Mathematics
We prove undecidability for every positive relevant logic extending the system axiomatized by hypothetical syllogism, prefixing, and suffixing and contained in the logic of the semilattice frame $(P_{\mathrm{fin}}(\mathbb{N}), \cup,…
The term Gibbons conjecture is widely used in connection with symmetry results for the Allen-Cahn equation. However, its origin is less transparent than its frequent citation suggests. In this note, we revisit its emergence, tracing it to a…
This paper introduces a Laws of Form version of the Quaternions. We call this the Q-Calculus, a 16-valued extension of Laws of Form (LoF) which is closely related to the BF Calculus (where we have a single square root of the mark) and the…
We prove some results about the theory of independence in $\mathrm{NSOP}_{3}$ theories that do not hold in $\mathrm{NSOP}_{4}$ theories. We generalize Chernikov's work on simple and co-simple types in $\mathrm{NTP}_{2}$ theories to types…
In chapter 9 of his book "The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal", Woodin shows how to force the Strong Chang Conjecture over models of determinacy using $\mathbb{P}_{\mathrm{max}}$. We show here how a…
In Grayson's combinatorial description of higher K-groups, the generators are bounded acyclic binary multi-complexes of arbitrary size. Generalising work by Kasprowski, Winges and the author, we show in this paper that multi-complexes of…
In a previous paper we introduced a version of associativity for a partial infinitary operation. We prove here that if $\gamma$ is an infinite ordinal and some associative infinitary operation is defined for all sequences indexed by…
We investigate B\"uchi Arithmetic $\mathsf{BA}_k$ -- the elementary theory of the natural numbers equipped with addition and the function mapping a number $x$ to the greatest power of $k$ dividing $x$. $\mathsf{BA}_k$ is known to be…
We analyze a construction of Cherlin, van den Dries, and Macintyre to code graphs in PAC fields. We show that, in many cases, model-theoretic properties of the graph are preserved in the passage from the graph to the field. As a corollary,…
We study the geodesic flow on the unit cotangent bundle $M=S^{*}\mathcal{N}$ of a closed hyperbolic surface $\mathcal{N}$, using the representation theory of $SL_{2}(\mathbb{R})$. We construct explicit $X$-adapted Hilbert spaces, obtained…
In one of his posthumous papers, conserved in G\"ottingen, Riemann considers the derivatives of $\log\zeta(s)$ at the point $1/2$, giving explicit values for them. Around 2010 we shared Riemann's value of the second derivative with some…
In this paper, we develop a mixed quantization technique for graph vector bundles and apply it to several asymptotic spectral problems, including the Alon-Boppana bound, the Kesten-McKay law, asymptotic determinant, quantum ergodicity, zero…
We extend a property of Mazzola's theory of cadential sets in relation to the modulation between minor and major tonalities from triadic to tetradic harmony, using the PLRQ group of Cannas et al. (2017) as the analogue of the classical PLR…
Unbounded {\L}ukasiewicz logic is a substructural logic that combines features of infinite-valued {\L}ukasiewicz logic with those of abelian logic. The logic is finitely strongly complete w.r.t.~the additive $\ell$-group on the reals…
For an $\omega$-categorical theory $T$ and model $\mathcal{M}$ of $T$ we define a hierarchy of ranks, the $n$-ranks for $n < \omega$ which only care about imaginary elements ``up to level $n$'', where level $n$ contains every element of $M$…
In this paper, we explore the connections between Christiaan Huygens and Niels Henrik Abel through the tautochrone problem. The problem -- determining the curve along which a particle descends under gravity in the same time, regardless of…
Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain. For any $\epsilon\in (0,1)$ we show that for any Dirichlet eigenvalue $\lambda_k(\Omega)>\Lambda(\epsilon,\Omega)$, it holds \begin{align*} k&\le…
We prove separation and excision results in functor homology. These results explain how the global Steinberg decomposition of functors proved by Djament, Touz{\'e} and Vespa behaves in Ext and Tor computations.
In 1964 Shepherdson \cite{shepherdson:1964} proved that a discretely ordered semiring $\mathcal{M}^+$ satisfies $\sf{IOpen}$ (quantifier free induction) iff the corresponding ring $\mathcal{M}$ is an integer part of the real closure of the…
We consider a logic with truth values in the unit interval and which uses aggregation functions instead of quantifiers, and we describe a general approach to asymptotic elimination of aggregation functions and, indirectly, of asymptotic…