentrated in the bottom sections of the aquarium; when the amplitude levels of anthropogenic sources were higher than biological ones, the specimens increased the time spent swimming. Moreover, their spatial distribution highlighted a tendency to occupy the least noisy sections of the aquarium. In conclusion, this study highlighted that anthropogenic noise is able to affect behaviour of catshark specimens and the impact depends on acoustic amplitude levels. [ABSTRACT FROM AUTHOR]

When dealing with elementary algorithms in the classroom, we almost always limit ourselves to considering the purely procedural aspects. In so doing, we run the risk of providing students with a distorted and reductive vision of mathematics, which could foster an attitude of disaffection towards the discipline among students. This work aims to show that we can overturn this situation by changing our approach and thus make the study of some elementary algorithms an opportunity to increase interest in mathematics and encourage the development of skills and competences in students. Our idea is to use the inquiry-based learning methodology in the study of algorithms in order to re-evaluate their role in school mathematics classes. Below we will describe an experiment on the teaching of elementary algorithms conducted between 2016 and 2018 as part of the 'Math High School' project of the University of Salerno and based on thorough historical and scientific research. The project involved a group of 120 first-year high school students and the skills that the students were able to develop suggest that the experiment should be extended to the use of IBL methodology in handling other algorithms. The inquiry-based activities proposed can be applied in the classroom by teachers and can offer practical strategies and tools for teaching algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we give possible suggestions for a classroom lesson about an application of probability using basic mathematical notions. We will approach to some combinatoric results without using ‘induction’, ‘polynomial identities’ nor ‘generating functions’, and will give a proof of the ‘Vandermonde Identity’ using elementary notions of probability. [ABSTRACT FROM AUTHOR]

In line with the latest positions of Gottlob Frege, this article puts forward the hypothesis that the cognitive bases of mathematics are geometric in nature. Starting from the geometry axioms of theElementsof Euclid, we introduce a geometric theory of proportions along the lines of the one introduced by Grassmann inAusdehnungslehrein 1844. Assuming as axioms, the cognitive contents of the theorems of Pappus and Desargues, through their configurations, in an Euclidean plane a natural field structure can be identified that reveals the purely geometric nature of complex numbers. Reasoning based on figures is becoming a growing interdisciplinary field in logic, philosophy and cognitive sciences, and is also of considerable interest in the field of education, moreover, recently, it has been emphasized that the mutual assistance that geometry and complex numbers give is poorly pointed out in teaching and that a unitary vision of geometrical aspects and calculation can be clarifying. [ABSTRACT FROM PUBLISHER]

Let X1,...,Xk be quasinormed spaces with quasinorms | ⋅ |j, j = 1,...,k, respectively. For any f = (f1,⋯,fk) ∈ X1 ×⋯× Xk let ρ( f) be the unique non-negative root of the Cauchy polynomial . We prove that ρ(⋅) (which in general cannot be expressed by radicals when k ≥ 5) is a quasinorm on X1 ×⋯× Xk, which we call root quasinorm, and we find a characterization of this quasinorm as limit of ratios of consecutive terms of a linear recurrence relation. If X1,...,Xk are normed, Banach or Banach function spaces, then the same construction gives respectively a normed, Banach or a Banach function space. Norms obtained as roots of polynomials are already known in the framework of the variable Lebesgue spaces, in the case of the exponent simple function with values 1,...,k. We investigate the properties of the root quasinorm and we establish a number of inequalities, which come from a rich literature of the past century. [ABSTRACT FROM AUTHOR]

Abstract: We extend to FC⁎, the class of generalized FC-groups introduced in [F. de Giovanni, A. Russo, G. Vincenzi, Groups with restricted conjugacy classes, Serdica Math. J. 28 (2002) 241–254], some results known for FC-groups. The main theorem involves the extended residually finite property (ERF), i.e., all subgroups are closed in the profinite topology. The ERF-groups belonging to several large classes of groups have been determined, for example nilpotent groups [M. Menth, Nilpotent groups with every quotient residually finite, J. Group Theory 5 (2002) 199–217] and FC-groups [D.J.S. Robinson, A. Russo, G. Vincenzi, On groups whose subgroups are closed in the profinite topology, J. Pure Appl. Algebra 213 (2009) 421–429]. Here we generalize these results by classifying completely the ERF-groups belonging to FC⁎. [ABSTRACT FROM AUTHOR]

Abstract: A group is called extended residually finite (ERF) if every subgroup is closed in the profinite topology. The ERF-property is studied for nilpotent groups, soluble groups, locally finite groups and FC-groups. A complete characterization is given of FC-groups which are ERF. [Copyright &y& Elsevier]

A group is called HNN-free if it has no subgroups that are nontrivial HNN-extensions. We prove that finitely generated HNN-free implies virtually polycyclic for a large class of groups. We also consider finitely generated groups with no free subsemigroups of rank 2 and show that in many situations such groups are virtually nilpotent. Finally, as an application of our results, we determine the structure of locally graded groups in which every subgroup is pronormal, thus generalizing a theorem of Kuzennyi and Subbotin. [ABSTRACT FROM AUTHOR]

Rough-toothed dolphin's abundance and distribution is largely unknown worldwide and evaluation of its conservation status in the Mediterranean Sea is necessary. A rough-toothed dolphin was sighted offshore Eastern Sicily (Mediterranean Sea) in July 2017 and acoustic data were acquired in the same area of Watkins, Tyack, Moore, and Notarbartolo di Sciara [(1987). Mar. Mamm. Sci. 3, 78–82]. An automatic detection algorithm was developed to identify the echolocation clicks recorded within both datasets and a recurrent inter-click interval value was identified during the new encounter. Distinctive whistle classes were also identified with similar contour shapes within both datasets. [ABSTRACT FROM AUTHOR]

This study examined the effects of lab-generated acoustic signals on the behaviour and biochemistry of Mediterranean mussels ( Mytilus galloprovincialis ). The experiment was carried out in a tank equipped with a video-recording system using six groups of five mussels exposed to five acoustic treatments (each treatment was replicated three times) for 30 min. The acoustic signals, with a maximum sound pressure level of 150 dB rms re 1 μPa, differed in frequency range as follows: low (0.1–5 kHz), mid-low (5–10 kHz), mid (10–20 kHz), mid-high (20–40 kHz) and high (40–60 kHz). The exposure to sweeps did not produce any significant changes in the mussels' behaviour. Conversely, the specimens exposed to the low frequency band treatment showed significantly higher values of the following biochemical stress parameters measured in their plasma and tissues: glucose, total proteins, total haemocyte number (THC), heat shock protein 70 (Hsp70) expression, and Acetylcholinesterase (AChE) activity. The responses observed in the mussels exposed to low frequency sweeps enable us to suppose a biological and ecological role for this sound, which contains the main frequencies produced by both shipping traffic and the acoustic emissions of fish. [ABSTRACT FROM AUTHOR]

This study examined the effects of recorded boat noises on the behaviour and biochemistry of the common prawn ( Palaemon serratus ) in laboratory settings. The experiment was carried out in a tank equipped with a video-recording system using six groups (three control and three tested) of eight common prawns (48 animals in total). After habituation for 1 h, the behaviour of the prawns was monitored for 1 h. During the last 30 min, the animals in the test groups were exposed to noise resembling a marine area with high anthropogenic acoustic pollution. The exposure to noise produced significant changes in locomotor patterns, presence inside or outside a shelter, total protein concentrations in the haemolymph and brain, DNA integrity, and the expression protein levels of HSP 27 and 70 in brain tissues, revealing a stressful effect of noise on this crustacean species. This study clearly reveals that invertebrates are likely to be susceptible to acoustic stimuli from boat's noise and that they are potential valuable targets for further and detailed investigations into the effects of the anthropogenic noise pollution. [ABSTRACT FROM AUTHOR]