Ryotaro's theories
I'm interested in mixed-integer nonlinear programmings (and generalized disjunctive programmings) which are mainly applied to engineering areas but unfamiliar with physicists and chemists. I search for the applications of the mathematics to the theoretical physics and chemistry. For example, mathematical crystal chemistry indicates that introducing integer variables to extract the “combinatorial backbone” of an original continuous optimization problem accelerates visiting every local optimum by iteratively optimizing the continuous and integer variables like the Logic Based Outer-Approximation and remembering infeasible subsets of integer variables to avoid precise optimization of continous variables with infeasible integer variables like the GDP Branch and Bound method. I hope to share my curiosity and future prospect with you through this homepage.
Mathematical crystal chemistry
Inorganic structural chemistry describes the spatial orders of atoms by simple concepts such as packings of atomic spheres (continuous aspect) and linking of coordination polyhedra (discrete aspect). Mathematical crystal chemistry formulates the simple concepts as objective or constraint functions of generalized disjunctive programming (GDP) which is formulated using continuous and Boolean variables to involve the algebraic equations, disjunctions, and logic propositions. GDP is one of the specific forms of mixed-integer nonlinear programming (MINLP), where all the integer variables are only used for indicating whether certain constraints on real variables are enforced or not. MINLP is an optimization problem of real and integer variables with equality and/or inequality constraints, where a zero-or-one variable is called "logical variables". Mathematical crystal chemistry combines the continuous and discrete aspects of crystal structures elucidated by inorganic structural chemistry: The continuous variables represent the packing of atomic spheres consisting of several kinds of atomic radii, while the Boolean variables represent graphs describing crystal structures with clear chemical meanings. Mathematical crystal chemistry clarifies that the simple concepts are enough to design prototypes of crystal structures with small computations. Feel free to use software MARICI, since I think it gives a kind introduction to mathematical crystal chemistry.