94,496 research outputs found
Counting Proper Mergings of Chains and Antichains
A proper merging of two disjoint quasi-ordered sets and is a
quasi-order on the union of and such that the restriction to and
yields the original quasi-order again and such that no elements of and
are identified. In this article, we consider the cases where and
are chains, where and are antichains, and where is an antichain and
is a chain. We give formulas that determine the number of proper mergings
in all three cases, and introduce two new bijections from proper mergings of
two chains to plane partitions and from proper mergings of an antichain and a
chain to monotone colorings of complete bipartite digraphs. Additionally, we
use these bijections to count the Galois connections between two chains, and
between a chain and a Boolean lattice respectively.Comment: 36 pages, 15 figures, 5 table
Mergeable Dictionaries
A data structure is presented for the Mergeable Dictionary
abstract data type, which supports the following operations on
a collection of disjoint sets of totally ordered data: Predecessor-Search, Split and Union. While Predecessor-Search and Split
work in the normal way, the novel operation is Union. While in
a typical mergeable dictionary (e.g. 2-4 Trees), the Union operation can only be performed on sets that span disjoint intervals in
keyspace, the structure here has no such limitation, and permits
the merging of arbitrarily interleaved sets. Our data structure
supports all operations, including Union, in O(log n) amortized
time, thus showing that interleaved Union operations can be supported at no additional cost vis-a-vis disjoint Union operations
Complexity of union-split-find problems
Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 45-46).In this thesis, we investigate various interpretations of the Union-Split-Find problem, an extension of the classic Union-Find problem. In the Union-Split Find problem, we maintain disjoint sets of ordered elements subject to the operations of constructing singleton sets, merging two sets together, splitting a set by partitioning it around a specified value, and finding the set that contains a given element. The different interpretations of this problem arise from the different assumptions made regarding when sets can be merged and any special properties the sets may have. We define and analyze the Interval, Cyclic, Ordered, and General Union-Split-Find problems. Previous work implies optimal solutions to the Interval and Ordered Union-Split-Find problems and an (log n/ log log n) lower bound for the Cyclic Union-Split-Find problem in the cell-probe model. We present a new data structure that achieves a matching upper bound of (log n/ log log n) for Cyclic Union-Split Find in the word RAM model. For General Union-Split-Find, no o(n) bound is known. We present a data structure which has an [Omega](log2 n) amortized lower bound in the worst case that we conjecture has polylogarithmic amortized performance. This thesis is the product of joint work with Erik Demaine.by Katherine Jane Lai.M.Eng
Proper Mergings of Stars and Chains are Counted by Sums of Antidiagonals in Certain Convolution Arrays -- The Details
A proper merging of two disjoint quasi-ordered sets and is a
quasi-order on the union of and such that the restriction to or
yields the original quasi-order again and such that no elements of and
are identified. In this article, we determine the number of proper mergings in
the case where is a star (i.e. an antichain with a smallest element
adjoined), and is a chain. We show that the lattice of proper mergings of
an -antichain and an -chain, previously investigated by the author, is a
quotient lattice of the lattice of proper mergings of an -star and an
-chain, and we determine the number of proper mergings of an -star and an
-chain by counting the number of congruence classes and by determining their
cardinalities. Additionally, we compute the number of Galois connections
between certain modified Boolean lattices and chains.Comment: 27 pages, 7 figures, 1 table. Jonathan Farley has solved Problem
4.18; added Section 4.4 to describe his solutio
Branch merging on continuum trees with applications to regenerative tree growth
We introduce a family of branch merging operations on continuum trees and
show that Ford CRTs are distributionally invariant. This operation is new even
in the special case of the Brownian CRT, which we explore in more detail. The
operations are based on spinal decompositions and a regenerativity preserving
merging procedure of -strings of beads, that is, random
intervals equipped with a random discrete measure
arising in the limit of ordered -Chinese restaurant
processes as introduced recently by Pitman and Winkel. Indeed, we iterate the
branch merging operation recursively and give an alternative approach to the
leaf embedding problem on Ford CRTs related to -regenerative tree growth processes.Comment: 40 pages, 5 figure
Matching Tree-Level Matrix Elements with Interleaved Showers
We present an implementation of the so-called CKKW-L merging scheme for
combining multi-jet tree-level matrix elements with parton showers. The
implementation uses the transverse-momentum-ordered shower with interleaved
multiple interactions as implemented in PYTHIA8. We validate our procedure
using e+e--annihilation into jets and vector boson production in hadronic
collisions, with special attention to details in the algorithm which are
formally sub-leading in character, but may have visible effects in some
observables. We find substantial merging scale dependencies induced by the
enforced rapidity ordering in the default PYTHIA8 shower. If this rapidity
ordering is removed the merging scale dependence is almost negligible. We then
also find that the shower does a surprisingly good job of describing the
hardness of multi-jet events, as long as the hardest couple of jets are given
by the matrix elements. The effects of using interleaved multiple interactions
as compared to more simplistic ways of adding underlying-event effects in
vector boson production are shown to be negligible except in a few sensitive
observables. To illustrate the generality of our implementation, we also give
some example results from di-boson production and pure QCD jet production in
hadronic collisions.Comment: 44 pages, 23 figures, as published in JHEP, including all changes
recommended by the refere
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