1,012 research outputs found
Canonical Representatives of Morphic Permutations
An infinite permutation can be defined as a linear ordering of the set of
natural numbers. In particular, an infinite permutation can be constructed with
an aperiodic infinite word over as the lexicographic order
of the shifts of the word. In this paper, we discuss the question if an
infinite permutation defined this way admits a canonical representative, that
is, can be defined by a sequence of numbers from [0, 1], such that the
frequency of its elements in any interval is equal to the length of that
interval. We show that a canonical representative exists if and only if the
word is uniquely ergodic, and that is why we use the term ergodic permutations.
We also discuss ways to construct the canonical representative of a permutation
defined by a morphic word and generalize the construction of Makarov, 2009, for
the Thue-Morse permutation to a wider class of infinite words.Comment: Springer. WORDS 2015, Sep 2015, Kiel, Germany. Combinatorics on
Words: 10th International Conference. arXiv admin note: text overlap with
arXiv:1503.0618
Minimal complexity of equidistributed infinite permutations
An infinite permutation is a linear ordering of the set of natural numbers.
An infinite permutation can be defined by a sequence of real numbers where only
the order of elements is taken into account. In the paper we investigate a new
class of {\it equidistributed} infinite permutations, that is, infinite
permutations which can be defined by equidistributed sequences. Similarly to
infinite words, a complexity of an infinite permutation is defined as a
function counting the number of its subpermutations of length . For infinite
words, a classical result of Morse and Hedlund, 1938, states that if the
complexity of an infinite word satisfies for some , then the
word is ultimately periodic. Hence minimal complexity of aperiodic words is
equal to , and words with such complexity are called Sturmian. For
infinite permutations this does not hold: There exist aperiodic permutations
with complexity functions growing arbitrarily slowly, and hence there are no
permutations of minimal complexity. We show that, unlike for permutations in
general, the minimal complexity of an equidistributed permutation is
. The class of equidistributed permutations of minimal
complexity coincides with the class of so-called Sturmian permutations,
directly related to Sturmian words.Comment: An old (weaker) version of the paper was presented at DLT 2015. The
current version is submitted to a journa
Graphettes: Constant-time determination of graphlet and orbit identity including (possibly disconnected) graphlets up to size 8.
Graphlets are small connected induced subgraphs of a larger graph G. Graphlets are now commonly used to quantify local and global topology of networks in the field. Methods exist to exhaustively enumerate all graphlets (and their orbits) in large networks as efficiently as possible using orbit counting equations. However, the number of graphlets in G is exponential in both the number of nodes and edges in G. Enumerating them all is already unacceptably expensive on existing large networks, and the problem will only get worse as networks continue to grow in size and density. Here we introduce an efficient method designed to aid statistical sampling of graphlets up to size k = 8 from a large network. We define graphettes as the generalization of graphlets allowing for disconnected graphlets. Given a particular (undirected) graphette g, we introduce the idea of the canonical graphette [Formula: see text] as a representative member of the isomorphism group Iso(g) of g. We compute the mapping [Formula: see text], in the form of a lookup table, from all 2k(k - 1)/2 undirected graphettes g of size k ≤ 8 to their canonical representatives [Formula: see text], as well as the permutation that transforms g to [Formula: see text]. We also compute all automorphism orbits for each canonical graphette. Thus, given any k ≤ 8 nodes in a graph G, we can in constant time infer which graphette it is, as well as which orbit each of the k nodes belongs to. Sampling a large number N of such k-sets of nodes provides an approximation of both the distribution of graphlets and orbits across G, and the orbit degree vector at each node
Morphic words and equidistributed sequences
The problem we consider is the following: Given an infinite word on an
ordered alphabet, construct the sequence , equidistributed on
and such that if and only if ,
where is the shift operation, erasing the first symbol of . The
sequence exists and is unique for every word with well-defined positive
uniform frequencies of every factor, or, in dynamical terms, for every element
of a uniquely ergodic subshift. In this paper we describe the construction of
for the case when the subshift of is generated by a morphism of a
special kind; then we overcome some technical difficulties to extend the result
to all binary morphisms. The sequence in this case is also constructed
with a morphism.
At last, we introduce a software tool which, given a binary morphism
, computes the morphism on extended intervals and first elements of
the equidistributed sequences associated with fixed points of
On non-abelian homomorphic public-key cryptosystems
An important problem of modern cryptography concerns secret public-key
computations in algebraic structures. We construct homomorphic cryptosystems
being (secret) epimorphisms f:G --> H, where G, H are (publically known) groups
and H is finite. A letter of a message to be encrypted is an element h element
of H, while its encryption g element of G is such that f(g)=h. A homomorphic
cryptosystem allows one to perform computations (operating in a group G) with
encrypted information (without knowing the original message over H).
In this paper certain homomorphic cryptosystems are constructed for the first
time for non-abelian groups H (earlier, homomorphic cryptosystems were known
only in the Abelian case). In fact, we present such a system for any solvable
(fixed) group H.Comment: 15 pages, LaTe
Commutative combinatorial Hopf algebras
We propose several constructions of commutative or cocommutative Hopf
algebras based on various combinatorial structures, and investigate the
relations between them. A commutative Hopf algebra of permutations is obtained
by a general construction based on graphs, and its non-commutative dual is
realized in three different ways, in particular as the Grossman-Larson algebra
of heap ordered trees.
Extensions to endofunctions, parking functions, set compositions, set
partitions, planar binary trees and rooted forests are discussed. Finally, we
introduce one-parameter families interpolating between different structures
constructed on the same combinatorial objects.Comment: 29 pages, LaTEX; expanded and updated version of math.CO/050245
Determinantal Characterization of Canonical Curves and Combinatorial Theta Identities
We characterize genus g canonical curves by the vanishing of combinatorial
products of g+1 determinants of Brill-Noether matrices. This also implies the
characterization of canonical curves in terms of (g-2)(g-3)/2 theta identities.
A remarkable mechanism, based on a basis of H^0(K_C) expressed in terms of
Szego kernels, reduces such identities to a simple rank condition for matrices
whose entries are logarithmic derivatives of theta functions. Such a basis,
together with the Fay trisecant identity, also leads to the solution of the
question of expressing the determinant of Brill-Noether matrices in terms of
theta functions, without using the problematic Klein-Fay section sigma.Comment: 35 pages. New results, presentation improved, clarifications added.
Accepted for publication in Math. An
- …