It was first demonstrated mathematically by economists Kenneth Arrow and Gérard Debreu.

This analysis mathematically demonstrated that subtleties of evanescent waves, restriction to a finite slab and absorption had led to inconsistencies and divergencies that contradict the basic mathematical properties of scattered wave fields.

These basis states, referred to as spin-up and spin-down, are hence eigenvalues of the perturbed Hamiltonian, so this level splitting is both easy to demonstrate mathematically and intuitively evident.

To demonstrate mathematically how a nonlinear component can multiply signals and generate heterodyne frequencies, the nonlinear function F can be expanded in a power series (MacLaurin series):

Together with Bernard Jeune, he demonstrated mathematically a continuum of plant forms that spans not only organ categories such as root, stem, and leaf, but also different hierarchical levels of organ systems, organs, and tissues.

This process accelerates because it is, as Shannon demonstrated mathematically, logarithmic.

I shall develop the inevitability of the breakdown of the capitalist system, and I shall demonstrate mathematically why it must break down.

Using geometry, Ibn al-Haytham mathematically demonstrated that place (al-makan) is the imagined three-dimensional void between the inner surfaces of a containing body.

Early this century, the Yugoslav astronomer Milutin Milankovitch demonstrated mathematically that these and other variations, however slight, could have dire climatic consequences, presumably accounting for a succession of ice ages in the recent geological past.

Ricardo demonstrated mathematically that the gains from trade could outweigh the perceived advantages of protectionist policy.