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Interconnected dynamic systems are a pervasive component of our modern infrastructures. The complexity of such systems can be staggering, which motivates simplified representations for their manipulation and analysis. This work introduces…
We study the number of factorizations of a positive integer, where the parts of the factorization are of l different colors (or kinds). Recursive or explicit formulas are derived for the case of unordered and ordered, distinct and…
In this paper we study the behavior of the fractions of a factorial design under permutations of the factor levels. We focus on the notion of regular fraction and we introduce methods to check whether a given symmetric orthogonal array can…
Problem solvers often need to choose between adapting a current solution and exchanging it for a new one. How do such decisions depend on structural and surface features of the task? The present study investigated the interplay between the…
The constraints on the models for the structure formation arising from various cosmological observations at different length scales are reviewed. The status of different models for structure formation is examined critically in the light of…
An equivalence structure is a set with a single binary relation, satisfying sentences stating that the relation is an equivalence relation. A computable structure A is said to be $\Delta^0_\alpha$ categorical if for any computable structure…
Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider…
A general theme of computable structure theory is to investigate when structures have copies of a given complexity $\Gamma$. We discuss such problem for the case of equivalence structures and preorders. We show that there is a $\Pi^0_1$…
Learning arguably involves the discovery and memorization of abstract rules. The aim of this paper is to study associative memory mechanisms. Our model is based on high-dimensional matrices consisting of outer products of embeddings, which…
We report on recent analyses of different pieces of data, which exhibit Geometrical Scaling (GS) and its breaking. GS is a consequence of the existence of an intermediate energy scale, called saturation momentum, and allows to relate data…
When are all positions of a game numbers? We show that two properties are necessary and sufficient. These properties are consequences of that, in a number, it is not an advantage to be the first player. One of these properties implies the…
We define and study complex structures and generalizations on spaces consisting of geodesics or harmonic maps that are compatible with the symmetries of these spaces. The main results are about existence and uniqueness of such structures.
Errors quoted on results are often given in asymmetric form. An account is given of the two ways these can arise in an analysis, and the combination of asymmetric errors is discussed. It is shown that the usual method has no basis and is…
Morphisms, structure preserving maps, are everywhere in Mathematics as useful tools for thinking and problem solving, or as objects to study. Here, we argue that the idea of operations being compatible across two domains goes beyond its…
Shapley value has recently become a popular way to explain the predictions of complex and simple machine learning models. This paper is discusses the factors that influence Shapley value. In particular, we explore the relationship between…
A dilatation structure is a concept in between a group and a differential structure. In this article we study fundamental properties of dilatation structures on metric spaces. This is a part of a series of papers which show that such a…
Scaling problems have a rich and diverse history, and thereby have found numerous applications in several fields of science and engineering. For instance, the matrix scaling problem has had applications ranging from theoretical computer…
Values are things that are important to us. Actions activate values - they either go against our values or they promote our values. Values themselves can either be conforming or conflicting depending on the action that is taken. In this…
Building of some isomorphic classes for noncanonical hypercomplex number systems o dimension 2 is described. In general case, such systems with specific constraints to structural constants can be isomorphic to complex, dual or double number…
Data analysts commonly utilize statistics to summarize large datasets. While it is often sufficient to explore only the summary statistics of a dataset (e.g., min/mean/max), Anscombe's Quartet demonstrates how such statistics can be…