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According to the classification scheme of the generalized random matrix ensembles, we present various kinds of concrete examples of the generalized ensemble, and derive their joint density functions in an unified way by one simple formula…
We propose a novel Bayesian model framework for discrete ordinal and count data based on conditional transformations of the responses. The conditional transformation function is estimated from the data in conjunction with an a priori chosen…
We develop a representation theory of categories as a means to explore characteristic structures in algebra. Characteristic structures play a critical role in isomorphism testing of groups and algebras, and their construction and…
We introduce a new class of extensions of terms that consists in navigation strategies and insertion of contexts. We introduce an operation of combination on this class which is associative, admits a neutral element and so that each…
We present a categorical construction for modelling causal structures within a general class of process theories that include the theory of classical probabilistic processes as well as quantum theory. Unlike prior constructions within…
Seq2seq models have been shown to struggle with compositional generalisation, i.e. generalising to new and potentially more complex structures than seen during training. Taking inspiration from grammar-based models that excel at…
We generalize the Bia{\l}ynicki-Birula decomposition from actions of $G_m$ on smooth varieties to actions of linearly reductive group ${\bf G}$ on finite type schemes and algebraic spaces. We also provide a relative version and briefly…
The growing complexity of modern practical problems puts high demands on the mathematical modelling. Given that various models can be used for modelling one physical phenomenon, the role of model comparison and model choice becomes…
We describe a design-based framework for drawing causal inference in general randomized experiments. Causal effects are defined as linear functionals evaluated at unit-level potential outcome functions. Assumptions about the potential…
We introduce semiframes (an algebraic structure) and investigate their duality with semitopologies (a topological one). Both semitopologies and semiframes are relatively recent developments, arising from a novel application of topological…
In this paper we use topological techniques to construct generalized trace and modified dimension functions on ideals in certain ribbon categories. Examples of such ribbon categories naturally arise in representation theory where the usual…
We provide a motivated introduction to the theory of categorical actions of groups and the local geometric Langlands program. Along the way we emphasize applications, old and new, to the usual representation theory of reductive and affine…
This essay advocates the view that any problem that has a meaningful empirical content, can be formulated in constructive, more definitely, finite terms. We consider combinatorial models of dynamical systems and approaches to statistical…
We introduce a theory for encoding and manipulating algebraic data on categories via $\textit{concentration structures}$, which are equivalence relations on morphisms that satisfy certain axioms. For any category with a concentration…
Interest is growing in social learning models where users share opinions and adjust their beliefs in response to others. This paper introduces generalized-bias opinion models, an extension of the DeGroot model, that captures a broader range…
Valuation algebras abstract a large number of formalisms for automated reasoning and enable the definition of generic inference procedures. Many of these formalisms provide some notions of solutions. Typical examples are satisfying…
This paper investigates some issues arising in categorical models of reversible logic and computation. Our claim is that the structural (coherence) isomorphisms of these categorical models, although generally overlooked, have decidedly…
The paper presents a new algorithmic construction of a finite generating set of rational invariants for the rational action of an algebraic group on the affine space. The construction provides an algebraic counterpart of the moving frame…
In this paper we give a new foundational, categorical formulation for operations and relations and objects parameterizing them. This generalizes and unifies the theory of operads and all their cousins including but not limited to PROPs,…
Foundation models can be disruptive for future AI development by scaling up deep learning in terms of model size and training data's breadth and size. These models achieve state-of-the-art performance (often through further adaptation) on a…