9,170 research outputs found
Properties of Two-Dimensional Words
Combinatorics on words in one dimension is a well-studied subfield of theoretical computer science with its origins in the early 20th century. However, the closely-related study of two-dimensional words is not as popular, even though many results seem naturally extendable from the one-dimensional case. This thesis investigates various properties of these two-dimensional words.
In the early 1960s, Roger Lyndon and Marcel-Paul Schutzenberger developed two famous results on conditions where nontrivial prefixes and suffixes of a one-dimensional word are identical and on conditions where two one-dimensional words commute. Here, the theorems of Lyndon and Schutzenberger are extended in the one-dimensional case to include a number of additional equivalent conditions. One such condition is shown to be equivalent to the defect theorem from formal languages and coding theory. The same theorems of Lyndon and Schutzenberger are then generalized to the two-dimensional case.
The study of two-dimensional words continues by considering primitivity and periodicity in two dimensions, where a method is developed to enumerate two-dimensional primitive words. An efficient computer algorithm is presented to assist with checking the property of primitivity in two dimensions. Finally, borders in both one and two dimensions are considered, with some results being proved and others being offered as suggestions for future work. Another efficient algorithm is presented to assist with checking whether a two-dimensional word is bordered.
The thesis concludes with a selection of open problems and an appendix containing extensive data related to one such open problem
Properties of Two-Dimensional Words
Combinatorics on words in one dimension is a well-studied subfield of theoretical computer science with its origins in the early 20th century. However, the closely-related study of two-dimensional words is not as popular, even though many results seem naturally extendable from the one-dimensional case. This thesis investigates various properties of these two-dimensional words.
In the early 1960s, Roger Lyndon and Marcel-Paul Schutzenberger developed two famous results on conditions where nontrivial prefixes and suffixes of a one-dimensional word are identical and on conditions where two one-dimensional words commute. Here, the theorems of Lyndon and Schutzenberger are extended in the one-dimensional case to include a number of additional equivalent conditions. One such condition is shown to be equivalent to the defect theorem from formal languages and coding theory. The same theorems of Lyndon and Schutzenberger are then generalized to the two-dimensional case.
The study of two-dimensional words continues by considering primitivity and periodicity in two dimensions, where a method is developed to enumerate two-dimensional primitive words. An efficient computer algorithm is presented to assist with checking the property of primitivity in two dimensions. Finally, borders in both one and two dimensions are considered, with some results being proved and others being offered as suggestions for future work. Another efficient algorithm is presented to assist with checking whether a two-dimensional word is bordered.
The thesis concludes with a selection of open problems and an appendix containing extensive data related to one such open problem
On the presentation of pointed Hopf algebras
We give a presentation in terms of generators and relations of Hopf algebras
generated by skew-primitive elements and abelian group of group-like elements
with action given via characters. This class of pointed Hopf algebras has shown
great importance in the classification theory and can be seen as generalized
quantum groups. As a consequence we get an analog presentation of Nichols
algebras of diagonal type
The algebra of cell-zeta values
In this paper, we introduce cell-forms on , which are
top-dimensional differential forms diverging along the boundary of exactly one
cell (connected component) of the real moduli space
. We show that the cell-forms generate the
top-dimensional cohomology group of , so that there is a
natural duality between cells and cell-forms. In the heart of the paper, we
determine an explicit basis for the subspace of differential forms which
converge along a given cell . The elements of this basis are called
insertion forms, their integrals over are real numbers, called cell-zeta
values, which generate a -algebra called the cell-zeta algebra. By
a result of F. Brown, the cell-zeta algebra is equal to the algebra of
multizeta values. The cell-zeta values satisfy a family of simple quadratic
relations coming from the geometry of moduli spaces, which leads to a natural
definition of a formal version of the cell-zeta algebra, conjecturally
isomorphic to the formal multizeta algebra defined by the much-studied double
shuffle relations
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