Abstract
A growing literature provides evidence on multigenerational inequality -- the extent to which socio-economic advantages persist across three or more generations. This chapter reviews its main findings and implications. Most studies find that inequality is more persistent than a naive iteration of conventional parent-child correlations would suggest. We discuss potential interpretations of this new ``fact'' related to (i) latent, (ii) non-Markovian or (iii) non-linear transmission processes, empirical strategies to discriminate between them, and the link between multigenerational and assortative associations.

Abstract
Understanding the role of human behavior in shaping environmental outcomes is crucial for addressing global challenges such as climate change. Environmental systems are influenced not only by natural factors like temperature, but also by human decisions regarding mitigation efforts, which are often based on forecasts or predictions about future environmental conditions. Over time, different outcomes can emerge, including scenarios where the environment deteriorates despite efforts to mitigate, or where successful mitigation leads to environmental resilience. Additionally, fluctuations in the level of human participation in mitigation can occur, reflecting shifts in collective behavior. In this study, we consider a variety of human mitigation decisions, in addition to the feedback loop that is created by changes in human behavior because of environmental changes. While these outcomes are based on simplified models, they offer important insights into the dynamics of human decision-making and the factors that influence effective action in the context of environmental sustainability. This study aims to examine key social dynamics influencing society's response to a worsening climate. While others conclude that homophily prompts greater warming unconditionally, this model finds that homophily can prevent catastrophic effects given a poor initial environmental state. Assuming that poor countries have the resources to do so, a consensus in that class group to defect from the strategy of the rich group (who are generally incentivized to continue ``business as usual'') can frequently prevent the vegetation proportion from converging to 0.

Abstract
Motivated by previous work leveraging factorizations of second- and fourth-order differential operators, a general integral inequality involving higher order derivatives is proven by elementary means. It is then shown how this framework generalizes the notions of Hardy improving potentials and Bessel pairs. Numerous examples of inequalities both new and previously known in the literature are given that may be proven in this manner.

Abstract
This paper investigates the interactions among consumption/savings, investment, and retirement choices with income disaster. We consider low-income people who are exposed to income disaster so that they retire involuntarily when income disaster occurs. The government provides extra income support to low-income retirees who suffer from significant income gaps. We demonstrate that the decision to enter retirement in the event of income disaster depends crucially on the level of income support. In particular, we quantitatively identify a certain income support level below which the optimal decision is to delay retirement. This implies that availability of the government's extra income support can be particularly important for the low-income people to achieve optimal retirement with income disaster.

Abstract
We prove an averaged version of a claim suspected to be true by Alladi, Erd\"os, and Vaaler. Qualitatively, the result states that a divisor sum of a multiplicative function, which obeys certain size constraints, derives most of its value from its "small" divisors.

AI Insights

• The paper sharpens the average bound for multiplicative functions over squarefree integers, surpassing Soundararajan’s estimates.
• Using the Erdos‑Kac distribution of ω(m), the authors prove the large‑divisor contribution is o(x).
• A new product formula links f₀ and f₁, exposing a hidden symmetry in the divisor sum.
• Partial summation translates size constraints into precise asymptotics for the average.
• The authors suggest any further improvement will require a fundamentally new analytic idea.
• The bulk of the divisor sum originates from divisors below a logarithmic threshold, confirming the “small divisor” intuition.
• The bibliography cites classic texts like “Number Theory: An Introduction Through Problems” and key papers on multiplicative functions.