Laver and set theory

In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip

Dense subsets of products of finite trees

We prove a "uniform" version of the finite density Halpern-L\"{a}uchli Theorem. Specifically, we say that a tree $T$ is homogeneous if it is uniquely rooted and there is an integer $b\geq 2$, called the branching number of $T$, such that every $t\in T$ has exactly $b$ immediate successors. We show the following. For every integer $d\geq 1$, every $b_1,...,b_d\in\mathbb{N}$ with $b_i\geq 2$ for all $i\in\{1,...,d\}$, every integer k\meg 1 and every real $0<\epsilon\leq 1$ there exists an integer $N$ with the following property. If $(T_1,...,T_d)$ are homogeneous trees such that the branching number of $T_i$ is $b_i$ for all $i\in\{1,...,d\}$, $L$ is a finite subset of $\mathbb{N}$ of cardinality at least $N$ and $D$ is a subset of the level product of $(T_1,...,T_d)$ satisfying $$|D\cap \big(T_1(n)\times ...\times T_d(n)\big)| \geq \epsilon |T_1(n)\times ...\times T_d(n)|$$ for every $n\in L$, then there exist strong subtrees $(S_1,...,S_d)$ of $(T_1,...,T_d)$ of height $k$ and with common level set such that the level product of $(S_1,...,S_d)$ is contained in $D$. The least integer $N$ with this property will be denoted by $UDHL(b_1,...,b_d|k,\epsilon)$. The main point is that the result is independent of the position of the finite set $L$. The proof is based on a density increment strategy and gives explicit upper bounds for the numbers $UDHL(b_1,...,b_d|k,\epsilon)$.Comment: 36 pages, no figures; International Mathematics Research Notices, to appea

Physical (A)Causality: Determinism, Randomness and Uncaused Events

Physical indeterminism; Randomness in physics; Physical random number generators; Physical chaos; Self-reflexive knowledge; Acausality in physics; Irreducible randomnes

Permutations, moments, measures

We present a continued fraction with 13 permutation statistics, several of them new, connecting a great number of combinatorial structures to a wide variety of moment sequences and their measures from classical and noncommutative probability. The Hankel determinants of these moment sequences are a product of (p,q)-factorials, unifying several instances from the literature. The corresponding measures capture as special cases several classical laws, such as the Gaussian, Poisson, and exponential, along with further specializations of the orthogonalizing measures in the q-Askey scheme and several known noncommutative central limits. Statistics in our continued fraction generalize naturally to signed and colored permutations, and to the k-arrangements introduced here, permutations with k-colored fixed points

The reverse mathematics of elementary recursive nonstandard analysis: a robust contribution to the foundations of mathematics

Reverse Mathematics (RM) is a program in the Foundations of Mathematics founded by Harvey Friedman in the Seventies ([17, 18]). The aim of RM is to determine the minimal axioms required to prove a certain theorem of ‘ordinary’ mathematics. In many cases one observes that these minimal axioms are also equivalent to this theorem. This phenomenon is called the ‘Main Theme’ of RM and theorem 1.2 is a good example thereof. In practice, most theorems of everyday mathematics are equivalent to one of the four systems WKL0, ACA0, ATR0 and Π1-CA0 or provable in the base theory RCA0. An excellent introduction to RM is Stephen Simpson’s monograph [46]. Nonstandard Analysis has always played an important role in RM. ([32,52,53]). One of the open problems in the literature is the RM of theories of first-order strength I∆0 + exp ([46, p. 406]). In Chapter I, we formulate a solution to this problem in theorem 1.3. This theorem shows that many of the equivalences from theorem 1.2 remain correct when we replace equality by infinitesimal proximity ‘≈’ from Nonstandard Analysis. The base theory now is ERNA, a nonstandard extension of I∆0 + exp. The principle that corresponds to ‘Weak K ̈onig’s lemma’ is the Universal Transfer Principle (see axiom schema 1.57). In particular, one can say that the RM of ERNA+Π1-TRANS is a ‘copy up to infinitesimals’ of the RM of WKL0. This implies that RM is ‘robust’ in the sense this term is used in Statistics and Computer Science ([25,35]). Furthermore, we obtain applications of our results in Theoretical Physics in the form of the ‘Isomorphism Theorem’ (see theorem 1.106). This philosophical excursion is the first application of RM outside of Mathematics and implies that ‘whether reality is continuous or discrete is undecidable because of the way mathematics is used in Physics’ (see paragraph 3.2.4, p. 53). We briefly explore a connection with the program ‘Constructive Reverse Mathematics’ ([30,31]) and in the rest of Chapter I, we consider the RM of ACA0 and related systems. In particular, we prove theorem 1.161, which is a first step towards a ‘copy up to infinitesimals’ of the RM of ACA0. However, one major aesthetic problem with these results is the introduction of extra quantifiers in many of the theorems listed in theorem 1.3 (see e.g. theorem 1.94). To overcome this hurdle, we explore Relative Nonstandard Analysis in Chapters II and III. This new framework involves many degrees of infinity instead of the classical ‘binary’ picture where only two degrees ‘finite’ and ‘infinite’ are available. We extend ERNA to a theory of Relative Nonstandard Analysis called ERNAA and show how this theory and its extensions allow for a completely quantifier- free development of analysis. We also study the metamathematics of ERNAA, motivated by RM. Several ERNA-theorems would not have been discovered without considering ERNAA first

Variational methods and its applications to computer vision

Many computer vision applications such as image segmentation can be formulated in a ''variational'' way as energy minimization problems. Unfortunately, the computational task of minimizing these energies is usually difficult as it generally involves non convex functions in a space with thousands of dimensions and often the associated combinatorial problems are NP-hard to solve. Furthermore, they are ill-posed inverse problems and therefore are extremely sensitive to perturbations (e.g. noise). For this reason in order to compute a physically reliable approximation from given noisy data, it is necessary to incorporate into the mathematical model appropriate regularizations that require complex computations. The main aim of this work is to describe variational segmentation methods that are particularly effective for curvilinear structures. Due to their complex geometry, classical regularization techniques cannot be adopted because they lead to the loss of most of low contrasted details. In contrast, the proposed method not only better preserves curvilinear structures, but also reconnects some parts that may have been disconnected by noise. Moreover, it can be easily extensible to graphs and successfully applied to different types of data such as medical imagery (i.e. vessels, hearth coronaries etc), material samples (i.e. concrete) and satellite signals (i.e. streets, rivers etc.). In particular, we will show results and performances about an implementation targeting new generation of High Performance Computing (HPC) architectures where different types of coprocessors cooperate. The involved dataset consists of approximately 200 images of cracks, captured in three different tunnels by a robotic machine designed for the European ROBO-SPECT project.Open Acces

Limits Under Conjugacy of the Diagonal Cartan Subgroup in SL(n,R)

A conjugacy limit group is the limit of a sequence of conjugates of the positive diagonal Cartan subgroup, C ≤ SL(3,R). In chapter 6, we prove a variant of a theorem of Haettel, and show that up to conjugacy in SL(3,R), the positive diagonal Cartan subgroup has 5 possible conjugacy limit groups. Each conjugacy limit group is determined by a nonstandard triangle. We give a criterion for a sequence of conjugates of C to converge to each of the 5 conjugacy limit groups.In chapter 8, we give a quadratic lower bound on the dimension of the space of conjugacy classes of subgroups of SL(n,R) that are limits under conjugacy of the positive diagonal subgroup. We give the first explicit examples of abelian (n − 1)-dimensional subgroups of SL(n,R) which are not such a limit, however all such abelian groups are limits of the positive diagonal group iff n ≤ 4.In chapter 4, we classify all subgroups of PGL(4,R) isomorphic to (R^3,+), up to conjugacy, and Haettel shows each is a limit of the positive diagonal Cartan subgroup. By taking subgroups of these groups satisfying certain properties, we show there are 4 possible families of generalized cusps up to projective equivalence in dimension 3, and describe each cusp