Related papers: Module Theorem for The General Theory of Stable Mo…
Modularity is a general principle present in many fields. It offers attractive advantages, including, among others, ease of conceptualization, interpretability, scalability, module combinability, and module reusability. The deep learning…
One of the main reasons for topological persistence being useful in data analysis is that it is backed up by a stability (isometry) property: persistence diagrams of $1$-parameter persistence modules are stable in the sense that the…
Inspired by work done for systems of polynomial exponential equations, we study systems of equations involving the modular $j$ function. We show general cases in which these systems have solutions, and then we look at certain situations in…
Propositional formulas that are equivalent in intuitionistic logic, or in its extension known as the logic of here-and-there, have the same stable models. We extend this theorem to propositional formulas with infinitely long conjunctions…
Persistence modules stratify their underlying parameter space, a quality that make persistence modules amenable to study via invariants of stratified spaces. In this article, we extend a result previously known only for one-parameter…
Safe first-order formulas generalize the concept of a safe rule, which plays an important role in the design of answer set solvers. We show that any safe sentence is equivalent, in a certain sense, to the result of its grounding -- to the…
Datalog$^\neg$ is a central formalism used in a variety of domains ranging from deductive databases and abstract argumentation frameworks to answer set programming. Its model theory is the finite counterpart of the logical semantics…
Neural module networks (NMNs) are a popular approach for modeling compositionality: they achieve high accuracy when applied to problems in language and vision, while reflecting the compositional structure of the problem in the network…
The literature in persistent homology often refers to a "structure theorem for finitely generated graded modules over a graded principal ideal domain". We clarify the nature of this structure theorem in this context.
The notion of singular reduction modules, i.e., of singular modules of nonclassical (conditional) symmetry, of differential equations is introduced. It is shown that the derivation of nonclassical symmetries for differential equations can…
We study the modular representation theory of the symmetric and alternating groups. One of the most natural ways to label the irreducible representations of a given group or algebra in the modular case is to show the unitriangularity of the…
This is the first part in a series of papers in which we introduce and develop a natural, general tensor category theory for suitable module categories for a vertex (operator) algebra. This theory generalizes the tensor category theory for…
Multisorted modules, equivalently representations of quivers, equivalently additive functors on preadditive categories, encompass a wide variety of additive structures. In addition, every module has a natural and useful multisorted…
We develop a comprehensive theory of the stable representation categories of several sequences of groups, including the classical and symmetric groups, and their relation to the unstable categories. An important component of this theory is…
We construct a moduli space of stable pairs over a smooth projective variety, parametrizing morphisms from a fixed coherent sheaf to a varying sheaf of fixed topological type, subject to a stability condition. This generalizes the notion…
Scaling model capacity has been vital in the success of deep learning. For a typical network, necessary compute resources and training time grow dramatically with model size. Conditional computation is a promising way to increase the number…
Recent work has shown that memory modules are crucial for the generalization ability of neural networks on learning simple algorithms. However, we still have little understanding of the working mechanism of memory modules. To alleviate this…
It is well known that, under certain conditions, it is possible to split logic programs under stable model semantics, i.e. to divide such a program into a number of different "levels", such that the models of the entire program can be…
The pioneering work of Blok and J\'onsson and its further development by Galatos and Tsinakis initiated an abstract study of consequence relations using the tools of module theory, where consequence relations over all types of syntactic…
We find the model completion of the theory modules over $A$, where $A$ is a finitely generated commutative algebra over a field $K$. This is done in a context where the field $K$ and the module are represented by sorts in the theory, so…