We recall the experimental evidence according to which nuclei are composed by extended protons and neutrons in conditions of partial mutual penetration with ensuing interactions that are: linear, local and potential, thus variationally self-adjoint (SA) [1], as well as non-linear in the wave functions (as pioneered by W. Heisenberg), non-local because defined on volumes (as pioneered by L. de Broglie and D. Bohm) and non-derivable from a potential because of contact, thus zero range and variationally non-self-adjoint (NSA) type, as pioneered by R. M. Santilli in 1978 then at Harvard University under DOE support [1]. We then review the foundations of the time reversal invariant Lie-isotopic mathematics, also known as Santilli iso-mathematics, proposed in the volume [2] which is based on the preservation of the abstract axioms of 20th century applied mathematics and the use of their broadest possible realization, thus including the isotopy of [2]-[6]: 1) The quantum mechanical enveloping associative algebra of Hermitean operators ξ : {A, B, ...; AB = A × B, I} representing SA interactions via a Hamiltonian H(r, p) into the iso-associative enveloping algebra  with iso-product , iso-unit and the Santillian for the representation of NSA interactions; 2) Lie’s theory into the Lie-Santilli iso-theory; 3) Numeric fields into Santilli iso-fields of iso-numbers ; 4) Functional analysis into the iso-functional form; 5) Metric spaces over into iso-metric iso-spaces over and iso-metrics ; 6) Newton-Leibnitz local differential calculus into a non-linear, non-local and NSA form with iso-differential ; 7) Geometries on S over F into iso-geometries on isospaces over . Iso-mathematics characterizes the iso-mechanical branch of hadronic mechanics with Lie-isotopic generalization of Heisenberg equation for the time evolution of an observable in the infinitesimal and finite forms [2] [3]

where is the exponentiation in . It should be indicated that iso-mathematics and iso-mechanics can be constructed via the simple non-unitary transformation provided it is applied to the totality of conventional formalisms. We finally indicate that iso-mathematics and iso-mechanics have permitted the first and only known numerically exact and time invariant representation of experimental data for stable nuclei [7]-[8]. Tutoring lecture [9] may be a good introduction to iso-mathematics for physicists. A knowledge of this lecture is a necessary pre-requisite for the subsequent lecture on the broader, irreversible, Lie-admissible mathematics used for irreversible nuclear fusions.