The non-crystallographic root systems H2 and H4 are both governed by the golden ring Z[phi], but they are usually treated as separate topics: the planar Penrose pattern on the H2 side, and the icosian 600-cell on the H4 side. In this work we develop a single exact-arithmetic bridge that ties the decagonal cut-and-project benchmark to the icosian shell and, finally, to the trace-normalized Z-lattice K8 of the icosian order, which is isometric to the classical 5-modular icosian lattice Q8(1). The discriminant group (Z/5Z)4 of this lattice is classical; the point of the present work is its arithmetic origin. We show that the icosian order is unimodular over Z[phi] for the reduced trace form, so that the reciprocal lattice of K8 is exactly the codifferent multiple I/5 of the order, its discriminant group is the residue algebra of the icosians at the ramified prime 5 of the golden field, and the E8 lattice is the twist of the same Hermitian module by a totally positive generator of the codifferent. This is the precise sense in which the reciprocal module of icosian model-set labels, which is what indexes diffraction, is not preserved by the unimodular E8 normalization: the layer erased is the residue algebra of the icosians modulo 5, and the explicit denominator- 5 reciprocal generators are, up to a unit and the prime 5, the Z[phi]-dual basis of the order.
Abstract Mechanical desorption of an infinitely long macromolecule in the isometric ensemble, that is, by moving its end to a certain distance from the grafting surface, is studied theoretically. The approach is based on the generating functions method and the calculation of the average tail length of the desorbed macromolecule as a function of the end distance from the surface which gives access to the reaction force of the macromolecule to the desorption. The cases of a homopolymer and a regular multiblock copolymer are considered. For the homopolymer, the tail length is a linear function of the end distance and the reaction force is a constant independent of the end height above the surface which value is defined by the adsorption strength. In the case of a multiblock copolymer consisting of alternating adsorption-active and inert blocks, the reaction force is a non-monotonic oscillating function of the end distance from the surface, with decaying amplitude. The shape of the force-distance curve depends on the adsorption strength, blocks’ lengths, and monomer composition of the multiblock copolymer. These dependences perfectly describe the invariant part of the force-distance curves for mechanical desorption of finite multiblock copolymer chains grafted at one end to the surface calculated by using the transfer matrix approach.
Abstract For an integrable generalization of the Painlevé-II equation to a system of coupled equations with symmetry breaking terms, an asymptotically exact WKB analysis is applied to obtain connection formulas for solutions at different infinities. The analysis relies on an exact solution of the quantum mechanical Demkov–Osherov model, revealing a possible deeper relation between classical integrable systems and solvable multistate Landau–Zener models. An application of the connection formulas to the problem of unstable vacuum decay during a second-order phase transition provides precise scaling of the number of excitations, including subdominant contributions.
Abstract We present a method for associating labeled directed graphs to finite-dimensional Lie algebras that are graph-admissible . The proposed framework enables visual identification of key structural algebraic features. We first formalize this approach and establish several classes of Lie algebras that admit the categorization as graph-admissible Lie algebras, and then investigate suitable valid graphs for this approach by translating the antisymmetry of the Lie bracket and the Jacobi identity into rules that such graphs must satisfy. Furthermore, based on these foundations, we provide graph-theoretic criteria for solvability, nilpotency, presence of ideals, simplicity, semisimplicity, and reductiveness of an algebra. We also present practical algorithms for constructing such graphs and those associated with the lower central series and derived series via iterative pruning procedures. We further cover specific examples, such as the Schrödinger and Lorentz algebras, which illustrate the applicability of the developed tools to physically relevant cases. Moreover, we explore applications in physics, where the method facilitates computation of similarity transformations essential for determining quantum mechanical time evolution via the Lie-algebraic factorization method. We also discuss a possible generalization of the framework based on magma-gradings of Lie algebras and formulate related open questions. Our approach bridges algebraic and combinatorial perspectives, offering theoretical tools for this area of mathematical physics.
Abstract We develop an exact deformation calculus for single-mode Hermite–Gaussian states under the quadratic analytic map T ϕ = exp { 2 ϕ ( ∂ x ) } , ϕ ( z ) = c 1 + c 2 z + c 3 z 2 . The construction is not proposed as a general replacement for non-Hermitian quantum mechanics; rather, it provides an exactly solvable sector within the standard metric/quasi-Hermitian framework. For the literal similarity realization H ϕ = T ϕ − 1 H T ϕ , the physical inner product is generated by M ϕ = T ϕ † T ϕ , time evolution is governed by the Schrödinger equation with H ϕ , and physical observables are self-adjoint with respect to the metric inner product. Separately, the deformation-adapted Hermite–Gaussian realization leads to closed coordinate- and Fourier-space expressions for coherent, squeezed, and squeezed-coherent states. Using the Hermite generating function and Mehler kernel, we obtain explicit Gaussian formulae controlled by κ = e i θ tanh r , Δ ϕ = 1 − κ + 8 c 3 κ , and Γ ϕ = 1 + κ + 8 c 3 κ , together with their Wigner profiles. In these dimensionless coefficients, c 3 is understood in oscillator-length units, as specified in the main text. The pure quadratic sector gives an exactly solvable benchmark for non-unitary Gaussian filtering and effective resolution broadening in continuous-variable optical models.
