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We prove that if a parabolic Lipschitz (i.e., Lip(1,1/2)) graph domain has the property that its caloric measure is parabolic A ∞ with respect to surface measure (which property is in turn equivalent to L p solvability of the Dirichlet problem for some finite p), then the function defining the graph has a half-order time derivative in the space of (parabolic) bounded mean oscillation. Equivalently, we prove that the A ∞ property of caloric measure implies, in this case, that the boundary is parabolic uniformly rectifiable. Consequently, by combining our result with the work of Lewis and Murray we resolve a long standing open problem in the field by characterizing those parabolic Lipschitz graph domains for which one has L p solvability (for some p<∞) of the Dirichlet problem for the heat equation. The key idea of our proof is to view the level sets of the Green function as extensions of the original boundary graph for which we can prove (local) square function estimates of Littlewood-Paley type.

We obtain Morita invariant versions of Eilenberg-Watts type theorems, relating Deligne products of finite linear categories to categories of left exact as well as of right exact functors. This makes it possible to switch between different functor categories as well as Deligne products, which is often very convenient. For instance, we can show that applying the equivalence from left exact to right exact functors to the identity functor, regarded as a left exact functor, gives a Nakayama functor. The equivalences of categories we exhibit are compatible with the structure of module categories over finite tensor categories. This leads to a generalization of Radford’s S⁴-theorem to bimodule categories. We also explain the relation of our construction to relative Serre functors on module categories that are constructed via inner Hom functors.

Social norms can help solve pressing societal challenges, from mitigating climate change to reducing the spread of infectious diseases. Despite their relevance, how norms shape cooperation among strangers remains insufficiently understood. Influential theories also suggest that the level of threat faced by different societies plays a key role in the strength of the norms that cultures evolve. Still little causal evidence has been collected. Here we deal with this dual challenge using a 30-day collective-risk social dilemma experiment to measure norm change in a controlled setting. We ask whether a looming risk of collective loss increases the strength of cooperative social norms that may avert it. We find that social norms predict cooperation, causally affect behavior, and that higher risk leads to stronger social norms that are more resistant to erosion when the risk changes. Taken together, our results demonstrate the causal effect of social norms in promoting cooperation and their role in making behavior resilient in the face of exogenous change. Large-scale cooperation is needed to reduce existential risks like those posed by pandemics and climate change. Here the authors demonstrate that social norms can emerge and sustain cooperation in situations of collective risk and that the level of risk influences the strength of the norms.

Together with the fundamentals of probability, random processes and statistical analysis, this insightful book also presents a broad range of advanced topics and applications. There is extensive coverage of Bayesian vs. frequentist statistics, time series and spectral representation, inequalities, bound and approximation, maximum-likelihood estimation and the expectation-maximization (EM) algorithm, geometric Brownian motion and Ito process. Applications such as hidden Markov models (HMM), the Viterbi, BCJR, and Baum-Welch algorithms, algorithms for machine learning, Wiener and Kalman filters, and queueing and loss networks are treated in detail. The book will be useful to students and researchers in such areas as communications, signal processing, networks, machine learning, bioinformatics, econometrics and mathematical finance. With a solutions manual, lecture slides, supplementary materials and MATLAB programs all available online, it is ideal for classroom teaching as well as a valuable reference for professionals.

We prove that on a compact Kähler manifold, the non-pluripolar Monge-Ampère mass of a θ-psh function decreases as the singularities increase. This was conjectured by Boucksom-Eyssidieux-Guedj-Zeriahi, who proved it under the additional assumption of the functions having small unbounded locus. As a corollary, we get a comparison principle for θ-psh functions, analogous to the comparison principle for psh functions due to Bedford-Taylor.

The entropies of Shannon, Rényi and Kolmogorov are analyzed and compared together with their main properties. The entropy of some particular antennas with a pre-fractal shape, also called fractal antennas, is studied. In particular, their entropy is linked with the fractal geometrical shape and the physical performance.

This work presents an analysis of fractional derivatives and fractal derivatives, discussing their differences and similarities. The fractal derivative is closely connected to Haussdorff’s concepts of fractional dimension geometry. The paper distinguishes between the derivative of a function on a fractal domain and the derivative of a fractal function, where the image is a fractal space. Different continuous approximations for the fractal derivative are discussed, and it is shown that the q-calculus derivative is a continuous approximation of the fractal derivative of a fractal function. A similar version can be obtained for the derivative of a function on a fractal space. Caputo’s derivative is also proportional to a continuous approximation of the fractal derivative, and the corresponding approximation of the derivative of a fractional function leads to a Caputo-like derivative. This work has implications for studies of fractional differential equations, anomalous diffusion, information and epidemic spread in fractal systems, and fractal geometry.

A specific goal of the field of cultural evolution is to understand how processes of transmission and selection at the individual level lead to population-wide patterns of cultural diversity and change. Models of cultural evolution have typically assumed that traits are independent of one another and essentially exchangeable. But culture has a structure: traits bear relationships to one another that affect the transmission and selection process itself. Here, we introduce a modelling framework to explore the effect of interdependencies on the process of learning. Through simulations, we find that introducing a simple structure changes the cultural dynamics. Based on a basic filtering mechanism for parsing trait relationships, more elaborate cultural filters emerge. In a mostly incompatible cultural domain of traits, these filters organize culture into mostly (but not fully) consistent and stable systems. Incompatible domains produce small homogeneous cultures, while more compatibility increases size, diversity and group divergence. When individuals copy based on a trait's features (here, its compatibility relationships), they produce more homogeneous cultures than when they copy based on the agent carrying the cultural trait. We discuss the implications of considering cultural systems and filters in the dynamics of cultural change. This article is part of the theme issue 'Foundations of cultural evolution'.

Humans are characterized by an extreme dependence on culturally transmitted information. Such dependence requires the complex integration of social and asocial information to generate effective learning and decision making. Recent formal theory predicts that natural selection should favour adaptive learning strategies, but relevant empirical work is scarce and rarely examines multiple strategies or tasks. We tested nine hypotheses derived from theoretical models, running a series of experiments investigating factors affecting when and how humans use social information, and whether such behaviour is adaptive, across several computer-based tasks. The number of demonstrators, consensus among demonstrators, confidence of subjects, task difficulty, number of sessions, cost of asocial learning, subject performance and demonstrator performance all influenced subjects' use of social information, and did so adaptively. Our analysis provides strong support for the hypothesis that human social learning is regulated by adaptive learning rules.