Measuring definable sets in o-minimal fields

Maříková, Jana; Shiota, Masahiro

doi:10.1007/s11856-015-1234-0

Cited by 6 publications

(20 citation statements)

References 11 publications

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“…rOmpA has been associated with adherence to and invasion of human endothelial cells by interacting with integrin alpha2beta1 (Hillman et al, 2013). The tandem repeat structure was first analyzed in the rompA gene of R. rickettsii (Anderson et al, 1990), but the functions of the tandem repeats remain to be elucidated. Our analysis showed that the tandem repeat regions in rompA are composed of three very similar sequences of 225, 216, and 216 bp in the two species (Figure 4).…”

Section: Resultsmentioning

confidence: 99%

Genomic Features of Rickettsia heilongjiangensis Revealed by Intraspecies Comparison and Detailed Comparison With Rickettsia japonica

Kasama

Fujita²,

Yamamoto

et al. 2019

Front. Microbiol.

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Rickettsia heilongjiangensis is the causative agent of Far-Eastern spotted fever (FESF). In Japan, a human case of FESF was identified in Sendai in Miyagi Prefecture in 2008, and R. heilongjiangensis bacteria were isolated from Haemaphysalis concinna ticks collected in the suspected geographical area of infection. Although the intraspecies genome diversity of Rickettsia has been poorly investigated, our recent analysis revealed extremely low genomic diversity of R. japonica, the agent of Japanese spotted fever, which is a close relative of R. heilongjiangensis. In this study, to investigate the genomic diversity of R. heilongjiangensis and understand the genetic relationship between Japanese and Chinese isolates, we sequenced three isolates from H. concinna ticks collected in Sendai and one isolate from a H. concinna tick collected in Inner Mongolia, China, and performed genomic comparisons between these isolates and strain 054, the type strain isolated from a Dermacentor silvarum tick in Heilongjiang Province, China. Although the three Japanese strains were isolated in 2008, 2009, and 2012, their genome sequences were identical, indicating that H. concinna ticks carrying a single R. heilongjiangensis clone have been distributed in Sendai, Japan. Among the five R. heilongjiangensis isolates, only 81 SNPs and 13 insertion/deletion sites were identified, despite the significant differences in these isolates both geographically and temporally. A significant portion of the 81 SNPs (16/81) were found to be recombinogenic. These results indicate low genomic diversity of R. heilongjiangensis, as observed in R. japonica. We further performed a detailed genomic comparison of R. heilongjiangensis and R. japonica to accurately define conserved and species-specific genes. This analysis revealed that although notable variations were found in the genomic loci encoding RelA/SpoT family proteins and tandem repeats in major surface proteins, there was only a small difference in the gene repertoire between the two species, suggesting that SNPs and small InDels are responsible for the functional or physiological differences between the two species, if present. Through this analysis, several species-specific genomic regions that can serve as ideal PCR targets for distinguishing R. heilongjiangensis and R. japonica were also identified.

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Section: Resultsmentioning

confidence: 99%

Genomic Features of Rickettsia heilongjiangensis Revealed by Intraspecies Comparison and Detailed Comparison With Rickettsia japonica

Kasama

Fujita²,

Yamamoto

et al. 2019

Front. Microbiol.

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“…However, the argument about the saturated field is quite complicated, so we do not discuss it here. See for details.…”

Section: Semialgebraic Chains and Integrationsmentioning

confidence: 99%

“…Then, by using Theorem 1.1, we can show that v n is uniformly bounded in R, that is, there is c ∈ R such that v (d) n (X) < c for any n ∈ N. Moreover, the dth Hausdorff measure H d (X) (or the volume) of X is given by lim n→∞ v (d) n (X), which exists in some saturated field containing R. However, the argument about the saturated field is quite complicated, so we do not discuss it here. See [11] for details.…”

Section: Integrals Of Differential Formsmentioning

confidence: 99%

“…We remark that our definition of

\int_{X} ω

works over not only the real number field

double-struckR

but also any possibly non‐archimedian real closed field

R

. Then the value of the integral does not take values in the same field

R

but in a larger saturated field (see ). In particular, the volume of a bounded definable set in

R^{m}

is bounded within

R

(Remark ).…”

Section: Introductionmentioning

confidence: 99%

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C1-triangulations of semialgebraic sets

Ohmoto

Shiota

2017

Journal of Topology

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We show that every semialgebraic set admits a semialgebraic triangulation such that each closed simplex is C1 differentiable. As an application, we give a straightforward definition of the integration ∫Xω over a compact semialgebraic subset X of a differential form ω on an ambient semialgebraic manifold. This provides a significant simplification of the theory of semialgebraic singular chains and integrations without using geometric measure theory. Our results hold over every (possibly non‐archimedian) real closed field.

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“…Measure theory of real closed fields is delicate for these fields are generally not complete topological spaces (for their natural topology). Recently, T. Kaiser [4] introduced a generalized Lebesgue measure for semi-algebraic sets on real closed fields (see also [8]). It would be interesting to investigate the extent to which the estimate we give on the geometry of the set of generalized critical values is related to the Lebesgue measure introduced in [4].…”

Section: Introductionmentioning

confidence: 99%

A generalized Sard theorem on real closed fields

Valette¹,

Valette

2016

Mathematische Nachrichten

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Abstract. We work with semi-algebraic functions on arbitrary real closed fields. We generalize the notion of critical values and prove a Sard type theorem in our framework. IntroductionThe famous Sard's Theorem asserts that the set of critical values of a C ∞ map f : R n → R k is of Lebesgue measure zero. Let us recall that the set of critical values of f is the image under f of all the elements at which f fails to be a submersion. In this article, we focus on semi-algebraic functions on real closed fields. The notion of Lebesgue measure does not really make sense on a real closed field. However, the dimension of a semi-algebraic set can always be defined and it is well known that one can prove that the set of critical values of a mapping f : R n → R k has dimension less than k [BCR], if R stands for a real closed field. In this note, we establish a related result. We introduce a more general notion of critical values and show a Sard type theorem for these critical values, estimating the size of the set of our generalized critical values. There is no nice geometric measure theory on real closed fields. We therefore make use of the notion of v-thin sets introduced in [V]. Further features of these critical values will be developped in forthcoming articles.We then give an application of our Sard's theorem to the study of asymptotic critical values of real mappings f : X → R k , X ⊂ R n . We shall show that the set of asymptotic critical values of a semi-algebraic mapping f : X → R k has dimension less than k. A similar result was also obtained by K. Kurdyka, P. Orro and S. Simon [KOS] who showed that the set of generalized critical values at infinity for semi-algebraic mappings has dimension less than k. This result lead them to establish a generalization of Ehresmann's fibration theorem for nonproper mappings. This provided an effective way to ensure stability and yield that the set of bifurcation points of a semi-algebraic mapping is relatively small. By R we denote a real closed field. We refer to [BCR] for basic facts about semialgebraic sets. We denote by cl(X) and int(X) the closure and interior of a set and set δX := cl(X) \ int(X) as well as f r(X) := cl(X) \ X. We will write |x| for the Euclidean norm of x ∈ R n and d(·, ·) for the Euclidean distance, while B n (x, r) will stand for the ball of radius r centered at x for this metric. v-thin setsWe introduce the notion of v-thin sets as it was defined in [V].

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Measuring definable sets in o-minimal fields

Cited by 6 publications

References 11 publications

Genomic Features of Rickettsia heilongjiangensis Revealed by Intraspecies Comparison and Detailed Comparison With Rickettsia japonica

Genomic Features of Rickettsia heilongjiangensis Revealed by Intraspecies Comparison and Detailed Comparison With Rickettsia japonica

C1-triangulations of semialgebraic sets

A generalized Sard theorem on real closed fields

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