1,085,746 research outputs found
Nonautonomous saddle-node bifurcations: random and deterministic forcing
We study the effect of external forcing on the saddle-node bifurcation
pattern of interval maps. By replacing fixed points of unperturbed maps by
invariant graphs, we obtain direct analogues to the classical result both for
random forcing by measure-preserving dynamical systems and for deterministic
forcing by homeomorphisms of compact metric spaces. Additional assumptions like
ergodicity or minimality of the forcing process then yield further information
about the dynamics. The main difference to the unforced situation is that at
the critical bifurcation parameter, two alternatives exist. In addition to the
possibility of a unique neutral invariant graph, corresponding to a neutral
fixed point, a pair of so-called pinched invariant graphs may occur. In
quasiperiodically forced systems, these are often referred to as 'strange
non-chaotic attractors'. The results on deterministic forcing can be considered
as an extension of the work of Novo, Nunez, Obaya and Sanz on nonautonomous
convex scalar differential equations. As a by-product, we also give a
generalisation of a result by Sturman and Stark on the structure of minimal
sets in forced systems.Comment: 17 pages, 5 figure
Unrestricted State Complexity of Binary Operations on Regular and Ideal Languages
We study the state complexity of binary operations on regular languages over
different alphabets. It is known that if and are languages of
state complexities and , respectively, and restricted to the same
alphabet, the state complexity of any binary boolean operation on and
is , and that of product (concatenation) is . In
contrast to this, we show that if and are over different
alphabets, the state complexity of union and symmetric difference is
, that of difference is , that of intersection is , and
that of product is . We also study unrestricted complexity of
binary operations in the classes of regular right, left, and two-sided ideals,
and derive tight upper bounds. The bounds for product of the unrestricted cases
(with the bounds for the restricted cases in parentheses) are as follows: right
ideals (); left ideals ();
two-sided ideals (). The state complexities of boolean operations
on all three types of ideals are the same as those of arbitrary regular
languages, whereas that is not the case if the alphabets of the arguments are
the same. Finally, we update the known results about most complex regular,
right-ideal, left-ideal, and two-sided-ideal languages to include the
unrestricted cases.Comment: 30 pages, 15 figures. This paper is a revised and expanded version of
the DCFS 2016 conference paper, also posted previously as arXiv:1602.01387v3.
The expanded version has appeared in J. Autom. Lang. Comb. 22 (1-3), 29-59,
2017, the issue of selected papers from DCFS 2016. This version corrects the
proof of distinguishability of states in the difference operation on p. 12 in
arXiv:1609.04439v
High-rate self-synchronizing codes
Self-synchronization under the presence of additive noise can be achieved by
allocating a certain number of bits of each codeword as markers for
synchronization. Difference systems of sets are combinatorial designs which
specify the positions of synchronization markers in codewords in such a way
that the resulting error-tolerant self-synchronizing codes may be realized as
cosets of linear codes. Ideally, difference systems of sets should sacrifice as
few bits as possible for a given code length, alphabet size, and
error-tolerance capability. However, it seems difficult to attain optimality
with respect to known bounds when the noise level is relatively low. In fact,
the majority of known optimal difference systems of sets are for exceptionally
noisy channels, requiring a substantial amount of bits for synchronization. To
address this problem, we present constructions for difference systems of sets
that allow for higher information rates while sacrificing optimality to only a
small extent. Our constructions utilize optimal difference systems of sets as
ingredients and, when applied carefully, generate asymptotically optimal ones
with higher information rates. We also give direct constructions for optimal
difference systems of sets with high information rates and error-tolerance that
generate binary and ternary self-synchronizing codes.Comment: 9 pages, no figure, 2 tables. Final accepted version for publication
in the IEEE Transactions on Information Theory. Material presented in part at
the International Symposium on Information Theory and its Applications,
Honolulu, HI USA, October 201
Bipolar picture fuzzy sets and relations with applications
The notions of both the bipolar fuzzy sets and picture fuzzy sets have been studied by many authors, the bipolar picture
fuzzy set is the nice combination of these two notions. Basically, the concepts we present in our study are the direct extensions of
both the bipolar fuzzy sets and picture fuzzy sets. In this study, we add few more operations and results in the theory of the bipolar
picture fuzzy sets. We also initiate the notion of bipolar picture fuzzy relations along with their applications. We present numerous
basic operations along with the algebraic sums, bounded sums, algebraic product, bounded difference on bipolar picture fuzzy sets.
Different types of distances between two bipolar picture fuzzy sets are also addressed. We provide the application of bipolar picture
fuzzy sets towards decision making theory along with its algorithm. Afterward, we introduce different types of bipolar picture
fuzzy relations like bipolar picture fuzzy reflexive, symmetric and transitive relations. Subsequently, we introduce the concepts of
the bipolar picture fuzzy equivalence relation and partition. We also produce numerous interesting results based on these relations.
Finally, we establish the criteria for the detection of covid-19 at the base of bipolar picture fuzzy relations
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