arXiv subjects

Jonathan D. H. Smith

Publications and source records attributed to Jonathan D. H. Smith.

10 recordsLinked to original sources

arXiv2026

Cut-and-Project Density Functional Theory for Quasicrystals

Cut-and-project from a symmetric structure in a higher-dimensional space is a standard method for describing the structure of a large class of quasicrystals. By means of a novel localization procedure, we now show how local physical interactions within these quasicrystals are also accurately described by cut-and-project, from corresponding physical interactions in the higher-dimensional space. A density functional theory (DFT++) formulation allows the cut-and-project method to handle the Schroedinger equation for interactions in quasicrystals. The theory is both rigorous and computationally tractable. The resulting ab initio approach specifies quasicrystalline quantum states, in contrast to previous approaches which only worked with crystalline approximants of the quasi-periodic structures.

cond-mat.mtrl-sci↗

arXiv2026

On the intrinsic geometry of polyhedra: Convex polygon coordinates

There is a very extensive literature dealing with convex polytopes from the standpoints of combinatorics and numerical analysis. By contrast, the current paper adopts an alternative viewpoint that regards a polytope as an autonomous space in its own right, with its own intrinsic geometry. Our attention is focused on the complete set of all the coordinate systems that serve to locate a point of the polytope; divorced, for example, from the smoothness issues that are of concern for applications in numerical analysis. We use the efficient and appropriate algebraic language of barycentric algebras to elicit the convex structure of the set of polytope coordinate systems. Specializing to convex polygons, we examine the chordal coordinate systems that are determined by the triangulations of the polygon. An algorithm to compute the coordinates of a point within such a system is presented. The algorithm relies on a coalgebra structure that transports a probability distribution on one side of a triangle to distributions on each of the remaining two sides. The Catalan number enumeration of the polygon triangulations (well-known within combinatorics) is then obtained in a natural geometric fashion from the parsing trees of the coalgebra structure of our algorithm.

math.MG↗

arXiv2025

From affine to barycentric coordinates in polytopes

Each point of a simplex is expressed as a unique convex combination of the vertices. The coefficients in the combination are the barycentric coordinates of the point. For each point in a general convex polytope, there may be multiple representations, so its barycentric coordinates are not necessarily unique. There are various schemes to fix particular barycentric coordinates: Gibbs, Wachspress, cartographic, etc. In this paper, a method for producing sparse barycentric coordinates in polytopes will be discussed. It uses a purely algebraic treatment of affine spaces and convex sets, with barycentric algebras. The method is based on a certain decomposition of each finite-dimensional convex polytope into a union of simplices of the same dimension.

math.MG↗

arXiv2024

Accurate Machine Learning Predictions of Coercivity in High-Performance Permanent Magnets

Increased demand for high-performance permanent magnets in the electric vehicle and wind turbine industries has prompted the search for cost-effective alternatives.Discovering new magnetic materials with the desired intrinsic and extrinsic permanent magnet properties presents a significant challenge to researchers because of issues with the global supply of rare-earth elements, material stability, and a low maximum magnetic energy product BH$_{max}$.While first-principle density functional theory (DFT) predicts materials' magnetic moments, magneto-crystalline anisotropy constants, and exchange interactions, it cannot compute coercivity ($H_c$).Although it is possible to calculate $H_c$ theoretically with micromagnetic simulations, the predicted value is larger than the experiment by almost an order of magnitude, due to the Brown paradox.To circumvent these, we employ machine learning (ML) methods on an extensive database obtained from experiments, DFT calculations, and micromagnetic modeling.The use of a large dataset enables realistic $H_c$ predictions for materials such as Ce-doped Nd$_2$Fe$_{14}$B, comparing favorably against micromagnetically simulated coercivities.Remarkably, our ML model accurately identifies uniaxial magneto-crystalline anisotropy as the primary contributor to $H_c$. With DFT calculations, we predict the Nd-site dependent magnetic anisotropy behavior in Nd$_2$Fe$_{14}$B, confirming that Nd $4g$-sites mainly contribute to uniaxial magneto-crystalline anisotropy, and also calculate Curie temperature (T$_{C}$).Both calculated results are in good agreement with experiment.The coupled experimental dataset and ML modeling with DFT input predict $H_c$ with far greater accuracy and speed than was previously possible using micromagnetic modeling.Further, we reverse-engineer the inter-grain exchange coupling with micromagnetic simulations by employing the ML predictions.

cond-mat.mtrl-sci↗

arXiv2024

Bilayer Ion Trap Design for 2D Arrays

Junctions are fundamental elements that support qubit locomotion in two-dimensional ion trap arrays and enhance connectivity in emerging trapped-ion quantum computers. In surface ion traps they have typically been implemented by shaping radio frequency (RF) electrodes in a single plane to minimize the disturbance to the pseudopotential. However, this method introduces issues related to RF lead routing that can increase power dissipation and the likelihood of voltage breakdown. Here, we propose and simulate a novel two-layer junction design incorporating two perpendicularly rotoreflected (rotated, then reflected) linear ion traps. The traps are vertically separated, and create a trapping potential between their respective planes. The orthogonal orientation of the RF electrodes of each trap relative to the other provides perpendicular axes of confinement that can be used to realize transport in two dimensions. While this design introduces manufacturing and operating challenges, as now two separate structures have to be precisely positioned relative to each other in the vertical direction and optical access from the top is obscured, it obviates the need to route RF leads below the top surface of the trap and eliminates the pseudopotential bumps that occur in typical junctions. In this paper the stability of idealized ion transfer in the new configuration is demonstrated, both by solving the Mathieu equation analytically to identify the stable regions and by numerically modeling ion dynamics. Our novel junction layout has the potential to enhance the flexibility of microfabricated ion trap control to enable large-scale trapped-ion quantum computing.

quant-ph↗

arXiv2023

Octonions as Clifford-like algebras

The associative Cayley-Dickson algebras over the field of real numbers are also Clifford algebras. The alternative but nonassociative real Cayley-Dickson algebras, notably the octonions and split octonions, share with Clifford algebras an involutary anti-automorphism and a set of mutually anticommutative generators. On the basis of these similarities, we introduce Kingdon algebras: alternative Clifford-like algebras over vector spaces equipped with a symmetric bilinear form. Over three-dimensional vector spaces, our construction quantizes an alternative non-associative analogue of the exterior algebra. The octonions and split octonions, along with other real generalized Cayley-Dickson algebras in Albert's sense, arise as Kingdon algebras. Our construction gives natural characterizations of the octonion and split octonion algebras by a universality property endowing them with a selected superalgebra structure.

math.RA↗

arXiv2019

Quasigroup words and reversible automata

This paper examines two related topics: the linearization of the reversible automata of Gvaramiya and Plotkin, and the problem of finding a faithful representation of the words in a central quasigroup that respects the triality symmetry of the language of quasigroups.

math.GR↗

arXiv2019

On the Enumeration and Asymptotic Growth of Free Quasigroup Words

The paper counts the number of reduced quasigroup words of a particular length in a certain number of generators. Taking account of the relationship with the Catalan numbers, counting words in a free magma, we introduce the term peri-Catalan number for the free quasigroup word counts. The main result of the paper is an exact recursive formula for the peri-Catalan numbers, structured by the Euclidean Algorithm. The Euclidean Algorithm structure does not readily lend itself to standard techniques of asymptotic analysis. However, conjectures for the asymptotic behavior of the peri-Catalan numbers, substantiated by numerical data, are presented. A remarkable aspect of the observed asymptotic behavior is the so-called asymptotic irrelevance of quasigroup identities, whereby cancelation resulting from quasigroup identities has a negligible effect on the asymptotic behavior of the peri-Catalan numbers for long words in a large number of generators.

math.CO↗

arXiv2019

Isomorphism Invariants for Linear Quasigroups

For a unital ring $S$, an $S$-linear quasigroup is a unital $S$-module, with automorphisms $ρ$ and $λ$ giving a (nonassociative) multiplication $x\cdot y=x^ρ+y^λ$. If $S$ is the field of complex numbers, then ordinary characters provide a complete linear isomorphism invariant for finite-dimensional $S$-linear quasigroups. Over other rings, it is an open problem to determine tractably computable isomorphism invariants. The paper investigates this isomorphism problem for $\mathbb{Z}$-linear quasigroups. We consider the extent to which ordinary characters classify $\mathbb{Z}$-linear quasigroups and their representations of the free group on two generators. We exhibit non-isomorphic $\mathbb{Z}$-linear quasigroups with the same ordinary character. For a subclass of $\mathbb{Z}$-linear quasigroups, equivalences of the corresponding ordinary representations are realized by permutational intertwinings. This leads to a new equivalence relation on $\mathbb{Z}$-linear quasigroups, namely permutational similarity. Like the earlier concept of central isotopy, permutational similarity is intermediate between isomorphism and isotopy.

math.GR↗

arXiv2008

A General Framework for Recursive Decompositions of Unitary Quantum Evolutions

Decompositions of the unitary group U(n) are useful tools in quantum information theory as they allow one to decompose unitary evolutions into local evolutions and evolutions causing entanglement. Several recursive decompositions have been proposed in the literature to express unitary operators as products of simple operators with properties relevant in entanglement dynamics. In this paper, using the concept of grading of a Lie algebra, we cast these decompositions in a unifying scheme and show how new recursive decompositions can be obtained. In particular, we propose a new recursive decomposition of the unitary operator on $N$ qubits, and we give a numerical example.

quant-ph↗