Algorithmic computation of principal posets using Maple and Python

We present symbolic and numerical algorithms for a computer search in the Coxeter spectral classification problems. One of the main aims of the paper is to study finite posets I that are principal, i.e., the rational symmetric Gram matrix GI : = 1/2[CI+CItr] ∈ MI(Q) of I is positive semi-definite of corank one, where CI ∈ MI(Z) is the incidence matrix of I. With any such a connected poset I, we associate a simply laced Euclidean diagram DI ∈ {A˜n, D˜n, E˜₆, E˜₇, E˜₈}, the Coxeter matrix CoxI := −CI ⋅ C−trI, its complex Coxeter spectrum speccI, and a reduced Coxeter number cI. One of our aims is to show that the spectrum speccI of any such a poset I determines the incidence matrix CI (hence the poset I) uniquely, up to a Z-congruence. By computer calculations, we find a complete list of principal one-peak posets I (i.e., I has a unique maximal element) of cardinality ≤ 15, together with speccI, cI, the incidence defect ∂I : ZI → Z, and the Coxeter-Euclidean type DI. In case when DI ∈ {A˜n, D˜n, E˜₆, E˜₇, E˜₈} and n := |I| is relatively small, we show that given such a principal poset I, the incidence matrix CI is Z-congruent with the non-symmetric Gram matrix GˇDI of DI, speccI = speccDI and cˇI = cˇDI. Moreover, given a pair of principal posets I and J, with |I| = |J| ≤ 15, the matrices CI and CJ are Z-congruent if and only if speccI = speccJ

A Framework for Coxeter Spectral Classification of Finite Posets and Their Mesh Geometries of Roots

Following our paper [Linear Algebra Appl. 433(2010), 699–717], we present a framework and computational tools for the Coxeter spectral classification of finite posets J≡(J,⪯). One of the main motivations for the study is an application of matrix representations of posets in representation theory explained by Drozd [Funct. Anal. Appl. 8(1974), 219–225]. We are mainly interested in a Coxeter spectral classification of posets J such that the symmetric Gram matrix GJ:=(1/2)[CJ+CJtr]∈J(ℚ) is positive semidefinite, where CJ∈J(ℤ) is the incidence matrix of J. Following the idea of Drozd mentioned earlier, we associate to J its Coxeter matrix CoxJ:=-CJ·CJ-tr, its Coxeter spectrum speccJ, a Coxeter polynomial coxJ(t)∈ℤ[t], and a Coxeter number  cJ. In case GJ is positive semi-definite, we also associate to J a reduced Coxeter number   čJ, and the defect homomorphism ∂J:ℤJ→ℤ. In this case, the Coxeter spectrum speccJ is a subset of the unit circle and consists of roots of unity. In case GJ is positive semi-definite of corank one, we relate the Coxeter spectral properties of the posets J with the Coxeter spectral properties of a simply laced Euclidean diagram DJ∈{̃n,̃6,̃7,̃8} associated with J. Our aim of the Coxeter spectral analysis of such posets J is to answer the question when the Coxeter type CtypeJ:=(speccJ,cJ,  čJ) of J determines its incidence matrix CJ (and, hence, the poset J) uniquely, up to a ℤ-congruency. In connection with this question, we also discuss the problem studied by Horn and Sergeichuk [Linear Algebra Appl. 389(2004), 347–353], if for any ℤ-invertible matrix A∈n(ℤ), there is B∈n(ℤ) such that Atr=Btr·A·B and B2=E is the identity matrix

Intelligent Systems

This book is dedicated to intelligent systems of broad-spectrum application, such as personal and social biosafety or use of intelligent sensory micro-nanosystems such as "e-nose", "e-tongue" and "e-eye". In addition to that, effective acquiring information, knowledge management and improved knowledge transfer in any media, as well as modeling its information content using meta-and hyper heuristics and semantic reasoning all benefit from the systems covered in this book. Intelligent systems can also be applied in education and generating the intelligent distributed eLearning architecture, as well as in a large number of technical fields, such as industrial design, manufacturing and utilization, e.g., in precision agriculture, cartography, electric power distribution systems, intelligent building management systems, drilling operations etc. Furthermore, decision making using fuzzy logic models, computational recognition of comprehension uncertainty and the joint synthesis of goals and means of intelligent behavior biosystems, as well as diagnostic and human support in the healthcare environment have also been made easier

A complex systems approach to education in Switzerland

The insights gained from the study of complex systems in biological, social, and engineered systems enables us not only to observe and understand, but also to actively design systems which will be capable of successfully coping with complex and dynamically changing situations. The methods and mindset required for this approach have been applied to educational systems with their diverse levels of scale and complexity. Based on the general case made by Yaneer Bar-Yam, this paper applies the complex systems approach to the educational system in Switzerland. It confirms that the complex systems approach is valid. Indeed, many recommendations made for the general case have already been implemented in the Swiss education system. To address existing problems and difficulties, further steps are recommended. This paper contributes to the further establishment complex systems approach by shedding light on an area which concerns us all, which is a frequent topic of discussion and dispute among politicians and the public, where billions of dollars have been spent without achieving the desired results, and where it is difficult to directly derive consequences from actions taken. The analysis of the education system's different levels, their complexity and scale will clarify how such a dynamic system should be approached, and how it can be guided towards the desired performance

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282

Divergence in Architectural Research

ConCave Ph.D. Symposium 2020: Divergence in Architectural Research, March 5-6, 2020, Georgia Institute of Technology, Atlanta, GA.The essays in this volume have come together under the theme “Divergence in Architectural Research” and present a snapshot of Ph.D. research being conducted in over thirty architectural research institutions, representing fourteen countries around the world. These essays also provide a window into the presentations and discussions that took place March 5-6, 2020, during the ConCave Ph.D. Symposium “Divergence in Architectural Research,” under the auspices of the School of Architecture, Georgia Institute of Technology, in Atlanta, Georgia. On a preliminary reading, the essays respond to the call of divergence by doing just that; they present the great diversity of research topics, methodologies, and practices currently found under the umbrella of “architectural research.” They inform inquiry within architectural programs and across disciplinary concentrations, and also point to the ways that the academy, research methodologies, and the design profession are evolving and encroaching upon one another, with the unspoken hope of encouraging new relationships, reconfiguring previous assumptions about the discipline, and interweaving research and practice