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We consider the elastic scattering problem by multiple disjoint arcs or \emph{cracks} in two spatial dimensions. A key aspect of our approach lies in the parametric description of each arc's shape, which is controlled by a potentially high-dimensional, possibly countably infinite, set of parameters. We are interested in the efficient approximation of the parameter-to-solution map employing model order reduction techniques, specifically the reduced basis method. Initially, we utilize boundary potentials to transform the boundary value problem, originally posed in an unbounded domain, into a system of boundary integral equations set on the parametrically defined open arcs. Our aim is to construct a rapid surrogate for solving this problem. To achieve this, we adopt the two-phase paradigm of the reduced basis method. In the offline phase, we compute solutions for this problem under the assumption of complete decoupling among arcs for various shapes. Leveraging these high-fidelity solutions and Proper Orthogonal Decomposition (POD), we construct a reduced-order basis tailored to the single arc problem. Subsequently, in the online phase, when computing solutions for the multiple arc problem with a new parametric input, we utilize the aforementioned basis for each individual arc. To expedite the offline phase, we employ a modified version of the Empirical Interpolation Method (EIM) to compute a precise and cost-effective affine representation of the interaction terms between arcs. Finally, we present a series of numerical experiments demonstrating the advantages of our proposed method in terms of both accuracy and computational efficiency.

Here, the synthesis of polyglobalide (PGl) by enzymatic ring-opening polymerization (e-ROP) is investigated, using pressurized carbon dioxide (CO2), pressurized CO2 + dichloromethane (DCM), and pressurized propane as solvents. Particularly, the effects of phase equilibrium on the course of e-ROP and PGl final properties are discussed. The partition coefficients of CO2, DCM, propane, globalide and PGl were calculated with help of thermodynamic models, providing proper understanding of monomer partitioning in the reaction system. Reactions performed in pure CO2 resulted in monomer conversions of 100%. Besides, when only one liquid phase was present inside the reactor, PGl samples presented low polydispersities and high average molecular weights. When carried out in CO2 + DCM, e-ROP resulted in lower monomer conversions and PGl samples with higher polydispersities and lower average molecular weights. Finally, reactions carried out in pressurized propane (200 bar) produced PGl samples with the highest average molecular weights among the analyzed products., Comment: 41 pages, 9 figures, 8 tables

Contemporary process industries are constantly confronted with volatile market conditions that jeopardise their financial sustainability. While mature markets transition to oligopoly structures, the supply chain operation should adapt to a more customer-centric focus. Key issues related to the modelling and impact of the related contractual agreements between firms and customers remain largely unexplored. In the present work, we examine the problem of fair customer allocation in oligopolies under different contractual agreements within a multi-period setting. We consider an ensemble of contract types that vary in terms of pricing mechanisms and duration. The role of fairness is examined following the social welfare and Nash bargaining scheme. In the latter case, the overall problem is formulated as an MINLP. For its efficient solution we employ a piecewise linearisation strategy based on special-ordered sets. The impact of the different fairness schemes on the optimal customer allocation is evaluated via two case studies from the industrial gases market.

We establish shape holomorphy results for general weakly- and hyper-singular boundary integral operators arising from second-order partial differential equations in unbounded two-dimensional domains with multiple finite-length open arcs. After recasting the corresponding boundary value problems as boundary integral equations, we prove that their solutions depend holomorphically upon perturbations of the arcs' parametrizations. These results are key to prove the shape (domain) holomorphy of the domain-to-solution maps for the associated boundary integral equations with applications in uncertainty quantification, inverse problems and deep learning.

We investigate the local fluctuations of filamentous actin (F-actin), with focus on the skeletal thin filament, using single-particle optical trapping interferometry. This experimental technique allows us to detect the Brownian motion of a tracer bead immersed in a complex fluid with nanometric resolution at the microsecond time-scale. The mean square displacement, loss modulus, and velocity autocorrelation function (VAF) of the trapped microprobes in the fluid follow power-law behaviors, whose exponents can be determined in the short-time/high-frequency regime along several decades. We obtain 7/8 subdiffusive power-law exponents for polystyrene depleted microtracers at low optical trapping forces. Microrheologically, the elastic modulus of these suspensions is observed to be constant up to the limit of high frequencies, confirming the origin of this subdiffusive exponent on the local longitudinal fluctuations of the polymers. Deviations from this value are measured and discussed in relation to the characteristic lengths scales of these F-actin networks and probes' properties, and also in connection with the different power-law exponents detected in the VAFs. Finally, we observe that the thin filament, composed by tropomyosin (Tm) and troponin (Tn) coupled to F-actin in the presence of Ca$^{2+}$, returns exponent values less dispersed than F-actin alone, which we interpret as a micro-measurement of the filament stabilization., Comment: Published in Soft Matter in 2023

We study the high-frequency micro-mechanical response of suspensions composed by cardiac and skeletal muscle myosin by optical trapping interferometry. We observe that in low ionic strength solutions upon the addition of magnesium adenosine triphosphate (MgATP$^{2-}$), myosin suspensions radically change their micro-mechanics properties, generating a viscoelastic fluid characterized by a complex modulus analogous to a suspension of worm-like micelles. This transduction of energy, from chemical to mechanical, may be related to the relaxed states of myosin, which regulate muscle contractility and can be involved in the etiology of many myopathies. Within a similar generic mechanical response, cardiac and skeletal myosin suspensions provide different stress relaxation times, elastic modulus values, and characteristic lengths. These discrepancies probably rely on the dissimilar physiological functions of cardiac and skeletal muscle, on the different MgATPase hydrolysis rates of cardiac and skeletal myosins, and on the observed distinct cooperative behavior of their myosin heads in the super-relaxed state. \textit{In vitro} studies like these allow to understand the foundations of muscle cells mechanics on the micro-scale, and may contribute to the engineering of biological materials whose micro-mechanics can be activated by energy regulators.

We study the elastic time-harmonic wave scattering problems on unbounded domains with boundaries composed of finite collections of disjoints finite open arcs (or cracks) in two dimensions. Specifically, we present a fast spectral Galerkin method for solving the associated weakly- and hyper-singular boundary integral equations (BIEs) arising from Dirichlet and Neumann boundary conditions, respectively. Discretization bases of the resulting BIEs employ weighted Chebyshev polynomials that capture the solutions' edge behavior. We show that these bases guarantee exponential convergence in the polynomial degree when assuming analyticity of sources and arcs geometries. Numerical examples demonstrate the accuracy and robustness of the proposed method with respect to number of arcs and wavenumber., Comment: 28