A discrete ordered set of observed data that changes over time is called a time series and denoted as (y(t) = { y(t_{1}), y(t_{2}), ldots,y(t_{i}), ldots }), where (y(t_{i})) is the observed data at the moment (t _{i}).
Figure 1 shows a schematic: translation determines a series of amino acid sequences {.., Seqt−1, Seqt, Seqt+1,.} = {Seqt}t∈Z, each Seqt for one protein chain, ordered by a discrete temporal order t∈Z of corresponding tertiary structures {.., Strt−1, Strt, Strt+1,.} = {Strt}t∈Z, where Z = {..,−1, 0, 1,.} is the set of integers.
In 1986, Nagaraja [28] explored the behavior of higher order conditional probabilities of order statistics in an attempt to understand the structure of discrete order statistics.
The discrete switching sequence may also be viewed as a discrete strictly ordered set of real or integer nonnegative numbers of first element (i.e., first switching point) in the sense that such that.
The multi-dimensional characteristics of perceived pain is evident by the diversity of operational definitions of pain and pain-related disability ranging from assessments on one or more rating scales with a discrete number of ordered alternatives or on visual analogue scales to electrical pain threshold assessments [ 8].
An example of such a function is Ψ u)=uα/2, 0<α<2, which describes a discrete Laplacian of fractional order α/2.
Each gene was thought to be a discrete piece of DNA; the order of its DNA bases the well-known "letter" molecules that are the building blocks of DNA were thought to code for a particular protein.
In fact, system (5) is also a discrete analog of the second order Hamiltonian system of the form U^{primeprime} ( t ) +BU ( t ) +nabla H ( t,U ) =0,quad tin R. (8) The existence of periodic solutions of (8) have also been extensively discussed since Poincaré [10].
In first-order form, a discrete set of equations is obtained which is both conservative and positive.
System (1.1) is a discrete form of classical second-order Hamiltonian systems, which has been paid much attention to by many mathematicians in the past 30 years; see [17 24] for example.
We propose to use transformations in order to implement a discrete version of some differential operators.
