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This paper is about the study of F-transforms based on overlap and grouping maps, residual and co-residual implicator over complete lattice from both constructive and axiomatic approaches. Further, the duality, basic properties, and the…
Computational methods that automatically extract knowledge from data are critical for enabling data-driven materials science. A reliable identification of lattice symmetry is a crucial first step for materials characterization and…
For a many-to-many matching market, we study the lattice structure of the set of random stable matchings. We define a partial order on the random stable set and present two intuitive binary operations to compute the least upper bound and…
Understanding the drift motion and dynamical locking of crystalline clusters on patterned substrates is important for the diffusion and manipulation of nano- and micro-scale objects on surfaces. In a previous work, we studied the…
Recently, we presented a framework for understanding protein structure based on the idea that simple constructs of holding hands or touching of objects can be used to rationalize the common characteristics of globular proteins. We developed…
In this paper, we present an algorithmic approach to design and construct planar truss structures based on symmetric lattices using modular elements. The method of assembly is similar to Leonardo grids as they both rely on the property of…
Using the framework of pastures and foundations of matroids developed by Baker-Lorscheid, we give algorithms to: (i) compute the foundation of a matroid, and (ii) compute all morphisms between two pastures. Together, these provide an…
Artificial intelligence for scientific discovery has recently generated significant interest within the machine learning and scientific communities, particularly in the domains of chemistry, biology, and material discovery. For these…
I propose a mathematical framework for embedding an unshaped discrete lattice $L$ on a smooth manifold $M$. This framework simplifies complex concepts in pure mathematics and physics by connecting discrete lattice structures with continuous…
Lattice-based Cryptography is considered to have the characteristics of classical computers and quantum attack resistance. We will design various graphic lattices and matrix lattices based on knowledge of graph theory and topological…
Sketching is used as a ubiquitous tool of expression by novices and experts alike. In this thesis I explore two methods that help a system provide a geometric machine-understanding of sketches, and in-turn help a user accomplish a…
The closure $\textrm{cl}(R)$ of a consistent set $R$ of triples (rooted binary trees on three leaves) provides essential information about tree-like relations that are shown by any supertree that displays all triples in $R$. In this…
Fix two lattice paths $P$ and $Q$ from $(0,0)$ to $(m,r)$ that use East and North steps with $P $ never going above $Q$. Bonin et al. show that the lattice paths that go from $(0,0)$ to $(m,r)$ and remain bounded by $P$ and $Q$ can be…
We study rank-three matroids, known as point-line configurations, and their associated matroid varieties, defined as the Zariski closures of their realization spaces. Our focus is on determining finite generating sets of defining equations…
A general model for geometric structures on differentiable manifolds is obtained by deforming infinitesimal symmetries. Specifically, this model consists of a Lie algebroid, equipped with an affine connection compatible with the Lie…
Recently $C^m$-conforming finite elements on simplexes in arbitrary dimension are constructed by Hu, Lin and Wu. The key in the construction is a non-overlapping decomposition of the simplicial lattice in which each component will be used…
We give a combinatorial characterization of generic frameworks that are minimally rigid under the additional constraint of maintaining symmetry with respect to a finite order rotation or a reflection. To establish these results we develop a…
Many innovative applications require establishing correspondences among 3D geometric objects. However, the countless possible deformations of smooth surfaces make shape matching a challenging task. Finding an embedding to represent the…
A concise theoretical framework, the $\textit{partial}$ Gauss-Hermite quadrature (pGHQ), is established for constructing on-node lattices of the lattice Boltzmann (LB) method under a Cartesian coordinate system. Comparing with existing…
We prove that the prescription for construction of supersymmetric lattice gauge theories by orbifolding and deconstruction directly leads to Catterall's geometrical discretization scheme in general. These two prescriptions always give the…