Abstract In this paper, we introduce the concept of soft GE-algebras within the framework of GE-algebra theory, extending existing approaches by incorporating principles of soft set theory. We investigate the structural properties of soft GE-algebras, define operations on these structures, and establish criteria for identifying soft subalgebras. Furthermore, we present key theorems that relate the properties of soft GE-algebras to operations such as intersection, union, and mapping, particularly emphasizing the role of GE-morphisms. These findings lay the groundwork for further exploration of soft structures in the context of GE-algebras and their potential applications in decision-making and computational logic.
Abstract Within the lower Wumishan Formation at the eastern edge of the Tai-hang Mountains in North China, a ~ 10 m stratigraphic interval contains of alternately “bright and dark” laminites, with enigmatic, cross-sectional preserved loop structures (2.5 ~ 27.5 cm in length and 0.6 ~ 12 cm in height), named as “loopites” by this study. Based on the different morphologies and formations, the loopites, composed of the cores and annulate laminations, can be divided into three different types which are type Ⅰ, Ⅱ and Ⅲ. Despite the loopites are similar to the loop beddings of soft-sediment deformation structures, we suggest that they are previously undescribed microbial mat structures (MMS). The formation of type Ⅰ is interpreted as the initial microbial mat, grows on the micro-highland of carbonate deposits, are wrapped by the subsequent microbial mats. The core is the initial microbial mat, but the micro-highland, which are formed by increased carbonate sedimentation rate, are not wrapped together. In contrast, the formation of type Ⅱ and Ⅲ is interpreted that the micro-highlands, which also can be treated as the cores and may be formed by the thrombolites, rock debris and the fragments of microbial mats lie on the flat microbial mats, are wrapped by the top-covered and underlaid microbial mats. In consequence, differing from the earthquake-induced loop beddings, the formation of loopites is due to the growth, wrapping and deposition of the microbial mats. Furthermore, the discovery and the possible formation of the loopites may provide a new type of MMS and indicate a stable, anoxic and carbonate supersaturated environment with a relatively weak hydrodynamics for microbial mats to form the annulate structures which is controlled by illumination, microtopography and hydrodynamics.
We define a probabilistic programming language for Gaussian random variables with a first-class exact conditioning construct. We give operational, denotational and equational semantics for this language, establishing convenient properties like exchangeability of conditions. Conditioning on equality of continuous random variables is nontrivial, as the exact observation may have probability zero; this is Borel’s paradox. Using categorical formulations of conditional probability, we show that the good properties of our language are not particular to Gaussians, but can be derived from universal properties, thus generalizing to wider settings. We define the Cond construction, which internalizes conditioning as a morphism, providing general compositional semantics for probabilistic programming with exact conditioning.
Brazier, J, Antrobus, M, Stebbings, GK, Day, SH, Callus, P, Erskine, RM, Bennett, MA, Kilduff, LP, and Williams, AG. Anthropometric and physiological characteristics of elite male rugby athletes. J Strength Cond Res XX(X): 000-000, 2018-This is the first article to review the anthropometric and physiological characteristics required for elite rugby performance within both rugby union (RU) and rugby league (RL). Anthropometric characteristics such as height and body mass, and physiological characteristics such as speed and muscular strength, have previously been advocated as key discriminators of playing level within rugby. This review aimed to identify the key anthropometric and physiological properties required for elite performance in rugby, distinguishing between RU and RL, forwards and backs and competitive levels. There are differences between competitive standards such that, at the elite level, athletes are heaviest (RU forwards ∼111 kg, backs ∼93 kg; RL forwards ∼103 kg, backs ∼90 kg) with lowest % body fat (RU forwards ∼15%, backs ∼12%; RL forwards ∼14%, backs ∼11%), they have most fat-free mass and are strongest (back squat: RU forwards ∼176 kg, backs ∼157 kg; RL forwards ∼188 kg, backs ∼168 kg; bench press: RU forwards ∼131 kg, backs ∼118 kg; RL forwards ∼122 kg, backs ∼113 kg) and fastest (10 m: RU forwards ∼1.87 seconds, backs ∼1.77 seconds; 10 m: RL forwards ∼1.9 seconds, backs ∼1.83 seconds). We also have unpublished data that indicate contemporary RU athletes have less body fat and are stronger and faster than the published data suggest. Regardless, well-developed speed, agility, lower-body power, and strength characteristics are vital for elite performance, probably reflect both environmental (training, diet, etc.) and genetic factors, distinguish between competitive levels, and are therefore important determinants of elite status in rugby.