279 research outputs found
On the Coloring of Pseudoknots
Pseudodiagrams are diagrams of knots where some information about which
strand goes over/under at certain crossings may be missing. Pseudoknots are
equivalence classes of pseudodiagrams, with equivalence defined by a class of
Reidemeister-type moves. In this paper, we introduce two natural extensions of
classical knot colorability to this broader class of knot-like objects. We use
these definitions to define the determinant of a pseudoknot (i.e. the
pseudodeterminant) that agrees with the classical determinant for classical
knots. Moreover, we extend Conway notation to pseudoknots to facilitate the
investigation of families of pseudoknots and links. The general formulae for
pseudodeterminants of pseudoknot families may then be used as a criterion for
p-colorability of pseudoknots.Comment: 22 pages, 24 figure
A bijection between unicellular and bicellular maps
In this paper we present a combinatorial proof of a relation between the
generating functions of unicellular and bicellular maps. This relation is a
consequence of the Schwinger-Dyson equation of matrix theory. Alternatively it
can be proved using representation theory of the symmetric group. Here we give
a bijective proof by rewiring unicellular maps of topological genus
into bicellular maps of genus and pairs of unicellular maps of lower
topological genera. Our result has immediate consequences for the folding of
RNA interaction structures, since the time complexity of folding the
transformed structure is , where are the lengths of the
respective backbones, while the folding of the original structure has
time complexity, where is the length of the longer sequence.Comment: 18 pages, 13 figure
Prediction of RNA pseudoknots by Monte Carlo simulations
In this paper we consider the problem of RNA folding with pseudoknots. We use
a graphical representation in which the secondary structures are described by
planar diagrams. Pseudoknots are identified as non-planar diagrams. We analyze
the non-planar topologies of RNA structures and propose a classification of RNA
pseudoknots according to the minimal genus of the surface on which the RNA
structure can be embedded. This classification provides a simple and natural
way to tackle the problem of RNA folding prediction in presence of pseudoknots.
Based on that approach, we describe a Monte Carlo algorithm for the prediction
of pseudoknots in an RNA molecule.Comment: 22 pages, 14 figure
Psybrackets, Pseudoknots and Singular Knots
We introduce algebraic structures known as psybrackets and use them to define
invariants of pseudoknots and singular knots and links. Psybrackets are
Niebrzydowski tribrackets with additional structure inspired by the
Reidemeister moves for pseudoknots and singular knots. Examples and
computations are provided.Comment: 12 page
Benchmarking Implementations of Functional Languages with ``Pseudoknot'', a Float-Intensive Benchmark
Over 25 implementations of different functional languages are benchmarked using the same program, a floatingpoint intensive application taken from molecular biology. The principal aspects studied are compile time and execution time for the various implementations that were benchmarked. An important consideration is how the program can be modified and tuned to obtain maximal performance on each language implementation.\ud
With few exceptions, the compilers take a significant amount of time to compile this program, though most compilers were faster than the then current GNU C compiler (GCC version 2.5.8). Compilers that generate C or Lisp are often slower than those that generate native code directly: the cost of compiling the intermediate form is normally a large fraction of the total compilation time.\ud
There is no clear distinction between the runtime performance of eager and lazy implementations when appropriate annotations are used: lazy implementations have clearly come of age when it comes to implementing largely strict applications, such as the Pseudoknot program. The speed of C can be approached by some implemtations, but to achieve this performance, special measures such as strictness annotations are required by non-strict implementations.\ud
The benchmark results have to be interpreted with care. Firstly, a benchmark based on a single program cannot cover a wide spectrum of 'typical' applications.j Secondly, the compilers vary in the kind and level of optimisations offered, so the effort required to obtain an optimal version of the program is similarly varied
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