A Generalised Twinning Property for Minimisation of Cost Register Automata

Weighted automata (WA) extend finite-state automata by associating with transitions weights from a semiring S, defining functions from words to S. Recently, cost register automata (CRA) have been introduced as an alternative model to describe any function realised by a WA by means of a deterministic machine. Unambiguous WA over a monoid (M, ⊗) can equivalently be described by cost register automata whose registers take their values in M, and are updated by operations of the form x: = y ⊗ c, with c ∈ M. This class is denoted by CRA⊗c(M). We introduce a twinning property and a bounded variation property parametrised by an integer k, such that the corresponding notions introduced originally by Choffrut for finite-state transducers are obtained for k = 1. Given an unambiguous weighted automaton W over an infinitary group (G, ⊗) realizing some function f, we prove that the three following properties are equivalent: i) W satisfies the twinning property of order k, ii) f satisfies the k-bounded variation property, and iii) f can be described by a CRA⊗c(G) with at most k registers. In the spirit of tranducers, we actually prove this result in a more general setting by considering machines over the semiring of finite sets of elements from (G, ⊗): the three properties are still equivalent for such finite-valued weighted automata, that is the ones associating with words subsets of G of cardinality at most ℓ, for some natural ℓ. Moreover, we show that if the operation ⊗ of G is commutative and computable, then one can decide whether a WA satisfies the twinning property of order k. As a corollary, this allows to decide the register minimisation problem for the class CRA⊗c(G). Last, we prove that a similar result holds for finite-valued finite-state transducers, and that the register minimisation problem for the class CRA.c (B*) is Pspace-complete

Degree of Sequentiality of Weighted Automata

Weighted automata (WA) are an important formalism to describe quantitative properties. Obtaining equivalent deterministic machines is a longstanding research problem. In this paper we consider WA with a set semantics, meaning that the semantics is given by the set of weights of accepting runs. We focus on multi-sequential WA that are defined as finite unions of sequential WA. The problem we address is to minimize the size of this union. We call this minimum the degree of sequentiality of (the relation realized by) the WA. For a given positive integer k, we provide multiple characterizations of relations realized by a union of k sequential WA over an infinitary finitely generated group: a Lipschitz-like machine independent property, a pattern on the automaton (a new twinning property) and a subclass of cost register automata. When possible, we effectively translate a WA into an equivalent union of k sequential WA. We also provide a decision procedure for our twinning property for commutative computable groups thus allowing to compute the degree of sequentiality. Last, we show that these results also hold for word transducers and that the associated decision problem is PSPACE -complete

Quantum field theory in generalised Snyder spaces

We discuss the generalisation of the Snyder model that includes all possible deformations of the Heisenberg algebra compatible with Lorentz invariance and investigate its properties. We calculate peturbatively the law of addition of momenta and the star product in the general case. We also undertake the construction of a scalar field theory on these noncommutative spaces showing that the free theory is equivalent to the commutative one, like in other models of noncommutative QFT.Comment: 12 pages. arXiv admin note: substantial text overlap with arXiv:1608.0620

Poland’s Integration with the European Union

The present book “Poland – History, Culture and Society. Selected Readings” is the third edition of a collection of academic texts written with the intention to accompany the module by providing incoming students with teaching materials that will assist them in their studies of the course module and encourage further search for relevant information and data. The papers collected in the book have been authored by academic teachers from the University of Łódź, specialists in such fields as history, geography, literature, sociology, ethnology, cultural studies, and political science. Each author presents one chapter related to a topic included in the module or extending its contents. The book contains the extensive bibliography

Twist for Snyder space

We construct the twist operator for the Snyder space. Our starting point is a non-associative star product related to a Hermitian realisation of the noncommutative coordinates originally introduced by Snyder. The corresponding coproduct of momenta is non-coassociative. The twist is constructed using a general definition of the star product in terms of a bi-differential operator in the Hopf algebroid approach. The result is given by a closed analytical expression. We prove that this twist reproduces the correct coproducts of the momenta and the Lorentz generators. The twisted Poincar\'{e} symmetry is described by a non-associative Hopf algebra, while the twisted Lorentz symmetry is described by the undeformed Hopf algebra. This new twist might be important in the construction of different types of field theories on Snyder space.Comment: 15 pages, references added, matches published versio

Gauging as constraining: the universal generalised geometry action in two dimensions

One of the central concepts in modern theoretical physics, gauge symmetry, is typically realised by lifting a finite-dimensional global symmetry group of a given functional to an infinite-dimensional local one by extending the functional to include gauge fields. In this contribution we review the construction of gauged actions for two-dimensional sigma models, considering a more general notion to be gauged, namely that of a (possibly singular) foliation. In particular, the original action does not need to have any global symmetry for this purpose. Moreover, reformulating the ungauged theory by means of auxiliary 1-form fields taking values in the generalised tangent bundle over the target, all possible such gauge theories result from restriction of these fields to take values in (possibly small) Dirac structures. This turns all the remaining 1-form fields into gauge fields and leads to the presence of a local symmetry. We recall all needed mathematical notions, those of (higher) Lie algebroids, Courant algebroids, and Dirac structures.Comment: 20 pages; proceedings of "Recent Developments in Strings and Gravity", Corfu Summer Institute 201

Still vacant after all these years – Evaluating the efficiency of property-led urban regeneration

The author(s) received no financial support for the research, authorship, and/or publication of this article.Peer reviewedPostprin

Analysis of a moving mask approximation for martensitic transformations

In this work we introduce a moving mask approximation to describe the dynamics of austenite to martensite phase transitions at a continuum level. In this framework, we prove a new type of Hadamard jump condition, from which we deduce that the deformation gradient must be of the form $\mathsf{1} +\mathbf{a}\otimes \mathbf{n}$ a.e. in the martensite phase. This is useful to better understand the complex microstructures and the formation of curved interfaces between phases in new ultra-low hysteresis alloys such as Zn45Au30Cu25, and provides a selection mechanism for physically-relevant energy-minimising microstructures. In particular, we use the new type of Hadamard jump condition to deduce a rigidity theorem for the two well problem. The latter provides more insight on the cofactor conditions, particular conditions of supercompatibility between phases believed to influence reversibility of martensitic transformations

A generalisation of the Hill's quadratic yield function for planar plastic anisotropy to consider loading direction

© 2017 The Authors In this work, a new generalised quadratic yield function for plane stress analysis that is able to describe the plastic anisotropy of metals and also the asymmetric behaviour in tension-compression typical of the Hexagonal Closed-Pack (HCP) materials, is developed. The new yield function has a quadratic form in the stress tensor and it simultaneously predicts the r-values and directional flow stresses, which is shown to agree very well with experimental results. It also accurately describes the biaxial symmetric stress state which is fundamental for the accurate modelling of aluminium alloys. The new quadratic yield function represents the non-symmetric biaxial stress state by performing a linear interpolation from pure uniaxial loading to a biaxial symmetric stress state. The main advantages of this new yield function is that it can be used for the modelling of metals with any crystallographic structure (BCC, FCC or HCP), it only has five anisotropic coefficients and also that it is a simple quadratic yield criterion that is able to accurately reproduce the plastic anisotropy of metals whilst using an associated flow rule