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We study the approximate fixed point property (AFPP) for continuous single-valued functions and for continuous multivalued functions in digital topology. We extend what is known about these notions and discuss errors that have appeared in…
Deep neural networks, despite their success in numerous applications, often function without established theoretical foundations. In this paper, we bridge this gap by drawing parallels between deep learning and classical numerical analysis.…
We investigate the expressive power of neural networks from the point of view of descriptive complexity. We study neural networks that use floating-point numbers and piecewise polynomial activation functions from two perspectives: 1) the…
In this paper we investigate fixed-point numbers of endomorphisms on complex tori. Specifically, motivated by the asymptotic perspective that has turned out in recent years to be so fruitful in Algebraic Geometry, we study how the number of…
Motivated by the scaling limits of the connected components of the configuration model, we study uniform connected multigraphs with fixed degree sequence $\mathcal{D}$ and with surplus $k$. We call those random graphs…
We associate all small subgraph counting problems with a systematic graph encoding/representation system which makes a coherent use of graphlet structures. The system can serve as a unified foundation for studying and connecting many…
Let $\mathbb F_q$ be the finite field with $q$ elements and, for each positive integer $n$, let $A_q(n)$ be the number of non isomorphic functional graphs arising from $\mathbb F_q$-linear maps $T:\mathbb F_{q}^n\to \mathbb F_{q}^n$. In…
Graph convolutional networks (GCNs) and its variants are designed for unsigned graphs containing only positive links. Many existing GCNs have been derived from the spectral domain analysis of signals lying over (unsigned) graphs and in each…
This paper catalogues a variety of examples concerning a type of function of a $p$-adic integer variable defined by a formal series expression we have dubbed "$\mathcal{F}$-series". These series exhibit a new, previously undocumented form…
Let $\sigma$ denote an endomorphism of a smooth algebraic group $G$ over the algebraic closure of a finite field, and assume all iterates of $\sigma$ have finitely many fixed points. Steinberg gave a formula for the number of fixed points…
In this paper, we study the existence of fixed points for mappings defined on complete (compact) metric space (X, d) satisfying a general contractive (contraction) inequality depended on another function. These conditions are analogous to…
We analyze the topological properties of the set of functions that can be implemented by neural networks of a fixed size. Surprisingly, this set has many undesirable properties. It is highly non-convex, except possibly for a few exotic…
It turned out that the set of the fixed points is not necessarily the same as the set of the local minima of the energy functional. It depends on the diagonal elements of the connection matrix. The simple method which allows to cut off…
Let $G$ be a connected undirected graph on $n$ vertices with no loops but possibly multiedges. Given an arithmetical structure $(\textbf{r}, \textbf{d})$ on $G$, we describe a construction which associates to it a graph $G'$ on $n-1$…
In this paper, we establish a new bound tying together the effective length and the maximum correlation between the outputs of an arbitrary pair of Boolean functions which operate on two sequences of correlated random variables. We derive a…
We analyze the problem of network identifiability with nonlinear functions associated with the edges. We consider a static model for the output of each node and by assuming a perfect identification of the function associated with the…
Problems related to network coding for acyclic, instantaneous networks (where the edges of the acyclic graph representing the network are assumed to have zero-delay) have been extensively dealt with in the recent past. The most prominent of…
Let $C$ be a convex subset of a locally convex space. We provide optimal approximate fixed point results for sequentially continuous maps $f\colon C\to\bar{C}$. First we prove that if $f(C)$ is totally bounded, then it has an approximate…
We give a quick survey of the various fixed point theorems in computability theory, partial combinatory algebra, and the theory of numberings, as well as generalizations based on those. We also point out several open problems connected to…
We consider several coding discretizations of continuous functions which reflect their variation at some given precision. We study certain statistical and combinatorial properties of the sequence of finite words obtained by coding a typical…