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In this paper, we study a class of binary sequences with two-valued non-zero periodic autocorrelation sum and good periodic crosscorrelation sum as well as balanced properties. We make use of the sequences obtained in (No, J. et al., IEEE Trans. Inform. Theory 44(3), 1278-1282 2001) and adopt the extraction method similar to (Lüke, H. IEEE Trans. Inform. Theory 43(1) 1997). The new sequences are proven to be balanced or almost balanced. Based on these correlation and balanced properties, an important application is to construct Hadamard matrices of order p + 1 for p ≡ 3 ( mod 4) and 2 p + 2 for p ≡ 1 ( mod 4). Some examples are shown to verify the theoretical results. [ABSTRACT FROM AUTHOR]

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]

Background: The World Health Organization (WHO) has established recommended daily intakes for sodium and potassium. However, there is currently some controversy regarding the association between sodium intake, potassium intake, the sodium-to-potassium ratio, and overall mortality. To assess the correlations between sodium intake, potassium intake, the sodium-to-potassium ratio, and overall mortality, as well as the potential differences in sodium and potassium intake thresholds among different population groups, we analyzed data from NHANES 2003–2018. Methods: NHANES is an observational cohort study that estimates sodium and potassium intake through one or two 24-h dietary recalls. Hazard ratios (HR) for overall mortality were calculated using multivariable adjusted Cox models accounting for sampling design. A total of 13855 out of 26288 participants were included in the final analysis. Restricted cubic spline analyses were used to examine the relationship between sodium intake, potassium intake, and overall mortality. If non-linearity was detected, we employed a recursive algorithm to calculate inflection points. Results: Based on one or two 24-h dietary recalls, the sample consisted of 13,855 participants, representing a non-institutionalized population aged 40–80 years, totaling 11,348,771 person-months of mean follow-up 99.395 months. Daily sodium intake and daily potassium intake were inversely associated with all-cause mortality. Restrictive cubic spline analysis showed non-linear relationships between daily sodium intake, potassium intake, sodium–potassium ratio, and total mortality. The inflection point for daily sodium intake was 3133 mg/d, and the inflection point for daily potassium intake was 3501 mg/d, and the inflection point for daily sodium–potassium ratio intake was 1.203 mg/mg/d. In subgroup analyses, a significant interaction was found between age and high sodium intake, which was further confirmed by the smooth curves that showed a U-shaped relationship between sodium intake and all-cause mortality in the elderly population, with a inflection point of 3634 mg/d. Conclusion: Nonlinear associations of daily sodium intake, daily potassium intake and daily sodium–potassium ratio intake with all-cause mortality were observed in American individuals. The inflection point for daily sodium intake was 3133 mg/d. And the inflection point for daily sodium intake was 3634 mg/d in elderly population. The inflection point for daily potassium intake was 3501 mg/d. The inflection point for daily sodium–potassium ratio intake was 1.203 mg/mg/d, respectively, A healthy diet should be based on reasonable sodium intake and include an appropriate sodium-to-potassium ratio. [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]

This correction notice is for an article titled "Sodium, potassium intake, and all-cause mortality: confusion and new findings" published in BMC Public Health. The correction addresses an incorrect author name, with the correct name being Ping Zeng instead of Ping Zen. The original article has been updated to reflect this correction. The publisher, Springer Nature, remains neutral in terms of jurisdictional claims and institutional affiliations. The authors of the article are Donghao Liu, Yuqing Tian, Rui Wang, Tianyue Zhang, Shuhui Shen, Ping Zeng, and Tong Zou. [Extracted from the article]