D

Danilevski method of matrix transformation A method for computing eigenvalues of a matrix M , involving application of row and column operations that produce the companion matrix of M .

decimal number system The base 10 positional system for representing real numbers. Every real number x has a representation of the form

dn−1dn−2 · · · d1d0 · d−1d−2 · · · , in which the di are the digits of x; d0 and d−1 are separated by the decimal point (.). The digits of x are determined recursively by the formulas dn−1 =

and for k > 1,

dn−k = ⌊ x −∑k−1j=1(dn−j10n−j )

⌋

,

where n is the unique integer that satisfies 10n−1 ≤ x < 10n and t is the floor function (the greatest integer≤ t). For example, ifx = 2383

4 ,

then n = 3, d2 = 2, d1 = 3, d0 = 8, d−1 = 7, d−2 = 5. The only possible digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The digit d0 is called the ones digit, d1 the tens digit, d2 the hundreds digit, d3 the thousands digit, etc.; also, d−1 is the tenths digit, d−2 the hundredths digit, etc. It can be shown that, precisely when x is rational, the sequence of digits of x eventually repeats. That is, there is a smallest integer p for which di−p = di for every i less than some fixed index. Here we call the string of digits di−1di−2 · · · di−p a repeating block and p the period of the representation of x. In the special case that the repeating block is the single digit 0, the convention is to drop all the trailing zeros from the representation and say that x has a finite or terminating decimal expansion. Further, it is possible in this case to give a second, distinct expansion of x: if x has finite decimal expansion with final nonzero digit

di , then another representation of x can be obtained by replacing di with di − 1 and defining di−1 = di−2 = · · · = 9. For example, 238.75 = 238.74999· · · . decomposable operator A bounded linear operator T , on the separable Hilbert space L2

(, μ;H) of square-integrable, measurable, Hvalued functions on some measure space (, μ) where H is also a Hilbert space, so that for each measurable ξ(γ ), the function γ → T (γ )ξ(γ ) is measurable, and so that, for each ξ ∈ L2(, μ;H), T can be represented as the direct integral

T ξ = ∫

⊕T (γ )ξ(γ )dμ(γ ) .