Combinatorics, Probability and Computing

The main theme of this workshop was the use of probabilistic methods in combinatorics and theoretical computer science. Although these methods have been around for decades, they are being refined all the time: they are getting more and more sophisticated and powerful. Another theme was the study of random combinatorial structures, either for their own sake, or to tackle extremal questions. The workshop also emphasized connections between probabilistic combinatorics and discrete probability

Graph Theory

Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem

A graph-theoretic analysis of the semantic paradoxes

We introduce a framework for a graph-theoretic analysis of the semantic paradoxes. Similar frameworks have been recently developed for infinitary propositional languages by Cook [5, 6] and Rabern, Rabern, and Macauley [16]. Our focus, however, will be on the language of first-order arithmetic augmented with a primitive truth predicate. Using Leitgeb’s [14] notion of semantic dependence, we assign reference graphs (rfgs) to the sentences of this language and define a notion of paradoxicality in terms of acceptable decorations of rfgs with truth values. It is shown that this notion of paradoxicality coincides with that of Kripke [13]. In order to track down the structural components of an rfg that are responsible for paradoxicality, we show that any decoration can be obtained in a three-stage process: first, the rfg is unfolded into a tree, second, the tree is decorated with truth values (yielding a dependence tree in the sense of Yablo [21]), and third, the decorated tree is re-collapsed onto the rfg. We show that paradoxicality enters the picture only at stage three. Due to this we can isolate two basic patterns necessary for paradoxicality. Moreover, we conjecture a solution to the characterization problem for dangerous rfgs that amounts to the claim that basically the Liar- and the Yablo graph are the only paradoxical rfgs. Furthermore, we develop signed rfgs that allow us to distinguish between ‘positive’ and ‘negative’ reference and obtain more fine-grained versions of our results for unsigned rfgs

Algebraic Structures of Neutrosophic Triplets, Neutrosophic Duplets, or Neutrosophic Multisets

Neutrosophy (1995) is a new branch of philosophy that studies triads of the form (, , ), where is an entity {i.e. element, concept, idea, theory, logical proposition, etc.}, is the opposite of , while is the neutral (or indeterminate) between them, i.e., neither nor .Based on neutrosophy, the neutrosophic triplets were founded, which have a similar form (x, neut(x), anti(x)), that satisfy several axioms, for each element x in a given set.This collective book presents original research papers by many neutrosophic researchers from around the world, that report on the state-of-the-art and recent advancements of neutrosophic triplets, neutrosophic duplets, neutrosophic multisets and their algebraic structures – that have been defined recently in 2016 but have gained interest from world researchers. Connections between classical algebraic structures and neutrosophic triplet / duplet / multiset structures are also studied. And numerous neutrosophic applications in various fields, such as: multi-criteria decision making, image segmentation, medical diagnosis, fault diagnosis, clustering data, neutrosophic probability, human resource management, strategic planning, forecasting model, multi-granulation, supplier selection problems, typhoon disaster evaluation, skin lesson detection, mining algorithm for big data analysis, etc

Partitions of R^n with Maximal Seclusion and their Applications to Reproducible Computation

We introduce and investigate a natural problem regarding unit cube tilings/partitions of Euclidean space and also consider broad generalizations of this problem. The problem fits well within a historical context of similar problems and also has applications to the study of reproducibility in randomized computation. Given $k\in\mathbb{N}$ and $\epsilon\in(0,\infty)$, we define a $(k,\epsilon)$-secluded unit cube partition of $\mathbb{R}^{d}$ to be a unit cube partition of $\mathbb{R}^{d}$ such that for every point $\vec{p}\in\R^d$, the closed $\ell_{\infty}$ $\epsilon$-ball around $\vec{p}$ intersects at most $k$ cubes. The problem is to construct such partitions for each dimension $d$ with the primary goal of minimizing $k$ and the secondary goal of maximizing $\epsilon$. We prove that for every dimension $d\in\mathbb{N}$, there is an explicit and efficiently computable $(k,\epsilon)$-secluded axis-aligned unit cube partition of $\mathbb{R}^d$ with $k=d+1$ and $\epsilon=\frac{1}{2d}$. We complement this construction by proving that for axis-aligned unit cube partitions, the value of $k=d+1$ is the minimum possible, and when $k$ is minimized at $k=d+1$, the value $\epsilon=\frac{1}{2d}$ is the maximum possible. This demonstrates that our constructions are the best possible. We also consider the much broader class of partitions in which every member has at most unit volume and show that $k=d+1$ is still the minimum possible. We also show that for any reasonable $k$ (i.e. $k\leq 2^{d}$), it must be that $\epsilon\leq\frac{\log_{4}(k)}{d}$. This demonstrates that when $k$ is minimized at $k=d+1$, our unit cube constructions are optimal to within a logarithmic factor even for this broad class of partitions. In fact, they are even optimal in $\epsilon$ up to a logarithmic factor when $k$ is allowed to be polynomial in $d$. We extend the techniques used above to introduce and prove a variant of the KKM lemma, the Lebesgue covering theorem, and Sperner\u27s lemma on the cube which says that for every $\epsilon\in(0,\frac12]$, and every proper coloring of $[0,1]^{d}$, there is a translate of the $\ell_{\infty}$ $\epsilon$-ball which contains points of least $(1+\frac23\epsilon)^{d}$ different colors. Advisers: N. V. Vinodchandran & Jamie Radcliff

Adolescents' understanding of limits and infinity

AIM To investigate mathematically able adolescents' conceptions of the basic notions behind the Calculus: infinity (including the infinitely large, the infinitely small and infinite aggregates); limits (of sequences, series and functions); and real numbers. To observe the effect, if any, on these conceptions, of a one year calculus course. EXPERIMENTS Pilot interviews and questionnaires helped identify areas on which to focus the study. A questionnaire was administered to Lower Sixth Form students with 0-level mathematics passes. The questionnaire was administered twice, once in September and again the following May. The A-level mathematicians had received instruction in most of the techniques of the Calculus by May. Interviews, to clarify ambiguities, elicit reasoning behind the responses and probe typicality and atypicality, were conducted in the month following each administration. A second questionnaire, an amended version of the first, was administered to a larger but similar audience. The responses were analysed in the light of hypotheses formulated in the analysis of data from the first 5ample. PRINCIPAL FINDINGS Subjects have a concept of infinity. It exists mainly as a process, anything that goes on and on. It may exist as an object, as a large number or the cardinality of a set, but in these forms it is a vague and indeterminate form. The concept of infinity is inherently contradictory and labile. Recurring decimals are perceived as dynamic, not static, entities and are not proper numbers. Similar attitudes exist towards infinitesimals when they are seen to exist. Subjects' conception of the continuum do not conform to classical or nonstandard paradigms. Convergence / divergence properties are generally noted with infinite sequences and functions. With infinite series, however, convergence / divergence properties, when observed, are seen as secondary to the fact that any infinite series goes on indefinitely and is thus similar to any other infinite series. The concept that the hut is the saue type of entitiy as the finite tens is strong in subjects' thoughts. We coin the term generic hiuit for this phenomenon. The generic limit of 0.9, 0.99, is 0.9, not 1. Similarly the reasoning scheme that whatever holds for the finite holds for the infinite has widespread application. We coin the term generic law for this scheme. Many of the phrases used in calculus courses (in particular hut, tends to, approaches and converges) have everyday meanings that conflict with their mathematical definitions. Numeric/geometric, counting/measuring and static/dynamic contextual influences were observed in some areas. The first year of a calculus course has a negligible effect on students conceptions of limits, infinity and real numbers. IMPLICATIONS FOR TEACHING On introducing limits teachers should encourage full class discussion to ensure that potential cognitive obstacles are brought out into the open. Teachers should take great care that their use of language is understood. A-level courses should devote more of their time to studying the continuum. Nonstandard analysis is an unsuitable tool for introducing elementary calculus