arXiv subjects

Manuel Joseph C. Loquias

Publications and source records attributed to Manuel Joseph C. Loquias.

18 recordsLinked to original sources

arXiv2026

A Matrix Analogue of Rational Number Systems

Let $P,Q \in \mathbb{Z}^{d\times d}$ be invertible coprime matrices such that all the eigenvalues of $Q^{-1}P$ have modulus greater than 1, and $\mathbb{Z}^d[Q^{-1}P]$ be the smallest non-trivial $Q^{-1}P$-invariant $\mathbb{Z}$-module containing $\mathbb{Z}^d$. Suppose there is a finite digit set $\mathcal{D}\subseteq \mathbb{Z}^d + P\mathbb{Z}^d[Q^{-1}P]$ for which every vector $x \in \mathbb{Z}^d + P\mathbb{Z}^d[Q^{-1}P]$ can be represented in the form \[ x = \sum_{i=0}^{\ell-1} (Q^{-1}P)^i Q^{-1}d_i, \] where the digits $d_i \in \mathcal{D}$ for all $i \in \{0,1,\ldots,\ell-1\}$. We call such a representation a $P/Q$-expansion of $x$, and we say that the digit system $(P,Q,\mathcal{D})$ has the finiteness property. If, in addition, $\mathcal{D}$ is a complete set of residues of the quotient group $(\mathbb{Z}^d + P\mathbb{Z}^d[Q^{-1}P])/P\mathbb{Z}^d[Q^{-1}P]$, then the digits $d_0, d_1, \dots, d_{\ell-1}$ in the $P/Q$-expansion of $x$ are unique whenever $\ell \in \mathbb{Z}^+$ is minimal, and the resulting digit system is said to have the uniqueness property. We present sufficient conditions for the existence of a digit set $\mathcal{D}$ in which $(P,Q,\mathcal{D})$ has the finiteness property. For $d=2$, we make use of finite automata to construct digit systems $(P,Q,\mathcal{D})$ having both the finiteness and uniqueness properties.

math.NT↗

arXiv2025

Addition Automata and Attractors of Digit Systems Corresponding to Expanding Rational Matrices

Let $A$ be an expanding $2 \times 2$ matrix with rational entries and $\mathbb{Z}^2[A]$ be the smallest $A$-invariant $\mathbb{Z}$-module containing $\mathbb{Z}^2$. Let $\mathcal{D}$ be a finite subset of $\mathbb{Z}^2[A]$ which is a complete residue system of $\mathbb{Z}^2[A]/A\mathbb{Z}^2[A]$. The pair $(A,\mathcal{D})$ is called a {\em digit system} with {\em base} $A$ and {\em digit set} $\mathcal{D}$. It is well known that every vector $x \in \mathbb{Z}^2[A]$ can be written uniquely in the form \[ x = d_0 + Ad_1 + \cdots + A^kd_k + A^{k+1}p, \] with $k\in \mathbb{N}$ minimal, $d_0,\dots,d_k \in \mathcal{D}$, and $p$ taken from a finite set of {\em periodic elements}, the so-called {\em attractor} of $(A,\mathcal{D})$. If $p$ can always be chosen to be $0$ we say that $(A,\mathcal{D})$ has the {\em finiteness property}. In the present paper we introduce finite-state transducer automata which realize the addition of the vectors $\pm(1,0)^\top$ and $\pm(0,1)^\top$ to a given vector $x\in \mathbb{Z}^2[A]$ in a number system $(A,\mathcal{D})$ with collinear digit set. These automata are applied to characterize all pairs $(A,\mathcal{D})$ that have the finiteness property and, more generally, to characterize the attractors of these digit systems.

math.NT↗

arXiv2023

Perfect precise colorings of plane semiregular tilings

A coloring of a planar semiregular tiling $\mathcal{T}$ is an assignment of a unique color to each tile of $\mathcal{T}$. If $G$ is the symmetry group of $\mathcal{T}$, we say that the coloring is perfect if every element of $G$ induces a permutation on the finite set of colors. If $\mathcal{T}$ is $k$-valent, then a coloring of $\mathcal{T}$ with $k$ colors is said to be precise if no two tiles of $\mathcal{T}$ sharing the same vertex have the same color. In this work, we obtain perfect precise colorings of some families of $k$-valent semiregular tilings in the plane, where $k\leq 6$.

math.CO↗

arXiv2022

Multiple contractions of permutation arrays

Given a permutation $σ$ on $n$ symbols $\{0, 1, \ldots, n-1\}$ and an integer $1 \leq m \leq n-1$, the $m$th contraction of $σ$ is the permutation $σ^{{\sf CT}^m}$ on $n-m$ symbols obtained by deleting the symbols $n-1, n-2, \ldots, n-m$ from the cycle decomposition of $σ$. The Hamming distance ${\rm hd}(σ,τ)$ between two permutations $σ$ and $τ$ is the number of symbols $x$ such that $σ(x) \neq τ(x)$. In this paper we give a complete characterization of the effect of a single contraction on the Hamming distance between two permutations. This allows us to obtain sufficient conditions for ${\rm hd}(σ,τ)-{\rm hd}(σ^{{\sf CT}^m},τ^{{\sf CT}^m})\leq 2m$.

math.CO↗

arXiv2020

Similarity Isometries of Point Packings

A linear isometry $R$ of $\mathbb{R}^d$ is called a similarity isometry of a lattice $Γ\subseteq \mathbb{R}^d$ if there exists a positive real number $β$ such that $βRΓ$ is a sublattice of (finite index in) $Γ$. The set $βRΓ$ is referred to as a similar sublattice of $Γ$. A (crystallographic) point packing generated by a lattice $Γ$ is a union of $Γ$ with finitely many shifted copies of $Γ$. In this study, the notion of similarity isometries is extended to point packings. We provide a characterization for the similarity isometries of point packings and identify the corresponding similar subpackings. Planar examples will be discussed, namely, the $1 \times 2$ rectangular lattice and the hexagonal packing (or honeycomb lattice). Finally, we also consider similarity isometries of point packings about points different from the origin by studying similarity isometries of shifted point packings. In particular, similarity isometries of a certain shifted hexagonal packing will be computed and compared with that of the hexagonal packing.

math.MG↗

arXiv2019

Inverse Kinematics of Some General 6R/P Manipulators

We develop an algorithm that solves the inverse kinematics of general serial 2RP3R, 2R2P2R, 3RP2R and 6R manipulators based on the HuPf algorithm. We identify the workspaces of the 3-subchains of the manipulator with a quasi-projective variety in $\mathbb{P}^7$ via dual quaternions. This allows us to compute linear forms that describe linear spaces containing the workspaces of these 3-subchains. We present numerical examples that illustrate the algorithm and show the real solutions.

cs.RO↗

arXiv2019

Fundamental domains for rhombic lattices with dihedral symmetry of order 8

We show by construction that every rhombic lattice $Γ$ in $\mathbb{R}^{2}$ has a fundamental domain whose symmetry group contains the point group of $Γ$ as a subgroup of index $2$. This solves the last open case of a question raised in [3] on fundamental domains for planar lattices whose symmetry groups properly contain the point groups of the lattices.

math.CO↗

arXiv2018

Highly symmetric fundamental domains for lattices in R^2 and R^3

It is shown that most lattices $Γ$ in $\mathbb{R}^2$ and $\mathbb{R}^3$ possess a fundamental domain $F$ for the action of $Γ$ on $\mathbb{R}^2$, respectively $\mathbb{R}^3$, having more symmetries than the point group $P(Γ)$, i.e., the group $P (Γ) \subset O(d)$ fixing $Γ$. In particular, $P (Γ)$ is a subgroup of the symmetry group $S(F)$ of $F$ of index 2 in these cases. Exceptions are cubic lattices in the three-dimensional case, where such an $F$ does not exist. Possible exceptions are rhombic lattices in the plane case, where the constructions presented here do not seem to work.

math.CO↗

arXiv2018

The coincidence problem for shifted lattices and crystallographic point packings

A coincidence site lattice is a sublattice formed by the intersection of a lattice $Γ$ in $\mathbb{R}^d$ with the image of $Γ$ under a linear isometry. Such a linear isometry is referred to as a linear coincidence isometry of $Γ$. Here, we consider the more general case allowing any affine isometry. Consequently, general results on coincidence isometries of shifted copies of lattices, and of crystallographic point packings are obtained. In particular, we discuss the shifted square lattice and the diamond packing in detail.

math.MG↗

arXiv2016

Color groups of colorings of $N$-planar modules

A submodule of a $\mathbb{Z}$-module determines a coloring of the module where each coset of the submodule is associated to a unique color. Given a submodule coloring of a $\mathbb{Z}$-module, the group formed by the symmetries of the module that induces a permutation of colors is referred to as the color group of the coloring. In this contribution, a method to solve for the color groups of colorings of $N$-planar modules where $N=4$ and $N=6$ are given. Examples of colorings of rectangular lattices and of the vertices of the Ammann-Beenker tiling are given to exhibit how these methods may be extended to the general case.

math.MG↗

arXiv2016

Rational digit systems over finite fields and Christol's Theorem

Let $P, Q\in \mathbb{F}_q[X]\setminus\{0\}$ be two coprime polynomials over the finite field $\mathbb{F}_q$ with $\operatorname{deg}{P} > \operatorname{deg}{Q}$. We represent each polynomial $w$ over $\mathbb{F}_q$ by \[w=\sum_{i=0}^k\frac{s_i}{Q}{\left(\frac{P}{Q}\right)}^i\] using a rational base $P/Q$ and digits $s_i\in\mathbb{F}_q[X]$ satisfying $\operatorname{deg}{s_i} < \operatorname{deg}{P}$. Digit expansions of this type are also defined for formal Laurent series over $\mathbb{F}_q$. We prove uniqueness and automatic properties of these expansions. Although the $ω$-language of the possible digit strings is not regular, we are able to characterize the digit expansions of algebraic elements. In particular, we give a version of Christol's Theorem by showing that the digit string of the digit expansion of a formal Laurent series is automatic if and only if the series is algebraic over $\mathbb{F}_q[X]$. Finally, we study relations between digit expansions of formal Laurent series and a finite fields version of Mahler's $3/2$-problem.

math.NT↗

arXiv2015

Coincidence indices of sublattices and coincidences of colorings

Even though a lattice and its sublattices have the same group of coincidence isometries, the coincidence index of a coincidence isometry with respect to a lattice $Λ_1$ and to a sublattice $Λ_2$ may differ. Here, we examine the coloring of $Λ_1$ induced by $Λ_2$ to identify how the coincidence indices with respect to $Λ_1$ and to $Λ_2$ are related. This leads to a generalization of the notion of color symmetries of lattices to what we call color coincidences of lattices. Examples involving the cubic and hypercubic lattices are given to illustrate these ideas.

math.MG↗

arXiv2015

Symmetries and color symmetries of a family of tilings with a singular point

We obtain tilings with a singular point by applying conformal maps on regular tilings of the Euclidean plane, and determine its symmetries. The resulting tilings are then symmetrically colored by applying the same conformal maps on colorings of regular tilings arising from sublattice colorings of the centers of its tiles. In addition, we determine conditions so that the coloring of a tiling with singularity that is obtained in this manner is perfect.

math.MG↗

arXiv2014

Dually weighted Stirling-type sequences

We introduce a generalization of the Stirling numbers via symmetric functions involving two weight functions. The resulting extension unifies previously known Stirling-type sequences with known symmetric function forms, as well as other sequences such as the $p,q$-binomial coefficients. Recurrence relations, generating functions, orthogonality relations, convolution formulas, and determinants of certain matrices involving the obtained sequences are derived. We also give combinatorial interpretations of certain cases in terms of colored partitions and permutations.

math.CO↗

arXiv2013

Coincidences of a shifted hexagonal lattice and the hexagonal packing

A geometric study of twin and grain boundaries in crystals and quasicrystals is achieved via coincidence site lattices (CSLs) and coincidence site modules (CSMs), respectively. Recently, coincidences of shifted lattices and multilattices (i.e. finite unions of shifted copies of a lattice) have been investigated. Here, we solve the coincidence problem for a shifted hexagonal lattice. This result allows us to analyze the coincidence isometries of the hexagonal packing by viewing the hexagonal packing as a multilattice.

math.MG↗

arXiv2010

Colourings of lattices and coincidence site lattices

The relationship between the coincidence indices of a lattice $Γ_1$ and a sublattice $Γ_2$ of $Γ_1$ is examined via the colouring of $Γ_1$ that is obtained by assigning a unique colour to each coset of $Γ_2$. In addition, the idea of colour symmetry, originally defined for symmetries of lattices, is extended to coincidence isometries of lattices. An example involving the Ammann-Beenker tiling is provided to illustrate the results in the quasicrystal setting.

math.MG↗

arXiv2010

Coincidence isometries of a shifted square lattice

We consider the coincidence problem for the square lattice that is translated by an arbitrary vector. General results are obtained about the set of coincidence isometries and the coincidence site lattices of a shifted square lattice by identifying the square lattice with the ring of Gaussian integers. To illustrate them, we calculate the set of coincidence isometries, as well as generating functions for the number of coincidence site lattices and coincidence isometries, for specific examples.

math.MG↗

arXiv2010

Enumerating and identifying semiperfect colorings of symmetrical patterns

If $G$ is the symmetry group of an uncolored pattern then a coloring of the pattern is semiperfect if the associated color group $H$ is a subgroup of $G$ of index 2. We give results on how to identify and enumerate all inequivalent semiperfect colorings of certain patterns. This is achieved by treating a coloring as a partition $\{hJ_iY_i:i\in I,h\in H\}$ of $G$, where $H$ is a subgroup of index 2 in $G$, $J_i\leq H$ for $i\in I$, and $Y=\cup_{i\in I}{Y_i}$ is a complete set of right coset representatives of $H$ in $G$. We also give a one-to-one correspondence between inequivalent semiperfect colorings whose associated color groups are conjugate subgroups with respect to the normalizer of $G$ in the group of isometries of $\mathbf{R}^n$.

math.CO↗