2,110 research outputs found
Maple umbral calculus package
We are developing a Maple package of functions related to Rota's Umbral
Calculus. A Mathematica version of this package is being developed in parallel
Appell polynomials and their relatives II. Boolean theory
The Appell-type polynomial family corresponding to the simplest
non-commutative derivative operator turns out to be connected with the Boolean
probability theory, the simplest of the three universal non-commutative
probability theories (the other two being free and tensor/classical
probability). The basic properties of the Boolean Appell polynomials are
described. In particular, their generating function turns out to have a
resolvent-type form, just like the generating function for the free Sheffer
polynomials. It follows that the Meixner (that is, Sheffer plus orthogonal)
polynomial classes, in the Boolean and free theory, coincide. This is true even
in the multivariate case. A number of applications of this fact are described,
to the Belinschi-Nica and Bercovici-Pata maps, conditional freeness, and the
Laha-Lukacs type characterization.
A number of properties which hold for the Meixner class in the free and
classical cases turn out to hold in general in the Boolean theory. Examples
include the behavior of the Jacobi coefficients under convolution, the
relationship between the Jacobi coefficients and cumulants, and an operator
model for cumulants. Along the way, we obtain a multivariate version of the
Stieltjes continued fraction expansion for the moment generating function of an
arbitrary state with monic orthogonal polynomials
Boundary Algebra: A Simple Notation for Boolean Algebra and the Truth Functors
Boundary algebra [BA] is a simpler notation for Spencer-Brownâs (1969) primary algebra [pa], the Boolean algebra 2, and the truth functors. The primary arithmetic [PA] consists of the atoms â()â and the blank page, concatenation, and enclosure between â(â and â)â, denoting the primitive notion of distinction. Inserting letters denoting the presence or absence of () into a PA formula yields a BA formula. The BA axioms are "()()=()" (A1), and "(()) [=?] may be written or erased at willâ (A2). Repeated application of these axioms to a PA formula yields a member of B= {(),?} called its simplification. (a) has two intended interpretations: (a) ? a? (Boolean algebra 2), and (a) ? ~a (sentential logic). BA is self-dual: () ? 1 [dually 0] so that B is the carrier for 2, ab ? a?b [a?b], and (a)b [(a(b))] ? a=b, so that ?=() [()=?] follows trivially and B is a poset. The BA basis abc= bca (Dilworth 1938), a(ab)= a(b), and a()=() (Bricken 2002) facilitates clausal reasoning and proof by calculation. BA also simplifies normal forms and Quineâs (1982) truth value analysis. () ? true [false] yields boundary logic.G. Spencer Brown; boundary algebra; boundary logic; primary algebra; primary arithmetic; Boolean algebra; calculation proof; C.S. Peirce; existential graphs.
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