Binary patterns in the Prouhet-Thue-Morse sequence

We show that, with the exception of the words $a^2ba^2$ and $b^2ab^2$, all (finite or infinite) binary patterns in the Prouhet-Thue-Morse sequence can actually be found in that sequence as segments (up to exchange of letters in the infinite case). This result was previously attributed to unpublished work by D. Guaiana and may also be derived from publications of A. Shur only available in Russian. We also identify the (finitely many) finite binary patterns that appear non trivially, in the sense that they are obtained by applying an endomorphism that does not map the set of all segments of the sequence into itself.

Oil Spill Detection in SAR Images Using Online Extended Variational Learning of Dirichlet Process Mixtures of Gamma Distributions

In this paper, we propose a Dirichlet process (DP) mixture model of Gamma distributions, which is an extension of the finite Gamma mixture model to the infinite case. In particular, we propose a novel online nonparametric Bayesian analysis method based on the infinite Gamma mixture model where the determination of the number of clusters is bypassed via an infinite number of mixture components. The proposed model is learned via an online extended variational Bayesian inference approach in a flexible way where the priors of model’s parameters are selected appropriately and the posteriors are approximated effectively in a closed form. The online setting has the advantage to allow data instances to be treated in a sequential manner, which is more attractive than batch learning especially when dealing with massive and streaming data. We demonstrated the performance and merits of the proposed statistical framework with a challenging real-world application namely oil spill detection in synthetic aperture radar (SAR) images.

Global Solution of Semi-infinite Programs with Existence Constraints

We consider what we term existence-constrained semi-infinite programs. They contain a finite number of (upper-level) variables, a regular objective, and semi-infinite existence constraints. These constraints assert that for all (medial-level) variable values from a set of infinite cardinality, there must exist (lower-level) variable values from a second set that satisfy an inequality. Existence-constrained semi-infinite programs are a generalization of regular semi-infinite programs, possess three rather than two levels, and are found in a number of applications. Building on our previous work on the global solution of semi-infinite programs (Djelassi and Mitsos in J Glob Optim 68(2):227–253, 2017), we propose (for the first time) an algorithm for the global solution of existence-constrained semi-infinite programs absent any convexity or concavity assumptions. The algorithm is guaranteed to terminate with a globally optimal solution with guaranteed feasibility under assumptions that are similar to the ones made in the regular semi-infinite case. In particular, it is assumed that host sets are compact, defining functions are continuous, an appropriate global nonlinear programming subsolver is used, and that there exists a Slater point with respect to the semi-infinite existence constraints. A proof of finite termination is provided. Numerical results are provided for the solution of an adjustable robust design problem from the chemical engineering literature.

The Non-Commuting, Non-Generating Graph of a Nilpotent Group

For a nilpotent group $G$, let $\Xi(G)$ be the difference between the complement of the generating graph of $G$ and the commuting graph of $G$, with vertices corresponding to central elements of $G$ removed. That is, $\Xi(G)$ has vertex set $G \setminus Z(G)$, with two vertices adjacent if and only if they do not commute and do not generate $G$. Additionally, let $\Xi^+(G)$ be the subgraph of $\Xi(G)$ induced by its non-isolated vertices. We show that if $\Xi(G)$ has an edge, then $\Xi^+(G)$ is connected with diameter $2$ or $3$, with $\Xi(G) = \Xi^+(G)$ in the diameter $3$ case. In the infinite case, our results apply more generally, to any group with every maximal subgroup normal. When $G$ is finite, we explore the relationship between the structures of $G$ and $\Xi(G)$ in more detail.

m-Polar Generalization of Fuzzy T-Ordering Relations: An Approach to Group Decision Making

Recently, T-orderings, defined based on a t-norm T and infimum operator (for infinite case) or minimum operator (for finite case), have been applied as a generalization of the notion of crisp orderings to fuzzy setting. When this concept is extending to m-polar fuzzy data, it is questioned whether the generalized definition can be expanded for any aggregation function, not necessarily the minimum operator, or not. To answer this question, the present study focuses on constructing m-polar T-orderings based on aggregation functions A, in particular, m-polar T-preorderings (which are reflexive and transitive m-polar fuzzy relations w.r.t T and A) and m-polar T-equivalences (which are symmetric m-polar T-preorderings). Moreover, the construction results for generating crisp preference relations based on m-polar T-orderings are obtained. Two algorithms for solving ranking problem in decision-making are proposed and validated by an illustrative example.

Stiffer, Stronger and Centrosymmetrical Class of Pentamodal Mechanical Metamaterials

Pentamode metamaterials have been used as a crucial element to achieve elastical unfeelability cloaking devices. They are seen as potentially fragile and not simple for integration in anisotropic structures due to a non-centrosymmetric crystalline structure. Here, we introduce a new class of pentamode metamaterial with centrosymmetry, which shows better performances regarding stiffness, toughness, stability and size dependence. The phonon band structure is calculated based on the finite element method, and the pentamodal properties are evaluated by analyzing the single band gap and the ratio of bulk and shear modulus. The Poisson’s ratio becomes isotropic and close to 0.5 in the limit of small double-cone connections. Stability and scalability analysis results show that the critical load factor of this structure is obviously higher than the classical pentamode structure under the same static elastic properties, and the Young’s modulus gradually converges to a stable value (the infinite case) with an increasing number of unit cells.

STANDARD BAYES LOGIC IS NOT FINITELY AXIOMATIZABLE

In the article [2] a hierarchy of modal logics has been defined to capture the logical features of Bayesian belief revision. Elements in that hierarchy were distinguished by the cardinality of the set of elementary propositions. By linking the modal logics in the hierarchy to the modal logics of Medvedev frames it has been shown that the modal logic of Bayesian belief revision determined by probabilities on a finite set of elementary propositions is not finitely axiomatizable. However, the infinite case remained open. In this article we prove that the modal logic of Bayesian belief revision determined by standard Borel spaces (these cover probability spaces that occur in most of the applications) is also not finitely axiomatizable.

A Logical Theory of Truth-Makers and Falsity-Makers

We explicate the different ways that a first-order sentence can be true (resp., false) in a model M, as formal objects, called (M-relative) truth-makers (resp., falsity-makers). M-relative truth-makers and falsity-makers are co-inductively definable, by appeal to the “atomic facts” in M, and to certain rules of verification and of falsification, collectively called rules of evaluation. Each logical operator has a rule of verification, much like an introduction rule; and a rule of falsification, much like an elimination rule. Applications of the rules (∀) and (∃) involve infinite furcation when the domain of M is infinite. But even in the infinite case, truth-makers and falsity-makers are tree-like objects whose branches are at most finitely long. A sentence φ is true (resp., false) in a model M (in the sense of Tarski) if and only if there existsπ such that π is an M-relative truth-maker (resp., falsity-maker) for φ. With “ways of being true” explicated as these logical truthmakers, one can re-conceive logical consequence between given premises and a conclusion. It obtains just in case there is a suitable method for transforming M-relative truthmakers for the premises into an M-relative truthmaker for the conclusion, whatever the model M may be.

CONTINUOUS ASSOCIATION SCHEMES AND HYPERGROUPS

Classical finite association schemes lead to finite-dimensional algebras which are generated by finitely many stochastic matrices. Moreover, there exist associated finite hypergroups. The notion of classical discrete association schemes can be easily extended to the possibly infinite case. Moreover, this notion can be relaxed slightly by using suitably deformed families of stochastic matrices by skipping the integrality conditions. This leads to a larger class of examples which are again associated with discrete hypergroups. In this paper we propose a topological generalization of association schemes by using a locally compact basis space $X$ and a family of Markov-kernels on $X$ indexed by some locally compact space $D$ where the supports of the associated probability measures satisfy some partition property. These objects, called continuous association schemes, will be related to hypergroup structures on $D$ . We study some basic results for this notion and present several classes of examples. It turns out that, for a given commutative hypergroup, the existence of a related continuous association scheme implies that the hypergroup has many features of a double coset hypergroup. We, in particular, show that commutative hypergroups, which are associated with commutative continuous association schemes, carry dual positive product formulas for the characters. On the other hand, we prove some rigidity results in particular in the compact case which say that for given spaces $X,D$ there are only a few continuous association schemes.