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This paper is a sharp and focussed exploration of the Fibonacci substitution and the mathematical entity it gives rise to, the Fibonacci word. Our investigations are both of an algebraic and a geometric nature. Indeed, it is the combination of the two that gives this paper its overall character. The work is in four parts. Chapter 1 is a brisk tour of necessary basics; definitions, key theorems, and a number of techniques subsequently used extensively. A simple one dimensional drawing rule is investigated in chapter 2 with the aid of what is thought to be an original geometric figure that we will call a "deviation from zero diagram". A highlight of the chapter is its concluding elementary proof of a non-trivial result. Chapter 3 presents a two dimensional drawing rule. Although selected because it is amongst the simplest form possible, this time the object derived from the rule is a fractal. That this is so is proven and its fractal (Hausdorff) dimension calculated. This fractal is a (possibly previously unexplored) variant of that known in the literature as "The Fibonacci Fractal". By way of an overall conclusion, the last chapter, the fourth, suggests a few aspects of those preceding it worthy of further analysis., Comment: 63 pages, 43 figures

Civilizations have tried to make drinking water safe to consume for thousands of years. The process of determining water contaminants has evolved with the complexity of the contaminants due to pesticides and heavy metals. The routine procedure to determine water safety is to use targeted analysis which searches for specific substances from some known list; however, we do not explicitly know which substances should be on this list. Before experimentally determining which substances are contaminants, how do we answer the sampling problem of identifying all the substances in the water? Here, we present an approach that builds on the work of Jaanus Liigand et al., which used non-targeted analysis that conducts a broader search on the sample to develop a random-forest regression model, to predict the names of all the substances in a sample, as well as their respective concentrations[1]. This work utilizes techniques from dimensionality reduction and linear decompositions to present a more accurate model using data from the European Massbank Metabolome Library to produce a global list of chemicals that researchers can then identify and test for when purifying water., Comment: 12 pages, 24 figures

Discrete structure rules for validating molecular structures are usually limited to fulfillment of the octet rule or similar simple deterministic heuristics. We propose a model, inspired by language modeling from natural language processing, with the ability to learn from a collection of undirected molecular graphs, enabling fitting of any underlying structure rule present in the collection. We introduce an adaption to the popular Transformer model, which can learn relationships between atoms and bonds. To our knowledge, the Transformer adaption is the first model that is trained to solve the unsupervised task of recovering partially observed molecules. In this work, we assess how different degrees of information impact performance w.r.t. to fitting the QM9 dataset, which conforms to the octet rule, and to fitting the ZINC dataset, which contains hypervalent molecules and ions requiring the model to learn a more complex structure rule. More specifically, we test a full discrete graph with bond order information, a full discrete graph with only connectivity, a bag-of-neighbors, a bag-of-atoms, and a count-based unigram statistics. These results provide encouraging evidence that neural networks, even when only connectivity is available, can learn arbitrary molecular structure rules specific to a dataset, as the Transformer adaption surpasses a strong octet rule baseline on the ZINC dataset.

We present the open-source package openQ*D-1.0, which has been primarily, but not uniquely, designed to perform lattice simulations of QCD+QED and QCD, with and without C* boundary conditions, and O(a) improved Wilson fermions. The use of C* boundary conditions in the spatial direction allows for a local and gauge-invariant formulation of QCD+QED in finite volume, and provides a theoretically clean setup to calculate isospin-breaking and radiative corrections to hadronic observables from first principles. The openQ*D code is based on openQCD-1.6 and NSPT-1.4. In particular it inherits from openQCD-1.6 several core features, e.g. the highly optimized Dirac operator, the locally deflated solver, the frequency splitting for the RHMC, or the 4th order OMF integrator., Comment: RC* collaboration, 41 pages, 8 figures, 1 table, source code reference: https://dx.doi.org/10.20350/digitalCSIC/8591

A hypothesis is tested and validated for predicting the vortex strength induced by a vortex generator in wall-bounded flow by combining the knowledge of the Vortex Generator (VG) geometry and the approaching boundary layer velocity distribution. In this paper, the spanwise distribution of bound circulation on the vortex generator is computed from integrating the pressure force along the VG height calculated using CFD. It is then assumed that all this bound circulation is shed into the wake to fulfill Helmholtz's theorem and then curl up into one primary tip vortex. To validate this, the trailed circulation estimated from the distribution of the bound circulation is compared to the one in the wake behind the vortex generator determined directly from the wake velocities at some downstream distance. In practical situations, the pressure distribution on the vane is unknown and consequently other estimates of the spanwise force distribution on the VG must instead be applied, such as using 2D airfoil data corresponding to the VG geometry at each wall-normal distance. Such models have previously been proposed and used as an engineering tool to aid preliminary VG design and it is not the purpose of this paper to refine such engineering models, but to validate their assumptions such as applying a lifting line model on a VG that has a very low aspect ratio and placed in wall boundary layer. Herein, high Reynolds number boundary layer measurements of VG induced flow were used to validate the Reynolds Averaged Navier-Stokes (RANS) modeled circulation results and are used for further illustration and validation of the hypothesis., Comment: Accepted for publication

Patterns of vorticity in the wake of a single rectangular winglet (vortex generator) embedded in a turbulent boundary layer have been studied using Stereoscopic Particle Image Velocimetry (SPIV). The winglet was mounted normally to a flat surface with an angle to the oncoming flow. A parametric study varying the winglet height (constant aspect ratio) and angle has shown, contrary to the common classical single tip-vortex conception, that the wake generally consists of a complex system of multiple vortex structures. The primary vortex has previously been discovered to contain a direct coupling between the axial and the rotational flow. In the current work, even the longitudinal secondary structures detected from measured streamwise vorticity display similar behavior. A regime map depicting the observed stable far wake states of the multiple vortices as a function of winglet height and angle reveals complex patterns of the flow topologies not only with the primary tip vortex, but with the additional secondary structures as well. A bifurcation diagram shows distinct regimes of the various secondary structures as well as how the primary vortex is in some cases significantly affected by their presence. These data should serve as inspiration in the process of generating longitudinal vortices for enhancement of heat and mass transfer in industrial devices since the multiple vortex regimes can help improve the conditions for these exchanges. Further, these results point to a weakness in existing inviscid models not accounting for the possibility of multiple vortical structures in the wake.

The abrupt expansion of the slender vortex core with changes in flow topology is commonly known as vortex breakdown. We present new experimental observations of an alteration of the helical vortex core in wall bounded turbulent flow with abrupt growth in core size, but without change in flow topology. The helical symmetry as such is preserved, though the characteristic parameters of helical symmetry of the vortex core transfer from a smooth linear variation to a different trend under the influence of a non-uniform pressure gradient, causing an increase in helical pitch without changing its sign.

Embedded vortices in turbulent wall-bounded flow over a flat plate, generated by a passive rectangular vane-type vortex generator with variable angle $\beta$ to the incoming flow in a low-Reynolds number flow ($Re=2600$ based on the inlet grid mesh size $L=0.039\;$m and free stream velocity $U_{\infty} = 1.0\;$m s$^{-1}$) have been studied with respect to helical symmetry. The studies were carried out in a low-speed closed-circuit wind tunnel utilizing Stereoscopic Particle Image Velocimetry (SPIV). The vortices have been shown to possess helical symmetry, allowing the flow to be described in a simple fashion. Iso-contour maps of axial vorticity revealed a dominant primary vortex and a weaker secondary one for $20^{\circ} \leq \beta \leq 40^{\circ}$. For angles outside of this range, the helical symmetry was impaired due to the emergence of additional flow effects. A model describing the flow has been utilized, showing strong concurrence with the measurements, even though the model is decoupled from external flow processes that could perturb the helical symmetry. The pitch, vortex core size, circulation and the advection velocity of the vortex all vary linearly with the device angle $\beta$. This is important for flow control, since one thereby can determine the axial velocity induced by the helical vortex as well as the swirl redistributing the axial velocity component for a given device angle $\beta$. This also simplifies theoretical studies, \eg to understand and predict the stability of the vortex and to model the flow numerically.

We present work flows and a software module for machine learning model building in surface science and heterogeneous catalysis. This includes fingerprinting atomic structures from 3D structure and/or connectivity information, it includes descriptor selection methods and benchmarks, and it includes active learning frameworks for atomic structure optimization, acceleration of screening studies and for exploration of the structure space of nano particles, which are all atomic structure problems relevant for surface science and heterogeneous catalysis. Our overall goal is to provide a repository to ease machine learning model building for catalysis, to advance the models beyond the chemical intuition of the user and to increase autonomy for exploration of chemical space.

Hadronic spectral densities are important quantities whose non-perturbative knowledge allows for calculating phenomenologically relevant observables, such as inclusive hadronic cross-sections and non-leptonic decay-rates. The extraction of spectral densities from lattice correlators is a notoriously difficult problem because lattice simulations are performed in Euclidean time and lattice data are unavoidably affected by statistical and systematic uncertainties. In this paper we present a new method for extracting hadronic spectral densities from lattice correlators. The method allows for choosing a smearing function at the beginning of the procedure and it provides results for the spectral densities smeared with this function together with reliable estimates of the associated uncertainties. The same smearing function can be used in the analysis of correlators obtained on different volumes, such that the infinite volume limit can be studied in a consistent way. While the method is described by using the language of lattice simulations, in reality it is completely general and can profitably be used to cope with inverse problems arising in different fields of research., Comment: 15 pages, 14 figures. Updated to match published version

We present the incorporation of a surrogate Gaussian Process Regression (GPR) atomistic model to greatly accelerate the rate of convergence of classical Nudged Elastic Band (NEB) calculations. In our surrogate model approach, the cost of converging the elastic band no longer scales with the number of moving images on the path. This provides a far more efficient and robust transition state search. In contrast to a conventional NEB calculation, the algorithm presented here eliminates any need for manipulating the number of images to obtain a converged result. This is achieved by inventing a new convergence criteria that exploits the probabilistic nature of the GPR to use uncertainty estimates of all images in combination with the force of the transition state in the analytic potential. Our method is an order of magnitude faster in terms of function evaluations than the conventional NEB method with no accuracy loss for the converged energy barrier values., Comment: 10 pages, 4 figures, supplemental material (2 pages, 1 figure, 1 table)

We present an extension of chiral perturbation theory that explicitly includes an isosinglet scalar in the Lagrangian. The dynamical effects from the scalar state is of phenomenological relevance in theories where the mass of the isosinglet scalar is comparable to the mass of the pseudo-Goldstone bosons. This near-degeneracy of states is for example observed in certain near-conformal BSM models. From the Lagrangian we calculate the one-loop radiative corrections to the pion mass, the pion decay constant, and the scalar mass. We then proceed and use the results to fit numerical lattice data for an SU(3) gauge theory with $N_f=8$ light flavours in the fundamental representation., Comment: 11 pages, 3 figures, 1 table. Talk given at XIII Quark Confinement and the Hadron Spectrum, 31 July - 6 August 2018, Maynooth University, Ireland

In this paper we show, for the first time, that charged-hadron masses can be calculated on the lattice without relying on gauge fixing at any stage of the calculations. In our simulations we follow a recent proposal and formulate full QCD+QED on a finite volume, without spoiling locality, by imposing C-periodic boundary conditions in the spatial directions. Electrically charged states are interpolated with a class of operators, originally suggested by Dirac and built as functionals of the photon field, that are invariant under local gauge transformations. We show that the quality of the numerical signal of charged-hadron masses is the same as in the neutral sector and that charged-neutral mass splittings can be calculated with satisfactory accuracy in this setup. We also discuss how to describe states of charged hadrons with real photons in a fully gauge-invariant way by providing a first evidence that the proposed strategy can be numerically viable., Comment: 17 pages, 3 figures, 1 table, accepted for publication in JHEP

We present preliminary results on the lattice simulation of an SU(2) gauge theory with two fermion flavors and one strongly interacting scalar field, all in the fundamental representation of SU(2). The motivation for this study comes from the recent proposal of "fundamental" partial compositeness models featuring strongly interacting scalar fields in addition to fermions. Here we describe the lattice setup for our study of this class of models and a first exploration of the lattice phase diagram. In particular we then investigate how the presence of a strongly coupled scalar field affects the properties of light meson resonances previously obtained for the SU(2) model.

We present our final results for the SU(3) sextet model with the non-improved Wilson fermion discretization. We find evidence for several phases of the lattice model, including a bulk phase with broken chiral symmetry. We study the transition between the bulk and weak coupling phase which corresponds to a significant change in the qualitative behavior of spectral and scale setting observables. In particular the t0 and w0 observables seem to diverge in the chiral limit in the weak coupling phase. We then focus on the study of spectral observables in the chiral limit in the weak coupling phase at infinite volume. We consider the masses and decay constants for the pseudoscalar and vector mesons, the mass of the axial vector meson and the spin-1/2 baryon as a function of the quark mass, while controlling finite volume effects. We then test our data against both the IR conformal and the chirally broken hypotheses., Comment: 8 pages, 6 figures. Talk presented at the 35th International Symposium on Lattice Field Theory, 18-24 June 2017, Granada, Spain

We present exploratory results from dynamical simulations of QCD in isolation, as well as QCD coupled to QED, with C* boundary conditions. In finite volume, the use of C* boundary conditions allows for a gauge invariant and local formulation of QED without zero modes. In particular we show that the simulations reproduce known results and that masses of charged mesons can be extracted in a completely gauge invariant way. For the simulations we use a modified version of the HiRep code. The primary features of the simulation code are presented and we discuss some details regarding the implementation of C* boundary conditions and the simulated lattice action., Comment: 8 pages, 2 figures. Talk presented at the 35th International Symposium on Lattice Field Theory, 18-24 June 2017, Granada, Spain

The openQ*D code for the simulation of QCD+QED with C$^\star$ boundary conditions is presented. This code is based on openQCD-1.6, from which it inherits the core features that ensure its efficiency: the locally-deflated SAP-preconditioned GCR solver, the twisted-mass frequency splitting of the fermion action, the multilevel integrator, the 4th order OMF integrator, the SSE/AVX intrinsics, etc. The photon field is treated as fully dynamical and C$^\star$ boundary conditions can be chosen in the spatial directions. We discuss the main features of openQ*D, and we show basic test results and performance analysis. An alpha version of this code is publicly available and can be downloaded from http://rcstar.web.cern.ch/ ., Comment: 35th International Symposium on Lattice Field Theory, Granada, Spain

We investigate the spectrum and IR properties of the SU(3) "sextet" model with two Dirac fermions in the two-index symmetric representation via lattice simulations. This model is a prime candidate for a realization of Walking Technicolor, which features a minimal matter content and it is expected to be inside or very close to the lower boundary of the conformal window. We use the Wilson discretization for the fermions and map the phase structure of the lattice model. We study several spectral and gradient flow observables both in the bulk and the weak coupling phases. While in the bulk phase we find clear signs of chiral symmetry breaking, in the weak coupling phase there is no clear indication for it, and instead the chiral limit of the model seems compatible with an IR-conformal behavior., Comment: 32 pages, many figures

We calculate non-perturbatively the coefficient c_sw required for O(a) improvement of the SU(3) gauge theory with Nf = 2 fermions in the two-index symmetric (sextet) representation. For the calculations we impose the standard improvement condition in the Schr\"odinger functional framework, using two different discretisations for the gauge field i.e. the Wilson plaquette action and the tree-level improved Symanzik (L\"uscher-Weisz) action., Comment: 14 pages, 4 figures, 4 tables

We investigate the spectrum of the SU(2) gauge theory with $N_f$ = 2 flavors of fermions in the fundamental representation, in the continuum, using lattice simulations. This model provides a minimal template which has been used for different strongly coupled extensions of the Standard Model ranging from composite (Goldstone) Higgs models to intriguing types of dark matter candidates, such as the SIMPs. Here we will focus on the composite Goldstone Higgs paradigm, for which this model provides a minimal UV complete realization in terms of a new strong sector with fermionic matter. After introducing the relevant Lattice methods used in our simulations, we will discuss our numerical results. We show that this model features a SU(4)/Sp(4) $\sim$ SO(6)/SO(5) flavor symmetry breaking pattern, and estimate the value of its chiral condensate. Finally, we present our results for the mass spectrum of the lightest spin one and zero resonances, analogue to the QCD $\rho$, $a_1$, $\sigma$, $\eta'$, $a_0$ resonances, which are relevant for searches of new, exotic resonances at the LHC., Comment: 11 pages, 6 figures, prepared for the proceedings of the "XIIth Quark Confinement and the Hadron Spectrum" conference

We present new results from our ongoing study of the SU(3) sextet model with two flavors in the two-index symmetric representation of the gauge group. In the simulations use unimproved Wilson fermions to investigate the infrared properties of the model. We have previously presented results for the spectrum of the model in the weak coupling regime. Here, to better understand the overall behavior of the lattice model, we map its non-trivial phase structure in the space of bare parameters. At strong coupling, we observe a first order phase transition when decreasing the bare quark mass. This first order transition weakens when moving towards weaker couplings with an endpoint at a finite value of the bare coupling, after which it appears to be a continuous transition. We also investigate the behavior of the mass spectrum and scale-setting observable, as a function of the quark mass, and show that their qualitative behavior change significantly when moving from the strong coupling into the weak coupling phase., Comment: 7 pages, 6 figures, Contribution to the 34th International Symposium on Lattice Field Theory, 24-30 July 2016, University of Southampton, UK. Corrected plot in Fig. 6