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In this paper, we develop a novel method to construct Goethals–Seidel (GS) sequences with special structures. In the existing methods, utilizing Turyn sequences is an effective and convenient approach; however, this method cannot cover all GS sequences. Motivated by this, we are devoted to designing some sequences that can potentially construct all GS sequences. Firstly, it is proven that a quad of ± 1 polynomials can be considered a linear combination of eight polynomials with coefficients uniquely belonging to { 0 , ± 1 } . Based on this fact, we change the construction of a quad of Goethals–Seidel sequences to find eight sequences consisting of 0 and ± 1 . One more motivation is to obtain these sequences more efficiently. To this end, we make use of the k-block, of which some properties of (anti) symmetry are discussed. After this, we can then look for the sequences with the help of computers since the symmetry properties facilitate reducing the search range. Moreover, we find that one of the eight blocks, which we utilize to construct GS sequences directly, can also be combined with Williamson sequences to generate GS sequences with more order. Several examples are provided to verify the theoretical results. The main contribution of this work is in building a bridge linking the GS sequences and eight polynomials, and the paper also provides a novel insight through which to consider the existence of GS sequences. [ABSTRACT FROM AUTHOR]

Compared to fixed orthodontic appliances with brackets, thermoplastic invisible orthodontic aligners offer several advantages, such as high aesthetic performance, good comfort, and convenient oral health maintenance, and are widely used in orthodontic fields. However, prolonged use of thermoplastic invisible aligners may lead to demineralization and even caries in most patients' teeth, as they enclose the tooth surface for an extended period. To address this issue, we have created PETG composites that contain piezoelectric barium titanate nanoparticles (BaTiO3NPs) to obtain antibacterial properties. First, we prepared piezoelectric composites by incorporating varying amounts of BaTiO3NPs into PETG matrix material. The composites were then characterized using techniques such as SEM, XRD, and Raman spectroscopy, which confirmed the successful synthesis of the composites. We cultivated biofilms of Streptococcus mutans (S. mutans) on the surface of the nanocomposites under both polarized and unpolarized conditions. We then activated piezoelectric charges by subjecting the nanocomposites to 10 Hz cyclic mechanical vibration. The interactions between the biofilms and materials were evaluated by measuring the biofilm biomass. The addition of piezoelectric nanoparticles had a noticeable antibacterial effect on both the unpolarized and polarized conditions. Under polarized conditions, nanocomposites demonstrated a greater antibacterial effect than under unpolarized conditions. Additionally, as the concentration of BaTiO3NPs increased, the antibacterial rate also increased, with the surface antibacterial rate reaching 67.39% (30 wt% BaTiO3NPs). These findings have the potential for application in wearable, invisible appliances to improve clinical services and reduce the need for cleaning methods. [ABSTRACT FROM AUTHOR]

In this paper, we are devoted to finding Goethals–Seidel sequences by using k-partition, and based on the finite Parseval relation, the construction of Goethals–Seidel sequences could be transformed to the construction of the associated polynomials. Three different structures of Goethals–Seidel sequences will be presented. We first propose a method based on T-matrices directly to obtain a quad of Goethals–Seidel sequences. Next, by introducing the k-partition, we utilize two classes of 8-partitions to obtain a new class of polynomials still remaining the same (anti)symmetrical properties, with which a quad of Goethals–Seidel sequences could be constructed. Moreover, an adoption of the 4-partition together with a quad of four symmetrical sequences can also lead to a quad of Goethals–Seidel sequences. [ABSTRACT FROM AUTHOR]

In this paper, we provide the left and right Parseval relations over the quaternions. Moreover, we construct a new infinite family of sequences almost over $Q_{8} $ with period $p$ by a generalization of Legendre mapping, where $p\equiv 1 \pmod {8}$ is prime and the sequences have ideal autocorrelation. The sequences are symmetric when $p\equiv 1 \pmod {16}$ , and antisymmetric when $p\equiv 9 \pmod {16} $. [ABSTRACT FROM AUTHOR]