In the first part of the chapter, the status of the discipline of mathematical geosciences (MG) is reviewed and a new definition of MG as an interdisciplinary field of science is suggested. Similar to other disciplines such as geochemistry and geophysics, mathematical geosciences or geomathematics is the science of studying mathematical properties and processes of the Earth (and other planets) with prediction of its resources and changing environments. In the second part of the chapter, original research results are presented. The new concepts of fractal density and local singularity are introduced. In the context of fractal density and singularity a new power-law model is proposed to associate differential stress with depth increments at the phase transition zone in the Earth's lithosphere. A case study is utilized to demonstrate the application of local singularity analysis for modeling the clustering frequency-depth distribution of earthquakes from the Pacific subduction zones. Datasets of earthquakes with magnitudes of at least 3 were selected from the Ring of Fire, subduction zones of Pacific plates. The results show that datasets from the Pacific subduction zones except from northeastern zones depict a profound frequency -depth cluster around the Moho. Further it is demonstrated that the clusters of frequency-depth distributions of earthquakes in the colder and older southwestern boundaries of the Pacific plates generally depict stronger singularity than those obtained from the earthquakes in their hotter and younger eastern boundaries.
