Mathematics

We generalize the classical Ceva's and Menelaus's theorems to curvilinear triangles bounded by circular arcs. We introduce trilinear coordinates associated with such triangles and develop several geometric constructions. In particular, for…

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,…

In this paper we prove the WALA conjecture.

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…

In this paper, we study the dual Minkowski problem under group symmetry. For $0<q\le n$, we give a complete existence characterization in the framework of $G$-invariant convex bodies when the group $G\subset O(n)$ has no nonzero fixed…

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…

This paper gives a complete description of the solutions of the global positioning problem, emphasizing the under-determined case. We show that the solutions form a quadric, which may degenerate in various ways. Perhaps more surprisingly,…

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 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 define a distance analogous to the Gromov-Hausdorff distance that enables the comparison of arbitrary quasi-isometric spaces. We also investigate properties preserved under limits with respect to this distance, as well as properties of…

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 $M$…

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…

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…

This work studies spectral properties of Schr\"odinger operators in the context of aperiodic order, using weighted Delone sets to explore the interplay between the underlying dynamics and spectral properties. We study parameter-dependent…