102 research outputs found
Suffix conjugates for a class of morphic subshifts
Let A be a finite alphabet and f: A^* --> A^* be a morphism with an iterative
fixed point f^\omega(\alpha), where \alpha{} is in A. Consider the subshift (X,
T), where X is the shift orbit closure of f^\omega(\alpha) and T: X --> X is
the shift map. Let S be a finite alphabet that is in bijective correspondence
via a mapping c with the set of nonempty suffixes of the images f(a) for a in
A. Let calS be a subset S^N be the set of infinite words s = (s_n)_{n\geq 0}
such that \pi(s):= c(s_0)f(c(s_1)) f^2(c(s_2))... is in X. We show that if f is
primitive and f(A) is a suffix code, then there exists a mapping H: calS -->
calS such that (calS, H) is a topological dynamical system and \pi: (calS, H)
--> (X, T) is a conjugacy; we call (calS, H) the suffix conjugate of (X, T). In
the special case when f is the Fibonacci or the Thue-Morse morphism, we show
that the subshift (calS, T) is sofic, that is, the language of calS is regular
Pattern avoidance: themes and variations
AbstractWe review results concerning words avoiding powers, abelian powers or patterns. In addition we collect/pose a large number of open problems
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
Power Beacon-Assisted Millimeter Wave Ad Hoc Networks
Deployment of low cost power beacons (PBs) is a promising solution for
dedicated wireless power transfer (WPT) in future wireless networks. In this
paper, we present a tractable model for PB-assisted millimeter wave (mmWave)
wireless ad hoc networks, where each transmitter (TX) harvests energy from all
PBs and then uses the harvested energy to transmit information to its desired
receiver. Our model accounts for realistic aspects of WPT and mmWave
transmissions, such as power circuit activation threshold, allowed maximum
harvested power, maximum transmit power, beamforming and blockage. Using
stochastic geometry, we obtain the Laplace transform of the aggregate received
power at the TX to calculate the power coverage probability. We approximate and
discretize the transmit power of each TX into a finite number of discrete power
levels in log scale to compute the channel and total coverage probability. We
compare our analytical predictions to simulations and observe good accuracy.
The proposed model allows insights into effect of system parameters, such as
transmit power of PBs, PB density, main lobe beam-width and power circuit
activation threshold on the overall coverage probability. The results confirm
that it is feasible and safe to power TXs in a mmWave ad hoc network using PBs.Comment: This work has been submitted to the IEEE for possible publication.
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A powerful abelian square-free substitution over 4 letters
AbstractIn 1961, Paul Erdös posed the question whether abelian squares can be avoided in arbitrarily long words over a finite alphabet. An abelian square is a non-empty word uv, where u and v are permutations (anagrams) of each other. The case of the four letter alphabet Σ4={a,b,c,d} turned out to be the most challenging and remained open until 1992 when the author presented an abelian square-free (a-2-free) endomorphism g85 of Σ4∗. The size of this g85, i.e., |g85(abcd)|, is equal to 4×85 (uniform modulus). Until recently, all known methods for constructing arbitrarily long a-2-free words on Σ4 have been based on the structure of g85 and on the endomorphism g98 of Σ4∗ found in 2002.In this paper, a great many new a-2-free endomorphisms of Σ4∗ are reported. The sizes of these endomorphisms range from 4×102 to 4×115. Importantly, twelve of the new a-2-free endomorphisms, of modulus m=109, can be used to construct an a-2-free (commutatively functional) substitution σ109 of Σ4∗ with 12 image words for each letter.The properties of σ109 lead to a considerable improvement for the lower bound of the exponential growth of cn, i.e., of the number of a-2-free words over 4 letters of length n. It is obtained that cn>β−50βn with β=121/m≃1.02306. Originally, in 1998, Carpi established the exponential growth of cn by showing that cn>β−tβn with β=219/t=219/(853−85)≃1.000021, where t=853−85 is the modulus of the substitution that he constructs starting from g85
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