Electric network and Hirota type 44-simplex maps

Bazhanov--Stroganov (4-simplex) maps are set-theoretical solutions to the 4-simplex equation, namely the fourth member of the family of $n$-simplex equations, which are fundamental equations of mathematical physics. In this paper, we develop a method for constructing Bazhanov--Stroganov maps as extensions of tetrahedron maps which are set-theoretical solutions to the Zamolodchikov tetrahedron (3-simplex) equation. We employ this method to construct birarional Bazhanov--Stroganov maps which boil down to the famous electric network and Hirota tetrahedron maps at a certain limit.Comment: 12 pages, 1 figur

Birational solutions to the set-theoretical 4-simplex equation

The 4-simplex equation is a higher-dimensional analogue of Zamolodchikov's tetrahedron equation and the Yang--Baxter equation which are two of the most fundamental equations of mathematical physics. In this paper, we introduce a method for constructing 4-simplex maps, namely solutions to the set-theoretical 4-simplex equation, using Lax matrix refactorisation problems. Employing this method, we construct 4-simplex maps which at a certain limit give tetrahedron maps classified by Kashaev, Korepanov and Sergeev. Moreover, we construct a Kadomtsev--Petviashvili type of 4-simplex map. Finally, we introduce a method for constructing 4-simplex maps which can be restricted on level sets to parametric 4-simplex maps using Darboux transformations of integrable PDEs. We construct a nonlinear Schr\"odinger type parametric 4-simplex map which is the first parametric 4-simplex map in the literature.Comment: Revised version, published in Physica

Local Yang--Baxter correspondences and set-theoretical solutions to the Zamolodchikov tetrahedron equation

We study tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation, and their matrix Lax representations defined by the local Yang--Baxter equation. Sergeev [S.M. Sergeev 1998 Lett. Math. Phys. 45, 113--119] presented classification results on three-dimensional tetrahedron maps obtained from the local Yang--Baxter equation for a certain class of matrix-functions in the situation when the equation possesses a unique solution which determines a tetrahedron map. In this paper, using correspondences arising from the local Yang--Baxter equation for some simple $2\times 2$ matrix-functions, we show that there are (non-unique) solutions to the local Yang--Baxter equation which define tetrahedron maps that do not belong to the Sergeev list; this paves the way for a new, wider classification of tetrahedron maps. We present invariants for the derived tetrahedron maps and prove Liouville integrability for some of them. Furthermore, using the approach of solving correspondences arising from the local Yang--Baxter equation, we obtain several new birational tetrahedron maps, including maps with matrix Lax representations on arbitrary groups, a $9$-dimensional map associated with a Darboux transformation for the derivative nonlinear Schr\"odinger (NLS) equation, and a $9$-dimensional generalisation of the $3$-dimensional Hirota map.Comment: 18 pages. New results added (section 4), and also the references list was update

Redefining the Use of Sustainable Development Goals at the Organisation and Project Levels—A Survey of Engineers

The United Nations’ (UN) Sustainable Development Goals (SDGs) aim to deliver an improved future for people, planet and profit. However, they have not gained the required traction at the business and project levels. This article explores how engineers rate and use the SDGs at the organisational and project levels. It adopts the Realist Evaluation’s Context−Mechanism−Outcomes model to critically evaluate practitioners’ views on using SDGs to measure business and project success. The study addresses the thematic areas of sustainability and business models through the theoretical lens of Creating Shared Value and the Triple Bottom Line. A survey of 325 engineers indicated four primary shortfalls for measuring SDGs on infrastructure projects, namely (1) leadership, (2) tools and methods, (3) engineers’ business skills in measuring SDG impact and (4) how project success is too narrowly defined as outputs (such as time, cost and scope) and not outcomes (longer-term local impacts and stakeholder value). The research study is of value to researchers developing business models that address the SDGs and also practitioners in the construction industry who seek to link their investment decisions to the broader outcomes of people, planet and profit through the UN SDGs

Delivering UN Sustainable Development Goals’ Impact on Infrastructure Projects: An Empirical Study of Senior Executives in the UK Construction Sector

Achievement of the United Nations’ 2030 Sustainable Development Goals (SDG) is of paramount importance for both business and society. Across the construction sector, despite evidence that suggests 88% of those surveyed want to measure the SDG impact at both the business and project levels, there continues to be major challenge in achieving this objective. This paper shares the results of a qualitative research study of 40 interviews with executives from the United Kingdom (UK) construction industry. It was supported by a text-based content analysis to strengthen the findings. The results indicate that SDG measurement practices are embraced in principle but are problematic in practice and that rarely does action match rhetoric. While the research was completed in the UK, the findings have broader applicability to other countries since most construction firms have extensive global business footprints. Researchers can use the findings to extend the current understanding of measuring outcomes and impact at project level, and, for practitioners, the study provides insights into the contextual preconditions necessary to achieve the intended outcomes of adopting a mechanism for the measurement of SDGs. The international relevance of this research is inherently linked to the global nature of the SDGs and therefore the results could be used outside of UK

On the solutions of the local Zamolodchikov tetrahedron equation

We study the solutions of the local Zamolodhcikov tetrahedron equation in the form of correspondences derived by $3\times 3$ matrices. We present all the associated generators of 4-simplex maps satisfying the local tetrahedron equation. Moreover, we demonstrate that, from some of our solutions, we can recover the 4-simplex extensions of Kashaev--Korepanov--Sergeev and Hirota type tetrahedron maps. Finally, we construct several novel 4-simplex maps.Comment: 15 pages, 1 figur

Grassmann extensions of Yang–Baxter maps

In this paper we show that there are explicit Yang-Baxter (YB) maps with Darboux-Lax representation between Grassman extensions of algebraic varieties. Motivated by some recent results on noncommutative extensions of Darboux transformations, we first derive a Darboux matrix associated with the Grassmann-extended derivative nonlinear Schrödinger (DNLS) equation, and then we deduce novel endomorphisms of Grassmann varieties, which possess the YB property. In particular, we present ten-dimensional maps which can be restricted to eight-dimensional YB maps on invariant leaves, related to the Grassmann-extended NLS and DNLS equations. We consider their vector generalisations

A non-commutative extension of the Adler-Yamilov Yang-Baxter map

In this paper, we construct a noncommutative extension of the Adler-Yamilov Yang-Baxter map which is related to the nonlinear Schr�dinger equation. Moreover, we show that this map is partially integrable

Spatial modeling of epidermal nerve fiber patterns

Peripheral neuropathy is a condition associated with poor nerve functionality. Epidermal nerve fiber (ENF) counts per epidermal surface are dramatically reduced and the two-dimensional (2D) spatial structure of ENFs tends to become more clustered as neuropathy progresses. Therefore, studying the spatial structure of ENFs is essential to fully understand the mechanisms that guide those morphological changes. In this article, we compare ENF patterns of healthy controls and subjects suffering from mild diabetic neuropathy by using suction skin blister specimens obtained from the right foot. Previous analysis of these data has focused on the analysis and modeling of the spatial ENF patterns consisting of the points where the nerves enter the epidermis, base points, and the points where the nerve fibers terminate, end points, projected on a 2D plane, regarding the patterns as realizations of spatial point processes. Here, we include the first branching points, the points where the nerve trees branch for the first time, and model the three-dimensional (3D) patterns consisting of these three types of points. To analyze the patterns, spatial summary statistics are used and a new epidermal active territory that measures the volume in the epidermis that is covered by the individual nerve fibers is constructed. We developed a model for both the 2D and the 3D patterns including the branching points. Also, possible competitive behavior between individual nerves is examined. Our results indicate that changes in the ENFs spatial structure can more easily be detected in the later parts of the ENFs