Pure Σ2\Sigma_2-Elementarity beyond the Core

We display the entire structure ${\cal R}_2$ coding $\Sigma_1$- and $\Sigma_2$-elementarity on the ordinals. This will enable the analysis of pure $\Sigma_3$-elementary substructures.Comment: Extensive rewrite of the introduction. Mathematical content of sections 2 and 3 unchanged, extended introduction to section 2. Removed section 4. Theorem 4.3 to appear elsewhere with corrected proo

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

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

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

Measuring Intelligence and Growth Rate: Variations on Hibbard's Intelligence Measure

In 2011, Hibbard suggested an intelligence measure for agents who compete in an adversarial sequence prediction game. We argue that Hibbard’s idea should actually be considered as two separate ideas: first, that the intelligence of such agents can be measured based on the growth rates of the runtimes of the competitors that they defeat; and second, one specific (somewhat arbitrary) method for measuring said growth rates. Whereas Hibbard’s intelligence measure is based on the latter growth-rate-measuring method, we survey other methods for measuring function growth rates, and exhibit the resulting Hibbard-like intelligence measures and taxonomies. Of particular interest, we obtain intelligence taxonomies based on Big-O and Big-Theta notation systems, which taxonomies are novel in that they challenge conventional notions of what an intelligence measure should look like. We discuss how intelligence measurement of sequence predictors can indirectly serve as intelligence measurement for agents with Artificial General Intelligence (AGIs)

Exploring Unidimensional Proficiency Classification Accuracy From Multidimensional Data in a Vertical Scaling Context

When Item Response Theory (IRT) is operationally applied for large scale assessments, unidimensionality is typically assumed. This assumption requires that the test measures a single latent trait. Furthermore, when tests are vertically scaled using IRT, the assumption of unidimensionality would require that the battery of tests across grades measures the same trait, just at different levels of difficulty. Many researchers have shown that this assumption may not hold for certain test batteries and, therefore, the results from applying a unidimensional model to multidimensional data may be called into question. This research investigated the impact on classification accuracy when multidimensional vertical scaling data are estimated with a unidimensional model. The multidimensional compensatory two-parameter logistic model (MC2PL) was the data-generating model for two levels of a test administered to simulees of correspondingly different abilities. Simulated data from the MC2PL model was estimated according to a unidimensional two-parameter logistic (2PL) model and classification decisions were made from a simulated bookmark standard setting procedure based on the unidimensional estimation results. Those unidimensional classification decisions were compared to the "true" unidimensional classification (proficient or not proficient) of simulees in multidimensional space obtained by projecting a simulee's generating two-dimensional theta vector onto a unidimensional scale via a number correct transformation on the entire test battery (i.e. across both grades). Specifically, conditional classification accuracy measures were considered. That is, the proportion of truly not proficient simulees classified correctly and the proportion of truly proficient simulees classified correctly were the criterion variables. Manipulated factors in this simulation study included the confound of item difficulty with dimensionality, the difference in mean abilities on both dimensions of the simulees taking each test in the battery, the choice of common items used to link the exams, and the correlation of the two abilities. Results suggested that the correlation of the two abilities and the confound of item difficulty with dimensionality both had an effect on the conditional classification accuracy measures. There was little or no evidence that the choice of common items or the differences in mean abilities of the simulees taking each test had an effect

Estimation of the Axial and Lateral Capacity of Driven Piles from the Results of Cone Penetration Test and Finite Element Analysis

Piles play an important role in transportation and bridges. They are used to resist axial and lateral loads transferred to them from structures, earth pressures, incline loads, vehicles, etc. In this study, the capacity of piles for axial and lateral loads is investigated. The ultimate axial capacity of piles can be estimated using different approaches including static pile load tests, dynamic load tests, statnamic load tests, and static analysis based on laboratory tests (effective and total stress approaches) or in-situ tests (SPT, CPT, etc.). For each approach, different researchers have proposed different solutions for different soils and different piles. Mostly, engineers use their engineering judgement based on the available information to estimate the pile’s length and diameter (or width). In this study, different pile-CPT methods were evaluated to estimate the accuracy and precision of them for estimating the axial capacity of the piles. Based on the obtained results, the log-normal distribution of the estimated to measured pile capacity for top-ranked pile-CPT methods was adopted to develop combined pile-CPT methods that optimize the estimation accuracy of axial pile capacity in different soil categories. Also, a model for estimating axial pile capacity was developed based on the results of 10 instrumented piles and 80 piles driven in Louisiana. For analyzing the lateral capacity of the piles, finite element method was used to obtain p-y curves. p-y curves is a simple and accurate approach that considers a nonlinear function for soil reaction with pile displacement. Different parameters for sands and clays were studied to find the effects of each parameter on the p-y curve characteristics. Models were developed for clays and sands that consider these parameters. Using the results from the parametric study, numerical models for the ultimate resistance, initial slope, and characteristic shape function are verified and compared to the existing models