9,362 research outputs found
Ambiguity, Weakness, and Regularity in Probabilistic B\"uchi Automata
Probabilistic B\"uchi automata are a natural generalization of PFA to
infinite words, but have been studied in-depth only rather recently and many
interesting questions are still open. PBA are known to accept, in general, a
class of languages that goes beyond the regular languages. In this work we
extend the known classes of restricted PBA which are still regular, strongly
relying on notions concerning ambiguity in classical omega-automata.
Furthermore, we investigate the expressivity of the not yet considered but
natural class of weak PBA, and we also show that the regularity problem for
weak PBA is undecidable
Tensor Product Approximation (DMRG) and Coupled Cluster method in Quantum Chemistry
We present the Copupled Cluster (CC) method and the Density matrix
Renormalization Grooup (DMRG) method in a unified way, from the perspective of
recent developments in tensor product approximation. We present an introduction
into recently developed hierarchical tensor representations, in particular
tensor trains which are matrix product states in physics language. The discrete
equations of full CI approximation applied to the electronic Schr\"odinger
equation is casted into a tensorial framework in form of the second
quantization. A further approximation is performed afterwards by tensor
approximation within a hierarchical format or equivalently a tree tensor
network. We establish the (differential) geometry of low rank hierarchical
tensors and apply the Driac Frenkel principle to reduce the original
high-dimensional problem to low dimensions. The DMRG algorithm is established
as an optimization method in this format with alternating directional search.
We briefly introduce the CC method and refer to our theoretical results. We
compare this approach in the present discrete formulation with the CC method
and its underlying exponential parametrization.Comment: 15 pages, 3 figure
Formal Solutions of a Class of Pfaffian Systems in Two Variables
In this paper, we present an algorithm which computes a fundamental matrix of
formal solutions of completely integrable Pfaffian systems with normal
crossings in two variables, based on (Barkatou, 1997). A first step was set in
(Barkatou-LeRoux, 2006) where the problem of rank reduction was tackled via the
approach of (Levelt, 1991). We give instead a Moser-based approach. And, as a
complementary step, we associate to our problem a system of ordinary linear
singular differential equations from which the formal invariants can be
efficiently derived via the package ISOLDE, implemented in the computer algebra
system Maple.Comment: Keywords: Linear systems of partial differential equations, Pfaffian
systems, Formal solutions, Moser-based reduction, Hukuhara- Turritin normal
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