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Jacek Malecki

Publications and source records attributed to Jacek Malecki.

18 recordsLinked to original sources

Managing Ambiguity: A Proof of Concept of Human-AI Symbiotic Sense-making based on Quantum-Inspired Cognitive Mechanism of Rogue Variable Detection

Organizations increasingly operate in environments characterized by volatility, uncertainty, complexity, and ambiguity (VUCA), where early indicators of change often emerge as weak, fragmented signals. Although artificial intelligence (AI) is widely used to support managerial decision-making, most AI-based systems remain optimized for prediction and resolution, leading to premature interpretive closure under conditions of high ambiguity. This creates a gap in management science regarding how human-AI systems can responsibly manage ambiguity before it crystallizes into error or crisis. This study addresses this gap by presenting a proof of concept (PoC) of the LAIZA human-AI augmented symbiotic intelligence system and its patented process: Systems and Methods for Quantum-Inspired Rogue Variable Modeling (QRVM), Human-in-the-Loop Decoherence, and Collective Cognitive Inference. The mechanism operationalizes ambiguity as a non-collapsed cognitive state, detects persistent interpretive breakdowns (rogue variables), and activates structured human-in-the-loop clarification when autonomous inference becomes unreliable. Empirically, the article draws on a three-month case study conducted in 2025 within the AI development, involving prolonged ambiguity surrounding employee intentions and intellectual property boundaries. The findings show that preserving interpretive plurality enabled early scenario-based preparation, including proactive patent protection, allowing decisive and disruption-free action once ambiguity collapsed. The study contributes to management theory by reframing ambiguity as a first-class construct and demonstrates the practical value of human-AI symbiosis for organizational resilience in VUCA environments.

cs.HC

On squared Bessel particle systems

We study the existence and uniqueness of SDEs describing squared Bessel particles systems in full generality. We define non-negative and non-colliding squared Bessel particle systems and we study their properties.

math.PR

A Characterization of Wishart Processes and Wishart Distributions

A characterization of the existence of non-central Wishart distributions (with shape and non-centrality parameter) as well as the existence of solutions to Wishart stochastic differential equations (with initial data and drift parameter) in terms of their exact parameter domains is given. These two families are the natural extensions of the non-central chi-square distributions and the squared Bessel processes to the positive semidefinite matrices.

math.PR

Dirichlet heat kernel for the Laplacian in a ball

We provide sharp two-sided estimates on the Dirichlet heat kernel $k_1(t,x,y)$ for the Laplacian in a ball. The result accurately describes the exponential behaviour of the kernel for small times and significantly improves the qualitatively sharp results known so far. As a consequence we obtain the full description of the kernel $k_1(t,x,y)$ in terms of its global two-sided sharp estimates.

math.AP

Wallach sets and squared Bessel particle systems

We determine the classical and the non-central Wallach sets $W_0$ and $W$ by classical probabilistic methods. We prove the Mayerhofer conjecture on $W$. We exploit the fact that $(x_0,β)\in W$ if and only if $x_0$ is the starting point and $2β$ is the drift of a squared Bessel matrix process $X_t$ on the cone $\bar{Sym^+(\mathbf{R},p)}$. Our methods are based on the study of SDEs for the symmetric polynomials of $X_t$ and for the eigenvalues of $X_t$, i.e. the squared Bessel particle systems.

math.PR

Fourier-Bessel heat kernel estimates

We provide sharp two-sided estimates of the Fourier-Bessel heat kernel and we give sharp two-sided estimates of the transition probability density for the Bessel process in (0,1) killed at 1 and killed or reflected at 0.

math.CA

Sharp estimates of Green function of hyperbolic Brownian Motion

The main objective of the work is to provide sharp two-sided estimates of $λ$-Green function of hyperbolic Brownian motion of a half-space. We strongly rely on recent results obtained by K. Bogus and J. Malecki [3], regarding precise estimates of the Bessel heat kernel of half-lines.

math.PR

Heat kernel estimates for the Bessel differential operator in half-line

In the paper we consider the Bessel differential operator L^(μ)=\dfrac{d^2}{dx^2}+\dfrac{2μ+1}{x}\dfrac{d}{dx} in half-line (a,\infty), a>0, and its Dirichlet heat kernel p_a^(μ)(t,x,y). For μ=0, by combining analytical and probabilistic methods, we provide sharp two-sided estimates of the heat kernel for the whole range of the space parameters x,y>a and every t>0, which complements the recent results given in [1], where the case μ\neq 0 was considered.

math.AP

Strong solutions of non-colliding particle systems

We study systems of stochastic differential equations describing positions x_1,x_2,...,x_p of p ordered particles, with inter-particles repulsions of the form H_{ij}(x_i,x_j)/(x_i-x_j). We show the existence of strong and pathwise unique non-colliding solutions of the system with a colliding initial point x_1(0)\leq ...\leq x_p(0) in the whole generality, under natural assumptions on the coefficients of the equations.

math.PR

The asymptotic behavior of the density of the supremum of Lévy processes

Let us consider a real Lévy process X whose transition probabilities are absolutely continuous and have bounded densities. Then the law of the past supremum of X before any deterministic time t is absolutely continuous on (0,\infty). We show that its density f_t(x) is continuous on (0,\infty) if and only if the potential density h' of the upward ladder height process is continuous on (0,\infty). Then we prove that f_t behaves at 0 as h'. We also describe the asymptotic behaviour of f_t, when t tends to infinity. Then some related results are obtained for the density of the meander and this of the entrance law of the Lévy process conditioned to stay positive.

math.PR

Multidimensional Yamada-Watanabe theorem and its applications to particle systems

A multidimensional version of the Yamada-Watanabe theorem is proved. It implies a spectral matrix Yamada-Watanabe theorem. It is also applied to particle systems of squared Bessel processes, corresponding to matrix analogues of squared Bessel processes: Wishart and Jacobi matrix processes. The beta-versions of these particle systems are also considered.

math.PR

One-dimensional quasi-relativistic particle in the box

Two-term Weyl-type asymptotic law for the eigenvalues of one-dimensional quasi-relativistic Hamiltonian (-h^2 c^2 d^2/dx^2 + m^2 c^4)^(1/2) + V_well(x) (the Klein-Gordon square-root operator with electrostatic potential) with the infinite square well potential V_well(x) is given: the n-th eigenvalue is equal to (n pi/2 - pi/8) h c/a + O(1/n), where 2a is the width of the potential well. Simplicity of eigenvalues is proved. Some L^2 and L^infinity properties of eigenfunctions are also studied. Eigenvalues represent energies of a `massive particle in the box' quasi-relativistic model.

math-ph

First passage times for subordinate Brownian motions

Let X_t be a subordinate Brownian motion, and suppose that the Levy measure of the underlying subordinator has completely monotone density. Under very mild conditions, we find integral formulae for the tail distribution P(τ_x > t) of first passage times τ_x through a barrier at x > 0, and its derivatives in t. As a corollary, we examine the asymptotic behaviour of P(τ_x > t) and its t-derivatives, either as t goes to infinity or x goes to 0.

math.PR

Hitting half-spaces or spheres by the Ornstein-Uhlenbeck type diffusions

The purpose of the paper is to provide a general method for computing hitting distributions of some regular subsets D for Ornstein-Uhlenbeck type operators of the form 1/2Δ+ F\cdot\nabla, with F bounded and orthogonal to the boundary of D. As an important application we obtain integral representations of the Poisson kernel for a half-space and balls for hyperbolic Brownian motion and for the classical Ornstein-Uhlenbeck process. The method developed in the paper is based on stochastic calculus and on skew product representation of multidimensional Brownian motion and yields more complete results as those based on Feynmann-Kac technique.

math.PR

Hitting hyperbolic half-space

Let X^μ={X_t^μ;t>=0}, μ>0, be the n-dimensional hyperbolic Brownian motion with drift, that is a diffusion on the real hyperbolic space H^n having the Laplace-Beltrami operator with drift as its generator. We prove the reflection principle for X^μ, which enables us to study the process X^μkilled when exiting the hyperbolic half-space, that is the set D={x\in H^n: x_1>0}. We provide formulae, uniform estimates and describe asymptotic behavior of the Green function and the Poisson kernel of D for the process X^μ. Finally, we derive formula for the lambda-Poisson kernel of the set D.

math.PR

Hitting times of Bessel processes

Let $T_1^{(μ)}$ be the first hitting time of the point 1 by the Bessel process with index $μ\in \R$ starting from $x>1$. Using an integral formula for the density $q_x^{(μ)}(t)$ of $T_1^{(μ)}$, obtained in Byczkowski, Ryznar (Studia Math., 173(1):19-38, 2006), we prove sharp estimates of the density of $T_1^{(μ)}$ which exibit the dependence both on time and space variables. Our result provides optimal estimates for the density of the hitting time of the unit ball by the Brownian motion in $\mathbb{R}^n$, which improve existing bounds. Another application is to provide sharp estimates for the Poisson kernel for half-spaces for hyperbolic Brownian motion in real hyperbolic spaces.

math.PR

Spectral Properties of the Massless Relativistic Harmonic Oscillator

The spectral properties of the pseudo-differential operator $(-d^2/dx^2)^{1/2}+x^2$ are analyzed by a combination of functional integration methods and direct analysis. We obtain a representation of its eigenvalues and eigenfunctions, prove precise asymptotic formulae, and establish various analytic properties. We also derive trace asymptotics and heat kernel estimates.

math.SP