Tur\'an Graphs, Stability Number, and Fibonacci Index

The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory. In this paper, we establish tight upper bounds for the Fibonacci index in terms of the stability number and the order of general graphs and connected graphs. Tur\'an graphs frequently appear in extremal graph theory. We show that Tur\'an graphs and a connected variant of them are also extremal for these particular problems.Comment: 11 pages, 3 figure

Investigation of the Effect of a Diamine-Based Friction Modifier on Micropitting and the Properties of Tribofilms in Rolling-Sliding Contacts

The effect of N-Tallow-1,3-DiaminoPropane (TDP) on friction, rolling wear and micropitting has been investigated with the ultimate objective of developing lubricants with no or minimal environmental impact. A Mini Traction Machine (MTM-SLIM) has been utilised in order to generate tribofilms and observe the effect of TDP on anti-wear tribofilm formation and friction. Micropitting was induced on the surface of specimens using a MicroPitting Rig (MPR). The X-ray Photoelectron Spectroscopy (XPS) surface analytical technique has been employed to investigate the effect of TDP on the chemical composition of the tribofilm while Atomic Force Microscopy (AFM) was used to generate high resolution topographical images of the tribofilms formed on the MTM discs. Experimental and analytical results showed that TDP delays the Zinc DialkylDithioPhosphate (ZDDP) anti-wear tribofilm formation. TDP in combination with ZDDP induces a thinner and smoother anti-wear tribofilm with a modified chemical structure composed of mixed Fe/Zn (poly)phosphates. The sulphide contribution to the tribofilm and oxygen-to-phosphorous atomic concentration ratio are greater in the bulk of the tribofilm derived from a combination of TDP and ZDDP compared to a tribofilm derived from ZDDP alone. Surface analysis showed that utilising TDP effectively mitigates micropitting wear in the test conditions used in this study. Reduction of micropitting, relevant to rolling bearing applications, can be attributed to the improved running-in procedure, reduced friction, formation of a smoother tribofilm and modification of the tribofilm composition induced by TDP

Trees with Given Stability Number and Minimum Number of Stable Sets

We study the structure of trees minimizing their number of stable sets for given order $n$ and stability number $\alpha$. Our main result is that the edges of a non-trivial extremal tree can be partitioned into $n-\alpha$ stars, each of size $\lceil \frac{n-1}{n-\alpha} \rceil$ or $\lfloor \frac{n-1}{n-\alpha}\rfloor$, so that every vertex is included in at most two distinct stars, and the centers of these stars form a stable set of the tree.Comment: v2: Referees' comments incorporate

Mathematical and Statistical Techniques for Systems Medicine: The Wnt Signaling Pathway as a Case Study

The last decade has seen an explosion in models that describe phenomena in systems medicine. Such models are especially useful for studying signaling pathways, such as the Wnt pathway. In this chapter we use the Wnt pathway to showcase current mathematical and statistical techniques that enable modelers to gain insight into (models of) gene regulation, and generate testable predictions. We introduce a range of modeling frameworks, but focus on ordinary differential equation (ODE) models since they remain the most widely used approach in systems biology and medicine and continue to offer great potential. We present methods for the analysis of a single model, comprising applications of standard dynamical systems approaches such as nondimensionalization, steady state, asymptotic and sensitivity analysis, and more recent statistical and algebraic approaches to compare models with data. We present parameter estimation and model comparison techniques, focusing on Bayesian analysis and coplanarity via algebraic geometry. Our intention is that this (non exhaustive) review may serve as a useful starting point for the analysis of models in systems medicine.Comment: Submitted to 'Systems Medicine' as a book chapte

30 years of collaboration

We highlight some of the most important cornerstones of the long standing and very fruitful collaboration of the Austrian Diophantine Number Theory research group and the Number Theory and Cryptography School of Debrecen. However, we do not plan to be complete in any sense but give some interesting data and selected results that we find particularly nice. At the end we focus on two topics in more details, namely a problem that origins from a conjecture of Rényi and Erdős (on the number of terms of the square of a polynomial) and another one that origins from a question of Zelinsky (on the unit sum number problem). This paper evolved from a plenary invited talk that the authors gaveat the Joint Austrian-Hungarian Mathematical Conference 2015, August 25-27, 2015 in Győr (Hungary)

Opening the black box of energy modelling: Strategies and lessons learned

The global energy system is undergoing a major transition, and in energy planning and decision-making across governments, industry and academia, models play a crucial role. Because of their policy relevance and contested nature, the transparency and open availability of energy models and data are of particular importance. Here we provide a practical how-to guide based on the collective experience of members of the Open Energy Modelling Initiative (Openmod). We discuss key steps to consider when opening code and data, including determining intellectual property ownership, choosing a licence and appropriate modelling languages, distributing code and data, and providing support and building communities. After illustrating these decisions with examples and lessons learned from the community, we conclude that even though individual researchers' choices are important, institutional changes are still also necessary for more openness and transparency in energy research

Orthonormal basis selection for LPV system identification, the Fuzzy-Kolmogorov c-Max approach

A fuzzy clustering approach is developed to select pole locations for orthonormal basis functions (OBFs), used for identification of linear parameter varying (LPV) systems. The identification approach is based on interpolation of locally identified linear time invariant (LTI) models, using globally fixed OBFs. Selection of the optimal OBF structure, that guarantees the least worst-case local modelling error in an asymptotic sense, is accomplished through the fusion of the Kolmogorov n-width (KnW) theory and fuzzy c-means (FcM) clustering of observed sample system pole

A prediction-error identification framework for linear parameter-varying systems

Identification of Linear Parameter-Varying (LPV) models is often addressed in an Input-Output (IO) setting using particular extensions of classical Linear Time-Invariant (LTI) prediction-error methods. However, due to the lack of appropriate system-theoretic results, most of these methods are applied without the understanding of their statistical properties and the behavior of the considered noise models. Using a recently developed series expansion representation of LPV systems, the classical concepts of the prediction-error framework are extended to the LPV case and the statistical properties of estimation are analyzed in the LPV context. In the introduced framework it can be shown that under minor assumptions, the classical results on consistency, convergence, bias and asymptotic variance can be extended for LPV predictionerror models and the concept of noise models can be clearly understood. Preliminary results on persistency of excitation and identifiability can also established

LPV system identification with globally fixed orthonormal basis functions

A global and a local identification approach are developed for approximation of linear parameter-varying (LPV) systems. The utilized model structure is a linear combination of globally fixed (scheduling-independent) orthonormal basis functions (OBFs) with scheduling-parameter dependent weights. Whether the weighting is applied on the input or on the output side of the OBFs, the resulting models have different modeling capabilities. The local identification approach of these structures is based on the interpolation of locally identified LTI models on the scheduling domain where the local models are composed from a fixed set of OBFs. The global approach utilizes a priori chosen functional dependence of the parameter-varying weighting of a fixed set of OBFs to deliver global model estimation from measured I/O data. Selection of the OBFs that guarantee the least worst-case modeling error for the local behaviors in an asymptotic sense, is accomplished through the fuzzy Kolmogorov c-max approach. The proposed methods are analyzed in terms of applicability and consistency of the estimates