Dissimilar laser welding of a NiTi shape memory alloy to Ti2AlNb

Funding Information: Acknowledgments: This work was supported by National Natural Science Foundation of China (No. 51775091), Science and Technology Project of Sichuan Province (No. 2020ZDZX0015). J.P. Oliveira acknowledges Fundação para a Ciência e a Tecnologia (FCT—MCTES) for its financial support via the project UID/00667/2020 (UNIDEMI). Funding Information: Funding: This work was supported by National Natural Science Foundation of China (No. 51775091), Science and Technology Project of Sichuan Province (No. 2020ZDZX0015). J.P. Oliveira acknowledges Fundação para a Ciência e a Tecnologia (FCT—MCTES) for its financial support via the project UID/00667/2020 (UNIDEMI). Funding Information: This work was supported by National Natural Science Foundation of China (No. 51775091), Science and Technology Project of Sichuan Province (No. 2020ZDZX0015). J.P. Oliveira acknowledges Funda??o para a Ci?ncia e a Tecnologia (FCT?MCTES) for its financial support via the project UID/00667/2020 (UNIDEMI).NiTi-based shape memory alloys and the Ti2AlNb alloy have gained increasing importance in the aerospace field. The joining of these two materials can further increment the importance and usage of these relevant engineering materials and expand their potential applications. However, when joining NiTi-based shape memory alloys to Ti-based alloys, the formation of brittle Ti-rich intermetallic compounds often occurs, significantly limiting their functionality and use. Dissimilar joints between a NiTi shape memory alloy and Ti2AlNb alloy were obtained using a 0.1 mm thick Niobium (Nb) interlayer via laser welding. By process optimization, sound joints were obtained. The microstructure evolution was assessed by means of electron microscopy, whereas the mechanical strength of the joints was evaluated using lap shear tensile testing. The best performing joints were seen to fracture at maximum loads above 1230 N, thus allowing us to consider this dissimilar pair for structural applications.publishersversionpublishe

Some Operations on Quaternion Numbers

In this article, we give some equality and basic theorems about quaternion numbers, and some special operations.Li Bo - Qingdao University of Science and Technology, ChinaLiang Xiquan - Qingdao University of Science and Technology, ChinaWang Pan - Qingdao University of Science and Technology, ChinaZhuang Yanping - Qingdao University of Science and Technology, ChinaGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Fuguo Ge. Inner products, group, ring of quaternion numbers. Formalized Mathematics, 16(2):135-139, 2008, doi:10.2478/v10037-008-0019-x.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Xiquan Liang and Fuguo Ge. The quaternion numbers. Formalized Mathematics, 14(4):161-169, 2006, doi:10.2478/v10037-006-0020-1.Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25-34, 1990.Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990

Borel-Cantelli Lemma

This article is about the Borel-Cantelli Lemma in probability theory. Necessary definitions and theorems are given in [10] and [7].Ludwig Maximilians University of Munich, GermanyGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Fuguo Ge and Xiquan Liang. On the partial product of series and related basic inequalities. Formalized Mathematics, 13(3):413-416, 2005.Hans-Otto Georgii. Stochastik, Einführung in die Wahrscheinlichkeitstheorie und Statistik. deGruyter, Berlin, 2 edition, 2004.Adam Grabowski. On the Kuratowski limit operators. Formalized Mathematics, 11(4):399-409, 2003.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Achim Klenke. Wahrscheinlichkeitstheorie. Springer-Verlag, Berlin, Heidelberg, 2006.Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Jarosław Kotowicz. The limit of a real function at infinity. Formalized Mathematics, 2(1):17-28, 1991.Andrzej Nędzusiak. Probability. Formalized Mathematics, 1(4):745-749, 1990.Andrzej Nędzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401-407, 1990.Konrad Raczkowski and Andrzej Nędzusiak. Series. Formalized Mathematics, 2(4):449-452, 1991.Piotr Rudnicki and Andrzej Trybulec. Abian's fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255-263, 1998.Bo Zhang, Hiroshi Yamazaki, and Yatsuka Nakamura. Limit of sequence of subsets. Formalized Mathematics, 13(2):347-352, 2005.Bo Zhang, Hiroshi Yamazaki, and Yatsuka Nakamura. Set sequences and monotone class. Formalized Mathematics, 13(4):435-441, 2005

The Quaternion Numbers

Summary. In this article, we define the set H of quaternion numbers as the set of all ordered sequences q = 〈x, y,w, z 〉 where x,y,w and z are real numbers. The addition, difference and multiplication of the quaternion numbers are also defined. We define the real and imaginary parts of q and denote this by x = ℜ(q), y = ℑ1(q), w = ℑ2(q), z = ℑ3(q). We define the addition, difference, multiplication again and denote this operation by real and three imaginary parts. We define the conjugate of q denoted by q ∗ ′ and the absolute value of q denoted by |q|. We also give some properties of quaternion numbers

Assessment of the Suitability of Wintering Anatidae Habitats before and after Impoundment in the Three Gorges Reservoir Region

In this study, we aimed to understand the distribution of and changes in the habitats suitable for Anatidae wintering in the Three Gorges Reservoir Region (TGRR), China, and to explore the impact of the impoundment during different impoundment periods. Based on species occurrence data for four dominant species of Anatidae and environmental factors, a maximum entropy (MaxEnt) model was used to analyze the suitability of habitats during five impoundment periods. The results show that the main factors affecting Anatidae distribution were temperature and roads before the Three Gorges Project (TGP) and elevation after the TGP. After the TGP, the area of the suitable habitat declined rapidly and then gradually increased with increasing water level. After impoundment, the primary area of increased habitat suitability was the main stream of the Yangtze River from Changshou District to Yunyang County and its tributary in the Kaizhou area. Among the habitats, the central water regions were more suitable than the marginal shoal areas. Anatidae habitats in the TGRR were distributed mainly within the Yangtze River main stream and the surrounding areas before the TGP, and the surrounding areas largely disappeared after the TGP, particularly in Chongqing City and Jiangjin District. In this context, it is challenging to create new protected areas within the habitat suitable for Anatidae in the main stream of the Yangtze River; we propose adding the Anatidae as conservation targets within the existing conservation agencies and implementing a waterbird monitoring program for scientific waterbird conservation and the sustainable development of the reservoir