Mathematics

The present text is aimed to a thorough introduction to convex, concave set and function, which entails a powerful and elegant interaction between analysis and geometry. Convex functions are powerful tools for proving a large class of inequalities. They provide an elegant and unified treatment of the most important classical inequalities. A general inequality is proved using the definition of convex functions. Many major inequalities are deduced as applications.

Quantum annealing has recently been used to determine the Ramsey numbers R(m,2) for 3 < m < 9 and R(3,3) [Bian et al. (2013) PRL 111, 130505]. This was greatly celebrated as the largest experimental implementation of an adiabatic evolution algorithm to that date. However, in that computation, more than 66% of the qubits used were auxiliary qubits, so the sizes of the Ramsey number Hamiltonians used were tremendously smaller than the full 128-qubit capacity of the device used. The reason these auxiliary qubits were needed was because the best quantum annealing devices at the time (and still now) cannot implement multi-qubit interactions beyond 2-qubit interactions, and they are also limited in their capacity for 2-qubit interactions. We present a method which allows the full qubit capacity of a quantum annealing device to be used, by reducing multi-qubit and 2-qubit interactions. With our method, the device used in the 2013 Ramsey number quantum computation could have determined R(16,2) and R(4,3) with under 10 minutes of runtime.

We build the first analytic empirical potential for the most deeply bound Li2 state: b(13Πu). Our potential is based on experimental energy transitions covering v=0−34, and very high precision theoretical long-range constants. It provides high accuracy predictions up to v=100 which pave the way for high-precision long-range measurements, and hopefully an eventual resolution of the age old discrepancy between experiment and theory for the Li(22S)+Li(22P) C3 value. State of the art ab initio calculations predict vibrational energy spacings that are all in at most 0.8 cm−1 disagreement with the empirical potential.

En el siguiente artículo se explica un método, no original de los autores, para aproximar una curva bidimensional con splines cúbicos. Además, se da un argumento de carácter informal para justificar el uso de dicho método. Este artículo constituye la respuesta a una pregunta de  un examen de la materia Matemática Computacional en el ITAM, y no pretende introducir ideas nuevas, sino analizar una técnica ya existente.