From the Poisson–Dirichlet diffusions to the Z-measure diffusions, they all have explicit transition densities. We show that the transition densities of the Z-measure diffusions can also be expressed as a mixture of a sequence of probability measures on the Thoma simplex. The coefficients are the same as the coefficients in the Poisson–Dirichlet diffusions. This fact will be uncovered by a dual process method in a special case where the Z-measure diffusions are established through an up–down chain in the Young graph.

Human reasoning is often conceived as an interplay between a more intuitive and deliberate thought process. In the last 50 years, influential fast-and-slow dual-process models that capitalize on this distinction have been used to account for numerous phenomena – from logical reasoning biases, over prosocial behavior, to moral decision making. The present paper clarifies that despite the popularity, critical assumptions are poorly conceived. My critique focuses on two interconnected foundational issues: the exclusivity and switch feature. The exclusivity feature refers to the tendency to conceive intuition and deliberation as generating unique responses such that one type of response is assumed to be beyond the capability of the fast-intuitive processing mode. I review the empirical evidence in key fields and show that there is no solid ground for such exclusivity. The switch feature concerns the mechanism by which a reasoner can decide to shift between more intuitive and deliberate processing. I present an overview of leading switch accounts and show that they are conceptually problematic – precisely because they presuppose exclusivity. I build on these insights to sketch the groundwork for a more viable dual-process architecture and illustrate how it can set a new research agenda to advance the field in the coming years.

This paper proposes a relationship between design thinking and computational thinking. It describes design thinking and computational thinking as two prominent ways of understanding how people address design problems. It suggests that, currently, each of design thinking and computational thinking is defined and theorized in isolation from the other. A two-dimensional ontological space of the ways that people think in addressing problems is proposed, based on the orientation of the thinker towards problem and solution generality/specificity. Placement of design thinking and computational thinking within this space and discussion of their relationship leads to the suggestion of a dual process model for addressing design problems. It suggests that, in this model, design thinking and computational thinking are processes that are ontological mirror images of each other, and are the two processes by which thinkers address problems. Thinkers can move fluently between the two. The paper makes a contribution towards the theoretical foundations of design thinking and proposes questions about how design thinking and computational thinking might be both investigated and taught as constituent parts of a dual process.

We consider the asymptotic behaviour of the stationary tail probabilities in the discrete-time GI/G/1-type queue with countable background state space. These probabilities are presented in matrix form with respect to the background state space, and shown to be the solution of a Markov renewal equation. Using this fact, we consider their decay rates. Applying the Markov renewal theorem, it is shown that certain reasonable conditions lead to the geometric decay of the tail probabilities as the level goes to infinity. We exemplify this result using a discrete-time priority queue with a single server and two types of customer.

In this paper, we are concerned with preservation properties of first and second order by an operator L representable in terms of a stochastic process Z with non-decreasing right-continuous paths. We introduce the derived operator D of L and the derived process V of Z in order to characterize the preservation of absolute continuity and convexity. To obtain different characterizations of the preservation of convexity, we introduce two kinds of duality, the first referring to the process Z and the second to the derived process V. We illustrate the preceding results by considering some examples of interest both in probability and in approximation theory - namely, mixtures, centred subordinators, Bernstein polynomials and beta operators. In most of them, we find bidensities to describe the duality between the derived processes. A unified approach based on stochastic orders is given.

In a branching process with random environments, the probability of ultimate extinction is a function of the environment sequence, and is therefore a random variable. Explicit results about the distribution of this random variable are difficult to obtain in general. Here we assume independent and identically distributed environments and use the special properties of fractional linear generating functions to derive some explicit distributions, which may be singular or absolutely continuous, depending on the values of certain parameters. We also consider briefly tail behaviour close to 1, and provide an extension to cases where probability generating functions are not fractional linear.

Under some regularity conditions, in the supercritical Smith–Wilkinson branching process it is shown that as k, the starting population size, tends to infinity, the rate of convergence of qk, the corresponding extinction probability, to zero is similar to that of:

k–θ, if there exists at least one subcritical state in the random environment space; xkk–α, if there exist only supercritical states in the random environment space; exp , if there exists at least one critical state and the others are supercritical in the random environment space.

Here θ, x, α and c are positive constants determined by the process.