Scholarly sources, which are normally published in journals, as book chapters, or in conference proceedings, usually cover specific topics at a fairly deep level. In Mathematics, scholarly sources may include research papers, conference proceedings, or pre-print material (not yet peer reviewed).
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A scholarly database organizes the best quality, most relevant information for a field or discipline in one place so it can be searched. Because there is so much scholarly research and so many different fields, databases specialize in which kinds of information or tools they provide.
Scholarly databases take into account the important publications and research methods of a field or discipline. The database helps you make choices about which articles you want to see - by subject, dates of publication, overall topic, and more. Google and Google Scholar, while vast, do no quality control on the content they provide or give you good tools to narrow your focus.
MathSciNet is the database supported by the American Mathematical Society. It is usually search by author or journal title. Review the tutorial on the database tutorials page to learn how to search it by topic.
Project Euclid is a mathematics research database that small, independent scholarly journals.
arXiv is a database of pre-print research articles in math, physics, and other topics. Pre-print means that the content is not in its final, published form, so the full recommendations from peer reviewers are not present. This is one way in which Mathematics differs from other disciplines - arXiv is a valuable resource without full peer-review.