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Number systems over orders.

ABSTRACT: Let K be a number field of degree k and let O be an order in K . A generalized number system over O (GNS for short) is a pair (p,D) where p?O[x] is monic and D?O is a complete residue system modulo p(0) containing 0. If each a?O[x] admits a representation of the form a??j=0?-1djxj(modp) with ??N and d0,…,d?-1?D then the GNS (p,D) is said to have the finiteness property. To a given fundamental domain F of the action of Zk on Rk we associate a class GF:={(p,DF):p?O[x]} of GNS whose digit sets DF are defined in terms of F in a natural way. We are able to prove general results on the finiteness property of GNS in GF by giving an abstract version of the well-known "dominant condition" on the absolute coefficient p(0) of p. In particular, depending on mild conditions on the topology of F we characterize the finiteness property of (p(x±m),DF) for fixed p and large m?N . Using our new theory, we are able to give general results on the connection between power integral bases of number fields and GNS.

SUBMITTER: Petho A

PROVIDER: S-EPMC6190796 | biostudies-other | 2018

REPOSITORIES: biostudies-other

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Number systems over orders.

Pethő Attila A Thuswaldner Jörg J

Monatshefte fur Mathematik 20180518 4

Let K be a number field of degree k and let O be an order in K . A generalized number system over O (GNS for short) is a pair ( p , D ) where p ∈ O [ x ] is monic and D ⊂ O is a complete residue system modulo p(0) containing 0. If each a ∈ O [ x ] admits a representation of the form a ≡ ∑ j = 0 ℓ - 1 d j x j ( mod p ) with ℓ ∈ N and d 0 , … , d ℓ - 1 ∈ D then the GNS ( p , D ) is said to have the finiteness property. To a given fundamental domain F of th ...[more]

PMID: 30393396

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Project description:Background/objectivesMany older persons with chronic illness use Physician Orders for Life-Sustaining Treatment (POLST) to document portable medical orders for emergency care. However, some POLSTs contain combinations of orders that do not translate into a cohesive care plan (eg, cardiopulmonary resuscitation [CPR] without intensive care, or intensive care without antibiotics). This study characterizes the prevalence and predictors of POLSTs with conflicting orders.DesignRetrospective cohort study.SettingLarge academic health system.ParticipantsA total of 3,123 POLST users with chronic life-limiting illness who died between 2010 and 2015 (mean age = 69.7 years).MeasurementsIn a retrospective review of all POLSTs in participants' electronic health records, we describe the prevalence of POLSTs with conflicting orders for cardiac arrest and medical interventions, and use clustered logistic regression to evaluate potential predictors of conflicting orders. We also examine the prevalence of conflicts between POLST orders for antibiotics and artificial nutrition with orders for cardiac arrest or medical interventions.ResultsAmong 3,924 complete POLSTs belonging to 3,123 decedents, 209 (5.3%) POLSTs contained orders to "attempt CPR" paired with orders for "limited interventions" or "comfort measures only"; 745/3169 (23.5%) POLSTs paired orders to restrict antibiotics with orders to deliver non-comfort-only care; and, 170/3098 (5.5%) POLSTs paired orders to withhold artificial nutrition with orders to deliver CPR or intensive care. Among POLSTs with orders to avoid intensive care, orders to attempt CPR were more likely to be present in POLSTs completed earlier in the patient's illness course (adjusted odds ratio = 1.27 per twofold increase in days from POLST to death; 95% confidence interval = 1.18-1.36; P < .001).ConclusionAlthough most POLSTs are actionable by clinicians, 5% had conflicting orders for cardiac arrest and medical interventions, and 24% had one or more conflicts between orders for cardiac arrest, medical interventions, antibiotics, and artificial nutrition. These conflicting orders make implementation of POLST challenging for clinicians in acute care settings.

Project description:ObjectivesTo characterise adoption and explore specific clinical and patient factors that might influence pulse oximetry and oxygen use in low-income and middle-income countries (LMICs) over time; to highlight useful considerations for entities working on programmes to improve access to pulse oximetry and oxygen.DesignA multihospital retrospective cohort study.SettingsAll admissions (n=132 737) to paediatric wards of 18 purposely selected public hospitals in Kenya that joined a Clinical Information Network (CIN) between March 2014 and December 2020.OutcomesPulse oximetry use and oxygen prescription on admission; we performed growth-curve modelling to investigate the association of patient factors with study outcomes over time while adjusting for hospital factors.ResultsOverall, pulse oximetry was used in 48.8% (64 722/132 737) of all admission cases. Use rose on average with each month of participation in the CIN (OR: 1.11, 95% CI 1.05 to 1.18) but patterns of adoption were highly variable across hospitals suggesting important factors at hospital level influence use of pulse oximetry. Of those with pulse oximetry measurement, 7% (4510/64 722) had hypoxaemia (SpO2 <90%). Across the same period, 8.6% (11 428/132 737) had oxygen prescribed but in 87%, pulse oximetry was either not done or the hypoxaemia threshold (SpO2 <90%) was not met. Lower chest-wall indrawing and other respiratory symptoms were associated with pulse oximetry use at admission and were also associated with oxygen prescription in the absence of pulse oximetry or hypoxaemia.ConclusionThe adoption of pulse oximetry recommended in international guidelines for assessing children with severe illness has been slow and erratic, reflecting system and organisational weaknesses. Most oxygen orders at admission seem driven by clinical and situational factors other than the presence of hypoxaemia. Programmes aiming to implement pulse oximetry and oxygen systems will likely need a long-term vision to promote adoption, guideline development and adherence and continuously examine impact.

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Number systems over orders.

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Number systems over orders.

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