Chasing Puppies: Mobile Beacon Routing on Closed Curves

We solve an open problem posed by Michael Biro at CCCG 2013 that was inspired by his and others' work on beacon-based routing. Consider a human and a puppy on a simple closed curve in the plane. The human can walk along the curve at bounded speed and change direction as desired. The puppy runs with unbounded speed along the curve as long as the Euclidean straight-line distance to the human is decreasing, so that it is always at a point on the curve where the distance is locally minimal. Assuming that the curve is smooth (with some mild genericity constraints) or a simple polygon, we prove that the human can always catch the puppy in finite time.Comment: Full version of a SOCG 2021 paper, 28 pages, 27 figure

From segment to somite: segmentation to epithelialization analyzed within quantitative frameworks

One of the most visually striking patterns in the early developing embryo is somite segmentation. Somites form as repeated, periodic structures in pairs along nearly the entire caudal vertebrate axis. The morphological process involves short- and long-range signals that drive cell rearrangements and cell shaping to create discrete, epithelialized segments. Key to developing novel strategies to prevent somite birth defects that involve axial bone and skeletal muscle development is understanding how the molecular choreography is coordinated across multiple spatial scales and in a repeating temporal manner. Mathematical models have emerged as useful tools to integrate spatiotemporal data and simulate model mechanisms to provide unique insights into somite pattern formation. In this short review, we present two quantitative frameworks that address the morphogenesis from segment to somite and discuss recent data of segmentation and epithelialization

Mathematical models of cell migration and self-organization in embryogenesis

In this thesis we deal with mathematical models and numerical simulations for cell migration and self-organization in embryogenesis. The part of biology which studies the formation and development of the embryo from fertilization until birth is called embryology. Morphogenesis is then the part of embryology which is concerned with the development of patterns and forms. It is well known that although morphogenesis processes are controlled at the genetic scale, genes themselves cannot create the pattern. In general a series of biological mechanisms of self-organization intervene during the early development and the formation of particular biological structures can not be anticipated solely by genetic information. This needs to be taken into account in the choice of a suitable mathematical formulation of such phenomena. Two main main topics will be investigated: we will analyze and mathematically model the self-organizing cell migration in the morphogenesis of the lateral line in the zebrafish (Danio rerio); in a second part, starting from this model, we will propose, and will study both from the analytical and the numerical point of view, a mathematical model of collective motion under only alignment and chemotaxis effects. The present thesis is organized in four chapters. In Chapter 1 we will introduce biological elements about the morphogenetic process occurring in the development of the lateral line in a zebrafish. After a first discussion on the lateral line system and on its fundamental relevance in the current scientific research, we will focus on the main mechanisms of chemical signaling and collective cell migration that will be taken into account later in our mathematical formulation of the phenomenon. In Chapter 2 we will provide a mathematical-modelling background that, starting from the morphogenesis on the chemical scale, will gradually lead us to discuss the existing mathematical models, proposed in the last years to describe collective motion in living system and in particular in the biological field. Example of numerical simulations, and their comparison with experimental evidences will be briefly shown, taken from the recent modelling literature. In Chapter 3 we will introduce a mathematical model describing the self-organizing cell migration in the zebrafish lateral line primordium. We will discuss the derivation of the model, justifying our modelling choices and comparing them with the existing literature. The proposed model will adopt a hybrid “discrete in continuous” description, where cells are treated as discrete entities moving in a continuous space, and chemical signals at molecular level are described by continuous variables. On the chemical scale we will employ diffusion and chemotaxis equations, while on the cellular scale a Newtonian second order equation for each cell will take into account typical effects arising from collective dynamics models. Cell dimension will be recovered introducing suitable detection radii and nonlocal effects. Particular steady states, corresponding to emerging structures, said neuromasts, will then be investigated and their stability will be numerically assessed. Moreover, after a description of the designed numerical approximation scheme, some dynamical simulations will be proposed to show the powerful and the limit of our approach. Finally, we will discuss the estimate of the parameters of the model, derived in part by the biological and the modelling literature, in part by the stationary model or by a numerical data fitting. In Chapter 4 we will propose a Cucker and Smale-like mathematical model of collective motion. Our hybrid model will describe a system of interacting particles under an alignment and chemotaxis effect. From an analytical point of view local and global existence and uniqueness of the solution will be proved. Furthermore, the asymptotic behaviour of the model will be investigated on a linearized form of the system. From a numerical point of view, through an approximation scheme based on finite differences, the full nonlinear system will be simulated and some significant dynamical tests will be shown. Numerical results will be compared with those analytical, and new perspectives will be proposed

Mathematical models of cell migration and self-organization in embryogenesis

In this thesis we deal with mathematical models and numerical simulations for cell migration and self-organization in embryogenesis. The part of biology which studies the formation and development of the embryo from fertilization until birth is called embryology. Morphogenesis is then the part of embryology which is concerned with the development of patterns and forms. It is well known that although morphogenesis processes are controlled at the genetic scale, genes themselves cannot create the pattern. In general a series of biological mechanisms of self-organization intervene during the early development and the formation of particular biological structures can not be anticipated solely by genetic information. This needs to be taken into account in the choice of a suitable mathematical formulation of such phenomena. Two main main topics will be investigated: we will analyze and mathematically model the self-organizing cell migration in the morphogenesis of the lateral line in the zebrafish (Danio rerio); in a second part, starting from this model, we will propose, and will study both from the analytical and the numerical point of view, a mathematical model of collective motion under only alignment and chemotaxis effects. The present thesis is organized in four chapters. In Chapter 1 we will introduce biological elements about the morphogenetic process occurring in the development of the lateral line in a zebrafish. After a first discussion on the lateral line system and on its fundamental relevance in the current scientific research, we will focus on the main mechanisms of chemical signaling and collective cell migration that will be taken into account later in our mathematical formulation of the phenomenon. In Chapter 2 we will provide a mathematical-modelling background that, starting from the morphogenesis on the chemical scale, will gradually lead us to discuss the existing mathematical models, proposed in the last years to describe collective motion in living system and in particular in the biological field. Example of numerical simulations, and their comparison with experimental evidences will be briefly shown, taken from the recent modelling literature. In Chapter 3 we will introduce a mathematical model describing the self-organizing cell migration in the zebrafish lateral line primordium. We will discuss the derivation of the model, justifying our modelling choices and comparing them with the existing literature. The proposed model will adopt a hybrid “discrete in continuous” description, where cells are treated as discrete entities moving in a continuous space, and chemical signals at molecular level are described by continuous variables. On the chemical scale we will employ diffusion and chemotaxis equations, while on the cellular scale a Newtonian second order equation for each cell will take into account typical effects arising from collective dynamics models. Cell dimension will be recovered introducing suitable detection radii and nonlocal effects. Particular steady states, corresponding to emerging structures, said neuromasts, will then be investigated and their stability will be numerically assessed. Moreover, after a description of the designed numerical approximation scheme, some dynamical simulations will be proposed to show the powerful and the limit of our approach. Finally, we will discuss the estimate of the parameters of the model, derived in part by the biological and the modelling literature, in part by the stationary model or by a numerical data fitting. In Chapter 4 we will propose a Cucker and Smale-like mathematical model of collective motion. Our hybrid model will describe a system of interacting particles under an alignment and chemotaxis effect. From an analytical point of view local and global existence and uniqueness of the solution will be proved. Furthermore, the asymptotic behaviour of the model will be investigated on a linearized form of the system. From a numerical point of view, through an approximation scheme based on finite differences, the full nonlinear system will be simulated and some significant dynamical tests will be shown. Numerical results will be compared with those analytical, and new perspectives will be proposed

Introduction to Computational Mechanics of Discontinua

This book is mainly based on the material initially published in Serbian, in 2021, by the University of Belgrade, Faculty of Mining and Geology, under the title Mathematical Physics (Theory and Examples). For the purpose of this book the material from the Serbian edition was reviewed, amended, and translated, with new material added in two final chapters in the second volume. We have divided text into two separate volumes: Mathematics of Physics - Analytical Methods and Mathematics of Physics - Numerical Methods. The first volume consists of 8 chapters: - The first 7 chapters were written by Dragoslav Kuzmanović, Dobrica Nikolić and Ivan Obradović, and correspond to the text from Chapters 1-8 of the Serbian edition, translated by Ivan Obradović. - The material of Chapter 8, which is of a monographic character, corresponds to the material of Chapter 9 in the Serbian edition, but was thoroughly reviewed and rewritten in English by Mihailo Lazarević. The second volume consists of 6 chapters: - The first 3 chapters were written by Aleksandar Sedmak and correspond to Chapter 10 of the Serbian edition, restructured and reviewed, and then translated by Simon Sedmak. - Chapter 4 corresponds to the text of Chapter 11 of the Serbian edition, written and translated by Nikola Mladenović. - Chapters 5 and 6, written by Rade Vignjević and Sreten Mastilović, respectively, offer completely new material. Chapters 4,5 and 6 are of a monographic character

An investigation of stress wave propagation through rock joints and rock masses

Tese de doutoramento. Engenharia Civil. Faculdade de Engenharia. Universidade do Porto, Laboratório Nacional de Engenharia Civil. 201

Path Planning of Industrial Manipulators for Dynamic Obstacles using a New Sensory System

Industrial manipulators perform repetitive and dangerous tasks. They are widely used, however present a source for accidental collisions with human operators. Therefore, they require large isolated spaces heavily taxing factory real-estate. Thus, there exists a need to create a safe cooperative working space shared by both manipulators and humans. The purpose of this research is to provide such an environment by integrating a safety mat-style sensory system, with an implementation of a potential field trajectory planning algorithm. The safety mat sensor has been designed and constructed in a cost effective means acting as a proof of concept for future industrial applications. Both the safety mat and potential field algorithm have been integrated with a CRS F3 manipulator for conducting validation experiments. We have found that our implementation of the potential field algorithm can successfully avoid single, and multiple obstacles detected by the mat. Moreover, collision avoidance is achieved for both static and dynamic obstacles. Finally, our implementation of the potential field algorithm is capable of preventing local minima entrapment of the manipulator, a problem affecting past implementations

Liquid Crystal Mediated Colloidal Self-Assembly and Light-Driven Dynamics

Liquid crystals (LCs) have fascinating optical, mechanical and electrical properties that are intrinsically anisotropic. The relatively weak interactions between their constituting particles allows for manipulating these properties with low energy input, making them good candidates for engineering novel materials. In this work, we introduce colloidal particles of different material, shape, and surface functionalization into LCs and investigate a variety of colloidal interactions and dynamics mediated by LC director distortion. Firstly, noble metal nanoparticles of different shape and topology are synthesized re-dispersed in LC. Facile electric switching of these composites is reminiscent of that of pristine liquid crystals, but provides a means of reconfiguring the nanoparticle assembly and thus also the ensuing composite medium&rsquo;s plasmonic properties. In addition to using particle shape and topology to guide the self-assembly process in LC, our study also extends to discovering the role of chirality of colloidal inclusions in affecting these long-range interactions, where chiral springs and helices are fabricated using two-photon photopolymerization. Our study shows that chirality may be a potential tool to modulate the global orientated self-organization of liquid crystal colloids. Surface functionalization is also a concern: photo-responsive molecules on the surface of colloidal particles further alters the interplay between LC directors and colloidal inclusions. We design and fabricate a micromotor system in LC consisting of silica micro-platelets capped with azobenzene monolayer that are sensitive to polarized blue light. Combined, these result in a feedback mechanism that spontaneously yields a continuous opto-elastic cycle and drives unidirectional particle spinning, with the handedness and frequency robustly controlled by the polarization and intensity of the incident light. Finally, the same mechanism also changes the interaction between these platelets. Coulomb-like and multipolar interaction similar to that in electrostatics are discovered and can be readily switched by shining light of different polarization. To sum up, we demonstrate that combining the unique properties of colloidal particles and facile responses of liquid crystals can lead to richness of soft composite material behavior and potentials of engineering a host of technological applications ranging from colloidal motors to micro-actuators and smart windows.</p

Characterising lentiviral host interactions

Previously it was thought that the HIV-1M capsid undergoes rapid and spontaneous disassembly upon entering the cytoplasm. We now hypothesise that the HIV-1M capsid remains intact as it traverses the cytoplasm and plays a role in coordinating several stages in the life cycle via interactions with host cell proteins. These host proteins, which are essential for optimal infection, are termed cofactors. Here we investigate lentivirus-cofactor interactions, focusing on the cofactors IP6 and CypA. Recently it has been found that the abundant inositol deviate IP6 is able to drive immature particle formation and stabilises the mature capsid core for HIV-1M. Taking a comparative virology approach, in combination with phylogenetic and mutagenesis studies, we investigated whether the use of IP6 is conserved throughout the lentivirus lineage or whether it is a specific cofactor for HIV-1M. We provide evidence that IP6 usage is conserved throughout the lentiviral lineage, via the mutagenesis of the IP6 binding site within the capsid of diverse lentiviruses, and is essential for particle production and reverse transcription. Furthermore, we investigated CypA usage across the lentivirus lineage. We show that CypA usage is a feature which has evolved along the SIVCpz-HIV-1M branch, as replication of the parental SIVCpz virus to HIV-1M is independent of CypA. We also show that CypA usage in HIV-1M can be influenced by the position of the capsid NTD beta-hairpin. Finally, we investigate cellular factors which can influence HIV-1M CypA dependency in cell lines. Focusing on the cellular nuclease TREX-1, we manipulate its expression levels and measured CypA dependency of HIV-1M and the parental SIVCpz. We found that TREX-1 expression levels do no influence CypA dependency of either HIV-1M or SIVCpz