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We investigate two types of boundedness criteria for bilinear Fourier multiplier operators with symbols with bounded partial derivatives of all (or sufficiently many) orders. Theorems of the first type explicitly prescribe only a certain…
The bifurcation theory of ordinary differential equations (ODEs), and its application to deterministic population models, are by now well established. In this article, we begin to develop a complementary theory for diffusion-like…
Our main theorem is in the generality of the axioms of Hilbert space, and the theory of unbounded operators. Consider two Hilbert spaces such that their intersection contains a fixed vector space D. It is of interest to make a precise…
The free central-limit theorem, a fundamental theorem in free probability, states that empirical averages of freely independent random variables are asymptotically semi-circular. We extend this theorem to general dynamical systems of…
I provide a novel approach to characterizing the set of interim realizable allocations, in the spirit of Matthews (1984) and Border (1991). The approach allows me to identify precisely why exact characterizations are difficult to obtain in…
The ideal Bayesian agent reasons from a global probability model, but real agents are restricted to simplified models which they know to be adequate only in restricted circumstances. Very little formal theory has been developed to help…
Bayesian Optimization, the application of Bayesian function approximation to finding optima of expensive functions, has exploded in popularity in recent years. In particular, much attention has been paid to improving its efficiency on…
Random utility theory models an agent's preferences on alternatives by drawing a real-valued score on each alternative (typically independently) from a parameterized distribution, and then ranking the alternatives according to scores. A…
We develop a novel data-driven approach to the inverse problem of classical statistical mechanics: given experimental data on the collective motion of a classical many-body system, how does one characterise the free energy landscape of that…
Geoffrion's theorem is a fundamental result from mathematical programming assessing the quality of Lagrangian relaxation, a standard technique to get bounds for integer programs. An often implicit condition is that the set of feasible…
Joint state and parameter estimation is a core problem for dynamic Bayesian networks. Although modern probabilistic inference toolkits make it relatively easy to specify large and practically relevant probabilistic models, the silver…
Bayesian priors offer a compact yet general means of incorporating domain knowledge into many learning tasks. The correctness of the Bayesian analysis and inference, however, largely depends on accuracy and correctness of these priors.…
Machine learning tasks may admit multiple competing models that achieve similar performance yet produce conflicting outputs for individual samples -- a phenomenon known as predictive multiplicity. We demonstrate that fairness interventions…
In a configuration space whose boundary can be identified with a subset of its interior, a boundary condition can relate the behaviour of a function on the boundary and in the interior. Additionally, boundary values can appear as additive…
The allocation of resources among multiple agents is a fundamental problem in both economics and computer science. In these settings, fairness plays a crucial role in ensuring social acceptability and practical implementation of resource…
We study multidimensional mechanism design in a common scenario where players have private information about their willingness to pay and their ability to pay. We provide a complete characterization of dominant-strategy incentive-compatible…
Since its inception, artificial intelligence has relied upon a theoretical foundation centered around perfect rationality as the desired property of intelligent systems. We argue, as others have done, that this foundation is inadequate…
Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…
Clustering is a fundamental unsupervised learning problem where a dataset is partitioned into clusters that consist of nearby points in a metric space. A recent variant, fair clustering, associates a color with each point representing its…
We prove a Borel version of the local lemma, i.e. we show that, under suitable assumptions, if the set of variables in the local lemma has a structure of a Borel space, then there exists a satisfying assignment which is a Borel function.…