Pure patterns of order 2

We provide mutual elementary recursive order isomorphisms between classical ordinal notations, based on Skolem hulling, and notations from pure elementary patterns of resemblance of order $2$, showing that the latter characterize the proof-theoretic ordinal of the fragment $\Pi^1_1$-$\mathrm{CA}_0$ of second order number theory, or equivalently the set theory $\mathrm{KPl}_0$. As a corollary, we prove that Carlson's result on the well-quasi orderedness of respecting forests of order $2$ implies transfinite induction up to the ordinal of $\mathrm{KPl}_0$. We expect that our approach will facilitate analysis of more powerful systems of patterns.Comment: corrected Theorem 4.2 with according changes in section 3 (mainly Definition 3.3), results unchanged. The manuscript was edited, aligned with reference [14] (moving former Lemma 3.5 there), and argumentation was revised, with minor corrections in (the proof of) Theorem 4.2; results unchanged. Updated revised preprint; to appear in the APAL (2017

Tracking chains revisited

The structure ${\cal C}_2:=(1^\infty,\le,\le_1,\le_2)$, introduced and first analyzed in Carlson and Wilken 2012 (APAL), is shown to be elementary recursive. Here, $1^\infty$ denotes the proof-theoretic ordinal of the fragment $\Pi^1_1$-$\mathrm{CA}_0$ of second order number theory, or equivalently the set theory $\mathrm{KPl}_0$, which axiomatizes limits of models of Kripke-Platek set theory with infinity. The partial orderings $\le_1$ and $\le_2$ denote the relations of $\Sigma_1$- and $\Sigma_2$-elementary substructure, respectively. In a subsequent article we will show that the structure ${\cal C}_2$ comprises the core of the structure ${\cal R}_2$ of pure elementary patterns of resemblance of order $2$. In Carlson and Wilken 2012 (APAL) the stage has been set by showing that the least ordinal containing a cover of each pure pattern of order $2$ is $1^\infty$. However, it is not obvious from Carlson and Wilken 2012 (APAL) that ${\cal C}_2$ is an elementary recursive structure. This is shown here through a considerable disentanglement in the description of connectivity components of $\le_1$ and $\le_2$. The key to and starting point of our analysis is the apparatus of ordinal arithmetic developed in Wilken 2007 (APAL) and in Section 5 of Carlson and Wilken 2012 (JSL), which was enhanced in Carlson and Wilken 2012 (APAL) specifically for the analysis of ${\cal C}_2$.Comment: The text was edited and aligned with reference [10], Lemma 5.11 was included (moved from [10]), results unchanged. Corrected Def. 5.2 and Section 5.3 on greatest immediate $\le_1$-successors. Updated publication information. arXiv admin note: text overlap with arXiv:1608.0842

Alcohols, esters and heavy sulphur compounds production by pure and mixed cultures of apiculate wine yeasts

Strains of Hanseniaspora uvarum, Hanseniaspora guilliermondii and Saccharomyces cerevisiae were used as pure or mixed starter cultures in commercial medium, in order to compare their kinetic parameters and fermentation patterns. In pure and mixed cultures, yeasts presented similar ethanol yield and productivity. Pure cultures of H. uvarum and S. cerevisiae showed a specific growth rate of 0.38 h⁻¹; however, this value decreased when these yeasts were grown in mixed cultures with H. guilliermondii. The specific growth rate of pure cultures of H. guilliermondii was 0.41 h⁻⁻¹ and was not affected by growth of other yeasts. H. guilliermondii was found to be the best producer of 2-phenylethyl acetate and 2-phenylethanol in both pure and mixed cultures. In pure cultures, H. uvarum led to the highest contents of heavy sulphur compounds, but H. guilliermondii and S. cerevisiae produced similar levels of methionol and 2-methyltetrahydrothiophen-3-one. Growth of apiculate yeasts in mixed cultures with S. cerevisiae led to amounts of 3-methylthiopropionic acid, acetic acid-3-(methylthio)propyl ester and 2- methyltetrahydrothiophen-3-one similar to those obtained in a pure culture of S. cerevisiae; however, growth of apiculate yeasts increased methionol contents of fermented media

Patterns for computational effects arising from a monad or a comonad

This paper presents equational-based logics for proving first order properties of programming languages involving effects. We propose two dual inference system patterns that can be instanciated with monads or comonads in order to be used for proving properties of different effects. The first pattern provides inference rules which can be interpreted in the Kleisli category of a monad and the coKleisli category of the associated comonad. In a dual way, the second pattern provides inference rules which can be interpreted in the coKleisli category of a comonad and the Kleisli category of the associated monad. The logics combine a 3-tier effect system for terms consisting of pure terms and two other kinds of effects called 'constructors/observers' and 'modifiers', and a 2-tier system for 'up-to-effects' and 'strong' equations. Each pattern provides generic rules for dealing with any monad (respectively comonad), and it can be extended with specific rules for each effect. The paper presents two use cases: a language with exceptions (using the standard monadic semantics), and a language with state (using the less standard comonadic semantics). Finally, we prove that the obtained inference system for states is Hilbert-Post complete

Sources of Bias and Solutions to Bias in the CPI

Four sources of bias in the Consumer Prices Index (CPI) have been identified. The most discussed is substitution bias, which creates a second order bias in the CPI. Three other changes besides prices changes create first order effects on a correctly measured cost of living index (COLI). (1) Introduction of new goods creates a first order effect of new good bias' (2) Quality changes in existing goods will lead to quality' bias, which has first order effects (3) Shifts in shopping patterns to lower priced stores can create first order outlet bias'. I explain in this paper that a pure price' based approach of surveying prices to estimate a COLI cannot succeed in solving the 3 problems of first order bias. Neither the BLS nor the recent report C. Schultze and C. Mackie, eds., At What Price (AWP, 2002), recognizes that to solve these problems, which have been long known, both quantity and price data are necessary. I discuss economic and econometric approaches to measuring the first order bias effects as well as the availability of scanner data that would permit implementation of the techniques. Lastly, I review recent research that demonstrates that these sources of bias are large in relation to measured inflation in the CPI.

Minimum Ranks and Refined Inertias of Sign Pattern Matrices

A sign pattern is a matrix whose entries are from the set $\{+, -, 0\}$. This thesis contains problems about refined inertias and minimum ranks of sign patterns. The refined inertia of a square real matrix $B$, denoted \ri(B), is the ordered $4$-tuple $(n_+(B), \ n_-(B), \ n_z(B), \ 2n_p(B))$, where $n_+(B)$ (resp., $n_-(B)$) is the number of eigenvalues of $B$ with positive (resp., negative) real part, $n_z(B)$ is the number of zero eigenvalues of $B$, and $2n_p(B)$ is the number of pure imaginary eigenvalues of $B$. The minimum rank (resp., rational minimum rank) of a sign pattern matrix $\cal A$ is the minimum of the ranks of the real (resp., rational) matrices whose entries have signs equal to the corresponding entries of $\cal A$. First, we identify all minimal critical sets of inertias and refined inertias for full sign patterns of order 3. Then we characterize the star sign patterns of order $n\ge 5$ that require the set of refined inertias $\mathbb{H}_n=\{(0, n, 0, 0), (0, n-2, 0, 2), (2, n-2, 0, 0)\}$, which is an important set for the onset of Hopf bifurcation in dynamical systems. Finally, we establish a direct connection between condensed $m \times n$ sign patterns and zero-nonzero patterns with minimum rank $r$ and $m$ point-$n$ hyperplane configurations in ${\mathbb R}^{r-1}$. Some results about the rational realizability of the minimum ranks of sign patterns or zero-nonzero patterns are obtained

Void Analysis of Hadronic Density Fluctuations at Phase Transition

The event-to-event fluctuations of hadron multiplicities are studied for a quark system undergoing second-order phase transition to hadrons. Emphasis is placed on the search for an observable signature that is realistic for heavy-ion collisions. It is suggested that in the 2-dimensional y-phi space the produced particles selected in a very narrow p_T window may exhibit clustering patterns even when integrated over the entire emission time. Using the Ising model to simulate the critical phenomenon and taking into account a p_T distribution that depends on the emission time, we study in the framework of the void analysis proposed earlier and find scaling behavior. The scaling exponents turn out to be larger than the ones found before for pure configurations without mixing. The signature is robust in that it is insensitive to the precise scheme of simulating time evolution. Thus it should reveal whether or not the dense matter created in heavy-ion collisions is a quark-gluon plasma before hadronization.Comment: 11 pages in LaTeX + 6 figures in p