Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

Novel Discretization Schemes for the Numerical Simulation of Membrane Dynamics

Motivated by the demands of simulating flapping wings of Micro Air Vehicles, novel numerical methods were developed and evaluated for the dynamic simulation of membranes. For linear membranes, a mixed-form time-continuous Galerkin method was employed using trilinear space-time elements, and the entire space-time domain was discretized and solved simultaneously. For geometrically nonlinear membranes, the model incorporated two new schemes that were independently developed and evaluated. Time marching was performed using quintic Hermite polynomials uniquely determined by end-point jerk constraints. The single-step, implicit scheme was significantly more accurate than the most common Newmark schemes. For a simple harmonic oscillator, the scheme was found to be symplectic, frequency-preserving, and conditionally stable. Time step size was limited by accuracy requirements rather than stability. The spatial discretization scheme employed a staggered grid, grouping of nonlinear terms, and polygon shape functions in a strong-form point collocation formulation. Validation against existing experimental data showed the method to be accurate until hyperelastic effects dominate

International Congress of Mathematicians: 2022 July 6–14: Proceedings of the ICM 2022

Following the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library

Integral points on algebraic varieties, with special emphasis on complements of divisors

Viene introdotto il concetto di insieme di punti interi su varietà, insieme alle sue generalizzazioni e applicazioni. Viene poi studiato il caso in cui la varietà è data come complemento in uno spazio proiettivo di divisori di ramificazione di opportune proiezioni, seguendo il metodo proposto da G. Faltings e poi ripreso da U. Zannier. Infine, studiamo un caso particolare, in cui la proiezione è fatta da una ipersuperficie, e dimostriamo alcuni risultati in tale contesto. Sono date alcune applicazioni dei risultati ottenuti e vengono richiamate le principali nozioni utilizzate nelle dimostrazioni, con particolare attenzione ai risultati di approssimazione diofantea e all'equazione delle S-unità

Independence Models for Integer Points of Polytopes.

The integer points of a high-dimensional polytope P are generally difficult to count or sample uniformly. We consider a class of low-complexity random models for these points which arise from an entropy maximization problem. From these models, by way of "anti-concentration" results for sums of independent random variables, we derive general, efficiently computable upper bounds on the number of integer points of P. We make a detailed study of contingency tables with bounded entries, which are the integer points of a transportation polytope truncated by a cuboid. We provide efficiently computable estimates for the logarithm of the number of m by n tables with specified row and column sums r_1, ..., r_m, c_1, ..., c_n and bounds on the entries. These estimates are asymptotic as m and n go to infinity simultaneously, given that no r_i (resp., c_j) is allowed to exceed a fixed multiple of the average row sum (resp., column sum). As an application, we consider a random, uniformly selected table with entries in {0, 1, ..., kappa} having a given sum. Responding to questions raised by Diaconis and Efron in the context of statistical significance testing, we show that the occurrence of row sums r_1, ..., r_m is positively correlated with the occurrence of column sums c_1, ..., c_n when kappa > 1 and r_1, ..., r_m, c_1, ..., c_n are sufficiently extreme. We give evidence that the opposite is true for near-average values of r_1, ..., r_m, c_1, ..., c_n.Ph.D.MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/86295/1/auspex_1.pd

Invariant Measures, Geometry, and Control of Hybrid and Nonholonomic Dynamical Systems

Constraints are ubiquitous when studying mechanical systems and fall into two main categories: hybrid (1-sided, unilateral) and nonholonomic/holonomic (2-sided, bilateral) constraints. A hybrid constraint takes the form h(x)≥0. An example of a constraint of this nature is requiring a billiard ball to remain within the confines of a table-top. The notable feature of these constraints is that when the ball reaches the boundary of the table-top (i.e. when h(x)=0), an impact occurs; this is a discontinuous jump in the dynamics. Dynamical systems that have this phenomenon generally fall under the domain of hybrid dynamical systems. On the other hand, nonholonomic constraints take the form h(x)=0. Generally, h will depend on both the positions and velocities and cannot be integrated to only depend on the positions (when it can be integrated, the constraint is called holonomic). An example of a nonholonomic constraint is an ice skate: motion is not allowed perpendicular to the direction of the skate. It is common that these systems are studied using tools from differential geometry. This thesis studies both hybrid and nonholonomic constraints together using the language of differential (specifically symplectic) geometry. However, due to the exotic nature of hybrid dynamics, some auxiliary results are found that pertain to the asymptotic nature of these systems. These include the idea of a hybrid limit-set, Floquet theory, and a Poincaré-Bendixson theorem for planar systems. The bulk of this work focuses on finding (smooth) invariant measures for both nonholonomic and hybrid systems (as well as systems involving both types of constraints). Necessary and sufficient conditions are found which guarantee the existence of an invariant measure for nonholonomic systems in which the density depends only on the configuration variables. Extending this idea to hybrid nonholonomic systems requires that the impact preserves the measure as well. To build towards this, relatively simple conditions to test whether or not a differential form is hybrid-invariant are derived. In the cases where the density depends on only the configuration variables, the measure is still invariant under the hybrid dynamics independent of the choice of impacts. The billiard problem with a vertical rolling disk as the billiard ball is one such system and is therefore recurrent for any choice of compact table-top. This thesis concludes with optimal control of hybrid systems. First, Hamilton-Jacobi is extended to the hybrid setting (nonholonomic constraints are not considered here) and the idea of completely integrable hybrid systems is introduced. It is shown that the usual billiard problem on a circular table is completely integrable. Finally, the hybrid Hamilton-Jacobi theory is extended to a hybrid Hamilton-Jacobi-Bellman theory which allows for the study of optimal control problems.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/163071/1/wiclark_1.pd

Making up Numbers

"Making up Numbers: A History of Invention in Mathematics offers a detailed but accessible account of a wide range of mathematical ideas. Starting with elementary concepts, it leads the reader towards aspects of current mathematical research. The book explains how conceptual hurdles in the development of numbers and number systems were overcome in the course of history, from Babylon to Classical Greece, from the Middle Ages to the Renaissance, and so to the nineteenth and twentieth centuries. The narrative moves from the Pythagorean insistence on positive multiples to the gradual acceptance of negative numbers, irrationals and complex numbers as essential tools in quantitative analysis. Within this chronological framework, chapters are organised thematically, covering a variety of topics and contexts: writing and solving equations, geometric construction, coordinates and complex numbers, perceptions of ‘infinity’ and its permissible uses in mathematics, number systems, and evolving views of the role of axioms. Through this approach, the author demonstrates that changes in our understanding of numbers have often relied on the breaking of long-held conventions to make way for new inventions at once providing greater clarity and widening mathematical horizons. Viewed from this historical perspective, mathematical abstraction emerges as neither mysterious nor immutable, but as a contingent, developing human activity. Making up Numbers will be of great interest to undergraduate and A-level students of mathematics, as well as secondary school teachers of the subject. In virtue of its detailed treatment of mathematical ideas, it will be of value to anyone seeking to learn more about the development of the subject.

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum