Foundations of mathematics is the study of the most basic concepts and logical structure of mathematics, with an eye to the unity of human knowledge. Almost all of the problems studied in this book are motivated by an overriding foundational question: What are the appropriate axioms for mathematics? Through a series of case studies, these axioms are examined to prove particular theorems in core...

It is commonly argued that the spectacular increase in order which has occurred on Earth does not violate the second law of thermodynamics because the Earth is an open system, and anything can happen in an open system as long as the entropy increases outside the system compensate the entropy decreases inside the system. However, if we define ‘‘X-entropy’’ to be the entropy associated with any d...

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Various external stimuli regulate cellular Ca2+ fluxes. Information from these stimuli is transmitted to the cell interior by signal transducing systems that involve interactions between receptors, G-proteins, and enzymes that generate second messengers. The possible mechanisms by which "cross-talk" within and between these second messenger-generating systems provides an intricate molecular ave...

This paper examines Chinese residents’ experience in learning English in the 3-D virtual world of Second Life (SL). With an introduction to the current English as a Second Language (ESL) education in China, ESL students’ demand for practice, I analyzed an online BBS created and maintained by Chinese SL residents and conducted interviews in SL to argue that there is an urgent need among Chinese ...

The utility of a one-dimensional magnetic resonance (MR) sequence to rapidly and accurately measure wave velocity in vivo was evaluated. Studies were conducted in the thoracic aortas of 20 healthy subjects of varying age, and the MR method was validated in a compliant tube model. Aortic wave velocity ranged from 3.8 to 9.7 m/sec and demonstrated a positive correlation with subjects' age. Peak b...