We study when the adjacency matrix of a Cayley graph factors as the product of two adjacency matrices of Cayley graphs on the same group. Let G be a finite group and let U subset of G \ {e} be symmetric. Writing A(G; U) for the adjacency matrix of Cay(G; U), we prove that, for symmetric subsets S, T, U subset of G \ {e}, A(G; U) = A(G; S)A(G; T) if and only if U=ST and each u E U has a unique representation u = st with s E S and t E T. (& sum;)(& sum;) Equivalently, = & sum; sES stET tuEU u in the group algebra. We also formulate the problem using group convolution: the unique-product condition is equivalent to S-1S boolean AND TT-1 = {e}. For cyclic groups, we reformulate factorability via mask polynomials and give a Chinese Remainder Theorem reduction for subsets of product form. Finally, we analyze dihedral groups D2n, give non-exhaustive families of factorable triples, and relate strongly symmetric factorizations of D2n to symmetric factorizations of Z2n via the P & ecirc;cher map. (c) 2026 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
For q∈R﹨{0}, the generalized Laplacian of a graph X is the matrix L=Δ+qA, where Δ is the degree matrix and A is the adjacency matrix of X. This paper investigates perfect state transfer (PST) on graphs with possible loops equipped with non-trivial involutions, where the generalized Laplacian matrix is taken as the Hamiltonian of the underlying spin network. We establish an equivalence between the existence of PST between pair states (or plus states) in such a graph and PST between vertices in a subgraph induced by the involution. This allows us to prove that almost all simple unweighted planar graphs (resp., almost all simple unweighted trees) can be assigned loops of weight one to exactly two vertices to produce PST between pair states relative to L. We also construct an infinite family of graphs with loops admitting PST between pair or plus states at arbitrarily long distances. For paths on n vertices, we show that PST occurs between end vertices relative to L if and only if n=2, or (n,q)=(3,k2−ℓ28ℓ2), where k and ℓ are integers with k>ℓ and k≢ℓ(mod2). For cycles, we show that the addition of an edge does not yield vertex PST relative to Laplacian and signless Laplacian matrices. Finally, we show that the addition of a few suitable edges (including loops) in complete bipartite graphs, cycles, and paths yields PST between pair states.
Understanding snap-through instability and dynamic behavior is vital for designing lightweight, damage-tolerant components and preventing catastrophic failure in structural applications. This study presents a comprehensive analytical, experimental, as well as numerical investigation of the nonlinear snap-through instability and dynamic behavior of 3D-printed architected sandwich spherical meta-shells with bamboo fiber-reinforced polylactic acid (PLA) composite face-sheets. The proposed structure combines lightweight architected star-shaped cellular cores with sustainable bio-composite face-sheets to achieve enhanced stiffness, stability, and vibration performance. Using Hamilton's principle, a nonlinear shell model is established within the framework of the First-Order Shear Deformation Theory (FSDT), while geometric nonlinearity is represented through the von-K & aacute;rm & aacute;n formulation. The governing equations are solved through the Ritz method employing Chebyshev polynomial expansion, while the cylindrical arc-length technique is adopted to accurately trace complex snap-through equilibrium paths. The present work investigates the influence of two practically important loading conditions, namely concentrated loading (CL) and uniformly distributed loading (UDL), on the nonlinear snap-through response of architected spherical meta-shells. The analytical model is first verified against benchmark solutions available in the literature. Additionally, a 3D finite element model is developed to capture the true physical dynamic behavior of the complex architected geometry, which is evaluated experimentally through impact hammer modal testing on a baseline pure PLA fused deposition modeling (FDM)-fabricated specimen. Following these evaluations, an extensive parametric study is carried out to examine the influence of shell geometry, architected core characteristics, thickness ratio, fiber volume fraction, loading scenarios, and support conditions on nonlinear vibration and snap-through responses. The developed bio-inspired architected sandwich meta-shells offer an efficient lightweight alternative for multifunctional structural systems that demand enhanced stability, excellent energy absorption, and outstanding dynamic performance.
Residential electrification driven by electric vehicles (EVs) and heat pumps (HPs) is fundamentally reshaping electricity demand in distribution systems. EV charging is stochastic and behavior-driven, while HP demand is continuous and governed by thermodynamic processes, creating hybrid load dynamics that may require dependence-aware representation when coincident demand, flexibility, or network constraints are relevant. The reviewed evidence includes technology-specific EV and HP studies as well as studies addressing multiple electrification technologies. Their scope and validation differ: representing EVs and HPs in a common scenario does not by itself establish that their joint behavioral, thermal, temporal, and network interactions have been explicitly modeled and validated. This distinction limits the conclusions that can be drawn about coincident demand, cold-climate operation, uncertainty propagation, and network-constrained integration.This paper presents a critical and unified review of residential electrification load modeling, covering data-driven, simulation-based, physics-based, and hybrid approaches across EV, HP, and integrated EV-HP domains. A key contribution is the identification of residential electrification as a hybrid stochastic-dynamic system, where peak demand emerges from the coupling of behavioral and thermal processes. The paper further introduces a grid-aware evaluation framework and a model suitability matrix to guide application-driven model selection. The results highlight the need for integrated, physics-informed, and grid-constrained modeling frameworks to support accurate forecasting, system planning, and real-time control in highly electrified residential networks.
We find expressions for the Gateaux derivative of the matrix norms in operator spaces, and operator systems. Some applications of the results to quantum probability measures, states on C & lowast;-algebras, and Birkhoff-James orthogonality are also presented. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
