Nir Shvalb

We propose a Ramsey—theoretic framework for quantum mechanics in which observables are represented as vertices of a complete graph and their commutation relations are encoded by a bi-coloring of the edges. Edges connecting strongly commuting operators are colored red, while non-commuting pairs are colored green. This construction transforms the algebra of quantum observables into a combinatorial object—a quantum Ramsey graph. Because commutation relations are preserved under unitary conjugation, the resulting coloring and its Ramsey substructures are unitary invariants, providing a new class of unitarily invariant combinatorial characteristics of quantum systems. When the Hamiltonian is included as a vertex, the graph encodes dynamical information through its commutation relations, revealing conserved quantities and symmetry constraints. Universal results from classical Ramsey theory then impose unavoidable constraints on quantum systems. In particular, the Ramsey number R ( 3 , 3 ) = 6 guarantees that any set of six observables contains a monochromatic triangle. Physically, this implies that every sufficiently rich operator set contains either a classical subcontext of three mutually commuting observables or an irreducibly incompatible triple. We analyze representative systems, including free particles, particles in central potentials, harmonic oscillators, and interacting spin systems, illustrating how symmetries and dynamics determine which Ramsey-enforced structures are realized. Hamiltonian-centered star-shaped graphs visualize conserved quantities and the propagation of incompatibility, while cyclic compatibility subgraphs connect naturally to contextuality scenarios (e.g. Klyachko–Can–Binicioğlu–Shumovsky-type structures). We discuss algebraic constraints (e.g. Jacobi identity effects under Lie-closure) on non-commuting triangles and extend the framework to infinite operator sets, where infinite monochromatic cliques arise. Overall, the work shows that compatibility, incompatibility, and contextuality reflect not only physical postulates but also universal combinatorial constraints induced by the structure of commuting observables.

Mechanical linkages play a central role in applied robotics, and have been of interest to mathematicians since the nineteenth century. One important engineering concept in this context is that of the actuation of such a linkage Gamma: intuitively, this means determining the position of Gamma by partial control of some components. This notion naturally translates mathematically into a mapping from the configuration space of Gamma to that of a suitablesub-mechanism. However, in practice an actuation almost never completely determines the position x of $\Gamma$; at best, x is prescribed uniquely only in a neighborhood. Our notion of an actuation requires it to be a local isomorphism away from a singular locus, with continuous singularities of codimension >0.We show that all planar linkages admit such actuations, aside from specific forbidden types of linkages.In R^3 the situation is more complicated, due to the existence of closed chains with higher mobility. Nevertheless, we utilize work of Li, Schicho et al. to describe sufficient conditions for a spatial rotational linkage to have an allowable actuation.

We introduce a five degree-of-freedom continuum endoluminal robot that mitigates the trade-off between maneuverability and operational stiffness in confined workspaces. A segmented backbone actuated via a cable-and-pulley architecture enables differential stiffness modulation: the proximal segment remains highly compliant for navigation, while the distal segment exhibits increased effective stiffness, providing improved operational stability once the target is reached.. The prototype robot tracks canonical trajectories with a mean endpoint error of 3.8 mm, maintains a prescribed standoff distance within ±0.4 mm during wall-following, and exhibits increased distal stiffness of approximately 30 N/m versus 20 N/m proximally.

Magnetic levitation may offer an alternative to traditional actuation in robotics, particularly for applications requiring minimal mechanical contact, reduced friction, and operation in sterile or sensitive environments. This paper presents a novel parallel Delta robot enhanced by magnetic levitation technology, utilizing "polarity-free magnetic repulsion" (PFR) method. Unlike conventional Delta robots that rely on mechanical joints and motors, our design integrates magnetic levitation to reduce the robot’s weight and mechanical wear. Levitation is controlled by manipulating the rotation speed of the base magnetic field, which adjusts the height of the robot’s legs using force-based dynamics rather than traditional kinematics. A mechanical low-pass-filter is used to stabilize the system. An extensive set of experiments demonstrates the feasibility of this technology in achieving high precision and stability in operations.

Ramsey theory is applied to the analysis of operators acting on the functions belonging to the L^2 Hilbert space. The operators form the vertices of the bi-colored graph. If the operators commute, they are connected by a red link; if the operators do not commute they are connected with a green link. Thus, the complete, bi-colored graph emerges and the Ramsey theory becomes applicable. If the graph contains six vertices/operators, at least one monochromatic triangle will necessarily appear in the graph. Thus, the triad of operators forming the read triangle possesses the common set of eigenfunctions. The extension of introduced approach to infinite sets of operators is addressed. Applications of the introduced approach to problems of classical and quantum mechanics are suggested.

We propose the Ramsey approach for the analysis of behavior of isolated mechanical systems containing interacting particles. The total momentum of the system in the frame of the center of masses is zero. The mechanical system is described by the Ramsey, bi-colored, complete graph. Vectors of momenta of the particles pi serve as the vertices of the graph. We start from the graph representing the system in the frame of the center of masses, the momenta of the particles in this system are pcmi. If (pcmi(t)∙pcmj(t))≥0 is true, vectors of momenta of the particles numbered i and j are connected with the red link; if (pcmi(t)∙pcmj(t))<0 takes place, the vectors of momenta are connected with the green link. Thus, the complete, bi-colored graph emerges. Consider the isolated system built of six interacting particles. According to the Ramsey Theorem, the graph inevitably comprises at least one monochromatic triangle. The coloring procedure is invariant relatively rotations/translations of frames; thus, the graph representing the system contains at least one monochromatic triangle in any of frames emerging by rotation/translation of the original frame. This gives rise to the novel kind of a mechanical invariant. Similar coloring is introduced for the angular momenta of the particles. However, the coloring procedure is sensitive to Galilean/Lorenz transformations. Extensions of the suggested approach are discussed.

Ramsey theory enables re-shaping of the basic ideas of the quantum mechanics. Quantum observables, represented by linear Hermitian operators are seen as the vertices of the graph. Relation of commutation define coloring of the edges linking the vertices: if the operators commute, they are connected with the red link; if they do not commute they are connected with the green link. Thus, a bi-colored, complete, Ramsey graph emerges. According to the Ramsey theorem, complete, bi-colored graph built of six vertices, will inevitably contain at least one monochromatic triangle; in other words, the Ramsey number R(3,3)=6 . In our interpretation, this triangle represents the triad of observables, which could or, alternatively, could not be established simultaneously in a given quantum system. The Ramsey approach to the quantum mechanics is illustrated with the numerous examples, including the motion of a particle in a centrally symmetrical field.