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We study the possible positions of an inscribed equilateral triangle with vertices on the sides of a generic triangle. We show that infinitely many such configurations exist and examine some extremal properties. [ABSTRACT FROM AUTHOR]

In this article we consider the geometrical iterative process defined by the Kasner triangles with a fixed real weight. The main results about the convergence of this process are given in Section 2. The proofs are elementary and use the general theory of the second order linear recurrences. In the remark after the proof a higher level approach is presented. [ABSTRACT FROM AUTHOR]

Escape rooms are an increasingly popular form of live-action entertainment in which teams of players need to solve puzzles to escape or reach a narrative goal within a specified time limit. Current research suggests that this style of activity promotes the effective use of team skills in the classroom, which are desirable in the workplace, and also that students are more engaged in the hands-on learning environment. During this study we created a game designed to teach mathematical concepts related to finding missing side lengths in a right-angled triangle using trigonometry to secondary school students who had limited knowledge in this area. Our experiment tested the success of our escape room’s teaching capabilities through comparing student performance on a test completed immediately after finishing the game. We also evaluated whether students’ perception of the applicability of mathematics and their feelings towards studying it improves from playing. Finally, we intended to confirm the findings of other researchers on engagement; extend that to the effective use of team skills through examining if there is a relationship between completion time and the observed behaviours within each team. Results suggest that the game is successful in teaching and building familiarity with some aspects of the topic and that there is no correlation between effective team skills and completion time. We make suggestions for improvements to the game design that would warrant an additional experiment into the teaching potential of escape rooms.

For a positive integer n ≥ 1 the n-th cyclotomic polynomial is defined by Φn(z) = Пςn=1(z - ζ), where are the primitive n-th roots of unity. These polynomials are known to possess many interesting properties. In this article we establish an integral formula for the coefficients of the cyclotomic polynomial, we then discuss the direct and alternate sums of coefficients, as well as the mid-term of Φn(z). Finally, these results are used in the computation of certain trigonometric integrals. [ABSTRACT FROM AUTHOR]

Pseudoprimes are composite integers sharing behaviours of the prime numbers, often used in practical applications like public-key cryptography. Many pseudoprimality notions known in the literature are defined by recurrent sequences. In this paper we first establish new arithmetic properties of the generalized Lucas and Pell-Lucas sequences. Then we study the recent notion of generalized Pell and Pell-Lucas pseudo-primes of level k, and find inclusions between the sets of pseudoprimes on different levels. In this process we extend several results concerning Fibonacci, Lucas, Pell, and Pell-Lucas sequences.

In this paper we introduce the functions which count the number of generalized Lucas and Pell-Lucas sequence terms not exceeding a given value x and, under certain conditions, we derive exact formulae (Theorems 3 and 4) and establish asymptotic limits for them (Theorem 6). We formulate necessary and sufficient arithmetic conditions which can identify the terms of a-Fibonacci and a-Lucas sequences. Finally, using a deep theorem of Siegel, we show that the aforementioned sequences contain only finitely many perfect powers. During the process we also discover some novel integer sequences.

We deduce exact integral formulae for the coefficients of Gaussian, multinomial and Catalan polynomials. The method used by the authors in the papers [2, 3, 4] to prove some new results concerning cyclotomic and polygonal polynomials, as well as some of their extensions is applied.

The concept of generalized exponential trichotomy for linear time-varying systems is investigated in relationship with the classical notion of uniform exponential trichotomy. Some key properties of generalized exponential trichotomy are explored through supplementary projections. These results are also extended to the case of projection sequences, while certain applications for adjoint systems are suggested.

Let be a positive integer and , , , and real numbers satisfying and . It is proved that for the real numbers the maximum of the function is attained if and only if of the numbers are equal to and the other are equal to , while is one of the values , , where denotes the integer part and represents the Stolarsky mean of and of powers and Some asymptotic results concerning are also discussed.

Recent developments in cloud computing and the Internet of Things have enabled smart environments, in terms of both monitoring and actuation. Unfortunately, this often results in unsustainable cloud-based solutions, whereby, in the interest of simplicity, a wealth of raw (unprocessed) data are pushed from sensor nodes to the cloud. Herein, we advocate the use of machine learning at sensor nodes to perform essential data-cleaning operations, to avoid the transmission of corrupted (often unusable) data to the cloud. Starting from a public pollution dataset, we investigate how two machine learning techniques (kNN and missForest) may be embedded on Raspberry Pi to perform data imputation, without impacting the data collection process. Our experimental results demonstrate the accuracy and computational efficiency of edge-learning methods for filling in missing data values in corrupted data series. We find that kNN and missForest correctly impute up to 40% of randomly distributed missing values, with a density distribution of values that is indistinguishable from the benchmark. We also show a trade-off analysis for the case of bursty missing values, with recoverable blocks of up to 100 samples. Computation times are shorter than sampling periods, allowing for data imputation at the edge in a timely manner.

Data collected in criminal investigations may suffer from issues like: (i) incompleteness, due to the covert nature of criminal organizations; (ii) incorrectness, caused by either unintentional data collection errors or intentional deception by criminals; (iii) inconsistency, when the same information is collected into law enforcement databases multiple times, or in different formats. In this paper we analyze nine real criminal networks of different nature (i.e., Mafia networks, criminal street gangs and terrorist organizations) in order to quantify the impact of incomplete data, and to determine which network type is most affected by it. The networks are firstly pruned using two specific methods: (i) random edge removal, simulating the scenario in which the Law Enforcement Agencies fail to intercept some calls, or to spot sporadic meetings among suspects; (ii) node removal, modeling the situation in which some suspects cannot be intercepted or investigated. Finally we compute spectral distances (i.e., Adjacency, Laplacian and normalized Laplacian Spectral Distances) and matrix distances (i.e., Root Euclidean Distance) between the complete and pruned networks, which we compare using statistical analysis. Our investigation identifies two main features: first, the overall understanding of the criminal networks remains high even with incomplete data on criminal interactions (i.e., when 10% of edges are removed); second, removing even a small fraction of suspects not investigated (i.e., 2% of nodes are removed) may lead to significant misinterpretation of the overall network., Competing Interests: The authors have declared that no competing interests exist.

Compared to other types of social networks, criminal networks present particularly hard challenges, due to their strong resilience to disruption, which poses severe hurdles to Law-Enforcement Agencies (LEAs). Herein, we borrow methods and tools from Social Network Analysis (SNA) to (i) unveil the structure and organization of Sicilian Mafia gangs, based on two real-world datasets, and (ii) gain insights as to how to efficiently reduce the Largest Connected Component (LCC) of two networks derived from them. Mafia networks have peculiar features in terms of the links distribution and strength, which makes them very different from other social networks, and extremely robust to exogenous perturbations. Analysts also face difficulties in collecting reliable datasets that accurately describe the gangs' internal structure and their relationships with the external world, which is why earlier studies are largely qualitative, elusive and incomplete. An added value of our work is the generation of two real-world datasets, based on raw data extracted from juridical acts, relating to a Mafia organization that operated in Sicily during the first decade of 2000s. We created two different networks, capturing phone calls and physical meetings, respectively. Our analysis simulated different intervention procedures: (i) arresting one criminal at a time (sequential node removal); and (ii) police raids (node block removal). In both the sequential, and the node block removal intervention procedures, the Betweenness centrality was the most effective strategy in prioritizing the nodes to be removed. For instance, when targeting the top 5% nodes with the largest Betweenness centrality, our simulations suggest a reduction of up to 70% in the size of the LCC. We also identified that, due the peculiar type of interactions in criminal networks (namely, the distribution of the interactions' frequency), no significant differences exist between weighted and unweighted network analysis. Our work has significant practical applications for perturbing the operations of criminal and terrorist networks., Competing Interests: The authors have declared that no competing interests exist.