A growing number of social scientists have turned to differential equations as a tool for capturing the dynamic interdependence among a system of variables. Current tools for fitting differential equation models do not provide a straightforward mechanism for diagnosing evidence for qualitative shifts in dynamics, nor do they provide ways of identifying the timing and possible determinants of such shifts. In this paper, we discuss regime-switching differential equation models, a novel modeling framework for representing abrupt changes in a system of differential equation models. Estimation was performed by combining the Kim filter (Kim and Nelson State-space models with regime switching: classical and Gibbs-sampling approaches with applications, MIT Press, Cambridge, 1999) and a numerical differential equation solver that can handle both ordinary and stochastic differential equations. The proposed approach was motivated by the need to represent discrete shifts in the movement dynamics of $n=29$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n= 29$$\end{document} mother–infant dyads during the Strange Situation Procedure (SSP), a behavioral assessment where the infant is separated from and reunited with the mother twice. We illustrate the utility of a novel regime-switching differential equation model in representing children’s tendency to exhibit shifts between the goal of staying close to their mothers and intermittent interest in moving away from their mothers to explore the room during the SSP. Results from empirical model fitting were supplemented with a Monte Carlo simulation study to evaluate the use of information criterion measures to diagnose sudden shifts in dynamics.

We discuss the dyadic John–Nirenberg space that is a generalization of functions of bounded mean oscillation. A John–Nirenberg inequality, which gives a weak type estimate for the oscillation of a function, is discussed in the setting of medians instead of integral averages. We show that the dyadic maximal operator is bounded on the dyadic John–Nirenberg space and provide a method to construct nontrivial functions in the dyadic John–Nirenberg space. Moreover, we prove that the John–Nirenberg space is complete. Several open problems are also discussed.

We study a weighted maximal weak-type inequality for Haar multipliers that can be regarded as a dual problem of Muckenhoupt and Wheeden. More precisely, if Tε is the Haar multiplier associated with the sequence ε with values in [−1, 1], and is the r-maximal operator, then for any weight w and any function f on [0, 1) we have

for some constant Cr depending only on r. We also show that the analogous result does not hold if we replace by the dyadic maximal operator Md. The proof rests on the Bellman function method; using this technique we establish related weighted Lp estimates for p close to 1, and then deduce the main result by extrapolation arguments.

We prove that the extremal sequences for the Bellman function of the dyadic maximal operator behave approximately as eigenfunctions of this operator for a specific eigenvalue. We use this result to prove the analogous one with respect to the Hardy operator.

The current study addressed whether attachment, rumination, and trait depression were predictive of co-rumination in dyadic friendships. The final sample consisted of 205 pairs of same-sex friends (Mage = 18.93 years, SD = 1.31) and 135 (66%) of the dyads were female. Results revealed that an individual's attachment avoidance was significantly related to their own (actor effect) and their friend's (partner effect) co-rumination. Gender moderated the (1) effect of attachment avoidance on co-rumination at the actor and partner levels and (2) effect of attachment anxiety on co-rumination at the partner level. Furthermore, an interaction emerged between friendship duration and trait depression on co-rumination at the actor level. Considering the adjustment trade-offs of co-rumination (e.g., closer friendship quality accompanied with greater levels of depression and anxiety), it is important to understand the predictors of co-rumination and the consequences of it in order to effectively apply intervention efforts.

This study integrated the investment model and attachment theory to examine: (1) whether the link between friendship attachment and commitment would be mediated by relationship features of satisfaction, quality of alternatives, and investment size; and (2) whether these mediation processes would function at the intrapersonal (actor) and interpersonal (partner) levels. The sample included 120 pairs of same-sex friends (78 pairs of women). Data were analysed with the Actor-Partner Interdependence Mediation Model (APIMeM). Mediated actor effects suggest that individuals high in attachment avoidance are less satisfied with and invest less in their current friendship; these in turn, lead them to show less commitment. Furthermore, individuals high in attachment anxiety invest more in their current friendship, which in turn leads to higher commitment. Mediated partner effects suggest that individuals high in attachment avoidance have a friend who feels less satisfied with, invests less, and shows low commitment in their current friendship. The current study demonstrated that a dyadic design is useful in illustrating the dynamics of commitment between close friends.