203 research outputs found
Sofic-Dyck shifts
We define the class of sofic-Dyck shifts which extends the class of
Markov-Dyck shifts introduced by Inoue, Krieger and Matsumoto. Sofic-Dyck
shifts are shifts of sequences whose finite factors form unambiguous
context-free languages. We show that they correspond exactly to the class of
shifts of sequences whose sets of factors are visibly pushdown languages. We
give an expression of the zeta function of a sofic-Dyck shift
Iterated Binomial Sums and their Associated Iterated Integrals
We consider finite iterated generalized harmonic sums weighted by the
binomial in numerators and denominators. A large class of these
functions emerges in the calculation of massive Feynman diagrams with local
operator insertions starting at 3-loop order in the coupling constant and
extends the classes of the nested harmonic, generalized harmonic and cyclotomic
sums. The binomially weighted sums are associated by the Mellin transform to
iterated integrals over square-root valued alphabets. The values of the sums
for and the iterated integrals at lead to new
constants, extending the set of special numbers given by the multiple zeta
values, the cyclotomic zeta values and special constants which emerge in the
limit of generalized harmonic sums. We develop
algorithms to obtain the Mellin representations of these sums in a systematic
way. They are of importance for the derivation of the asymptotic expansion of
these sums and their analytic continuation to . The
associated convolution relations are derived for real parameters and can
therefore be used in a wider context, as e.g. for multi-scale processes. We
also derive algorithms to transform iterated integrals over root-valued
alphabets into binomial sums. Using generating functions we study a few aspects
of infinite (inverse) binomial sums.Comment: 62 pages Latex, 1 style fil
The Impatient May Use Limited Optimism to Minimize Regret
Discounted-sum games provide a formal model for the study of reinforcement
learning, where the agent is enticed to get rewards early since later rewards
are discounted. When the agent interacts with the environment, she may regret
her actions, realizing that a previous choice was suboptimal given the behavior
of the environment. The main contribution of this paper is a PSPACE algorithm
for computing the minimum possible regret of a given game. To this end, several
results of independent interest are shown. (1) We identify a class of
regret-minimizing and admissible strategies that first assume that the
environment is collaborating, then assume it is adversarial---the precise
timing of the switch is key here. (2) Disregarding the computational cost of
numerical analysis, we provide an NP algorithm that checks that the regret
entailed by a given time-switching strategy exceeds a given value. (3) We show
that determining whether a strategy minimizes regret is decidable in PSPACE
An efficient algorithm for computing the Baker-Campbell-Hausdorff series and some of its applications
We provide a new algorithm for generating the Baker--Campbell--Hausdorff
(BCH) series Z = \log(\e^X \e^Y) in an arbitrary generalized Hall basis of
the free Lie algebra generated by and . It is based
on the close relationship of with a Lie algebraic structure
of labeled rooted trees. With this algorithm, the computation of the BCH series
up to degree 20 (111013 independent elements in ) takes less
than 15 minutes on a personal computer and requires 1.5 GBytes of memory. We
also address the issue of the convergence of the series, providing an optimal
convergence domain when and are real or complex matrices.Comment: 30 page
A differential identity for Green functions
If P is a differential operator with constant coefficients, an identity is
derived to calculate the action of exp(P) on the product of two functions. In
many-body theory, P describes the interaction Hamiltonian and the identity
yields a hierarchy of Green functions. The identity is first derived for scalar
fields and the standard hierarchy is recovered. Then the case of fermions is
considered and the identity is used to calculate the generating function for
the Green functions of an electron system in a time-dependent external
potential.Comment: 14 page
Learning Rational Functions
International audienceRational functions are transformations from words to words that can be defined by string transducers. Rational functions are also captured by deterministic string transducers with lookahead. We show for the first time that the class of rational functions can be learned in the limit with polynomial time and data, when represented by string transducers with lookahead in the diagonal-minimal normal form that we introduce
Sturmian morphisms, the braid group B_4, Christoffel words and bases of F_2
We give a presentation by generators and relations of a certain monoid
generating a subgroup of index two in the group Aut(F_2) of automorphisms of
the rank two free group F_2 and show that it can be realized as a monoid in the
group B_4 of braids on four strings. In the second part we use Christoffel
words to construct an explicit basis of F_2 lifting any given basis of the free
abelian group Z^2. We further give an algorithm allowing to decide whether two
elements of F_2 form a basis or not. We also show that, under suitable
conditions, a basis has a unique conjugate consisting of two palindromes.Comment: 25 pages, 4 figure
The Hopf Algebra of Renormalization, Normal Coordinates and Kontsevich Deformation Quantization
Using normal coordinates in a Poincar\'e-Birkhoff-Witt basis for the Hopf
algebra of renormalization in perturbative quantum field theory, we investigate
the relation between the twisted antipode axiom in that formalism, the Birkhoff
algebraic decomposition and the universal formula of Kontsevich for quantum
deformation.Comment: 21 pages, 15 figure
Generalized shuffles related to Nijenhuis and TD-algebras
Shuffle and quasi-shuffle products are well-known in the mathematics
literature. They are intimately related to Loday's dendriform algebras, and
were extensively used to give explicit constructions of free commutative
Rota-Baxter algebras. In the literature there exist at least two other
Rota-Baxter type algebras, namely, the Nijenhuis algebra and the so-called
TD-algebra. The explicit construction of the free unital commutative Nijenhuis
algebra uses a modified quasi-shuffle product, called the right-shift shuffle.
We show that another modification of the quasi-shuffle product, the so-called
left-shift shuffle, can be used to give an explicit construction of the free
unital commutative TD-algebra. We explore some basic properties of TD-operators
and show that the free unital commutative Nijenhuis algebra is a TD-algebra. We
relate our construction to Loday's unital commutative dendriform trialgebras,
including the involutive case. The concept of Rota-Baxter, Nijenhuis and
TD-bialgebras is introduced at the end and we show that any commutative
bialgebra provides such objects.Comment: 20 pages, typos corrected, accepted for publication in Communications
in Algebr
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