Also, for any two ordinals X,Y either X ∈ Y or Y ∈ X. As we saw before with ordinals ∈ is simply < or ⊂. Every well-ordered set is isomorphic to an ordinal. The first ordinal is 0, the second 1 = {0}, . . . the n’th ordinal is {0, 1, 2, . . . ,n − 1}. The first infintite ordinal is ω = {0, 1, 2, . . . , n, n + 1, . . .}. The second infinite ordinal is ω + 1 = {0, 1, . . . , n, . . . , ω}. In ge...

Ordinal classification of imbalanced data is a challenging problem that appears in many real world applications. The challenge to simultaneously consider the order classes and class imbalance, which can notably improve performance metrics. Bayesian formulation allows deal with these two characteristics jointly: It takes into account prior probability each decision costs, be used include imbalan...

We investigated if the temporal and ordinal structures of sequences can be represented and learned independently. In Experiment 1, subjects learned three rhythmic sequences of key presses with the right index finger: Combined consisted of nine key presses with a corresponding temporal structure of eight intervals; Temporal had the temporal structure of Combined but was performed on one key; Ord...

‎‎In this paper‎, ‎we define fuzzy subnexuses over a nexus $N$‎. ‎Define and study the notions of the prime fuzzy subnexuses and the fractions‎‎induced by them‎. ‎Finally‎, ‎we show that if S is a meet‎‎closed subset of the set Fsub(N), ‎of fuzzy subnexuses of a nexus N‎, ‎and‎‎h= ⋀S ϵ S, ‎then the fractions S^-1 N and h^-1 N are isomorphic as meet-semilattices‎.

Background Ordinal outcomes are common; however, selecting an appropriate analysis is often problematic. Acute stroke trials routinely measure dependency using an ordinal scale as the primary outcome. Historically trials have dichotomised ordinal scales to compare the proportion dependent across groups, which limits statistical power to detect an effect. The OAST Collaboration (2007) showed ord...

We propose new concepts in order to analyze and model the dependence structure between two time series. Our methods rely exclusively on the order structure of the data points. Hence, the methods are stable under monotone transformations of the time series and robust against small perturbations or measurement errors. Ordinal pattern dependence can be characterized by four parameters. We propose ...

This paper introduces an evolutionary dynamics based on imitate the better realization (IBR) rule. Under this rule, agent in a population game imitates strategy of randomly chosen opponent whenever opponent’s realized payoff is higher than his or her own. Such behavior generates mean which depends order payoffs, but not their magnitudes, and polynomial utilization frequencies. We demonstrate th...

Ordinal systems are structures for describing ordinal notation systems, which extend the more predicative approaches to ordinal notation systems, like the Can-tor normal form, the Veblen function and the Sch utte Klammer symbols, up to the Bachmann-Howard ordinal.-ordinal systems, which are natural extensions of this approach, reach without the use of cardinals the strength of the theories for ...