On the recursive properties of one kind hybrid power mean involving two-term exponential sums and Gauss sums

The main purpose of this paper is to study the computational problem of one kind hybrid power mean involving two-term exponential sums and quartic Gauss sums using the analytic method and the properties of the classical Gauss sums, and to prove some interesting fourth-order linear recurrence formulae for this problem. As an application of our result, we can also obtain an exact computational formula for one kind congruence equation mod p, an odd prime

On Chebyshev polynomials and their applications

Abstract The main purpose of this paper is, using some properties of the Chebyshev polynomials, to study the power sum problems for the sinx and cosx functions and to obtain some interesting computational formulas

Bernoulli Polynomials and Their Some New Congruence Properties

The aim of this article is to use the fundamental modus and the properties of the Euler polynomials and Bernoulli polynomials to prove some new congruences related to Bernoulli polynomials. One of them is that for any integer h or any non-negative integer n, we obtain the congruence B 2 n + 1 ( 2 h ) ≡ 0 mod ( 2 n + 1 ) , where B n ( x ) are Bernoulli polynomials

Some Types of Identities Involving the Legendre Polynomials

In this paper, a new non-linear recursive sequence is firstly introduced. Then, using this sequence, a computational problem involving the convolution of the Legendre polynomial is studied using the basic and combinatorial methods. Finally, we give an interesting identity

The Generalized Quadratic Gauss Sum and Its Fourth Power Mean

In this article, our main purpose is to introduce a new and generalized quadratic Gauss sum. By using analytic methods, the properties of classical Gauss sums, and character sums, we consider the calculating problem of its fourth power mean and give two interesting computational formulae for it

First-Principles Calculations of the Mechanical Properties of Doped Cu₃P Alloys

In the quest to enhance the mechanical properties of CuP alloys, particularly focusing on the Cu3P phase, this study introduces a comprehensive investigation into the effects of various alloying elements on the alloy’s performance. In this paper, the first principle of density universal function theory and the projection-enhanced wave method under VASP 5.4.4 software are used to recalculate the lattice constants, evaluate the lattice stability, and explore the mechanical properties of selected doped elements such as In, Si, V, Al, Bi, Nb, Sc, Ta, Ti, Y and Zr, including shear, stiffness, compression, and plasticity. The investigation reveals that strategic doping with In and Si significantly enhances shear resistance and stiffness, while V addition notably augments compressive resistance. Furthermore, incorporating Al, Bi, Nb, Sc, Ta, Ti, V, Y, and Zr has substantially improved plasticity, indicating a broad spectrum of mechanical enhancement through precise alloying. Crucially, the validation of our computational models is demonstrated through hardness experiments on Si and Sn-doped specimens, corroborating the theoretical predictions. Additionally, a meticulous analysis of the states’ density further confirms our computational approach’s accuracy and reliability. This study highlights the potential of targeted alloying to tailor the mechanical properties of Cu3P alloys and establishes a robust theoretical framework for predicting the effects of doping in metallic alloys. The findings presented herein offer valuable insights and a novel perspective on material design and optimization, marking a significant stride toward developing advanced materials with customized mechanical properties

Screening for Atrial Fibrillation by Village Doctors in Rural Areas of China:The Jiangsu Province Rural Community AF Project

China has a large population of elderly in rural areas. Village doctors are acting as health-care gatekeepers for the rural elderly in China and are encouraged to provide more long-term care for patients with chronic diseases such as atrial fibrillation (AF). The data of AF registries from the rural elderly are limited. The present registry aims to provide contemporary data on the current AF-related health status of the rural elderly and the gaps in management of AF by village doctors. This study has two phases. The first phase is a cross-sectional study of AF screening in two rural towns of eastern China. All the residents aged ≥65 years are eligible and will be invited to attend a government-led health examination or an in-house AF screening program. The AF detection rate, the awareness of AF and the usage of oral anticoagulants and smartphones by AF patients, and the ability to diagnose and manage AF by village doctors will be assessed. Participants with AF detected in the first phase are eligible for the second phase. A variety of modes of intensified education will be provided to all AF patients and their family members to enrich their AF-related knowledge. Their village doctors will be offered a lecture-based training program focusing on Atrial fibrillation Better Care (ABC) pathway. Follow-up will be conducted for 1 year. The primary endpoint is the composite of all stroke and all-cause mortality. The first phase of AF screening was conducted between April 2019 and June 2019, and 18,712 participants with the mean age of 73.1±5.8 years were enrolled. The second phase that includes 810 patients with AF, started on 1 May 2019. This study will provide a perspective of primary care system and would indirectly reflect the current status of chronic disease care in rural China