scholarly journals Two Effective Properties of ω-Rational Functions

2021 ◽  
pp. 1-20
Author(s):  
Olivier Finkel
Keyword(s):  
Effective Properties ◽  
Rational Functions ◽  
Baire Class ◽  
Infinite Words ◽  
Finite State ◽  
First Baire Class ◽  
Büchi Automaton

We prove two new effective properties of rational functions over infinite words which are realized by finite state Büchi transducers. Firstly, for each such function [Formula: see text], one can construct a deterministic Büchi automaton [Formula: see text] accepting a dense [Formula: see text]-subset of [Formula: see text] such that the restriction of [Formula: see text] to [Formula: see text] is continuous. Secondly, we give a new proof of the decidability of the first Baire class for synchronous [Formula: see text]-rational functions from which we get an extension of this result involving the notion of Wadge classes of regular [Formula: see text]-languages.

Lexicographic products as compact spaces of the first Baire class

2019 ◽  
Vol 267 ◽  
pp. 106871
Author(s):  
Antonio Avilés ◽  
Stevo Todorcevic
Keyword(s):  
Baire Class ◽  
Compact Spaces ◽  
First Baire Class

FINITE STATE PROCESSES, Z-TEMPORAL LOGIC AND THE MONADIC THEORY OF THE INTEGERS

1992 ◽  
Vol 03 (03) ◽  
pp. 233-244 ◽  
Author(s):  
A. SAOUDI ◽  
D.E. MULLER ◽  
P.E. SCHUPP
Keyword(s):  
Temporal Logic ◽  
Linear Temporal Logic ◽  
Second Order ◽  
Regular Languages ◽  
Infinite Words ◽  
First Order ◽  
Monadic Theory ◽  
Finite State ◽  
Temporal Formula

We introduce four classes of Z-regular grammars for generating bi-infinite words (i.e. Z-words) and prove that they generate exactly Z-regular languages. We extend the second order monadic theory of one successor to the set of the integers (i.e. Z) and give some characterizations of this theory in terms of Z-regular grammars and Z-regular languages. We prove that this theory is decidable and equivalent to the weak theory. We also extend the linear temporal logic to Z-temporal logic and then prove that each Z-temporal formula is equivalent to a first order monadic formula. We prove that the correctness problem for finite state processes is decidable.

An invariant of representations of phase-type distributions and some applications

1996 ◽  
Vol 33 (2) ◽  
pp. 368-381 ◽  
Author(s):  
C. Commault ◽  
J. P. Chemla
Keyword(s):  
Markov Chains ◽  
Laplace Transform ◽  
Rational Functions ◽  
Absorbing State ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Finite State ◽  
The Laplace Transform ◽  
The Difference ◽  
Real Poles

In this paper we consider phase-type distributions, their Laplace transforms which are rational functions and their representations which are finite-state Markov chains with an absorbing state. We first prove that, in any representation, the minimal number of states which are visited before absorption is equal to the difference of degree between denominator and numerator in the Laplace transform of the distribution. As an application, we prove that when the Laplace transform has a denominator with n real poles and a numerator of degree less than or equal to one the distribution has order n. We show that, in general, this result can be extended neither to the case where the numerator has degree two nor to the case of non-real poles.

scholarly journals An integral equation method for the homogenization of unidirectional fibre-reinforced media; antiplane elasticity and other potential problems

2017 ◽  
Vol 473 (2201) ◽  
pp. 20170080 ◽  
Author(s):  
Duncan Joyce ◽  
William J. Parnell ◽  
Raphaël C. Assier ◽  
I. David Abrahams
Keyword(s):  
Cross Section ◽  
Volume Fraction ◽  
Effective Properties ◽  
Circular Cross Section ◽  
Rational Functions ◽  
High Volume ◽  
Cross Sectional ◽  
Antiplane Elasticity ◽  
Unidirectional Fibre ◽  
Explicit Expressions

In Parnell & Abrahams (2008 Proc. R. Soc. A 464 , 1461–1482. ( doi:10.1098/rspa.2007.0254 )), a homogenization scheme was developed that gave rise to explicit forms for the effective antiplane shear moduli of a periodic unidirectional fibre-reinforced medium where fibres have non-circular cross section. The explicit expressions are rational functions in the volume fraction. In that scheme, a (non-dilute) approximation was invoked to determine leading-order expressions. Agreement with existing methods was shown to be good except at very high volume fractions. Here, the theory is extended in order to determine higher-order terms in the expansion. Explicit expressions for effective properties can be derived for fibres with non-circular cross section, without recourse to numerical methods. Terms appearing in the expressions are identified as being associated with the lattice geometry of the periodic fibre distribution, fibre cross-sectional shape and host/fibre material properties. Results are derived in the context of antiplane elasticity but the analogy with the potential problem illustrates the broad applicability of the method to, e.g. thermal, electrostatic and magnetostatic problems. The efficacy of the scheme is illustrated by comparison with the well-established method of asymptotic homogenization where for fibres of general cross section, the associated cell problem must be solved by some computational scheme.

Functions of the First Baire Class

1988 ◽  
Vol s2-37 (3) ◽  
pp. 535-544 ◽  
Author(s):  
C. A. Rogers
Keyword(s):  
Baire Class ◽  
First Baire Class

scholarly journals BIDUAL OCTAHEDRAL RENORMINGS AND STRONG REGULARITY IN BANACH SPACES

2019 ◽  
pp. 1-17 ◽  
Author(s):  
Johann Langemets ◽  
Ginés López-Pérez
Keyword(s):  
Banach Space ◽  
Studia Math ◽  
Direct Consequence ◽  
Baire Class ◽  
Dual Norm ◽  
Separable Predual ◽  
Separable Case ◽  
Strongly Regular ◽  
First Baire Class

We prove that every separable Banach space containing an isomorphic copy of $\ell _{1}$ can be equivalently renormed so that the new bidual norm is octahedral. This answers, in the separable case, a question in Godefroy [Metric characterization of first Baire class linear forms and octahedral norms, Studia Math. 95 (1989), 1–15]. As a direct consequence, we obtain that every dual Banach space, with a separable predual and failing to be strongly regular, can be equivalently renormed with a dual norm to satisfy the strong diameter two property.

The complexity of weakly recognizing morphisms

2018 ◽  
Vol 53 (1-2) ◽  
pp. 1-17
Author(s):  
Lukas Fleischer ◽  
Manfred Kufleitner
Keyword(s):  
Computational Complexity ◽  
Regular Languages ◽  
Decision Problems ◽  
Membership Problem ◽  
Surjective Morphism ◽  
Infinite Words ◽  
Descriptional Complexity ◽  
Emptiness Problem ◽  
Free Semigroups ◽  
Büchi Automaton

Weakly recognizing morphisms from free semigroups onto finite semigroups are a classical way for defining the class of ω-regular languages, i.e., a set of infinite words is weakly recognizable by such a morphism if and only if it is accepted by some Büchi automaton. We study the descriptional complexity of various constructions and the computational complexity of various decision problems for weakly recognizing morphisms. The constructions we consider are the conversion from and to Büchi automata, the conversion into strongly recognizing morphisms, as well as complementation. We also show that the fixed membership problem is NC1-complete, the general membership problem is in L and that the inclusion, equivalence and universality problems are NL-complete. The emptiness problem is shown to be NL-complete if the input is given as a non-surjective morphism.

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