Related papers: Algebraic characterisation of one-way patterns
Based on a simple example, it is explained how the homological analysis may be applied for modeling of the electric circuits. The homological branch, mesh and nodal analyses are presented. Geometrical interpretations are given.
The study of representations is of fundamental importance to any form of communication, and our ability to exploit them effectively is paramount. This article presents a novel theory -- Representational Systems Theory -- that is designed to…
Let $V$ be a finite dimensional vector space over a field $\mathrm{k}$ of characteristic $0$. Let $A$ be a linear mapping of $V$ into itself. This paper gives a normal form for $A$, which gives a better description of the structure of $A$…
Matrix Graph Grammars (MGG) is a novel approach to the study of graph dynamics ([15]). In the present contribution we look at MGG as a formal grammar and as a model of computation, which is a necessary step in the more ambitious program of…
Nanophotonics finds ever broadening applications requiring complex component designs with a large number of parameters to be simultaneously optimized. Recent methodologies employing optimization algorithms commonly focus on a single design…
In this paper, we propose to consider various models of pattern recognition. At the same time, it is proposed to consider models in the form of two operators: a recognizing operator and a decision rule. Algebraic operations are introduced…
The iteration of rational maps is well-understood in dimension 1 but less so in higher dimensions. We study some maps on spaces of matrices which present a weak complexity with respect to the ring structure. First we give some properties of…
Universal learning machine is a theory trying to study machine learning from mathematical point of view. The outside world is reflected inside an universal learning machine according to pattern of incoming data. This is subjective pattern…
A diagrammatic method is presented for averaging over the circular ensemble of random-matrix theory. The method is applied to phase-coherent conduction through a chaotic cavity (a ``quantum dot'') and through the interface between a normal…
In this short note, we give a new sufficient condition for a linear map from a product of copies of a field to endomorphisms of a finite dimensional vector space over the same field to be an algebra homomorphism. We expect that this result…
We take a categorical approach to describe ternary derivations and ternary automorphisms of triangular algebras. New classes of automorphisms and derivations of triangular algebras are also introduced and studied.
The analytic properties of the eikonal and U-matrix unitarization schemes are examined. It is shown that the basic properties of these schemes are identical. Both can fill the full circle of unitarity, and both can lead to standard and…
A classical approach for surface classification is to find a compact algebraic representation for each surface that would be similar for objects within the same class and preserve dissimilarities between classes. We introduce Self…
Interpretability has become an important issue in the machine learning field, along with the success of layered neural networks in various practical tasks. Since a trained layered neural network consists of a complex nonlinear relationship…
Determining the range of complex maps plays a fundamental role in the study of several complex variables and operator theory. In particular, one is often interested in determining when a given holomorphic function is a self-map of the unit…
Piecewise-linear maps describe dynamical phenomena that switch between distinct states and readily generate complex bifurcation structures due to their strong nonlinearity. We show that two-dimensional continuous piecewise-linear maps near…
A finite-dimensional unital and associative algebra over $\mathbb{R}$, or what we shall call simply "an algebra" in this paper for short, generalities the construction by which we derive the complex numbers by "adjoining an element $i$" to…
We formulate a theory of shape valid for objects of arbitrary dimension whose contours are path connected. We apply this theory to the design and modeling of viable trajectories of complex dynamical systems. Infinite families of…
The degree of static indeterminacy and its spatial distribution characterize load-bearing structures independent of a specific load case. The redundancy matrix stores the distribution of the static indeterminacy on its main diagonal, and…
A graphical model is a statistical model that is associated to a graph whose nodes correspond to variables of interest. The edges of the graph reflect allowed conditional dependencies among the variables. Graphical models admit…