Let A be a gentle algebra. For every collection of string and band diagrams, we consider the constructible subset of the variety of representations containing all modules with this underlying diagram. We study degenerations of such sets. We show that these sets are defined by vectors of integers which we call h-vectors and which are related to a restricted version of the hom-order. We provide combinatorial criteria for the existence of a degeneration, involving the removal of an arrow or the resolving of a type of configuration called “reaching”.
In this article, we introduce a method to extend involutive nondegenerate set-theoretic solutions to the Yang–Baxter equation by means of equivariant mappings to graded modules, thus leading to the notion of a twisted extension. Furthermore, we define coprime extensions of solutions and prove that each coprime extension of indecomposable solutions can be obtained as a suitable twisted extension. We then apply our results to obtain a full description of indecomposable solutions of size pqr, where p,q,r are different primes, from a structure theorem of Cedó and Okniński. We close with some remarks on a cohomology theory for solutions developed by Lebed and Vendramin. We express our results in the language of cycle sets.
Lagrangian systems subject to fractional damping can be incorporated into a variational framework by doubling the state variables and introducing fractional derivatives. Fractional variational integrators based on backward-differentiation convolution quadrature (BD-FCQ), combined with higher-order Galerkin methods, saturate at second-order accuracy because the multistep structure of BDFCQ does not take into account the internal stages of the Galerkin discretization. The main objective of this paper is to develop fractional variational integrators (FVIs) by combining Runge-Kutta convolution quadrature (RKCQ) for the approximation of fractional derivatives with higher-order Galerkin methods. The RKCQ approach is naturally compatible with such stage-based discretizations and is therefore better suited for the construction of higher-order schemes. We are particularly interested in the CQ based on Lobatto IIIC. Preservation properties such as energy decay, as well as convergence properties, are investigated numerically and proved for second-order schemes. The presented schemes reach 2nd, 4th and 6th order of accuracy. A brief discussion on the midpoint fractional integrator is also included.
This study explains experiments and analysis of the chaotic behaviour of cavitation dynamics in a convergent-divergent nozzle. It shows that adding of air injection can help reduce the chaos and make the flow more stable. Water flow rates through the nozzle vary between 10,000 and 12,000 litres per hour (LPH), and air injection rates range from 10 to 15 litres per minute (LPM) at four points along the nozzle. At 10,000 LPH, the average cavitation area drops from 26.16 mm² to 8.58 mm² when air is injected, especially at the mid-divergent port (V2). The flow becomes less chaotic over time, and the high-frequency turbulence is reduced by 60% when air is added. Nonlinear time series analysis identifies reliable parameters (embedding dimension m = 5, optimal time lag τ =11–82 frames). Air injection reduces system complexity by 41.7%. The highest Lyapunov exponent decreases from positive, chaotic values (0.05–0.11) to near-zero or negative, confirming chaos is reduced. Additionally, short-term predictability improves by over 50%, lowering the RMSE from 10.5 mm² to 4.02 mm² at 10,000 LPH with air injection at the V2 location.
Numerical design of heat transfer and pressure loss in heat exchangers depends strongly on the thermophysical property models used for the working fluid. This is particularly relevant for zeotropic mixtures, where saturation behavior, temperature glide, and transport properties determine local driving temperature differences during phase change. This work analyzes the interplay between heat exchanger modeling and thermophysical property models from user and developer perspectives, examining how model choice affects engineering predictions and analyzing how uncertainties in individual fluid properties propagate into application-level results. A one-dimensional spatially resolved double-pipe condenser model is applied to a zeotropic n-butane/CO₂ mixture with water as secondary fluid. The equations of state GERG-2008, Peng–Robinson, and a molecular-simulation-based Helmholtz model are combined with transport-property models based on the extended corresponding states principle and residual entropy scaling. Local heat-transfer coefficients and pressure gradients are evaluated using correlations suitable for zeotropic mixtures. Results show that fluid-model choice substantially affects temperature profiles and heat-exchanger sizing, mainly through differences in saturation states. Predicted heat exchanger lengths vary from −7.3% to + 11.8% relative to the mean; in a limiting case with small temperature difference full condensation is not always reached. Sensitivity analyses show that uncertainties in thermal conductivity, dynamic viscosity, and isobaric heat capacity dominate fluid-property uncertainty propagation into heat-transfer predictions, whereas pressure-drop-related properties have only minor influence on exchanger length under present conditions with relatively large tube diameters. Including empirical correlation uncertainties for heat-transfer coefficients and pressure loss makes them dominate the total output uncertainty; nevertheless, fluid-property uncertainties remain a relevant secondary contribution.
