Embedding theorems and finiteness properties for residuated structures and substructural logics

Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2008.Paper 1. This paper establishes several algebraic embedding theorems, each of which asserts that a certain kind of residuated structure can be embedded into a richer one. In almost all cases, the original structure has a compatible involution, which must be preserved by the embedding. The results, in conjunction with previous findings, yield separative axiomatizations of the deducibility relations of various substructural formal systems having double negation and contraposition axioms. The separation theorems go somewhat further than earlier ones in the literature, which either treated fewer subsignatures or focussed on the conservation of theorems only. Paper 2. It is proved that the variety of relevant disjunction lattices has the finite embeddability property (FEP). It follows that Avron’s relevance logic RMImin has a strong form of the finite model property, so it has a solvable deducibility problem. This strengthens Avron’s result that RMImin is decidable. Paper 3. An idempotent residuated po-monoid is semiconic if it is a subdirect product of algebras in which the monoid identity t is comparable with all other elements. It is proved that the quasivariety SCIP of all semiconic idempotent commutative residuated po-monoids is locally finite. The lattice-ordered members of this class form a variety SCIL, which is not locally finite, but it is proved that SCIL has the FEP. More generally, for every relative subvariety K of SCIP, the lattice-ordered members of K have the FEP. This gives a unified explanation of the strong finite model property for a range of logical systems. It is also proved that SCIL has continuously many semisimple subvarieties, and that the involutive algebras in SCIL are subdirect products of chains. Paper 4. Anderson and Belnap’s implicational system RMO can be extended conservatively by the usual axioms for fusion and for the Ackermann truth constant t. The resulting system RMO is algebraized by the quasivariety IP of all idempotent commutative residuated po-monoids. Thus, the axiomatic extensions of RMO are in one-to-one correspondence with the relative subvarieties of IP. It is proved here that a relative subvariety of IP consists of semiconic algebras if and only if it satisfies x (x t) x. Since the semiconic algebras in IP are locally finite, it follows that when an axiomatic extension of RMO has ((p t) p) p among its theorems, then it is locally tabular. In particular, such an extension is strongly decidable, provided that it is finitely axiomatized

Some Locally Tabular Logics with Contraction and Mingle

Anderson and Belnap’s implicational system RMO can be extended conservatively by the usual axioms for fusion and for the Ackermann truth constant t. The resulting system RMO is algebraized by the quasivariety IP of all idempotent commutative residuated po-monoids. Thus, the axiomatic extensions of RMO are in one-to-one correspondence with the relative subvarieties of IP. An algebra in IP is called semiconic if it decomposes subdirectly (in IP) into algebras where the identity element t is order-comparable with all other elements. The semiconic algebras in IP are locally finite. It is proved here that a relative subvariety of IP consists of semiconic algebras if and only if it satisfies x ≈ (x → t) → x. It follows that if an axiomatic extension of RMO has ((p → t) → p) → p among its theorems then it is locally tabular. In particular, such an extension is strongly decidable, provided that it is finitely axiomatized

Five insights from the Global Burden of Disease Study 2019

The Global Burden of Diseases, Injuries, and Risk Factors Study (GBD) 2019 provides a rules-based synthesis of the available evidence on levels and trends in health outcomes, a diverse set of risk factors, and health system responses. GBD 2019 covered 204 countries and territories, as well as first administrative level disaggregations for 22 countries, from 1990 to 2019. Because GBD is highly standardised and comprehensive, spanning both fatal and non-fatal outcomes, and uses a mutually exclusive and collectively exhaustive list of hierarchical disease and injury causes, the study provides a powerful basis for detailed and broad insights on global health trends and emerging challenges. GBD 2019 incorporates data from 281 586 sources and provides more than 3.5 billion estimates of health outcome and health system measures of interest for global, national, and subnational policy dialogue. All GBD estimates are publicly available and adhere to the Guidelines on Accurate and Transparent Health Estimate Reporting. From this vast amount of information, five key insights that are important for health, social, and economic development strategies have been distilled. These insights are subject to the many limitations outlined in each of the component GBD capstone papers.Peer reviewe

Precision Measurement of Cosmic-Ray Nitrogen and its Primary and Secondary Components with the Alpha Magnetic Spectrometer on the International Space Station

A precision measurement of the nitrogen flux with rigidity (momentum per unit charge) from 2.2 GV to 3.3 TV based on 2.2 x 10(6) events is presented. The detailed rigidity dependence of the nitrogen flux spectral index is presented for the first time. The spectral index rapidly hardens at high rigidities and becomes identical to the spectral indices of primary He, C, and O cosmic rays above similar to 700 GV. We observed that the nitrogen flux Phi(N) can be presented as the sum of its primary component Phi(P)(N) and secondary component Phi(S)(N), Phi(N) = Phi(P)(N) + Phi(S)(N), and we found Phi(N) is well described by the weighted sum of the oxygen flux Phi(O) (primary cosmic rays) and the boron flux Phi(B) (secondary cosmic rays), with Phi(P)(N) = (0.090 +/- 0.002) x Phi(O) and Phi(S)(N) = (0.62 +/- 0.02) x Phi(B) over the entire rigidity range. This corresponds to a change of the contribution of the secondary cosmic ray component in the nitrogen flux from 70% at a few GV to < 30% above 1 TV

Observation of New Properties of Secondary Cosmic Rays Lithium, Beryllium, and Boron by the Alpha Magnetic Spectrometer on the International Space Station

We report on the observation of new properties of secondary cosmic rays Li, Be, and B measured in the rigidity (momentum per unit charge) range 1.9 GV to 3.3 TV with a total of 5.4 x 10(6) nuclei collected by AMS during the first five years of operation aboard the International Space Station. The Li and B fluxes have an identical rigidity dependence above 7 GV and all three fluxes have an identical rigidity dependence above 30 GV with the Li/Be flux ratio of 2.0 +/- 0.1. The three fluxes deviate from a single power law above 200 GV in an identical way. This behavior of secondary cosmic rays has also been observed in the AMS measurement of primary cosmic rays He, C, and O but the rigidity dependences of primary cosmic rays and of secondary cosmic rays are distinctly different. In particular, above 200 GV, the secondary cosmic rays harden more than the primary cosmic rays

Observation of Fine Time Structures in the Cosmic Proton and Helium Fluxes with the Alpha Magnetic Spectrometer on the International Space Station

We present the precision measurement from May 2011 to May 2017 (79 Bartels rotations) of the proton fluxes at rigidities from 1 to 60 GV and the helium fluxes from 1.9 to 60 GV based on a total of 1 x 109 events collected with the Alpha Magnetic Spectrometer aboard the International Space Station. This measurement is in solar cycle 24, which has the solar maximum in April 2014. We observed that, below 40 GV, the proton flux and the helium flux show nearly identical fine structures in both time and relative amplitude. The amplitudes of the flux structures decrease with increasing rigidity and vanish above 40 GV. The amplitudes of the structures are reduced during the time period, which started one year after solar maximum, when the proton and helium fluxes steadily increase. Above similar to 3 GV the p/He flux ratio is time independent. We observed that below similar to 3 GV the ratio has a long-term decrease coinciding with the period during which the fluxes start to rise

Observation of the Identical Rigidity Dependence of He, C, and O Cosmic Rays at High Rigidities by the Alpha Magnetic Spectrometer on the International Space Station

We report the observation of new properties of primary cosmic rays He, C, and O measured in the rigidity (momentum/charge) range 2 GV to 3 TV with 90 x 10(6) helium, 8.4 x 10(6) carbon, and 7.0 x 10(6) oxygen nuclei collected by the Alpha Magnetic Spectrometer ( AMS) during the first five years of operation. Above 60 GV, these three spectra have identical rigidity dependence. They all deviate from a single power law above 200 GV and harden in an identical way

Global, regional, and national burden of rheumatoid arthritis, 1990–2020, and projections to 2050: a systematic analysis of the Global Burden of Disease Study 2021

Background Rheumatoid arthritis is a chronic autoimmune inflammatory disease associated with disability and premature death. Up-to-date estimates of the burden of rheumatoid arthritis are required for health-care planning, resource allocation, and prevention. As part of the Global Burden of Diseases, Injuries, and Risk Factors Study (GBD) 2021, we provide updated estimates of the prevalence of rheumatoid arthritis and its associated deaths and disability-adjusted life-years (DALYs) by age, sex, year, and location, with forecasted prevalence to 2050. Methods Rheumatoid arthritis prevalence was estimated in 204 countries and territories from 1990 to 2020 using Bayesian meta-regression models and data from population-based studies and medical claims data (98 prevalence and 25 incidence studies). Mortality was estimated from vital registration data with the Cause of Death Ensemble model (CODEm). Years of life lost (YLL) were calculated with use of standard GBD lifetables, and years lived with disability (YLDs) were estimated from prevalence, a meta-analysed distribution of rheumatoid arthritis severity, and disability weights. DALYs were calculated by summing YLLs and YLDs. Smoking was the only risk factor analysed. Rheumatoid arthritis prevalence was forecast to 2050 by logistic regression with Socio-Demographic Index as a predictor, then multiplying by projected population estimates. Findings In 2020, an estimated 17·6 million (95% uncertainty interval 15·8–20·3) people had rheumatoid arthritis worldwide. The age-standardised global prevalence rate was 208·8 cases (186·8–241·1) per 100 000 population, representing a 14·1% (12·7–15·4) increase since 1990. Prevalence was higher in females (age-standardised female-to-male prevalence ratio 2·45 [2·40–2·47]). The age-standardised death rate was 0·47 (0·41–0·54) per 100 000 population (38 300 global deaths [33 500–44 000]), a 23·8% (17·5–29·3) decrease from 1990 to 2020. The 2020 DALY count was 3 060 000 (2 320 000–3 860 000), with an age-standardised DALY rate of 36·4 (27·6–45·9) per 100 000 population. YLDs accounted for 76·4% (68·3–81·0) of DALYs. Smoking risk attribution for rheumatoid arthritis DALYs was 7·1% (3·6–10·3). We forecast that 31·7 million (25·8–39·0) individuals will be living with rheumatoid arthritis worldwide by 2050. Interpretation Rheumatoid arthritis mortality has decreased globally over the past three decades. Global age-standardised prevalence rate and YLDs have increased over the same period, and the number of cases is projected to continue to increase to the year 2050. Improved access to early diagnosis and treatment of rheumatoid arthritis globally is required to reduce the future burden of the disease. Funding Bill & Melinda Gates Foundation, Institute of Bone and Joint Research, and Global Alliance for Musculoskeletal Health

Antiproton Flux, Antiproton-to-Proton Flux Ratio, and Properties of Elementary Particle Fluxes in Primary Cosmic Rays Measured with the Alpha Magnetic Spectrometer on the International Space Station

A precision measurement by AMS of the antiproton flux and the antiproton-to-proton flux ratio in primary cosmic rays in the absolute rigidity range from 1 to 450 GV is presented based on 3.49 x 10(5) antiproton events and 2.42 x 10(9) proton events. The fluxes and flux ratios of charged elementary particles in cosmic rays are also presented. In the absolute rigidity range similar to 60 to similar to 500 GV, the antiproton (p) over bar, proton p, and positron e(+) fluxes are found to have nearly identical rigidity dependence and the electron e(-) flux exhibits a different rigidity dependence. Below 60 GV, the ((p) over bar /p), ((p) over bar /e(+)), and (p/e(+)) flux ratios each reaches a maximum. From similar to 60 to similar to 500 GV, the ((p) over bar /p), ((p) over bar /e(+)), and (p/e(+)) flux ratios show no rigidity dependence. These are new observations of the properties of elementary particles in the cosmos