118 research outputs found
Intermediate arithmetic operations on ordinal numbers
There are two well-known ways of doing arithmetic with ordinal numbers: the
"ordinary" addition, multiplication, and exponentiation, which are defined by
transfinite iteration; and the "natural" (or Hessenberg) addition and
multiplication (denoted and ), each satisfying its own set of
algebraic laws. In 1909, Jacobsthal considered a third, intermediate way of
multiplying ordinals (denoted ), defined by transfinite iteration of
natural addition, as well as the notion of exponentiation defined by
transfinite iteration of his multiplication, which we denote
. (Jacobsthal's multiplication was later rediscovered by
Conway.) Jacobsthal showed these operations too obeyed algebraic laws. In this
paper, we pick up where Jacobsthal left off by considering the notion of
exponentiation obtained by transfinitely iterating natural multiplication
instead; we will denote this . We show that
and that
;
note the use of Jacobsthal's multiplication in the latter. We also demonstrate
the impossibility of defining a "natural exponentiation" satisfying reasonable
algebraic laws.Comment: 18 pages, 3 table
Determinant Formulas of Some Hessenberg Matrices with Jacobsthal Entries
In this paper, we evaluate determinants of several families of Hessenberg matrices having various subsequences of the Jacobsthal sequence as their nonzero entries. These identities may be written equivalently as formulas for certain linearly recurrent sequences expressed in terms of sums of products of Jacobsthal numbers with multinomial coefficients. Among the sequences that arise in this way include the Mersenne, Lucas and Jacobsthal-Lucas numbers as well as the squares of the Jacobsthal and Mersenne sequences. These results are extended to Hessenberg determinants involving sequences that are derived from two general families of linear second-order recurrences. Finally, combinatorial proofs are provided for several of our determinant results which make use of various correspondences between Jacobsthal tilings and certain restricted classes of binary words
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