282 research outputs found
Darboux transformations, discrete integrable systems and related Yang-Baxter maps
No full textDarboux transformations constitute a very important tool in the theory of integrable systems. They map trivial solutions of integrable partial differential equations to non-trivial ones and they link the former to discrete integrable systems. On the other hand, they can be used to construct Yang-Baxter maps which can be restricted to completely integrable maps (in the Liouville sense) on invariant leaves.
In this thesis we study the Darboux transformations related to particular Lax operators of NLS type which are invariant under the action of the so-called reduction group. Specifically, we study the cases of: 1) the nonlinear Schrödinger equation (with no reduction), 2) the derivative nonlinear Schrödinger equation, where the corresponding Lax operator is invariant under the action of the Z₂-reduction group and 3) a deformation of the derivative nonlinear Schrödinger equation, associated to a Lax operator invariant under the action of the dihedral reduction group. These reduction groups correspond to recent classification results of automorphic Lie algebras.
We derive Darboux matrices for all the above cases and we use them to construct novel discrete integrable systems together with their Lax representations. For these systems of difference equations, we discuss the initial value problem and, moreover, we consider their integrable reductions. Furthermore, the derivation of the Darboux matrices gives rise to many interesting objects, such as Bäcklund transformations for the corresponding partial differential equations as well as symmetries and conservation laws of their associated systems of difference equations.
Moreover, we employ these Darboux matrices to construct six-dimensional Yang-Baxter maps for all the afore-mentioned cases. These maps can be restricted to four-dimensional Yang-Baxter maps on invariant leaves, which are completely integrable; we also consider their vector generalisations.
Finally, we consider the Grassmann extensions of the Yang-Baxter maps corresponding to the nonlinear Schrödinger equation and the derivative nonlinear Schrödinger equation. These constitute the first examples of Yang-Baxter maps with noncommutative variables in the literature
Entwining Yang–Baxter maps related to NLS type equations
No full textWe construct birational maps that satisfy the parametric set-theoretical Yang–Baxter equation and its entwining generalisation. For this purpose, we employ Darboux transformations related to integrable nonlinear Schrödinger type equations and study the refactorisation problems of the product of their associated Darboux matrices. Additionally, we study various algebraic properties of the derived maps, such as invariants and associated symplectic or Poisson structures, and we prove their complete integrability in the Liouville sense
Algebraic and differential-geometric constructions of set-theoretical solutions to the Zamolodchikov tetrahedron equation
No full textWe present several algebraic and differential-geometric constructions of
tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov
tetrahedron equation. In particular, we obtain a family of new (nonlinear)
polynomial tetrahedron maps on the space of square matrices of arbitrary size,
using a matrix refactorisation equation, which does not coincide with the
standard local Yang--Baxter equation. Liouville integrability is established
for some of these maps.
Also, we show how to derive linear tetrahedron maps as linear approximations
of nonlinear ones, using Lax representations and the differentials of nonlinear
tetrahedron maps on manifolds. We apply this construction to two nonlinear
maps: a tetrahedron map obtained in [arXiv:1708.05694] in a study of soliton
solutions of vector KP equations and a tetrahedron map obtained in
[arXiv:2005.13574] in a study of a matrix trifactorisation problem related to a
Darboux matrix associated with a Lax operator for the NLS equation. We derive
parametric families of new linear tetrahedron maps (with nonlinear dependence
on parameters), which are linear approximations for these nonlinear ones.
Furthermore, we present (nonlinear) matrix generalisations of a tetrahedron
map from Sergeev's classification [arXiv:solv-int/9709006]. These matrix
generalisations can be regarded as tetrahedron maps in noncommutative
variables.
Besides, several tetrahedron maps on arbitrary groups are constructed.Comment: 21 pages; v3: minor corrections in Example 5.12, references update
Pesticide inputs from the sewage treatment plant of Agrinio to River Acheloos, western Greece: occurrence and removal
No full textPublisher‘s note. We regret that the published version of this article erroneously denoted the first author as corresponding author; in fact the formal corresponding author of this paper is Professor Ioannis Konstantinou, whose address is repeated below.</jats:p
Sustainable refurbishment for an adaptable built environment
No full textThe reconsideration of the existing building stock is motivated by society’s efforts towards sustainability and resilience. The building sector has a considerable role to play in doing so. The process of refurbishment is complex, since aspects such as design decisions, existing construction, energy efficiency, and user behaviour need to be considered. The motivation for refurbishing existing buildings is related to environmental, social, and economic aspects of their use or reuse, which are the three core aspects of sustainability. The key environmental motivation is to reduce energy consumption from fossil fuels and related greenhouse gases (GHG) emissions, and to include energy generation from renewables; the key economic motivation is to lessen the cost of energy used for heating, and the key social motivation is to reduce fuel poverty and improve the quality of life and well-being of the occupants.This chapter aims to explain the role of refurbishment of the building stock for sustainability and resilience. Firstly, definitions of the levels of building upgrades are given, and the motivations for refurbishment are discussed. Furthermore, the ecological, economic, and social aspects of refurbishment are deliberated on, together with the importance of the building stock for resilience. Finally, case studies of refurbishment projects are presented, providing insights into different aspects of refurbishment for sustainability and resilience
Environmental Design Principles for the Building Envelope and More _: Passive and Active Measures
No full textGiven the need to reduce building sector related energy consumption and greenhouse gases (GHG), passive and sustainable buildings are a focal point. Simple methods and techniques, which use appropriate building design, material and systems selection, and reflect consideration of the local environmental elements, such as air and sun, provide thermal and visual comfort with less non-renewable energy sources. These techniques are referred to as environmental or bioclimatic design. There are two types of measures to be taken: passive and active. Passive principles exploit the design and properties of the building envelope to minimise or maximise the heat losses and heat gains respectively, to reduce the energy demand. In addition to passive, active measures such as heating systems and solar power technologies are used to produce and distribute the energy needed to achieve comfort of the occupants.The present chapter aims at giving an overview of design principles that result in more comfortable and energy efficient buildings. Passive and active design principles are in line with the environmental design concepts. The environmental design principles can be beneficial to the building performance, whether the design ambition is to have a comfortable and functional building with reasonable energy demand or goes as far as achieving sustainable standards such as zero-energy or passive house.Building Product InnovationDesign of Construtio
A discrete Darboux-Lax scheme for integrable difference equations
No full textWe propose a discrete Darboux-Lax scheme for deriving auto-Backlund transformations and constructing solutions to quad-graph equations that do not necessarily possess the 3D consistency property. As an illustrative example we use the Adler-Yamilov type system which is related to the nonlinear Schroedinger (NLS) equation [21]. In particular, we construct an auto-Backlund transformation for this discrete system, its superposition principle, and we employ them in the construction of the one- and two-soliton solutions of the Adler-Yamilov system
Conference of Mathematical Physics Kezenoi-Am 2016
No full textThis volume, whose contributors include leading researchers in their field, covers a wide range of topics surrounding Integrable Systems, from theoretical developments to applications. Comprising a unique collection of research articles and surveys, the book aims to serve as a bridge between the various areas of Mathematics related to Integrable Systems and Mathematical Physics. Recommended for postgraduate students and early career researchers who aim to acquire knowledge in this area in preparation for further research, this book is also suitable for established researchers aiming to get up to speed with recent developments in the area, and may very well be used as a guide for further study
The Internet of Things for the circular transition in the façade sector
No full textNel settore delle facciate, la transizione ecologica e circolare impone l’adozione di nuovi modelli di business che sfruttino al massimo il valore della materia. In questo contesto, l’Internet of Things (IoT) è identificato quale poten-ziale driver tecnologico per la diffusione di approcci circolari. Scopo dell’articolo è chiarire il ruolo dell’IoT nell’abilitare cinque modelli di business circolari nel settore delle facciate. Attraverso una matrice che evidenzia la relazione tra po-tenziali informazioni prodotte dall’IoT e azioni chiave per il raggiungimento dei modelli di business, si evidenziano i benefici di un sistema di facciata IoT-based. La discussione dei risultati apre il dibattito sulle prospettive di componenti edilizi digitalmente integrati
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