László Márk

Angle Maps Are Direction Fields. The half-integer rulebook for plasma polarimetry (MSE and Faraday): wrap-free map processing, an exact equivariance property, and a disclination parity test that flags what a smooth magnetic field cannot produce

Plasma polarimetry measures directions, not vectors: the polarization angle of motional-Stark-effect (MSE) emission and the Faraday rotation angle are defined modulo 180deg. The measurement hardware knows this — photoelastic-modulator systems encode the angle in second-harmonic p...

One Identity, Three Languages: the Geometric Conservation Law, the Discrete Piola Identity, and Null Lagrangians

Three communities keep the same identity under three names. Nonlinear elasticity calls it the Piola identity: div cof grad(phi) = 0 for every sufficiently smooth map. The calculus of variations calls it the null-Lagrangian property: the determinant (and each of its minors) is a L...

Geometric and Spectral Desingularization

There are two seemingly independent paradigms for resolving singularities: the geometric route — Arnold's σ-process, or blow-up, which replaces the singular point by the space of directions through it — and the spectral route — the Möbius transform, which turns sign-flipping (two...

The Mathematics of Twisted Waves: How signals are decomposed into waves — and what happens when the world of the signal is a Möbius strip. (High-school level)

This document explains the mathematics of a real research project using high-school tools: trigonometric functions, addition formulas and the trigonometric form of complex numbers. The starting point is the Fourier decomposition as a "basis and coordinates" idea, followed by an o...

Geometric and Spectral Desingularization

There are two seemingly independent paradigms for resolving singularities: the geometric route — Arnold's σ-process, or blow-up, which replaces the singular point by the space of directions through it — and the spectral route — the Möbius transform, which turns sign-flipping (two...

The Mathematics of Twisted Waves: How signals are decomposed into waves — and what happens when the world of the signal is a Möbius strip. (High-school level)

This document explains the mathematics of a real research project using high-school tools: trigonometric functions, addition formulas and the trigonometric form of complex numbers. The starting point is the Fourier decomposition as a "basis and coordinates" idea, followed by an o...

Fourier Analysis on Twisted Spaces, An introduction to the Möbius and Klein transforms — lecture notes for university students

An introductory set of notes for university students (BSc/MSc) on the Möbius and Kleintransforms. The starting point is an everyday phenomenon: many engineering and physical quantities are notperiodic but antiperiodic — after half a period they go over into their negative (half-w...

Music from a Twisted World: How the Möbius strip and the Klein bottle become tools for signal processing — popular science version

A popular account of how the curiosity of the Möbius strip and the Klein bottle can be turned into a computable engineering tool. The starting point is familiar: the Fourier transform separates signals into pure frequencies the way a prism separates white light into colours — but...

A Unified System of Tensor Invariants and Their Integral Reduction Theorems

This paper organizes the complete operator set and the integral theorems of classical vector analysis into a single unified system. The starting point is algebraic: the second-order tensor invariant of an arbitrary third-order tensor, given by the general epsilon-dual formula (th...

The Unified System of Tensor Invariants — Complete Teaching Edition

This edition contains everything the academic article does – the full five-dimensional invariant table, all theorems, the ideas behind the proofs and the research directions – but in teaching language: before every formula it states what the formula will claim and why it is true....

Tensor Invariants and Integral Theorems — Made Understandable (lecture notes)

Short introductory notes for university students on the unified system of tensor invariants. The many formulas of a vector analysis course – grad, div, rot, the Gauss and Stokes theorems, the Laplacian, the Green identities, Maxwell's equations – are in fact different faces of a...

Tensor Invariants and Integral Theorems — Made Understandable (lecture notes)

Short introductory notes for university students on the unified system of tensor invariants. The many formulas of a vector analysis course – grad, div, rot, the Gauss and Stokes theorems, the Laplacian, the Green identities, Maxwell's equations – are in fact different faces of a...

The Unified System of Tensor Invariants — Complete Teaching Edition

This edition contains everything the academic article does – the full five-dimensional invariant table, all theorems, the ideas behind the proofs and the research directions – but in teaching language: before every formula it states what the formula will claim and why it is true....

Sector-Switching Curl on the Klein Bottle — numerical verification report

Verification of the sharp statement formulated at the meeting point of the tensor-invariant system (jet/jad/dev) and the Möbius-Klein transform framework: on the Klein bottle the curl of a genuine (sigma-equivariant) vector field lives in the other sector (it is a twisted scalar)...

One Twist, Two Views — the connection between the tensor-invariant system and the Möbius–Klein transform

Two independently built frameworks – the unified tensor-invariant system of vector analysis (jet/jad/dev operators, dimensional ladder) and the spectral theory of non-orientable surfaces (Möbius and Klein transforms, sector decomposition) – rest on a single common actor: orientat...

One Twist, Two Views — the connection between the tensor-invariant system and the Möbius–Klein transform

Two independently built frameworks – the unified tensor-invariant system of vector analysis (jet/jad/dev operators, dimensional ladder) and the spectral theory of non-orientable surfaces (Möbius and Klein transforms, sector decomposition) – rest on a single common actor: orientat...