16,966 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
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
Domains via approximation operators
In this paper, we tailor-make new approximation operators inspired by rough
set theory and specially suited for domain theory. Our approximation operators
offer a fresh perspective to existing concepts and results in domain theory,
but also reveal ways to establishing novel domain-theoretic results. For
instance, (1) the well-known interpolation property of the way-below relation
on a continuous poset is equivalent to the idempotence of a certain
set-operator; (2) the continuity of a poset can be characterized by the
coincidence of the Scott closure operator and the upper approximation operator
induced by the way below relation; (3) meet-continuity can be established from
a certain property of the topological closure operator. Additionally, we show
how, to each approximating relation, an associated order-compatible topology
can be defined in such a way that for the case of a continuous poset the
topology associated to the way-below relation is exactly the Scott topology. A
preliminary investigation is carried out on this new topology.Comment: 17 pages; 1figure, Domains XII Worksho
Modeling of Phenomena and Dynamic Logic of Phenomena
Modeling of complex phenomena such as the mind presents tremendous
computational complexity challenges. Modeling field theory (MFT) addresses
these challenges in a non-traditional way. The main idea behind MFT is to match
levels of uncertainty of the model (also, problem or theory) with levels of
uncertainty of the evaluation criterion used to identify that model. When a
model becomes more certain, then the evaluation criterion is adjusted
dynamically to match that change to the model. This process is called the
Dynamic Logic of Phenomena (DLP) for model construction and it mimics processes
of the mind and natural evolution. This paper provides a formal description of
DLP by specifying its syntax, semantics, and reasoning system. We also outline
links between DLP and other logical approaches. Computational complexity issues
that motivate this work are presented using an example of polynomial models
On mining complex sequential data by means of FCA and pattern structures
Nowadays data sets are available in very complex and heterogeneous ways.
Mining of such data collections is essential to support many real-world
applications ranging from healthcare to marketing. In this work, we focus on
the analysis of "complex" sequential data by means of interesting sequential
patterns. We approach the problem using the elegant mathematical framework of
Formal Concept Analysis (FCA) and its extension based on "pattern structures".
Pattern structures are used for mining complex data (such as sequences or
graphs) and are based on a subsumption operation, which in our case is defined
with respect to the partial order on sequences. We show how pattern structures
along with projections (i.e., a data reduction of sequential structures), are
able to enumerate more meaningful patterns and increase the computing
efficiency of the approach. Finally, we show the applicability of the presented
method for discovering and analyzing interesting patient patterns from a French
healthcare data set on cancer. The quantitative and qualitative results (with
annotations and analysis from a physician) are reported in this use case which
is the main motivation for this work.
Keywords: data mining; formal concept analysis; pattern structures;
projections; sequences; sequential data.Comment: An accepted publication in International Journal of General Systems.
The paper is created in the wake of the conference on Concept Lattice and
their Applications (CLA'2013). 27 pages, 9 figures, 3 table
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