3,807 research outputs found
Disjoint-union partial algebras
Disjoint union is a partial binary operation returning the union of two sets
if they are disjoint and undefined otherwise. A disjoint-union partial algebra
of sets is a collection of sets closed under disjoint unions, whenever they are
defined. We provide a recursive first-order axiomatisation of the class of
partial algebras isomorphic to a disjoint-union partial algebra of sets but
prove that no finite axiomatisation exists. We do the same for other signatures
including one or both of disjoint union and subset complement, another partial
binary operation we define.
Domain-disjoint union is a partial binary operation on partial functions,
returning the union if the arguments have disjoint domains and undefined
otherwise. For each signature including one or both of domain-disjoint union
and subset complement and optionally including composition, we consider the
class of partial algebras isomorphic to a collection of partial functions
closed under the operations. Again the classes prove to be axiomatisable, but
not finitely axiomatisable, in first-order logic.
We define the notion of pairwise combinability. For each of the previously
considered signatures, we examine the class isomorphic to a partial algebra of
sets/partial functions under an isomorphism mapping arbitrary suprema of
pairwise combinable sets to the corresponding disjoint unions. We prove that
for each case the class is not closed under elementary equivalence.
However, when intersection is added to any of the signatures considered, the
isomorphism class of the partial algebras of sets is finitely axiomatisable and
in each case we give such an axiomatisation.Comment: 30 page
Nominal Abstraction
Recursive relational specifications are commonly used to describe the
computational structure of formal systems. Recent research in proof theory has
identified two features that facilitate direct, logic-based reasoning about
such descriptions: the interpretation of atomic judgments through recursive
definitions and an encoding of binding constructs via generic judgments.
However, logics encompassing these two features do not currently allow for the
definition of relations that embody dynamic aspects related to binding, a
capability needed in many reasoning tasks. We propose a new relation between
terms called nominal abstraction as a means for overcoming this deficiency. We
incorporate nominal abstraction into a rich logic also including definitions,
generic quantification, induction, and co-induction that we then prove to be
consistent. We present examples to show that this logic can provide elegant
treatments of binding contexts that appear in many proofs, such as those
establishing properties of typing calculi and of arbitrarily cascading
substitutions that play a role in reducibility arguments.Comment: To appear in the Journal of Information and Computatio
Enumerating five families of pattern-avoiding inversion sequences; and introducing the powered Catalan numbers
The first problem addressed by this article is the enumeration of some
families of pattern-avoiding inversion sequences. We solve some enumerative
conjectures left open by the foundational work on the topics by Corteel et al.,
some of these being also solved independently by Lin, and Kim and Lin. The
strength of our approach is its robustness: we enumerate four families of pattern-avoiding inversion sequences
ordered by inclusion using the same approach. More precisely, we provide a
generating tree (with associated succession rule) for each family which
generalizes the one for the family .
The second topic of the paper is the enumeration of a fifth family of
pattern-avoiding inversion sequences (containing ). This enumeration is
also solved \emph{via} a succession rule, which however does not generalize the
one for . The associated enumeration sequence, which we call the
\emph{powered Catalan numbers}, is quite intriguing, and further investigated.
We provide two different succession rules for it, denoted and
, and show that they define two types of families enumerated
by powered Catalan numbers. Among such families, we introduce the \emph{steady
paths}, which are naturally associated with . They allow us to
bridge the gap between the two types of families enumerated by powered Catalan
numbers: indeed, we provide a size-preserving bijection between steady paths
and valley-marked Dyck paths (which are naturally associated with
).
Along the way, we provide several nice connections to families of
permutations defined by the avoidance of vincular patterns, and some
enumerative conjectures.Comment: V2 includes modifications suggested by referees (in particular, a
much shorter Section 3, to account for arXiv:1706.07213
Crossings and nestings in colored set partitions
Chen, Deng, Du, Stanley, and Yan introduced the notion of -crossings and
-nestings for set partitions, and proved that the sizes of the largest
-crossings and -nestings in the partitions of an -set possess a
symmetric joint distribution. This work considers a generalization of these
results to set partitions whose arcs are labeled by an -element set (which
we call \emph{-colored set partitions}). In this context, a -crossing or
-nesting is a sequence of arcs, all with the same color, which form a
-crossing or -nesting in the usual sense. After showing that the sizes of
the largest crossings and nestings in colored set partitions likewise have a
symmetric joint distribution, we consider several related enumeration problems.
We prove that -colored set partitions with no crossing arcs of the same
color are in bijection with certain paths in \NN^r, generalizing the
correspondence between noncrossing (uncolored) set partitions and 2-Motzkin
paths. Combining this with recent work of Bousquet-M\'elou and Mishna affords a
proof that the sequence counting noncrossing 2-colored set partitions is
P-recursive. We also discuss how our methods extend to several variations of
colored set partitions with analogous notions of crossings and nestings.Comment: 25 pages; v2: material revised and condensed; v3 material further
revised, additional section adde
The Structure of n-Point One-Loop Open Superstring Amplitudes
In this article we present the worldsheet integrand for one-loop amplitudes
in maximally supersymmetric superstring theory involving any number n of
massless open string states. The polarization dependence is organized into the
same BRST invariant kinematic combinations which also govern the leading string
correction to tree level amplitudes. The dimensions of the bases for both the
kinematics and the associated worldsheet integrals is found to be the unsigned
Stirling number S_3^{n-1} of first kind. We explain why the same combinatorial
structures govern on the one hand finite one-loop amplitudes of equal helicity
states in pure Yang Mills theory and on the other hand the color tensors at
quadratic alpha prime order of the color dressed tree amplitude.Comment: 75 pp, 8 figs, harvmac TeX, v2: published versio
Pattern avoidance for alternating permutations and reading words of tableaux
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student submitted PDF version of thesis.Includes bibliographical references (p. 67-69).We consider a variety of questions related to pattern avoidance in alternating permutations and generalizations thereof. We give bijective enumerations of alternating permutations avoiding patterns of length 3 and 4, of permutations that are the reading words of a "thickened staircase" shape (or equivalently of permutations with descent set {k, 2k, 3k, . . .}) avoiding a monotone pattern, and of the reading words of Young tableaux of any skew shape avoiding any of the patterns 132, 213, 312, or 231. Our bijections include a simple bijection involving binary trees, variations on the Robinson-Schensted-Knuth correspondence, and recursive bijections established via isomorphisms of generating trees.by Joel Brewster Lewis.Ph.D
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