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Combinatorial generation of expander families and Lindenmayer-style development models are both parallel in nature. Both can be handled within proposed parallel graph grammar formalism. Their first-order properties can then be checked by…
We introduce a notion of a filtered model structure and use this notion to produce various model structures on pro-categories. This framework generalizes several known examples. We give several examples, including a homotopy theory for…
We survey some recent results about the order structure of various kinds of ultrafilters. More precisely, we study Rudin-Keisler and Tukey reducibility in classes of selective, stable ordered-union, and P-point ultrafilters. Although these…
In this paper we explore the connection between the ranks of the magnitude homology groups of a graph and the structure of its subgraphs. To this end, we introduce variants of magnitude homology called eulerian magnitude homology and…
In this work, we introduce a filtration on temporal graphs based on $\delta$-temporal motifs (recurrent subgraphs), yielding a multi-scale representation of temporal structure. Our temporal filtration allows tools developed for filtered…
Let $(M,\scott X) \models \ACA$ be such that $P_\scott X$, the collection of all unbounded sets in $\scott X$, admits a definable complete ultrafilter and let $T$ be a theory extending first order arithmetic coded in $\scott X$ such that…
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.
This paper is devoted to systematic studies of some extensions of first-order G\"odel logic. The first extension is the first-order rational G\"odel logic which is an extension of first-order G\"odel logic, enriched by countably many…
The low-energy expansion of closed-string scattering amplitudes at genus one introduces infinite families of non-holomorphic modular forms called modular graph forms. Their differential and number-theoretic properties motivated Brown's…
We present an introductory survey to first order logic for metric structures and its applications to C*-algebras.
Two graphs are cospectral if their respective adjacency matrices have the same multiset of eigenvalues, and generalized cospectral if they are cospectral and so are their complements. We study generalized cospectrality in relation to…
We investigate various groupoids associated to an arbitrary inverse semigroup with zero. We show that the groupoid of filters with respect to the natural partial order is isomorphic to the groupoid of germs arising from the standard action…
Assuming an abstract comparison principle called the Ultrapower Axiom, which is motivated by the comparison process of inner model theory and generalizes the statement that the Mitchell order is linear on normal ultrafilters, we…
Local-order-invariant (first-order) logic is an extension of first-order logic where formulae have access to a ternary local order relation on the Gaifman graph, provided that the truth value does not depend on the specific order relation…
We deal with the existing problem of filtered multiplicative bases of finite-dimensional associative algebras. For an associative algebra A over a field, we investigate when the property of having a filtered multiplicative basis is…
An approach to universal (meta-)logical reasoning in classical higher-order logic is employed to explore and study simplifications of Kurt G\"odel's modal ontological argument. Some argument premises are modified, others are dropped, modal…
We extend the logical categories framework to first order modal logic. In our modal categories, modal operators are applied directly to subobjects and interact with the background factorization system. We prove a Joyal-style representation…
Given a fibration over the circle, we relate the eigenspace decomposition of the algebraic monodromy, the homological finiteness properties of the fiber, and the formality properties of the total space. In the process, we prove a more…
Recent advances in the integration of deep learning with automated theorem proving have centered around the representation of logical formulae as inputs to deep learning systems. In particular, there has been a growing interest in adapting…
E-graphs are a data structure for equational reasoning and optimization over ground terms. One of the benefits of e-graph rewriting is that it can declaratively handle useful but difficult to orient identities like associativity and…