9,362 research outputs found

    Ambiguity, Weakness, and Regularity in Probabilistic B\"uchi Automata

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    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

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    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

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    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 for
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