1,357 research outputs found
Error Graphs and the Reconstruction of Elements in Groups
Packing and covering problems for metric spaces, and graphs in particular,
are of essential interest in combinatorics and coding theory. They are
formulated in terms of metric balls of vertices. We consider a new problem in
graph theory which is also based on the consideration of metric balls of
vertices, but which is distinct from the traditional packing and covering
problems. This problem is motivated by applications in information transmission
when redundancy of messages is not sufficient for their exact reconstruction,
and applications in computational biology when one wishes to restore an
evolutionary process. It can be defined as the reconstruction, or
identification, of an unknown vertex in a given graph from a minimal number of
vertices (erroneous or distorted patterns) in a metric ball of a given radius r
around the unknown vertex. For this problem it is required to find minimum
restrictions for such a reconstruction to be possible and also to find
efficient reconstruction algorithms under such minimal restrictions.
In this paper we define error graphs and investigate their basic properties.
A particular class of error graphs occurs when the vertices of the graph are
the elements of a group, and when the path metric is determined by a suitable
set of group elements. These are the undirected Cayley graphs. Of particular
interest is the transposition Cayley graph on the symmetric group which occurs
in connection with the analysis of transpositional mutations in molecular
biology. We obtain a complete solution of the above problems for the
transposition Cayley graph on the symmetric group.Comment: Journal of Combinatorial Theory A 200
Spectral approach to linear programming bounds on codes
We give new proofs of asymptotic upper bounds of coding theory obtained
within the frame of Delsarte's linear programming method. The proofs rely on
the analysis of eigenvectors of some finite-dimensional operators related to
orthogonal polynomials. The examples of the method considered in the paper
include binary codes, binary constant-weight codes, spherical codes, and codes
in the projective spaces.Comment: 11 pages, submitte
Reconstruction of permutations distorted by single transposition errors
The reconstruction problem for permutations on elements from their
erroneous patterns which are distorted by transpositions is presented in this
paper. It is shown that for any an unknown permutation is uniquely
reconstructible from 4 distinct permutations at transposition distance at most
one from the unknown permutation. The {\it transposition distance} between two
permutations is defined as the least number of transpositions needed to
transform one into the other. The proposed approach is based on the
investigation of structural properties of a corresponding Cayley graph. In the
case of at most two transposition errors it is shown that
erroneous patterns are required in order to reconstruct an unknown permutation.
Similar results are obtained for two particular cases when permutations are
distorted by given transpositions. These results confirm some bounds for
regular graphs which are also presented in this paper.Comment: 5 pages, Report of paper presented at ISIT-200
Growth Estimates for the Numerical Range of Holomorphic Mappings and Applications
The numerical range of holomorphic mappings arises in many aspects of
nonlinear analysis, finite and infinite dimensional holomorphy, and complex
dynamical systems. In particular, this notion plays a crucial role in
establishing exponential and product formulas for semigroups of holomorphic
mappings, the study of flow invariance and range conditions, geometric function
theory in finite and infinite dimensional Banach spaces, and in the study of
complete and semi-complete vector fields and their applications to starlike and
spirallike mappings, and to Bloch (univalence) radii for locally biholomorphic
mappings.
In the present paper we establish lower and upper bounds for the numerical
range of holomorphic mappings in Banach spaces. In addition, we study and
discuss some geometric and quantitative analytic aspects of fixed point theory,
nonlinear resolvents of holomorphic mappings, Bloch radii, as well as radii of
starlikeness and spirallikeness.Comment: version 2: corrected misprints and simplified proofs in section
Efficient reconstruction of partitions
AbstractWe consider the problem of reconstructing a partition x of the integer n from the set of its t-subpartitions. These are the partitions of the integer n-t obtained by deleting a total of t from the parts of x in all possible ways. It was shown (in a forthcoming paper) that all partitions of n can be reconstructed from t-subpartitions if n is sufficiently large in relation to t. In this paper we deal with efficient reconstruction, in the following sense: if all partitions of n are t--reconstructible, what is the minimum number N=N-(n,t) such that every partition of n can be identified from any N+1 distinct subpartitions? We determine the function N-(n,t) and describe the corresponding algorithm for reconstruction. Superpartitions may be defined in a similar fashion and we determine also the maximum number N+(n,t) of t-superpartitions common to two distinct partitions of n
Finding a Mate With No Social Skills
Sexual reproductive behavior has a necessary social coordination component as
willing and capable partners must both be in the right place at the right time.
While there are many known social behavioral adaptations to support solutions
to this problem, we explore the possibility and likelihood of solutions that
rely only on non-social mechanisms. We find three kinds of social organization
that help solve this social coordination problem (herding, assortative mating,
and natal philopatry) emerge in populations of simulated agents with no social
mechanisms available to support these organizations. We conclude that the
non-social origins of these social organizations around sexual reproduction may
provide the environment for the development of social solutions to the same and
different problems.Comment: 8 pages, 5 figures, GECCO'1
Upper bounds for packings of spheres of several radii
We give theorems that can be used to upper bound the densities of packings of
different spherical caps in the unit sphere and of translates of different
convex bodies in Euclidean space. These theorems extend the linear programming
bounds for packings of spherical caps and of convex bodies through the use of
semidefinite programming. We perform explicit computations, obtaining new
bounds for packings of spherical caps of two different sizes and for binary
sphere packings. We also slightly improve bounds for the classical problem of
packing identical spheres.Comment: 31 page
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