Related papers: The Fractal Logic of Phi-adic Recursion

We establish several recurrence relations and an explicit formula for V(n), the number of factorizations of the length-n prefix of the Fibonacci word into a (not necessarily strictly) decreasing sequence of standard Fibonacci words. In…

Combinatorics, like computer science, often has to deal with large objects of unspecified (or unusable) structure. One powerful way to deal with such an arbitrary object is to decompose it into more usable components. In particular, it has…

The work is devoted to Computability Logic (CoL) -- the philosophical/mathematical platform and long-term project for redeveloping classical logic after replacing truth} by computability in its underlying semantics (see…

We show that numerous distinctive concepts of constructive mathematics arise automatically from an "antithesis" translation of affine logic into intuitionistic logic via a Chu/Dialectica construction. This includes apartness relations,…

Any finite set of linear operators on an algebra $A$ yields an operator algebra $B$ and a module structure on A, whose endomorphism ring is isomorphic to a subring $A^B$ of certain invariant elements of $A$. We show that if $A$ is a…

The most common methods in explainable artificial intelligence are post-hoc techniques which identify the most relevant features used by pretrained opaque models. Some of the most advanced post hoc methods can generate explanations that…

In this paper we use fractal geometry to investigate boundary aspects of the first homology group for finite coverings of the modular surface. We obtain a complete description of algebraically invisible parts of this homology group. More…

We present a lovely connection between the Fibonacci numbers and the sums of inverses of $(0,1)-$ triangular matrices, namely, a number $S$ is the sum of the entries of the inverse of an $n \times n$ $(n \geq 3)$ $(0,1)-$ triangular matrix…

We show that the Wynn recurrence (the missing identity of Frobenius of the Pad\'{e} approximation theory) can be incorporated into the theory of integrable systems as a reduction of the discrete Schwarzian Kadomtsev-Petviashvili equation.…

A Metis design is one for which v=r+k+1. This paper deals with Metis designs that are quasi-residual. The parameters of such designs and the corresponding symmetric designs can be expressed by Fibonacci numbers. Although the question of…

In this paper we investigate the Curry-Howard correspondence for constructive modal logic in light of the gap between the proof equivalences enforced by the lambda calculi from the literature and by the recently defined winning strategies…

We describe old and prove new results on properties of the Fibonacci Lie algebra in a self-contained exposition. First, we study the growth of this algebra in more details. So, we show that the polynomial behaviour of the growth function in…

In this short note, we establish some identities containing sums of binomials with coefficients satisfying third order linear recursive relations. As a result and in particular, we obtain general forms of earlier identities involving…

We propose a reinterpretation of the continuum grounded in the stratified structure of definability rather than classical cardinality. In this framework, a real number is not an abstract point on the number line, but an object expressible…

A way to add an extra dimension is briefly discussed.

In this paper, we introduce a general family of sequent-style calculi over the modal language and its fragments to capture the essence of all constructively acceptable systems. Calling these calculi \emph{constructive}, we show that any…

Proofs are traditionally syntactic, inductively generated objects. This paper reformulates first-order logic (predicate calculus) with proofs which are graph-theoretic rather than syntactic. It defines a combinatorial proof of a formula…

We introduce a non-wellfounded proof system for intuitionistic logic extended with inductive and co-inductive definitions, based on a syntax in which fixpoint formulas are annotated with explicit variables for ordinals. We explore the…

Let A be a finite-dimensional associative algebra and $\phi$ a symmetric linear function on $A$. In this note, we will show that the pseudotrace maps are obtained as special cases of well-known symmetric linear functions on the endomorphism…

We present a cohomological interpretation of the middle convolution functor MC and find an explicit Riemann-Hilbert correspondence for MC_\lambda. This leads to an algorithm for the construction of Fuchsian systems which correspond to…