An analogue of a theorem of Sierpinski about the classical operation ( ) provides the motivation for studying κ -Suslin logic, an extension of L κ + ω which is closed under a propositional connective based on ( ). This theorem is used to obtain a complete axiomatization for κ -Suslin logic and an upper bound on the κ -Suslin accessible ordinals (for κ = ω these results are due to Ellentuck [E]). It also yields a weak completeness theorem which we use to generalize a result of Barwise and Kunen [B-K] and show that the least ordinal not H ( κ + ) recursive is the least ordinal not κ -Suslin accessible. We assume familiarity with lectures 3, 4 and 10 of Keisler's Model theory for infinitary logic [Ke]. We use standard notation and terminology including the following. L κ + ω is the logic closed under negation, finite quantification, and conjunction and disjunction over sets of formulas of cardinality at most κ . For κ singular, conjunctions and disjunctions over sets of cardinality κ can be replaced by conjunctions and disjunctions over sets of cardinality less than κ so that we can (and will in §2) assume the formation rules of L κ + ω allow conjunctions and disjunctions only over sets of cardinality strictly less than κ whenever κ is singular.
Little data exists on physiological outcomes associated with bacterial changes that occur after fructan feeding in healthy adults. To this end, we studied potential relationships between changes in gut bacteria composition and host physiological parameters after feeding 3 × 5 g/d of β2‐1 fructan (BF) or maltodextrin to a group of 13 male and 17 female healthy adults in a placebo‐controlled, double‐blinded, randomised crossover‐study consisting of two 28‐d exposures separated by a 14‐d washout. Fasting blood and 1‐d faecal collections were obtained on d0 and d28 of each phase. Well‐being and general health, as well as gastrointestinal symptoms, were determined by questionnaire. Serum lipopolysaccharides (LPS), faecal SCFA, faecal bifidobacteria and indigestion were higher during the BF‐supplemented phase. Numbers of circulating lymphocytes and macrophages were unchanged but BF supplementation decreased serum IL‐10 and increased serum IL‐4. In addition, circulating percentages of CD282 + /TLR2 + myeloid dendritic cells increased as did ex vivo responses to a TLR2 agonist. Culture‐based analysis of feces showed that 87 bacterial species encompassing 30 genera and four phyla were able to use BF as the sole carbon source. Fecal 16S rRNA analysis (26 subjects) showed BF reduced community richness. Two response patterns were observed: in 17/26 subjects, the relative abundance of Bacteroides was reduced and phylotypes within the Bifidobacterium , Faecalibacterium and the family Lachnospiraceae increased during the BF phase. In the remaining subjects, phylotypes aligning within the genera Bacteroides, Prevotella and to a lesser extent Bifidobacterium increased in abundance while abundance of phylotypes within the Faecalibacterium and the family Lachnospiraceae decreased during the BF phase. Few relationships between the faecal community composition and measured parameters were noted: (a) increases in Bacteroidetes relative abundance correlated with increased faecal propionate; (b) subjects in whom Bacteroidetes increased during the BF phase excreted more caproic acid (during both phases); and (c) subjects in whom Bacteroidetes decreased during the BF phase had higher LPS and LPS binding protein (during both phases). The most likely explanation for the observed patterns of faecal community change associated with the BF phase is a distal gut fermentation driven by differences in peptidyl nitrogen. We found no links between bacterial community changes and physiological changes nor was there evidence of physiological changes that would indicate a health benefit in these healthy subjects. Support or Funding Information This trial was undertaken with financial support from Agriculture and Agri‐Food Canada (RPBI# 1501, MK, GDI, LJY), Health Canada (SB), General Mills (MK), Alberta Innovates Bio Solutions (GDI), the Advanced Food and Materials Network (MK), and the National Science and Engineering Research Council (JMJG).
Top Secret Rosies: The Female Computers of World War II is a documentary that focuses on four women who worked as “human computers”, computing individual ballistic trajectories for the Army at the University of Pennsylvania’s Moore School of Electrical Engineering. These trajectories were then compiled into tables at the Army’s Aberdeen Proving Ground (APG). Three men are also featured: two who were members of the Army Air Corps and one, Joseph Chapline, who worked with John Mauchly at the university. Several historians also appear giving commentary. The film gives a flavor of the wartime experiences of the seven principal interviewees and explores how they felt about their work. We learn from the women something of what it was like to work on the tables, and we learn from the men in the planes something of what it was like to use the tables for dropping bombs. We also learn that some of the calculations were done by hand, some were done using calculating machines, and some were done using the university’s differential analyzer, an analog electromechanical computing machine used to solve differential equations. Unfortunately, but understandably, we do not learn precisely what went into creating the tables or what sort of calculations the women were doing. Other technical details that would interest mathematicians are also not included. Nonetheless, the film is interesting and informative. It is particularly suitable for an audience that might not be aware of the pervasiveness of mathematics, in military applications and elsewhere. However, by mislabeling the women in the film as mathematicians, it does somewhat distort the role of women in the mathematical sciences in the midtwentieth century. Furthermore, while Top Secret Rosies shows a piece of history not usually seen, it does not show anything of the history of mathematics or the history of women in mathematics, as is claimed in some reviews of the film. The title of the film is clearly meant to evoke a comparison with “Rosie the Riveter”, a World War II symbol of women who worked in shipyards and aircraft factories and did other jobs previously done mainly by men. This comparison is not really appropriate since it was not unusual for women to do computations before the war. Furthermore, in the 1930s about 15 percent of all the American Ph.D.’s in mathematics were granted to women, and there were at least two hundred American women with Ph.D.’s in mathematics at the start of World War II.1 Many of these women, most notably Reviewed by Judy Green
Environmental enrichment (EE) is commonly included as an important component of animal housing to promote well being of laboratory animals; however, much remains to be learned about the impact of chewable forms of EE on experimental outcomes in the context of nutritional and microbiome-related studies, and whether outcomes differ between sexes. In the present study, nylon chew bones (gnaw sticks, GS) were evaluated for their effects on fermentation profiles, microbial community structure, and cytokine profiles of gastrointestinal and systemic tissues in pair-housed female and male Sprague Dawley (SD) rats. Food consumption and weight gain were not significantly altered by access to GS. Cecal short-chain fatty acid and branched-chain fatty acid profiles significantly differed between sexes in rats with access to GS, and alpha diversity of the microbiome decreased in females provided GS. Sex-related tissue cytokine profiles also significantly differed between rats with and without access to GS. These findings indicate that including GS can influence microbiota and immune-related parameters, in a sex dependent manner. This shows that environmental enrichment strategies need to be clearly reported in publications to properly evaluate and compare experimental results, especially with respect to the use of chewable EE in the context of studies examining diet, microbiome and immune parameters.
Consistency properties and their model existence theorems have provided an important method of constructing models for fragments of L ∞ω . In [E] Ellentuck extended this construction to Suslin logic. One of his extensions, the Borel consistency property, has its extra rule based not on the semantic interpretation of the extra symbols but rather on a theorem of Sierpinski about the classical operation . In this paper we extend that consistency property to the game logic L G and use it to show how one can extend results about and its countable fragments to L G and certain of its countable fragments. The particular formation of L G which we use will allow in the game quantifier infinite alternation of countable conjunctions and disjunctions as well as infinite alternation of quantifiers. In this way L G can be viewed as an extension of Suslin logic.
Let σ be any sequence B 0 , B 1 …, B n , … of transitive sets closed under pairs with for each n . In this paper we show that the smallest admissible set A σ with σ ∈ A σ is Σ 1 compact. Thus we have an entirely new class of explicitly describable uncountable Σ 1 compact sets. The search for uncountable Σ 1 compact languages goes back to Hanf's negative results on compact cardinals [7]. Barwise first showed that all countable admissible sets were Σ 1 compact [1] and then went on to give a characterization of the Σ 1 compact sets in terms of strict reflection [2]. While his characterization has been of interest in understanding the Σ 1 compactness phenomenon it has led to the identification of only one class of uncountable Σ 1 compact sets. In particular, Barwise showed [2], using the above notation, that if ⋃ n B n is power set admissible it satisfies the strict reflection principle and hence is Σ 1 compact. (This result was obtained independently by Karp using algebraic methods [9].) In proving our compactness theorem we follow Makkai's approach to the Barwise Compactness Theorem [12] and use a modified version of Smullyan's abstract consistency property [14]. A direct generalization of Makkai's method to the cofinality ω case yields a proof of the Barwise-Karp result mentioned above [6]. In order to obtain our new result we depart from the usual definition of language and use instead the indexed languages of Karp [9] in which a conjunction is considered to operate on a function whose range is a set of formulas rather than on a set of formulas itself.
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