A Set Theory with Support for Partial Functions

Partial functions can be easily represented in set theory as certain sets of ordered pairs. However, classical set theory provides no special machinery for reasoning about partial functions. For instance, there is no direct way of handling the application of a function to an argument outside its domain as in partial logic. There is also no utilization of lambda-notation and sorts or types as in type theory. This paper introduces a version of von-Neumann-Bernays-Gödel set theory for reasoning about sets, proper classes, and partial functions represented as classes of ordered pairs. The underlying logic of the system is a partial first-order logic, so class-valued terms may be nondenoting. Functions can be specified using lambda-notation, and reasoning about the application of functions to arguments is facilitated using sorts similar to those employed in the logic of the IMPS Interactive Mathematical Proof System. The set theory is intended to serve as a foundation for mechanized mathematics systems.     

A Necessary Relation Algebra for Mereotopology

The standard model for mereotopological structures are Boolean subalgebras of the complete Boolean algebra of regular closed subsets of a nonempty connected regular T ₀ topological space with an additional "contact relation" C defined by xCy x ØA (possibly) more general class of models is provided by the Region Connection Calculus (RCC) of Randell et al. We show that the basic operations of the relational calculus on a "contact relation" generate at least 25 relations in any model of the RCC, and hence, in any standard model of mereotopology. It follows that the expressiveness of the RCC in relational logic is much greater than the original 8 RCC base relations might suggest. We also interpret these 25 relations in the the standard model of the collection of regular open sets in the two-dimensional Euclidean plane.     

Priorean Strict Implication, Q and Related Systems

We introduce a system PSI for a strict implication operator called Priorean strict implication. The semantics for PSI is based on partial Kripke models without accessibility relations. PSI is proved sound and complete with respect to that semantics, and Prior's system Q and related systems are shown to be fragments of PSI or of a mild extension of it.     

声旁部分信息在儿童学习和记忆汉字中的作用

采用类似课堂教学的学习－测验任务,通过3个实验探讨了声旁提供的部分读音信息在儿童学习和记忆汉字中的作用。被试来自北京市两所小学260名四年级学生。实验中要求学生学习和记忆3种生字：（1）声旁提供汉字读音的全部信息,如规则一致字;（2）声旁提供汉字读音的部分信息,如声调不同字或声母不同字;（3）声旁不提供汉字读音的信息,如声旁不知字。要求学生学习所有的字一遍后回忆生字的读音,共学习3遍。实验发现,儿童学习和记忆汉字的正确率随声旁提供的整字读音信息不同而不同：当声旁提供全部信息时正确率最高,当声旁提供部分信息时其次,当声旁没有提供信息时最低;而且声旁提供的部分信息越多,正确率越高。结果表明,儿童对声旁提供的部分读音信息敏感,发展声旁意识对学习和记忆汉字有积极作用     

Division by zero in non-involutive meadows

Meadows have been proposed as alternatives for fields with a purely equational axiomatization. At the basis of meadows lies the decision to make the multiplicative inverse operation total by imposing that the multiplicative inverse of zero is zero. Thus, the multiplicative inverse operation of a meadow is an involution. In this paper, we study ‘non-involutive meadows’, i.e. variants of meadows in which the multiplicative inverse of zero is not zero, and pay special attention to non-involutive meadows in which the multiplicative inverse of zero is one.     

Robust Methods for Moderation Analysis with a Two-Level Regression Model

Moderation analysis has many applications in social sciences. Most widely used estimation methods for moderation analysis assume that errors are normally distributed and homoscedastic. When these assumptions are not met, the results from a classical moderation analysis can be misleading. For more reliable moderation analysis, this article proposes two robust methods with a two-level regression model when the predictors do not contain measurement error. One method is based on maximum likelihood with Student's t distribution and the other is based on M-estimators with Huber-type weights. An algorithm for obtaining the robust estimators is developed. Consistent estimates of standard errors of the robust estimators are provided. The robust approaches are compared against normal-distribution-based maximum likelihood (NML) with respect to power and accuracy of parameter estimates through a simulation study. Results show that the robust approaches outperform NML under various distributional conditions. Application of the robust methods is illustrated through a real data example. An R program is developed and documented to facilitate the application of the robust methods.     

Random effects and extended generalized partial credit models

In this paper it is shown that under the random effects generalized partial credit model for the measurement of a single latent variable by a set of polytomously scored items, the joint marginal probability distribution of the item scores has a closed-form expression in terms of item category location parameters, parameters that characterize the distribution of the latent variable in the subpopulation of examinees with a zero score on all items, and item-scaling parameters. Due to this closed-form expression, all parameters of the random effects generalized partial credit model can be estimated using marginal maximum likelihood estimation without assuming a particular distribution of the latent variable in the population of examinees and without using numerical integration. Also due to this closed-form expression, new special cases of the random effects generalized partial credit model can be identified. In addition to these new special cases, a slightly more general model than the random effects generalized partial credit model is presented. This slightly more general model is called the extended generalized partial credit model. Attention is paid to maximum likelihood estimation of the parameters of the extended generalized partial credit model and to assessing the goodness of fit of the model using generalized likelihood ratio tests. Attention is also paid to person parameter estimation under the random effects generalized partial credit model. It is shown that expected a posteriori estimates can be obtained for all possible score patterns. A simulation study is carried out to show the usefulness of the proposed models compared to the standard models that assume normality of the latent variable in the population of examinees. In an empirical example, some of the procedures proposed are demonstrated.     

Efficacy of Dialectical Behavior Therapy for Adolescents in a Partial Hospitalization Program

This study evaluated dialectical behavior therapy for adolescents (DBT‐A) vs. treatment as usual within a 6‐week partial hospitalization program. The 103 adolescent participants (mean age = 15.27 years) were predominantly girls (n = 63, 61%) with a variety of primary mental health diagnoses. Results indicated that DBT‐A was superior for decreasing symptoms of depression and interpersonal sensitivity, but no statistically significant difference was detected for anxiety or hostility. Implications for treating youth with transdiagnostic identities are discussed.     

Deformation of nanograined Ni–60Co alloy with low stacking fault energy

The evolution of deformation texture in a Ni–60Co alloy with low stacking fault energy and a grain size in the nanometre range has been investigated. The analyses of texture and microstructure suggest different mechanisms of deformation in nanocrystalline as compared to microcrystalline Ni–60Co alloy. In nanocrystalline material, the mechanism responsible for texture formation has been identified as partial slip, whereas in microcrystalline material, a characteristic texture forms due to twinning and shear banding.