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One essential ingredient in many machine learning (ML) based methods for atomistic modeling of materials and molecules is the use of locality. While allowing better system-size scaling, this systematically neglects long-range (LR) effects,…
We prove two results intended to streamline proofs about cellularity that pass through mutual algebraicity. First, we show that a countable structure $M$ is cellular if and only if $M$ is $\omega$-categorical and mutually algebraic. Second,…
Circuit topology refers to the arrangement of interactions between objects belonging to a linearly ordered object set. Linearly ordered set of objects are common in nature and occur in a wide range of applications in economics, computer…
We discuss three mathematical structures which arise in topologically massive abelian gauge theory. First, the euclidean topologically massive abelian gauge theory defines a contact structure on a manifold. We briefly discuss three…
Mathematical morphology contributes many profitable tools to image processing area. Some of these methods considered to be basic but the most important fundamental of data processing in many various applications. In this paper, we modify…
In this paper, we study confoliations in dimensions higher than three mostly from the perspective of symplectic fillability. Our main result is that Massot-Niederkr\"uger-Wendl's bordered Legendrian open book, an object that obstructs the…
If $\eta$ is a contact form on a manifold $M$ such that the orbits of the Reeb vector field form a simple foliation $\mathcal{F}$ on $M$, then the presymplectic 2-form $d\eta$ on $M$ induces a symplectic structure $\omega$ on the quotient…
Faithful representations of regular $\ast$-rings and modular complemented lattices with involution within orthosymmetric sesquilinear spaces are studied within the framework of Universal Algebra. In particular, the correspondence between…
We describe a class calculus that is expressive enough to describe and improve its own learning process. It can design and debug programs that satisfy given input/output constraints, based on its ontology of previously learned programs. It…
We study a generalized nonlocal theory of gravity which, in specific limits, can become either the curvature non-local or teleparallel non-local theory. Using the Noether Symmetry Approach, we find that the coupling functions coming from…
Deep metric learning (DML) involves training a network to learn a semantically meaningful representation space. Many current approaches mine n-tuples of examples and model interactions within each tuplets. We present a novel, compositional…
We exhibit an adjunction between a category of abstract algebras of partial functions that we call difference-restriction algebras and a category of Hausdorff \'etale spaces. Difference-restriction algebras are those algebras isomorphic to…
Module is effective representation of ring in Abelian group. Linear map of module over commutative ring is morphism of corresponding representation. This definition is the main subject of the book. To consider this definition from more…
Our main result is that any topological algebra based on a Boolean space is the extended Stone dual space of a certain associated Boolean algebra with additional operations. A particular case of this result is that the profinite completion…
We construct the augmentation representation. It is a representation of the fundamental group of the link complement associated to an augmentation of the framed cord algebra. This construction connects representations of two link invariants…
One of the prime motivation for topology was Homotopy theory, which captures the general idea of a continuous transformation between two entities, which may be spaces or maps. In later decades, an algebraic formulation of topology was…
Coordination geometries describe how the neighbours of a central particle are arranged around it. Such geometries can be thought to lie in an abstract topological space; a model of this space could provide a mathematical basis for…
The study of the relation between Lie algebras and groups, and especially the derivation of new algebras from them, is a problem of great interest in mathematics and physics, because finding a new Lie group from an already known one also…
The first goal of this paper is to construct examples of higher dimensional contact manifolds with specific properties. Our main results in this direction are the existence of tight virtually overtwisted closed contact manifolds in all…
The problem of decomposing non-manifold object has already been studied in solid modeling. However, the few proposed solutions are limited to the problem of decomposing solids described through their boundaries. In this thesis we study the…