哪位高手能帮我把这个专业数学的东西翻译成英文？多谢！！

来源：百度知道编辑：UC知道时间：2024/05/14 15:15:16

引言：
现今，各种高等教育教科书及参考书对于凹向的定义多种多样，但未能完整地记录不同定义之间的等价性。鉴于此点，我们总结了五种凹向的定义，并且系统地证明了五种定义之间的等价性。同时，我们强调了凹向的实用价值。希望通过这篇文章，使同学们对凹向有更加深刻的理解。

Introduction :
Nowadays, the definitions to Concave of all higher education textbooks and reference books diversity are diverse , but they failed to record the equivalence between the different definitions. In light of this point, we summed up the five definitions to Concave and systematically proved the equivalence between the five definitions to Concave. Meanwhile, we have emphasized the practical value of Concave. We hope that through this article , the students could have a more profound understanding of Concave.
ps:自己翻译的，不知道凹向怎么定义，只好用Concave，可能不是很完善。还希望见谅。