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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.

Summary: This paper deals with the extended design of Mittag‐Leffler state estimator and adaptive synchronization for fractional‐order bidirectional associative memory neural networks with time delays. By the aid of Lyapunov direct approach and Razumikhin‐type method, a suitable fractional‐order Lyapunov functional is constructed and a new set of novel sufficient condition are derived to estimate the neuron states via available output measurements such that the ensuring estimator error system is globally Mittag‐Leffler stable. Then, the adaptive feedback control rule is designed, under which the considered FBNNs can achieve Mittag‐Leffler adaptive synchronization by means of some fractional‐order inequality techniques. Moreover, the adaptive feedback control may be utilized even when there is no ideal information from the system parameters. Finally, two numerical simulations are given to reveal the effectiveness of the theoretical consequences. [ABSTRACT FROM AUTHOR]

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.