arXiv subjects

Andrzej Odrzywołek

Publications and source records attributed to Andrzej Odrzywołek.

4 recordsLinked to original sources

arXiv2026

Diversity of EML-type operators

The discovery of the EML operator, sufficient to evaluate the standard explicit purely transcendental elementary functions, has led to considerable interest and discussion across multiple scientific disciplines. However, most authors have focused on the binary EML itself, while numerous similar variants with slightly different properties are now known. This article attempts to close this gap by enumerating and classifying them. We also take this opportunity to clarify common misconceptions related to the EML operator. The principal goal, symbolic regression within an architecture as close as possible to proven neural networks which combine matrix multiplication with a single univariate non-linear activation function, remains beyond reach. Instead, we propose a Möbius layer, with rational functions replacing matrix operations, and showcase the recently discovered activation function eml(x,1/x), which allows exp(x) and ln(x) to be recovered separately, and hence all elementary functions to be evaluated within a rational generalization of the neural network.

cs.SC↗

arXiv2026

All elementary functions from a single binary operator

A single two-input gate suffices for all of Boolean logic in digital hardware. No comparable primitive has been known for continuous mathematics: computing elementary functions such as sin, cos, sqrt, and log has always required multiple distinct operations. Here I show that a single binary operator, eml(x,y)=exp(x)-ln(y), together with the constant 1, generates the standard repertoire of a scientific calculator. This includes constants such as e, pi, and i; arithmetic operations including addition, subtraction, multiplication, division, and exponentiation as well as the usual transcendental and algebraic functions. For example, exp(x)=eml(x,1), ln(x)=eml(1,eml(eml(1,x),1)), and likewise for all other operations. That such an operator exists was not anticipated; I found it by systematic exhaustive search and established constructively that it suffices for the concrete scientific-calculator basis. In EML (Exp-Minus-Log) form, every such expression becomes a binary tree of identical nodes, yielding a grammar as simple as S -> 1 | eml(S,S). This uniform structure also enables gradient-based symbolic regression: using EML trees as trainable circuits with standard optimizers (Adam), I demonstrate the feasibility of exact recovery of closed-form elementary functions from numerical data at shallow tree depths up to 4. The same architecture can fit arbitrary data, but when the generating law is elementary, it may recover the exact formula.

cs.SC↗

arXiv2025

Elastic Planetoids

Modeling the internal structure of self-gravitating solid and liquid bodies presents a challenge, as existing approaches are often limited to either overly simplistic constant-density approximations or more complex numerical equations of state. We present a detailed analysis of a tractable and physically motivated model for perfectly elastic, spherically symmetric self-gravitating bodies in hydrostatic equilibrium. The model employs a logarithmic equation of state (logotropic EOS) with a non-zero initial density and constant bulk modulus. Importantly, scaling properties of the model allow all solutions to be derived from a single, universal solution of an ordinary differential equation, resembling the Lane-Emden and Chandrasekhar models. The model provides new insights into stability issues and reveals oscillatory asymptotic behavior in the mass-radius relation, including the existence of both a maximum mass and a maximum radius. We derive useful, simple analytical approximations for key properties, such as central overdensity, moment of inertia, binding energy, and gravitational potential, applicable to small, metallic bodies like asteroids and moons. These new approximations could aid future research, including space mining and the scientific characterization of small Solar System bodies.

astro-ph.EP↗

arXiv2023

Equatorial accretion on the Kerr black hole

We investigate stationary accretion of the collisionless Vlasov gas onto the Kerr black hole, occurring in the equatorial plane. At infinity the gas obeys the Maxwell-Jüttner distribution, restricted to the equatorial plane. In the vicinity of the black hole, the motion of the gas is governed by the spacetime geometry. We compute accretion rates of the rest-mass, the energy, and the angular momentum, as well as the particle number surface density, focusing on the dependence of these quantities on the asymptotic temperature of the gas and the black hole spin. The accretion slows down the rotation of the black hole. We present preliminary results for a Vlasov gas accretion onto a Kerr black hole moving with a velocity parallel to the equatorial plane.

gr-qc↗