arXiv subjects

Alessio Basti

Publications and source records attributed to Alessio Basti.

8 recordsLinked to original sources

arXiv2026

The Asayama-Matsumoto conjecture and a refined discrepancy bound

Every $n$-vertex plane triangulation admits a polychromatic red--blue vertex coloring with discrepancy at most $n-2\ceil{n/3}\le\floor{n/3}$, resolving a conjecture of Asayama and Matsumoto. The proof follows by combining Loyola et al. 2026 and Kawarabayashi et al. 2026. For $n\ge6$ and $n\not\equiv5\pmod6$, we prove the sharper bound $n-2\ceil{(n+2)/3}$. An exhaustive census of all $9{,}150$ non-isomorphic sphere triangulations with $4\le n\le12$ verifies the bounds and also confirms the sharper estimate at $n=11$, the first order in the remaining open residue class.

math.CO↗

arXiv2026

Markov Chain Approximation of Sticky Diffusions and Hamilton-Jacobi-Bellman Equations on Networks

We propose a discrete Markov-chain approximation of diffusion processes on networks with both Kirchhoff and sticky vertex conditions. Stickiness is modeled by a probabilistic residence mechanism at the vertex, while the motion along the edges follows an Euler-Maruyama-type update at the diffusive scale. We prove that the associated time-interpolated chain converges in distribution to the limiting diffusion in the Skorokhod space using the Ethier-Kurtz framework. Based on this construction, we derive a fully discrete semi-Lagrangian scheme for Hamilton-Jacobi-Bellman equations on networks and establish its convergence using viscosity solution techniques.

math.NA↗

arXiv2026

Coevolution of self-replication and function in a digital primordial soup

While traditional evolutionary algorithms hard-code reproduction, self-replication can emerge spontaneously within digital ``primordial soups''. This paper investigates the coevolution of such emergent self-replication alongside problem-solving capabilities. We initialize a population of random 32-byte Z80 assembly programs, requiring self-replication to arise purely through random assembly-level mutations and pairwise program interactions. To couple computation with reproduction, we introduce a task-based validation step: correctly evaluating a polynomial raises a program's interaction probability above a baseline rate. Our experiments yield four primary findings. First, self-replication and mathematical problem-solving successfully coevolve from initial randomness. Second, the pressure to compute accelerates the emergence of compact, robust reproductive architectures that preserve memory for task execution. Third, applying metabolic constraints that penalize runtime promotes the emergence of sophisticated conditional execution patterns that reduce energy use. Finally, partitioning programs into interconnected task niches generates an emergent learning curriculum that utilizes simple solutions as stepping stones toward more complex tasks. Altogether, these results demonstrate an interactive feedback loop: environmental task demands actively shape the physical architecture of self-replication, while spontaneous replication alters the evolutionary trajectory of functional problem-solving.

cs.NE↗

arXiv2026

Kolmogorov $\varepsilon$-entropy of numerical solutions for scalar conservation laws with convex flux

Building on the information-theoretic perspective of P.~D.~Lax [\textit{Proc.\ Sympos., Math.\ Res.\ Center, Univ.\ Wisconsin}, 1978], we establish a two-sided quantitative compactness estimate for numerical solutions of scalar conservation laws with a uniformly convex flux, expressed in terms of Kolmogorov $\varepsilon$-entropy. We prove that, under specific grid constraints, conservative, monotone finite-difference schemes satisfying a discrete one-sided Lipschitz condition (OSLC) preserve the $1/\varepsilon$ Kolmogorov entropy scaling of the corresponding exact entropy solution set, matching the bounds obtained by De~Lellis and Golse [\textit{Comm.\ Pure Appl.\ Math.}\ \textbf{58} (2005)] and by Ancona, Glass, and Nguyen [\textit{Comm.\ Pure Appl.\ Math.}\ \textbf{65} (2012)]. Specifically, the upper bound follows from the discrete OSLC, while the lower bound relies on a uniform approximation argument on a bounded-variation precursor class. Our results show that prototypical first-order methods are high-resolution in Lax's sense. Finally, we abstract the lower bound mechanism into a general transfer principle, discuss implications for information recovery via post-processing, and indicate directions for future work.

math.NA↗

arXiv2026

A Generalized Framework of Antisymmetric Polyspectral Indices for Identifying High-Order Neural Interactions

Cross-frequency interactions are fundamental brain mechanisms for integrating information across temporal scales. However, accurate identification of these couplings is hindered by complex multi-frequency nonlinearities and by spurious, zero-lag artifacts caused by volume conduction. To our knowledge, conventional metrics lack a robust framework to characterize genuine interactions among multiple time series where a frequency of interest $f_N$ arises from the combination of $N-1$ components such that $f_N = \sum_{i=1}^{N-1} f_i$. We introduce a general family of antisymmetric cross-polyspectral indices designed to quantify these harmonic dependencies while being intrinsically robust to instantaneous mixing. We derive the theoretical properties of these quantities and validate them through simulations of cubic nonlinearities. As a proof of concept, we apply the indices to empirical EEG recordings; the results reveal significant higher-order dependencies that elude standard analytical approaches. We further discuss how these indices can inform novel, personalized multi-site transcranial magnetic stimulation (mTMS) protocols by enabling the selective monitoring and modulation of specific multi-frequency network interactions.

q-bio.NC↗

arXiv2026

Bridging Theory and Practice in Crafting Robust Spiking Reservoirs

Spiking reservoir computing provides an energy-efficient approach to temporal processing, but reliably tuning reservoirs to operate at the edge-of-chaos is challenging due to experimental uncertainty. This work bridges abstract notions of criticality and practical stability by introducing and exploiting the robustness interval, an operational measure of the hyperparameter range over which a reservoir maintains performance above task-dependent thresholds. Through systematic evaluations of Leaky Integrate-and-Fire (LIF) architectures on both static (MNIST) and temporal (synthetic Ball Trajectories) tasks, we identify consistent monotonic trends in the robustness interval across a broad spectrum of network configurations: the robustness-interval width decreases with presynaptic connection density $\beta$ (i.e., directly with sparsity) and directly with the firing threshold $\theta$. We further identify specific $(\beta, \theta)$ pairs that preserve the analytical mean-field critical point $w_{\text{crit}}$, revealing iso-performance manifolds in the hyperparameter space. Control experiments on Erd\H{o}s-R\'enyi graphs show the phenomena persist beyond small-world topologies. Finally, our results show that $w_{\text{crit}}$ consistently falls within empirical high-performance regions, validating $w_{\text{crit}}$ as a robust starting coordinate for parameter search and fine-tuning. To ensure reproducibility, the full Python code is publicly available.

cs.LG↗

arXiv2026

A Mean Field Games Perspective on Evolutionary Clustering

We propose a control-theoretic framework for evolutionary clustering based on quasi-stationary Mean Field Games. Each cluster is represented by a probability density whose evolution is governed by a Fokker--Planck equation, while the associated velocity field is determined through a stationary Hamilton--Jacobi equation. The general formulation does not prescribe a finite-dimensional statistical shape for the component densities, although the number of components is fixed. In the Gaussian specialization, we show that suitable affine dynamics reproduce the mean and covariance trajectories generated by the classical Expectation--Maximization procedure. To improve temporal coherence in the presence of noise and temporary cluster overlaps, we introduce causal and non-causal time-averaged log-likelihood objectives. We also develop a fully density-based numerical implementation for non-Gaussian components. The proposed formulations are assessed on synthetic and real time-dependent datasets and compared with independent snapshot Expectation--Maximization and with the same method applied to temporally smoothed observations. In the two-dimensional benchmark, an established evolutionary $k$-means method is also included as an external dynamic-clustering baseline.

math.NA↗

arXiv2025

$L^p$ Estimates for Numerical Approximation of Convex Hamilton-Jacobi Equations

We establish $L^p$ error estimates for monotone numerical schemes approximating convex Hamilton-Jacobi equations on the $d$-dimensional torus. Using the adjoint method and semiconcavity estimates, we first prove an $L^1$ error bound of order one for classical monotone schemes of Crandall-Lions type and semi-Lagrangian schemes under standard convexity assumptions on the Hamiltonian and semiconcavity assumptions on the initial datum. By interpolation with the classical $L^\infty$ estimate, we obtain $L^p$ estimates for every $1\le p<+\infty$.

math.AP↗