Connecting the two worlds: well-partial-orders and ordinal notation systems

Kruskal claims in his now-classical 1972 paper [47] that well-partial-orders are among the most frequently rediscovered mathematical objects. Well partial-orders have applications in many fields outside the theory of orders: computer science, proof theory, reverse mathematics, algebra, combinatorics, etc. The maximal order type of a well-partial-order characterizes that order’s strength. Moreover, in many natural cases, a well-partial-order’s maximal order type can be represented by an ordinal notation system. However, there are a number of natural well-partial-orders whose maximal order types and corresponding ordinal notation systems remain unknown. Prominent examples are Friedman’s well-partial-orders of trees with the gap-embeddability relation [76]. The main goal of this dissertation is to investigate a conjecture of Weiermann [86], thereby addressing the problem of the unknown maximal order types and corresponding ordinal notation systems for Friedman’s well-partial orders [76]. Weiermann’s conjecture concerns a class of structures, a typical member of which is denoted by T (W ), each are ordered by a certain gapembeddability relation. The conjecture indicates a possible approach towards determining the maximal order types of the structures T (W ). Specifically, Weiermann conjectures that the collapsing functions #i correspond to maximal linear extensions of these well-partial-orders T (W ), hence also that these collapsing functions correspond to maximal linear extensions of Friedman’s famous well-partial-orders

An order-theoretic characterization of the Howard-Bachmann-hierarchy

In this article we provide an intrinsic characterization of the famous Howard-Bachmann ordinal in terms of a natural well-partial-ordering by showing that this ordinal can be realized as a maximal order type of a class of generalized trees with respect to a homeomorphic embeddability relation. We use our calculations to draw some conclusions about some corresponding subsystems of second order arithmetic. All these subsystems deal with versions of light-face Π₁¹-comprehension

Ordinal notation systems corresponding to Friedman's linearized well-partial-orders with gap-condition

In this article we investigate whether the following conjecture is true or not: does the addition-free theta functions form a canonical notation system for the linear versions of Friedman’s well-partial-orders with the so-called gap-condition over a finite set of n labels. Rather surprisingly, we can show this is the case for two labels, but not for more than two labels. To this end, we determine the order type of the notation systems for addition-free theta functions in terms of ordinals less than ε0ε0 . We further show that the maximal order type of the Friedman ordering can be obtained by a certain ordinal notation system which is based on specific binary theta functions

How to compare Buchholz-style ordinal notation systems with Gordeev-style notation systems

By a syntactical construction we define an order-preserving mapping of Gordeev’s ordinal notation system PRJ(P) into Buchholz ordinal nota- tion system OT(P) where P represents a limit ordinal. Since Gordeev already showed that OT(P ) can be considered as a subsystem of PRJ(P), we obtain a direct proof of the equality of the order types of both systems. We expect that our result will contribute to the general program of determining the maximal order types of those well-quasi orders which are provided by gap-embeddability relations considered by Friedman, Gordeev and Kriz

Calculating maximal order types for finite rooted unstructured labeled trees

Diana Schmidt, in her Habilitationsschrift in 1979, completely classified the maximal order types of the natural tree embeddability relations for finite rooted structured labeled trees. Her results since have found interesting applications in proof theory and reverse mathematics. The question concerning the maximal order types of unstructured trees has been left open for years, and a conclusive answer will be given in this article. Moreover, we provide an answer to a question of Harvey Friedman regarding Γ0 and binary two-labeled unstructured trees

Antimicrobial prescribing behaviour in dogs and cats by Belgian veterinarians

The objective of this study is to survey general prescribing behaviour by Belgian companion animal veterinarians and to assess agreement of these practices with current treatment guidelines. Therefore an online survey was administered with five realistic and frequently occurring first-line cases to primary-care veterinary practitioners. For each case a predefined pattern of questions were asked about whether or not they would prescribe antimicrobials, if they would prescribe a non-antimicrobial treatment and if they would perform additional diagnostic steps. The responses were compared with recommendations in national guidelines and recent literature. The overall most prescribed antimicrobials were potentiated amoxicillin (43.0 per cent), fluoroquinolones (14.7 per cent), third-generation and fourth-generation cephalosporins (10.9 per cent) and tetracyclines (10.9 per cent). Only 48.3 per cent of the veterinarians complied with the guidelines in nearly all of the clinical scenarios (ie, prescribing antimicrobials when indicated, not prescribing antimicrobials when it is not indicated). Moreover, when prescribing highest priority critically important antimicrobials, susceptibility testing on bacterial cultures was performed in only 12.4 per cent of the prescriptions. The results showed that the prescribing behaviour of antimicrobial compounds by primary-care veterinary practitioners in dogs and cats is often not in agreement with national guidelines. Focus in improvement of this prescribing behaviour should be on performing the appropriate diagnostic steps and decreasing the use of highest priority critically important antimicrobials

The type-1 ribosome-inactivating protein OsRIP1 triggers caspase-independent apoptotic-like death in HeLa cells

Ribosome-inactivating proteins (RIPs) are capable of removing a specific adenine from 285 ribosomal RNA, thus inhibiting protein biosynthesis in an irreversible manner. In this study, recombinant OsRIP1, a type 1 RIP from rice (Oryza saliva L.), was investigated for its anti-proliferative properties. Human cervical cancer HeLa cells were incubated in the presence of OsRIP1 for 24-72 h. OsRIP1 treatment yielded an anti-proliferation response of the HeLa cells and resulted in apoptotic-like blebbing of the plasma membrane without causing DNA fragmentation. OsRIP1 labeled with FITC accumulated at the cell surface. Pull-down assays identified ASPP1 (Apoptosis-Stimulating Protein of p53 1) and IFITM3 (interferon-induced transmembrane protein 3) as potential interaction partners for OsRIP1. Transcript levels for several critical genes related to different signaling pathways were quantified by RT-qPCR. OsRIP1 provoked HeLa cells to undergo caspase-independent cell death, associated with a significant transcriptional upregulation of the apoptotic gene PUMA, interferon regulatory factor 1 (IRF1) and the autophagy-related marker LC3. No changes in caspase activities were observed. Together, these data suggest that apoptotic-like events were involved in OsRIP1-driven caspase-independent cell death that might trigger the IRF1 signaling pathway and LC3-mediated autophagy