215 research outputs found
Primitive digraphs with large exponents and slowly synchronizing automata
We present several infinite series of synchronizing automata for which the
minimum length of reset words is close to the square of the number of states.
All these automata are tightly related to primitive digraphs with large
exponent.Comment: 23 pages, 11 figures, 3 tables. This is a translation (with a
slightly updated bibliography) of the authors' paper published in Russian in:
Zapiski Nauchnyh Seminarov POMI [Kombinatorika i Teorija Grafov. IV], Vol.
402, 9-39 (2012), see ftp://ftp.pdmi.ras.ru/pub/publicat/znsl/v402/p009.pdf
Version 2: a few typos are correcte
NANOTECHNOLOGY APPLICATION IN COMPOSITE REBAR PRODUCTION
The paper deals with the influence of metal/carbon nanocomposite additives on the physical properties of composite polymer reinforcement samples. Novel methods of composite rebar production have been suggested and tested. The overall positive effect of polymer composite material modification by selected additives has been found and analyzed
On the interplay between Babai and Cerny's conjectures
Motivated by the Babai conjecture and the Cerny conjecture, we study the
reset thresholds of automata with the transition monoid equal to the full
monoid of transformations of the state set. For automata with states in
this class, we prove that the reset thresholds are upper-bounded by
and can attain the value . In addition, we study diameters
of the pair digraphs of permutation automata and construct -state
permutation automata with diameter .Comment: 21 pages version with full proof
On Synchronizing Colorings and the Eigenvectors of Digraphs
An automaton is synchronizing if there exists a word that sends all states of the automaton to a single state. A coloring of a digraph with a fixed out-degree k is a distribution of k labels over the edges resulting in a deterministic finite automaton. The famous road coloring theorem states that every primitive digraph has a synchronizing coloring. We study recent conjectures claiming that the number of synchronizing colorings is large in the worst and average cases.
Our approach is based on the spectral properties of the adjacency matrix A(G) of a digraph G. Namely, we study the relation between the number of synchronizing colorings of G and the structure of the dominant eigenvector v of A(G). We show that a vector v has no partition of coordinates into blocks of equal sum if and only if all colorings of the digraphs associated with v are synchronizing. Furthermore, if for each b there exists at most one partition of the coordinates of v into blocks summing up to b, and the total number of partitions is equal to s, then the fraction of synchronizing colorings among all colorings of G is at least (k-s)/k. We also give a combinatorial interpretation of some known results concerning an upper bound on the minimal length of synchronizing words in terms of v
The k-center Problem for Classes of Cyclic Words
The problem of finding k uniformly spaced points (centres) within a metric space is well known as the k-centre selection problem. In this paper, we introduce the challenge of k-centre selection on a class of objects of exponential size and study it for the class of combinatorial necklaces, known as cyclic words. The interest in words under translational symmetry is motivated by various applications in algebra, coding theory, crystal structures and other physical models with periodic boundary conditions. We provide solutions for the centre selection problem for both one-dimensional necklaces and largely unexplored objects in combinatorics on words - multidimensional combinatorial necklaces. The problem is highly non-trivial as even verifying a solution to the k-centre problem for necklaces can not be done in polynomial time relative to the length of the cyclic words and the alphabet size unless P= NP. Despite this challenge, we develop a technique of centre selection for a class of necklaces based on de-Bruijn Sequences and provide the first polynomial O(k· n) time approximation algorithm for selecting k centres in the set of 1D necklaces of length n over an alphabet of size q with an approximation factor of O(1+logq(k·n)n-logq(k·n)). For the set of multidimensional necklaces of size n1× n2× … × nd we develop an O(k· N2) time algorithm with an approximation factor of O(1+logq(k·N)N-logq(k·N)) in O(k· N2) time, where N= n1· n2· … · nd by approximating de Bruijn hypertori technique
STRUCTURE AND MECHANICAL PROPERTIES OF POLYMERIC COMPOSITES WITH CARBON NANOTUBES
Experimental investigations of single-wall carbon nanotubes (CNT) effect on the mechanical properties of polymeric composite materials based on epoxy matrix have been carried out. It has been found that addition of CNT at low concentration dramatically increases tensile strength (20 – 30 per cent growth) and Young’s modulus of the samples under study. Structure of polymeric composites with CNT was characterized by atomic force microscopy (AFM) and scanning electron microscopy (SEM). AFM images of the samples under study confirm strong interaction between polymeric matrix and nano-additives, demonstrating intimate contact between CNT and epoxy surroundings which is of great importance for composite material reinforcement. Dependences of tensile strength and those of Young’s modulus on CNT concentration are discussed using micromechanics models for nanocomposites
Cluster Exploration using Informative Manifold Projections
Dimensionality reduction (DR) is one of the key tools for the visual
exploration of high-dimensional data and uncovering its cluster structure in
two- or three-dimensional spaces. The vast majority of DR methods in the
literature do not take into account any prior knowledge a practitioner may have
regarding the dataset under consideration. We propose a novel method to
generate informative embeddings which not only factor out the structure
associated with different kinds of prior knowledge but also aim to reveal any
remaining underlying structure. To achieve this, we employ a linear combination
of two objectives: firstly, contrastive PCA that discounts the structure
associated with the prior information, and secondly, kurtosis projection
pursuit which ensures meaningful data separation in the obtained embeddings. We
formulate this task as a manifold optimization problem and validate it
empirically across a variety of datasets considering three distinct types of
prior knowledge. Lastly, we provide an automated framework to perform iterative
visual exploration of high-dimensional data
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