methods are used to prove elementary statements (typically analytical methods in number theory, like for Dirichlet’s theorem) These methods may use abstract existence statements, “ideal” objects, that may fail to exist effectively (typically, Hilbert’s proof of the basis theorem) Hilbert’s program: if one proves a concrete statement, one can always eliminate the use of these ideal objects, and ...

A definition is proposed to give precise meaning to the counterfac-tual statements that often appear in discussions of the implications of quantum mechanics. Of particular interest are counterfactual statements which involve events occurring at space-like separated points, which do not have an absolute time ordering. Some consequences of this definition are discussed.

Mathematicians reason about abstract objects, or at least they seem to do so. In the course of doing so, they routinely make statements on mathematical statements. In fact, every inference, such as “From equations (5.17) and (6.2) one obtains the following equation.”, is a statement on mathematical statements. Other examples of statements on mathematical statements are: “With the obvious modifi...

Welcome to the course page for Business Finance for Life Scientists This content is designed provide an overview of financial statements, and basic finance principles used in financial planning and decision making. The goal of this workshop is to help prepare scientists who want to start their own labs, and inform scientists who want to go into industry. This page includes the lecture videos, c...

Welcome to the course page for Business Finance for Life Scientists This content is designed provide an overview of financial statements, and basic finance principles used in financial planning and decision making. The goal of this workshop is to help prepare scientists who want to start their own labs, and inform scientists who want to go into industry. This page includes the lecture videos, c...

This entry provides all cardinality theorems of the Twelvefold Way. The Twelvefold Way [1, 5, 6] systematically classifies twelve related combinatorial problems concerning two finite sets, which include counting permutations, combinations, multisets, set partitions and number partitions. This development builds upon the existing formal developments [2, 3, 4] with cardinality theorems for those ...