László Márk

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 a third-order tensor, given by the general epsilon -dual formula (the dual is...

The Codimension-Three Rung. Twisted sections over a non-orientable link: the sector split of the Kondratiev indicial roots, a root-free window that triples, and the character ladder of lens rungs

The blow-up ladder of the framework's desingularization article was carried out for codimension-two centers, where the exceptional divisor is a circle; Section 9.10 explicitly left open the codimension-three rung and the full Kondratiev asymptotics. This article closes both. The...

The Codimension-Three Rung. Twisted sections over a non-orientable link: the sector split of the Kondratiev indicial roots, a root-free window that triples, and the character ladder of lens rungs

The blow-up ladder of the framework's desingularization article was carried out for codimension-two centers, where the exceptional divisor is a circle; Section 9.10 explicitly left open the codimension-three rung and the full Kondratiev asymptotics. This article closes both. The...

Sum Rules and Sign Thresholds for Twisted Sector Sums. An exact sum rule over all 2d twist sectors, and a proved threshold: the regularized sum turns positive exactly when the twisted directions are at least as many as the untwisted ones

Assign to each direction of a d-dimensional lattice a twist label t_i\{0,1\} and consider the regularized alternating lattice sum Z_t(s)='_{n Z^d}(-1)^{n· t}|n|^{-s} — the family of 2^d sector sums that the framework's sector calculus produces on a d-torus with periodic or antipe...

The Curl Spectrum on Non-Orientable Geometry: Sector Doublets, Vanishing Helicity, and Beltrami Pairs

The spectral theory of the curl operator on closed 3-manifolds is a developed and active field — but exclusively on oriented manifolds, because the definition of curl requires an orientation. This article opens the non-orientable column using the sector calculus. The key is a one...

The Weight Ledger of Metasurface Susceptibilities. Slice-adapted irreducible parametrization of dipole–quadrupole GSTCs: closed-form Neumann counting, incidence selection rules, exact rotation-equivariant fitting — and why C3 and C4 first part ways at the quadrupole

Metasurface modeling by generalized sheet transition conditions (GSTCs) suffers from a parameter explosion: the dipolar susceptibility tensors already carry 36 components, and the quadrupolar extension adds rank-3 objects of 27 components each. The established cure is Neumann's p...

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...

The Curl Spectrum on Non-Orientable Geometry: Sector Doublets, Vanishing Helicity, and Beltrami Pairs

The spectral theory of the curl operator on closed 3-manifolds is a developed and active field — but exclusively on oriented manifolds, because the definition of curl requires an orientation. This article opens the non-orientable column using the sector calculus. The key is a one...

The Weight Ledger of Metasurface Susceptibilities. Slice-adapted irreducible parametrization of dipole–quadrupole GSTCs: closed-form Neumann counting, incidence selection rules, exact rotation-equivariant fitting — and why C3 and C4 first part ways at the quadrupole

Metasurface modeling by generalized sheet transition conditions (GSTCs) suffers from a parameter explosion: the dipolar susceptibility tensors already carry 36 components, and the quadrupolar extension adds rank-3 objects of 27 components each. The established cure is Neumann's p...

Vortex and Fluxoid Selection Rules on Non-Orientable Superconductors. The loop dichotomy, the closure prohibition, and the pi-shifted Little–Parks ladder: which loops can carry a vortex — with an exact channel-discriminator identity and machine verification

Half-quantum fluxoid physics is measured reality: pi-shifted Little–Parks oscillations in beta-Bi2Pd rings, half-quantum features in Sr2RuO4 microrings, and a Ginzburg–Landau theory of the superconducting Möbius strip that found novel states near half-odd flux. What the literatur...

The Pairing Rulebook of the Brillouin Klein Bottle. Mirror circle, forbidden s-wave, and half-integer phase winding: the sector selections of Bogoliubov–de Gennes theory on non-orientable momentum space, with the Klein transform as the native mode machinery

A Brillouin zone can be a Klein bottle: under Z2 gauge fields the algebra of crystal symmetries becomes projective, a nonsymmorphic glide reflection appears in momentum space, and the fundamental domain is a Klein bottle instead of a torus — the experimentally realized Brillouin...

Which Loops Can Carry a Burgers Vector? Selection Rules on Non-Orientable Geometry

The strength of a dislocation is its Burgers vector: the circulation of the distortion field along a loop encircling the defect line. This article pursues a single question: what happens to this circulation when the carrier geometry is non-orientable? The answer reduces to one me...

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...

The Möbius and Klein Transforms: Spectral Analysis on Non-Orientable Manifolds

The classical Fourier transform is the harmonic analysis of the circle (and torus) – an orientable manifold. This article develops a concept: in analogy with the Fourier transform, we define two named, closed-form, unitary transform pairs on the two simplest non-orientable manifo...

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...

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...

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...

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...

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...

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...