arXiv subjects

Jacek Małecki

Publications and source records attributed to Jacek Małecki.

15 recordsLinked to original sources

arXiv2026

Collision types and times in interacting particle systems

We consider a system of stochastic interacting particles with general diffusion coefficient and drift functions and we study the types of collisions that arise in them. In particular, interactions between particles are inversely proportional to their separation, and the coupling function of interaction is also considered in great generality. Our main result shows that, under positivity and the stated variance compatibility conditions, no two distinct positive-root hyperplanes are reached simultaneously at a positive time. In type $A_{N-1}$, this excludes both collisions involving three or more particles and simultaneous collisions of disjoint pairs. In order to obtain our results we make use of symmetric polynomials in squared root projections; the degree of these polynomials indicates the type of collision, and by a locality argument we show that polynomials indicating a non-simple collision almost surely do not cancel. We use this result to obtain upper and lower bounds for the Hausdorff dimension of the set of collision times in terms of interaction-to-variance ratios. These bounds coincide in particular constant-ratio cases. Our results cover many of the most well-known particle systems, such as the Dyson model and Wishart processes and their extensions to non-constant diffusion coefficients and background drifts.

math.PR↗

arXiv2026

Just on Time: Token-Level Early Stopping for Diffusion Language Models

Diffusion language models generate text through iterative refinement, a process that is often computationally inefficient because many tokens reach stability long before the final denoising step. We introduce a training-free, token-level early stopping approach that identifies convergence independently at each position. Our method leverages lightweight signals derived from the model's predictions and local context to dynamically determine when individual tokens can be finalized. This yields adaptive per-token freezing without task-specific fine-tuning, substantially reducing the total number of diffusion steps required. Across diverse benchmarks, spanning mathematical reasoning, general question answering, and scientific understanding, our approach achieves substantial efficiency gains while preserving generation quality.

cs.LG↗

arXiv2026

Generalized Bessel-Dunkl diffusions

We develop a general theory of Bessel-Dunkl type diffusions in Weyl chambers associated with classical root systems. The class considered here allows time-dependent and configuration-dependent diffusion and drift coefficients, as well as state-dependent singular repulsion coefficients which may vanish on the walls of the chamber. This includes, in a unified framework, Dyson-type logarithmic particle systems, radial Dunkl processes, squared Bessel and Wishart particle systems, and non-colliding diffusion models. Using the geometry of the underlying root system and a symmetric-polynomial approach, we establish weak existence in both the strictly positive repulsion and the degenerate regimes. In the strictly positive regime, we prove that positive repulsion prevents positive-time multiple collisions. In the degenerate case, we identify geometric conditions which exclude boundary sticking and allow one to recover the genuine singular equation from its interior form. We also prove non-explosion under a radial linear-growth condition, obtain pathwise uniqueness and strong existence in the non-sticky class under natural Yamada-Watanabe type or locally Lipschitz assumptions, and establish a mean-field convergence theorem for systems of types $A_{N-1}$, $B_N$, and $D_N$. Together, these results build the structural foundations for a systematic theory of generalized Bessel-Dunkl diffusions with non-constant and possibly degenerate coefficients.

math.PR↗

arXiv2026

A Decentralized Frontier AI Architecture Based on Personal Instances, Synthetic Data, and Collective Context Synchronization

Recent progress in artificial intelligence has been driven largely by the scaling of centralized large language models through increased parameters, datasets, and computational resources. While effective, this paradigm introduces structural constraints related to compute concentration, energy consumption, data availability, and governance. This paper proposes an alternative architectural approach through the H3LIX Decentralized Frontier Model Architecture (DFMA), a distributed AI framework in which locally operating AI instances generate synthetic learning signals derived from reasoning processes and interactions. These signals are aggregated within a shared contextual substrate termed the Collective Context Field (CCF), which conditions reasoning behavior across the network without requiring direct parameter synchronization. By enabling contextual signal propagation rather than centralized retraining at every iteration, the architecture can be designed to support privacy-preserving collective learning under explicit assumptions, while facilitating distributed sharing of learned abstractions. The system further integrates Energy-Adaptive Model Evolution, aligning learning activities with renewable energy availability to support more sustainable AI infrastructure. Conceptually, the architecture reframes artificial intelligence as a distributed cognitive system analogous to biological neural networks, in which intelligence emerges from the interaction of many locally adaptive agents within a shared contextual environment. Together, these mechanisms suggest a new scaling pathway for artificial intelligence systems based on distributed contextual learning and collective experience accumulation.

cs.HC↗

arXiv2026

Rogue Variable Theory: A Quantum-Compatible Cognition Framework with a Rosetta Stone Alignment Algorithm

Many of the most consequential dynamics in human cognition occur \emph{before} events become explicit: before decisions are finalized, emotions are labeled, or meanings stabilize into narrative form. These pre-event states are characterized by ambiguity, contextual tension, and competing latent interpretations. Rogue Variable Theory (RVT) formalizes such states as \emph{Rogue Variables}: structured, pre-event cognitive configurations that influence outcomes while remaining unresolved or incompatible with a system's current representational manifold. We present a quantum-consistent information-theoretic implementation of RVT based on a time-indexed \emph{Mirrored Personal Graph} (MPG) embedded into a fixed graph Hilbert space, a normalized \emph{Quantum MPG State} (QMS) constructed from node and edge metrics under context, Hamiltonian dynamics derived from graph couplings, and an error-weighted `rogue operator'' whose principal eigenvectors identify rogue factor directions and candidate Rogue Variable segments. We further introduce a \emph{Rosetta Stone Layer} (RSL) that maps user-specific latent factor coordinates into a shared reference Hilbert space to enable cross-user comparison and aggregation without explicit node alignment. The framework is fully implementable on classical systems and does not assume physical quantum processes; \emph{collapse} is interpreted as informational decoherence under interaction, often human clarification.

q-bio.NC↗

arXiv2025

Person-AI Bidirectional Fit - A Proof-Of-Concept Case Study Of Augmented Human-Ai Symbiosis In Management Decision-Making Process

This article develops the concept of Person-AI bidirectional fit, defined as the continuously evolving, context-sensitive alignment-primarily cognitive, but also emotional and behavioral-between a human decision-maker and an artificial intelligence system. Grounded in contingency theory and quality theory, the study examines the role of P-AI fit in managerial decision-making through a proof-of-concept case study involving a real hiring process for a Senior AI Lead. Three decision pathways are compared: (1) independent evaluations by a CEO, CTO, and CSO; (2) an evaluation produced by an augmented human-AI symbiotic intelligence system (H3LIX-LAIZA); and (3) an assessment generated by a general-purpose large language model. The results reveal substantial role-based divergence in human judgments, high alignment between H3LIX-LAIZA and the CEOs implicit decision model-including ethical disqualification of a high-risk candidate and a critical false-positive recommendation from the LLMr. The findings demonstrate that higher P-AI fit, exemplified by the CEO H3LIX-LAIZA relationship, functions as a mechanism linking augmented symbiotic intelligence to accurate, trustworthy, and context-sensitive decisions. The study provides an initial verification of the P-AI fit construct and a proof-of-concept for H3LIX-LAIZA as an augmented human-AI symbiotic intelligence system.

cs.HC↗

arXiv2021

Archimedes' principle for ideal gas

We prove Archimedes' principle for a macroscopic ball in ideal gas consisting of point particles with non-zero mass. The main result is an asymptotic theorem, as the number of point particles goes to infinity and their total mass remains constant. We also show that, asymptotically, the gas has an exponential density as a function of height. We find the asymptotic inverse temperature of the gas. We derive an accurate estimate of the volume of the phase space using the local central limit theorem.

math.PR↗

arXiv2019

Density behaviour related to Lévy processes

Let $p_t(x)$, $f_t(x)$ and $q_t^*(x)$ be the densities at time $t$ of a real Lévy process, its running supremum and the entrance law of the reflected excursions at the infimum. We provide relationships between the asymptotic behaviour of $p_t(x)$, $f_t(x)$ and $q_t^*(x)$, when $t$ is small and $x$ is large. Then for large $x$, these asymptotic behaviours are compared to this of the density of the Lévy measure. We show in particular that, under mild conditions, if $p_t(x)$ is comparable to $tν(x)$, as $t\rightarrow0$ and $x\rightarrow\infty$, then so is $f_t(x)$.

math.PR↗

arXiv2019

The entrance law of the excursion measure of the reflected process for some classes of Lévy processes

We provide integral formulae for the Laplace transform of the entrance law of the reflected excursions for symmetric Lévy processes in terms of their characteristic exponent. For subordinate Brownian motions and stable processes we express the density of the entrance law in terms of the generalized eigenfunctions for the semigroup of the process killed when exiting the positive half-line. We use the formulae to study in-depth properties of the density of the entrance law such as asymptotic behavior of its derivatives in time variable.

math.PR↗

arXiv2019

Universality classes for general random matrix flows

We consider matrix-valued processes described as solutions to stochastic differential equations of very general form. We study the family of the empirical measure-valued processes constructed from the corresponding eigenvalues. We show that the family indexed by the size of the matrix is tight under very mild assumptions on the coefficients of the initial SDE. We characterize the limiting distributions of its subsequences as solutions to an integral equation. We use this result to study some universality classes of random matrix flows. These generalize the classical results related to Dyson Brownian motion and squared Bessel particle systems. We study some new phenomenons as the existence of the generalized Marchenko-Pastur distributions supported on the real line. We also introduce universality classes related to generalized geometric matrix Brownian motions and Jacobi processes. Finally we study, under some conditions, the convergence of the empirical measure-valued process of eigenvalues associated to matrix flows to the law of a free diffusion.

math.PR↗

arXiv2019

Reflection principles for functions of Neumann and Dirichlet Laplacians on open reflection invariant subsets of $\mathbb{R}^d$

For an open subset $Ω$ of $\mathbb R^d$, symmetric with respect to a hyperplane and with positive part $Ω_+$, we consider the Neumann/Dirichlet Laplacians $-Δ_{N/D,Ω}$ and $-Δ_{N/D,Ω_+}$. Given a Borel function $Φ$ on $[0,\infty)$ we apply the spectral functional calculus and consider the pairs of operators $Φ(-Δ_{N,Ω})$ and $Φ(-Δ_{N,Ω_+})$, or $Φ(-Δ_{D,Ω})$ and $Φ(-Δ_{D,Ω_+})$. We prove relations between the integral kernels for the operators in these pairs, which in particular cases of $Ω_+=\mathbb{R}^{d-1}\times(0,\infty)$ and $Φ_{t}(u)=\exp(-tu)$, $u \geq 0$, $t>0$, were known as reflection principles for the Neumann/Dirichlet heat kernels. These relations are then generalized to the context of symmetry with respect to a finite number of mutually orthogonal hyperplanes.

math.CA↗

arXiv2018

Short proofs in extrema of spectrally one sided Lévy processes

We provide short and simple proofs of the continuous time ballot theorem for processes with cyclically interchangeable increments and Kendall's identity for spectrally positive Lévy processes. We obtain the later result as a direct consequence of the former. The ballot theorem is extended to processes having possible negative jumps. Then we prove through straightforward arguments based on the law of bridges and Kendall's identity, Theorem 2.4 in \cite{mpp} which gives an expression for the law of the supremum of spectrally positive Lévy processes. An analogous formula is obtained for the supremum of spectrally negative Lévy processes.

math.PR↗

arXiv2015

Asymptotic estimate of eigenvalues of pseudo-differential operators in an interval

We prove a two-term Weyl-type asymptotic law, with error term O(1/n), for the eigenvalues of the operator psi(-Delta) in an interval, with zero exterior condition, for complete Bernstein functions psi such that x psi'(x) converges to infinity as x goes to infinity. This extends previous results obtained by the authors for the fractional Laplace operator (psi(x) = x^{alpha/2}) and for the Klein-Gordon square root operator (psi(x) = (1+x^2)^{1/2} - 1). The formula for the eigenvalues in (-a,a) is of the form lambda_n = psi(mu_n^2) + O(1/n), where mu_n is the solution of mu_n = (n pi)/(2 a) - theta(mu_n)/a, and theta(mu) is given as an integral involving psi.

math.SP↗

arXiv2013

Suprema of Lévy processes

In this paper we study the supremum functional $M_t=\sup_{0\le s\le t}X_s$, where $X_t$, $t\ge0$, is a one-dimensional Lévy process. Under very mild assumptions we provide a simple, uniform estimate of the cumulative distribution function of $M_t$. In the symmetric case we find an integral representation of the Laplace transform of the distribution of $M_t$ if the Lévy-Khintchin exponent of the process increases on $(0,\infty)$.

math.PR↗

arXiv2009

Spectral properties of the Cauchy process

We study the spectral properties of the transition semigroup of the killed one-dimensional Cauchy process on the half-line (0,infty) and the interval (-1,1). This process is related to the square root of one-dimensional Laplacian A = -sqrt(-d^2/dx^2) with a Dirichlet exterior condition (on a complement of a domain), and to a mixed Steklov problem in the half-plane. For the half-line, an explicit formula for generalized eigenfunctions psi_lambda of A is derived, and then used to construct spectral representation of A. Explicit formulas for the transition density of the killed Cauchy process in the half-line (or the heat kernel of A in (0,infty)), and for the distribution of the first exit time from the half-line follow. The formula for psi_lambda is also used to construct approximations to eigenfunctions of A in the interval. For the eigenvalues lambda_n of A in the interval the asymptotic formula lambda_n = n pi/2 - pi/8 + O(1/n) is derived, and all eigenvalues lambda_n are proved to be simple. Finally, efficient numerical methods of estimation of eigenvalues lambda_n are applied to obtain lower and upper numerical bounds for the first few eigenvalues up to 9th decimal point.

math.SP↗