Mass spectra and isotope distributions

we need some mathematical formulations that help us to model mass spectra and the related entities. 

1 Definition (low resolution mass spectrum) A low resolution mass spectrum i is a mapping 

Furthermore a spectrum is typically normalized to a certain maximum intensity. To simplify mathematical expressions we will normalize the spectrum to a maximum intensity 1 There exists an m such that 

In this manner we cannot only describe experimental spectra, but also theoretical isotope distributions and calculated spectra. The atoms of a chemical element X e are not necessarily all of the same mass. The mass of an atom is essentially the mass of its nucleus, which is composed of two types of elementary particles of unit mass, protons and neutrons. In the atoms of a given element X, the number of positively charged protons is fixed, while the number of uncharged neutrons may vary. Such atoms of different mass are isotopes of X (notation for the isotope of mass m X, e.g. C). Natural isotope distributions are known and almost constant, such that mass spectrometry provides information on the elemental composition of an unknown compound through isotope patterns. 

3 Definition (Natural isotope distribution) LetX be a chemical element. The nofora/isotope d/s-tribution of X is a mapping lx - tn - IxM that associates an intensity with each mass m depending on X,such that 

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