Related papers: Categorical Diffusion of Weighted Lattices

We introduce a pointfree theory of convergence on lattices and coframes. A convergence lattice is a lattice $L$ with a monotonic map $\lim_L$ from the lattice of filters on $L$ to $L$, meant to be an abstract version of the map sending…

We develop a categorical foundation for belief propagation on factor graphs. We construct the free hypergraph category \(\Syn_\Sigma\) on a typed signature and prove its universal property, yielding compositional semantics via a unique…

Mimicking the idea of the generalized Hamming weight of linear codes, we introduce a new lattice invariant, the generalized theta series. Applications range from identifying stable lattices to the lattice isomorphism problem. Moreover, we…

For a quantaloid $\mathcal{Q}$, considered as a bicategory, Walters introduced categories enriched in $\mathcal{Q}$. Here we extend the study of monad-quantale-enriched categories of the past fifteen years by introducing…

We present a variational theory of integrable differential-difference equations (semi-discrete integrable systems). This is a natural extension of the ideas known by the names "Lagrangian multiforms" and "Pluri-Lagrangian systems", which…

We investigate structure-preserving finite element discretizations of the steady-state Stefan--Maxwell diffusion problem which governs diffusion within a phase consisting of multiple species. An approach inspired by augmented Lagrangian…

In the effective topos there exists a chain-complete distributive lattice with a monotone and progressive endomap which does not have a fixed point. Consequently, the Bourbaki-Witt theorem and Tarski's fixed-point theorem for chain-complete…

We develop a theoretical framework for the diffusion of a single unconstrained species of atoms on a crystal lattice that provides a generalization of the classical theories of atomic diffusion and diffusion-induced phase separation to…

High-energy collisions at the Large Hadron Collider (LHC) provide valuable insights into open questions in particle physics. However, detector effects must be corrected before measurements can be compared to certain theoretical predictions…

This paper contributes to a theory of the behaviour of "finite-state" systems that is generic in the system type. We propose that such systems are modeled as coalgebras with a finitely generated carrier for an endofunctor on a locally…

We use the framework of generalized entanglement wedges to revisit the connected wedge theorem (CWT). This construction identifies an entanglement wedge associated for any bulk region and allows us to rephrase the CWT in terms of the…

We associate to a decorated liability network a liability sheaf on a directed hypergraph whose hyperedges separate the distribution of payments from the collection of receipts. Clearing configurations are precisely the global sections of…

We propose a unified four-dimensional (4D) spatiotemporal formulation for time-dependent convection-diffusion problems that preserves underlying physical structures. By treating time as an additional space-like coordinate, the evolution…

The representation theorem for odd or even involutive FLe-chains by bunches of layer groups, as discussed in [10], is redefined to demonstrate a more straightforward constructional relationship between odd or even involutive FLe-chains and…

Diffusion (score-based) generative models have been widely used for modeling various types of complex data, including images, audios, and point clouds. Recently, the deep connection between forward-backward stochastic differential equations…

We study the mixed formulation of the abstract Hodge Laplacian on axisymmetric domains with general data through Fourer-finite-element-methods in weighted functions spaces. Closed Hilbert complexes and commuting projectors are used through…

We restate a process presented by Stanley as a technique to prove that there exists exactly one $d$-differential distributive lattice for any positive integer $d$. This process can be trivially extended to apply to distributive finitary…

We study expansive dynamical systems in the setting of distributive lattices and their automorphisms, the usual notion of expansiveness for a homeomorphism of a compact metric space being the particular case when the lattice is the topology…

We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, $\mathcal{D}_{h}$. In fact, we prove that every sublattice of any hyperarithmetic lattice…

While diffusion models excel at generating continuous data such as images, adapting them to discrete tasks has relied on indirect approaches that either operate in continuous embedding spaces or use token masking mechanisms, both of which…