Intuitionistic sequential compactness?

Perhaps because the classical notion of sequential compactness fails to apply constructively even to $\{0,1\}$, Brouwer and his successors have paid little attention to the possibility of a constructive counterpart that is classically equivalent to sequential compactness and has serious potential for applications in analysis. We discuss such a notion – the anti-Specker property – and its equivalence, over Bishop-style constructive mathematics, to Brouwer’s fan theorem for $c$-bars.     

Tarski's theorem on intuitionistic logic,for polyhedra

In 1938, Tarski proved that a formula is not intuitionistically valid if, and only if, it has a counter-model in the Heyting algebra of open sets of some topological space. In fact, Tarski showed that any Euclidean space ${\mathbb{R}}^n$ with $n?1$ suffices, as does e.g. the Cantor space. In particular, intuitionistic logic cannot detect topological dimension in the Heyting algebra of all open sets of a Euclidean space. By contrast, we consider the lattice of open subpolyhedra of a given compact polyhedron $P?{\mathbb{R}}^n$, prove that it is a locally finite Heyting subalgebra of the (non-locally-finite) algebra of all open sets of P, and show that intuitionistic logic is able to capture the topological dimension of P through the bounded-depth axiom schemata. Further, we show that intuitionistic logic is precisely the logic of formulæ valid in all Heyting algebras arising from polyhedra in this manner. Thus, our main theorem reconciles through polyhedral geometry two classical results: topological completeness in the style of Tarski, and Ja?kowski's theorem that intuitionistic logic enjoys the finite model property. Several questions of interest remain open. E.g., what is the intermediate logic of all closed triangulable manifolds?     

Lewis meets Brouwer: Constructive strict implication

C.I. Lewis invented modern modal logic as a theory of “strict implication” $?$. Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than $\square$ and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic $\mathsf{K}$ becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the “strength” axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction has been discovered by the functional programming community in their study of “arrows” as contrasted with “idioms”. Our particular focus is on arithmetical interpretations of intuitionistic $?$ in terms of preservativity in extensions of $\mathsf{HA}$, i.e., Heyting’s Arithmetic.     

On the longest block in Lüroth expansion

In this paper, for a finite subset $A?\{2,3,?\}$, we introduce the notion of longest block function $L_n(x,A)$ for the Lüroth expansion of $x\in [0,1)$ with respect to A and consider the asymptotic behavior of $L_n(x,A)$ as n tends to ∞. We also obtain the Hausdorff dimensions of the level sets and exceptional set arising from the longest block function.     

Ghost circles in lattice Aubry–Mather theory

Monotone lattice recurrence relations such as the Frenkel–Kontorova lattice, arise in Hamiltonian lattice mechanics, as models for ferromagnetism and as discretization of elliptic PDEs. Mathematically, they are a multi-dimensional counterpart of monotone twist maps.Such recurrence relations often admit a variational structure, so that the solutions $x:{\mathbb{Z}}^d\to \mathbb{R}$ are the stationary points of a formal action function $W(x)$. Given any rotation vector $\omega \in {\mathbb{R}}^d$, classical Aubry–Mather theory establishes the existence of a large collection of solutions of $\mathrm{?}W(x)=0$ of rotation vector ω. For irrational ω, this is the well-known Aubry–Mather set. It consists of global minimizers and it may have gaps.In this paper, we study the parabolic gradient flow $\frac{dx}{dt}=?\mathrm{?}W(x)$ and we will prove that every Aubry–Mather set can be interpolated by a continuous gradient-flow invariant family, the so-called ‘ghost circle’. The existence of these ghost circles is known in dimension $d=1$, for rational rotation vectors and Morse action functions. The main technical result of this paper is therefore a compactness theorem for lattice ghost circles, based on a parabolic Harnack inequality for the gradient flow. This implies the existence of lattice ghost circles of arbitrary rotation vectors and for arbitrary actions.As a consequence, we can give a simple proof of the fact that when an Aubry–Mather set has a gap, then this gap must be filled with minimizers, or contain a non-minimizing solution.     

The omega-rule interpretation of transfinite provability logic

Given a computable ordinal Λ, the transfinite provability logic ${\mathsf{GLP}}_{\mathrm{\Lambda }}$ has for each $\xi <\mathrm{\Lambda }$ a modality $[\xi ]$ intended to represent a provability predicate within a chain of increasing strength. One possibility is to read $[\xi ]?$ as ? is provable in T using ω-rules of depth at most ξ, where T is a second-order theory extending ${\mathrm{ACA}}_0$.In this paper we will formalize such iterations of ω-rules in second-order arithmetic and show how it is a special case of what we call uniform provability predicates. Uniform provability predicates are similar to Ignatiev's strong provability predicates except that they can be iterated transfinitely. Finally, we show that ${\mathsf{GLP}}_{\mathrm{\Lambda }}$ is sound and complete for any uniform provability predicate.     

Contracting an element from a cocircuit

We consider the situation that M and N are 3-connected matroids such that $|E(N)|⩾4$ and $C^1$ is a cocircuit of M with the property that $M/x_0$ has an N-minor for some $x_0\in C^1$. We show that either there is an element $x\in C^1$ such that $\mathrm{si}(M/x)$ or $\mathrm{co}(\mathrm{si}(M/x))$ is 3-connected with an N-minor, or there is a four-element fan of M that contains two elements of $C^1$ and an element x such that $\mathrm{si}(M/x)$ is 3-connected with an N-minor.     

Tilting modules under special base changes

Given a non-unit, non-zero-divisor, central element x of a ring Λ, it is well known that many properties or invariants of Λ determine, and are determined by, those of $\mathrm{\Lambda }/x\mathrm{\Lambda }$ and ${\mathrm{\Lambda }}_x$. In the present paper, we investigate how the property of “being tilting” behaves in this situation. It turns out that any tilting module over Λ gives rise to tilting modules over ${\mathrm{\Lambda }}_x$ and $\mathrm{\Lambda }/x\mathrm{\Lambda }$ after localization and passing to quotient respectively. On the other hand, it is proved that under some mild conditions, a module over Λ is tilting if its corresponding localization and quotient are tilting over ${\mathrm{\Lambda }}_x$ and $\mathrm{\Lambda }/x\mathrm{\Lambda }$ respectively.     

The quartet spaces of G. ’t Hooft

In 1964, G. ’t Hooft postulated three axioms, and proved that every nonempty finite model of them has $4^n$ elements. This note confirms this by showing that every nonempty model can be made into a vector space over the field with four elements. For every pair of different elements $x$ and $y$, the quartet of $x$ and $y$ is the affine line through $x$ and $y$ in this vector space.     

A way to construct independence equivalent graphs

Let us denote the independence polynomial of a graph by $I_G\left(x\right)$. If $I_G\left(x\right)=I_H\left(x\right)$ implies that $G?H$ then we say $G$ is independence unique. For graph $G$ and $H$ if $I_G\left(x\right)=I_H\left(x\right)$ but $G$ and $H$ are not isomorphic, then we say $G$ and $H$ are independence equivalent. In [7], Brown and Hoshino gave a way to construct independent equivalent graphs for circulant graphs. In this work we give a way to construct the independence equivalent graphs for general simple graphs and obtain some properties of the independence polynomial of paths and cycles.     

Tangent cones and C1 regularity of definable sets

Let $X?{\mathbb{R}}^n$ be a connected locally closed set which is definable in an o-minimal structure. We prove that the following three statements are equivalent: (i) X is a $C^1$ manifold, (ii) the tangent cone and the paratangent cone of X coincide at every point in X, (iii) for every $x\in X$, the tangent cone of X at the point x is a k-dimensional linear subspace of ${\mathbb{R}}^n$ (k does not depend on x) varies continuously in x, and the density $\theta (X,x)<3/2$.