Order algebraizable logics

AbstractThis paper develops an order-theoretic generalization of Blok and Pigozziʼs notion of an algebraizable logic. Unavoidably, the ordered model class of a logic, when it exists, is not unique. For uniqueness, the definition must be relativized, either syntactically or semantically. In sentential systems, for instance, the order algebraization process may be required to respect a given but arbitrary polarity on the signature. With every deductive filter of an algebra of the pertinent type, the polarity associates a reflexive and transitive relation called a Leibniz order, analogous to the Leibniz congruence of abstract algebraic logic (AAL). Some core results of AAL are extended here to sentential systems with a polarity. In particular, such a system is order algebraizable if the Leibniz order operator has the following four independent properties: (i) it is injective, (ii) it is isotonic, (iii) it commutes with the inverse image operator of any algebraic homomorphism, and (iv) it produces anti-symmetric orders when applied to filters that define reduced matrix models. Conversely, if a sentential system is order algebraizable in some way, then the order algebraization process naturally induces a polarity for which the Leibniz order operator has properties (i)–(iv)

Extreme Ultra-Violet Spectroscopy of the Lower Solar Atmosphere During Solar Flares

The extreme ultraviolet portion of the solar spectrum contains a wealth of diagnostic tools for probing the lower solar atmosphere in response to an injection of energy, particularly during the impulsive phase of solar flares. These include temperature and density sensitive line ratios, Doppler shifted emission lines and nonthermal broadening, abundance measurements, differential emission measure profiles, and continuum temperatures and energetics, among others. In this paper I shall review some of the advances made in recent years using these techniques, focusing primarily on studies that have utilized data from Hinode/EIS and SDO/EVE, while also providing some historical background and a summary of future spectroscopic instrumentation.Comment: 34 pages, 8 figures. Submitted to Solar Physics as part of the Topical Issue on Solar and Stellar Flare

Joint modeling of longitudinal outcomes and survival using latent growth modeling approach in a mesothelioma trial

Joint modeling of longitudinal and survival data can provide more efficient and less biased estimates of treatment effects through accounting for the associations between these two data types. Sponsors of oncology clinical trials routinely and increasingly include patient-reported outcome (PRO) instruments to evaluate the effect of treatment on symptoms, functioning, and quality of life. Known publications of these trials typically do not include jointly modeled analyses and results. We formulated several joint models based on a latent growth model for longitudinal PRO data and a Cox proportional hazards model for survival data. The longitudinal and survival components were linked through either a latent growth trajectory or shared random effects. We applied these models to data from a randomized phase III oncology clinical trial in mesothelioma. We compared the results derived under different model specifications and showed that the use of joint modeling may result in improved estimates of the overall treatment effect

Relative congruence formulas and decompositions in quasivarieties

Quasivarietal analogues of uniform congruence schemes are discussed, and their relationship with the equational definability of principal relative congruences (EDPRC) is established, along with their significance for relative congruences on subalgebras of products. Generalizing the situation in varieties, we prove that a quasivariety is relatively ideal iff it has EDPRC; it is relatively filtral iff it is relatively semisimple with EDPRC. As an application, it is shown that a finitary sentential logic, algebraized by a quasivariety K, has a classical inconsistency lemma if and only if K is relatively filtral and the subalgebras of its nontrivial members are nontrivial. A concrete instance of this result is exhibited, in which K is not a variety. Finally, for quasivarieties M⊆K, we supply some conditions under which M is the restriction to K of a variety, assuming that K has EDPRC.The second author was supported in part by the National Research Foundation of South Africa (UID 85407).https://link.springer.com/journal/122482018-11-27hj2017Mathematics and Applied Mathematic