9,041 research outputs found
The enumeration spectrum hierarchy of n-families
© 2016 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimWe introduce a hierarchy of sets which can be derived from the integers using countable collections. Such families can be coded into countable algebraic structures preserving their algorithmic properties. We prove that for different finite levels of the hierarchy the corresponding algebraic structures have different classes of possible degree spectra
The enumeration spectrum hierarchy of α-families and low<inf>α</inf> degrees
© J.UCS.In this paper we introduce a hierarchy of families which can be derived from the integers using countable collections. This hierarchy coincides with the von Neumann hierarchy of hereditary countable sets in the ZFC-theory with urelements from N. The families from the hierarchy can be coded into countable algebraic structures preserving their algorithmic properties. We prove that there is no maximal level of the hierarchy and that the collection of non-lowα degrees for every computable ordinal α is the enumeration spectrum of a family from the hierarchy. In particular, we show that the collection of non-lowα degrees for every computable limit ordinal α is a degree spectrum of some algebraic structure
Multispecies Weighted Hurwitz Numbers
The construction of hypergeometric Toda -functions as generating
functions for weighted Hurwitz numbers is extended to multispecies families.
Both the enumerative geometrical significance of multispecies weighted Hurwitz
numbers, as weighted enumerations of branched coverings of the Riemann sphere,
and their combinatorial significance in terms of weighted paths in the Cayley
graph of are derived. The particular case of multispecies quantum
weighted Hurwitz numbers is studied in detail.Comment: this is substantially enhanced version of arXiv:1410.881
Enumeration of chord diagrams on many intervals and their non-orientable analogs
Two types of connected chord diagrams with chord endpoints lying in a
collection of ordered and oriented real segments are considered here: the real
segments may contain additional bivalent vertices in one model but not in the
other. In the former case, we record in a generating function the number of
fatgraph boundary cycles containing a fixed number of bivalent vertices while
in the latter, we instead record the number of boundary cycles of each fixed
length. Second order, non-linear, algebraic partial differential equations are
derived which are satisfied by these generating functions in each case giving
efficient enumerative schemes. Moreover, these generating functions provide
multi-parameter families of solutions to the KP hierarchy. For each model,
there is furthermore a non-orientable analog, and each such model likewise has
its own associated differential equation. The enumerative problems we solve are
interpreted in terms of certain polygon gluings. As specific applications, we
discuss models of several interacting RNA molecules. We also study a matrix
integral which computes numbers of chord diagrams in both orientable and
non-orientable cases in the model with bivalent vertices, and the large-N limit
is computed using techniques of free probability.Comment: 23 pages, 7 figures; revised and extended versio
On the Minimal Pseudo-Codewords of Codes from Finite Geometries
In order to understand the performance of a code under maximum-likelihood
(ML) decoding, it is crucial to know the minimal codewords. In the context of
linear programming (LP) decoding, it turns out to be necessary to know the
minimal pseudo-codewords. This paper studies the minimal codewords and minimal
pseudo-codewords of some families of codes derived from projective and
Euclidean planes. Although our numerical results are only for codes of very
modest length, they suggest that these code families exhibit an interesting
property. Namely, all minimal pseudo-codewords that are not multiples of a
minimal codeword have an AWGNC pseudo-weight that is strictly larger than the
minimum Hamming weight of the code. This observation has positive consequences
not only for LP decoding but also for iterative decoding.Comment: To appear in Proc. 2005 IEEE International Symposium on Information
Theory, Adelaide, Australia, September 4-9, 200
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