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Units of measure with prefixes and conversion rules are given a formal semantic model in terms of categorial group theory. Basic structures and both natural and contingent semantic operations are defined. Conversion rules are represented as…
A kind of unstable homotopy theory on the category of associative rings (without unit) is developed. There are the notions of fibrations, homotopy (in the sense of Karoubi), path spaces, Puppe sequences, etc. One introduces the notion of a…
A quantum Matter-Photonics framework is adapted to help scrutinize chemical reaction mechanisms and used to explore a process mapped from chemical tree topological model. The chemical concept of bond knitting/breaking is reformulated via…
By adding or removing appropriate structures to Gauss diagram, one can create useful objects related to virtual links. In this paper few objects of this kind are studied: twisted virtual links generalizing virtual links; signed chord…
These are exciting times for quantum physics as new quantum technologies are expected to soon transform computing at an unprecedented level. Simultaneously network science is flourishing proving an ideal mathematical and computational…
Networks are a fundamental model of complex systems throughout the sciences, and network datasets are typically analyzed through lower-order connectivity patterns described at the level of individual nodes and edges. However, higher-order…
A book Chapter consisting of some of the main areas of research in graph theory applied to physics. It includes graphs in condensed matter theory, such as the tight-binding and the Hubbard model. It follows the study of graph theory and…
This paper develops a methodology for representing machine learning models as models of formal theories, grounded in the perspective that machine learning models are a form of database and that databases are models of theories in coherent…
In this thesis we develop the cohomology of diagrams of algebras and then apply this to the cases of the $\lambda$-rings and the $\Psi$-rings. A diagram of algebras is a functor from a small category to some category of algebras. For an…
To a considerable extent, the continuing importance and popularity of complex networks as models of real-world structures has been motivated by scale free degree distributions as well as the respectively implied hubs. Being related to…
Polynomial functors are useful in the theory of data types, where they are often called containers. They are also useful in algebra, combinatorics, topology, and higher category theory, and in this broader perspective the polynomial aspect…
K-theory provides a framework for classifying Ramond-Ramond (RR) charges and fields. K-theory of manifolds has a natural extension to K-theory of noncommutative algebras, such as the algebra considered in noncommutative Yang-Mills theory or…
The way the topological structure goes from a decoupled state into a coupled one in multiplex networks has been widely studied by means of analytical and numerical studies, involving models of artificial networks. In general, these…
Graphical Models have various applications in science and engineering which include physics, bioinformatics, telecommunication and etc. Usage of graphical models needs complex computations in order to evaluation of marginal functions,so…
Higher-order connectivity patterns such as small induced sub-graphs called graphlets (network motifs) are vital to understand the important components (modules/functional units) governing the configuration and behavior of complex networks.…
Modelling compositionality has been a longstanding area of research in the field of vector space semantics. The categorical approach to compositionality maps grammar onto vector spaces in a principled way, but comes under fire for requiring…
Despite the extreme popularity of deep learning in science and industry, its formal understanding is limited. This thesis puts forth notions of rank as key for developing a theory of deep learning, focusing on the fundamental aspects of…
By using K-theory, we construct a map from the tangent space to the Hilbert scheme at a point Y to the local cohomology group. And we use this map to answer affirmatively(after slight modification) a question by Mark Green and Phillip…
Many dynamical systems -- from robots interacting with their surroundings to large-scale multiphysics systems -- involve a number of interacting subsystems. Toward the objective of learning composite models of such systems from data, we…
To a singular knot K with n double points, one can associate a chord diagram with n chords. A chord diagram can also be understood as a 4-regular graph endowed with an oriented Euler circuit. L. Traldi introduced a polynomial invariant for…