arXiv subjects

Małgorzata Nowak-Kępczyk

Publications and source records attributed to Małgorzata Nowak-Kępczyk.

6 recordsLinked to original sources

arXiv2026

Frobenius Revivals in Laplacian Cellular Automata: Replication, Entropy, and Information Recovery

We investigate Frobenius revivals in prime-modulus Laplacian cellular automata. The Frobenius identity yields exact superpositions of translated, coefficient-weighted seed copies at prime-power times; we distinguish this algebraic replication from their geometric separation. We investigate seed recovery from separated replicas and evaluate the correction of independent symbol errors introduced during transmission of encoded configurations. For the diagonal neighborhood, we determine the exact periods of moving corner operators on square windows and establish sufficient conditions for recovering any binary seed under a binary--ternary--binary schedule. The results connect prime-power algebraic structure with spatial pattern formation and precisely delimited forms of information recovery.

nlin.CG↗

arXiv2026

Stack Constructions and Equality Cases for Laffey's Bound for Maximal Commutative Matrix Algebras

Let $A\subseteq M_n(K)$ be a local maximal commutative algebra whose radical satisfies $J^3=0$. Laffey obtained a sharp lower bound for $\dim A$ in terms of the three Loewy-layer dimensions and constructed an infinite family of equality cases. We introduce a stacking operation which combines self-centralizing cube-zero algebras with common outer Loewy layers and preserves maximal commutativity. This gives, for every $a,c\ge2$, a standard equality brick \[B_{a,c}=(a,ac+1,c;a+c)\] and reduces the construction of further equality cases to the study of primitive integral points on the corresponding equality rays. Over fields of characteristic zero, we determine all minimal equality bricks on the balanced branch. In particular, for every odd $k\ge3$, \[E_{k,k} =\left(k,\frac{k^2+1}{2},k;k\right),\qquad E_{k,k}^{\star2}\sim_{\rm type}B_{k,k}.\] We also determine the balanced--nearly-balanced envelope of the minimized Laffey profile, construct the first nontrivial nearly balanced root $E_{4,5}=(4,7,5;3)$, and solve the complete $(4,5)$-sector for $b\ge4$. As consequences, the minimized cube-zero profile is attained for every $14\le n\le52$, for every $93\le n\le114$, and along several infinite families of matrix dimensions. The finite exceptional bricks used in the constructions are accompanied by exact centralizer certificates.

math.RA↗

arXiv2026

Long-Lived Carpet-Like Transients in Time-Dependent Modular Discrete Laplacian Dynamics

Binary modular Laplacian dynamics is organized by repeated replication, growth, and collapse on dyadic scales. We study how isolated and periodically repeated non-binary updates modify this organization and whether they can sustain densely occupied geometric structures. Computational experiments across multiple finite seeds, neighborhood masks, inserted moduli, and periodic schedules show that constant prime moduli retain the expected $p$-adic replication hierarchy. A single non-binary insertion mainly acts as an effective-seed replacement: subsequent evolution remains binary-like, but may resume at an integer phase shift of the collapse--recovery cycle, with only modest finite-time densification. A qualitatively different response occurs for periodic schedules $[2,k,2^s]^\infty$, where $2^s$ denotes $s$ consecutive binary updates. Odd-modulus insertions can sustain long-lived, spatially coherent carpet-like regimes with comparatively high and stable occupation, whereas even-modulus insertions remain binary-like. Ternary insertion gives the broadest and most robust high-density response. For $k=3$, density depends non-monotonically on the binary-tail length, with troughs near $s=7,15,23$ that reveal phase-sensitive interaction with the binary epoch structure. Cross-modulus synchronization also occurs: for degree-four diagonal Neumann and von Neumann masks, $k=5,7,9$ repeatedly collapse onto the same complete configuration at subsequent binary phases. A parity argument explains this loss of memory. Increasing seed extent generally raises occupation while reducing tail-length sensitivity. Overall, periodic modular insertions act as phase-selective, geometry-dependent mechanisms transforming $p$-adically organized replication into long-lived carpet-like dynamics.

math.DS↗

arXiv2026

On the minimal dimension of maximal commutative subalgebras of $M_6(k)$

We study the minimal dimension of maximal commutative subalgebras of the matrix algebra $M_n(k)$ over an algebraically closed field. While examples with dimension strictly smaller than n are known for $n \geq 14$, no such examples are known in smaller dimensions. In this paper, we show that for n = 6 every maximal commutative subalgebra $A\subset M_6(k)$ satisfies $\dim A \geq 6$. The proof is based on a detailed analysis of local algebras and their module structure, combined with explicit estimates of the dimension of the centralizer.

math.RA↗

arXiv2026

Fractal Patterns in Discrete Laplacians: Iterative Construction on 2D Square Lattices

We investigate the iterative construction of discrete Laplacians on 2D square lattices, revealing emergent fractal-like patterns shaped by modular arithmetic. While classical 2222-style iterations reproduce known structures such as the Sierpinski triangle, our alternating binary-ternary (2322-style) process produces a novel class of aperiodic figures. These display low density variance, minimal connectivity loss, and non-repetitive organization reminiscent of Dekking's sequences. Fourier and autocorrelation analyses confirm their quasi-periodic nature, suggesting applications in self-assembly, sensor networks, and biological modeling. The findings open new paths toward structured randomness and fractal dynamics in discrete systems. These findings also open avenues for exploring higher-dimensional Laplacian constructions and their implications in quasicrystals, aperiodic tilings, and stochastic processes.

math.DS↗

arXiv2025

Stabilization and Regaining Periodicity in Modular Laplacian Dynamics

We study discrete Laplacians on two-dimensional lattices under modular iterations, focusing on the emergence of nontrivial large-scale patterns. While purely binary or constant modular sequences quickly collapse into strict periodicity, the insertion of a single non-binary step k yields qualitatively new behavior. Through extensive computer-assisted exploration we identify a taxonomy of long-lived figures - rugs, quasi-carpets, and carpets - whose occurrence depends systematically on seed symmetry, neighborhood mask, and sequence structure. In particular, we show that mixed families of the form [2,k,2 to power s] can stabilize high-density carpets beyond the universal decay time characteristic of binary dynamics. Our approach combines algebraic replication laws with large-scale simulations and density tracking, producing both theoretical conditions (periodicity via Lucas theorem, non-overlap criteria) and experimental evidence of persistent quasi-aperiodic architectures. The results highlight how minimal modifications in discrete local rules generate unexpectedly rich multiscale geometries, bridging rigorous analysis with computer-assisted discovery.

math.DS↗