We regret that the paper by J. M. Greenberg, ‘Models of elastic–perfectly plastic materials’, vol. 1, part 2, pp. 131–150, was repeated in error as pp. 225–244 in part 3.

The publisher apologizes to the author and readers for this unfortunate error.

In Chen and Liang previous work, a model, together with its well-posedness, was established for credit rating migrations with different upgrade and downgrade thresholds (i.e. a buffer zone, also called dead band in engineering). When positive dividends are introduced, the model in Chen and Liang (SIAM Financ. Math. 12, 941–966, 2021) may not be well-posed. Here, in this paper, a new model is proposed to include the realistic nonzero dividend scenarios. The key feature of the new model is that partial differential equations in Chen and Liang (SIAM Financ. Math. 12, 941–966, 2021) are replaced by variational inequalities, thereby creating a new free boundary, besides the original upgrading and downgrading free boundaries. Well-posedness of the new model, together with a few financially meaningful properties, is established. In particular, it is shown that when time to debt paying deadline is long enough, the underlying dividend paying company is always in high grade rating, that is, only when time to debt paying deadline is within a certain range, there can be seen the phenomenon of credit rating migration.

The time-global unique classical positive solutions to the reaction–diffusion equations for prey–predator models with dormancy of predators are constructed. The feature appears on the nonlinear terms of Holling type $\rm I\!I$ functional response. The crucial step is to establish time-local positive classical solutions by using a new approximation associated with time-evolution operators. Although the system does not equip usual comparison principle for solutions to partial differential equation, a priori bounds are derived by enclosing and renormalising arguments of solutions to the corresponding ordinary differential equations. Furthermore, time-global existence, invariant regions and asymptotic behaviours of solutions follow from such a priori bounds.

The paper presents the results of an asymptotic theory of axially symmetric cavity flows at small and zero cavitation number. The results have been obtained on the basis of the variational Riabouchinsky principle and the asymptotic theory of slender bodies. This variational-asymptotic approach has been applied to deduce asymptotic expansions for the shape of the cavity and the force exerted on the cavitator at small cavitation number. At the zero cavitation number an integro-differential equation for the shape of the free streamline has been obtained. An exact integral of the equation has been found and a one-parameter family of solutions has been constructed and which has refined earlier asymptotics of Levinson and Gurevich. The equation and asymptotic expansion are independent of the cavitator shape and in this sense are maximally accurate. Any further amendment of the equation of higher order of accuracy would be connected with the shape of the cavitator.

We study the geodesic interpolating spline with a biharmonic regulariser for solving the landmark image registration problem. We show existence of solutions, discuss uniqueness and show how the problem can be efficiently solved numerically. The main advantage of the geodesic interpolating spline is that it provides a diffeomorphism and we show this is preserved under our numerical approximation.

We present a stability result for a class of non-trivial self-similarly vanishing solutions to a 3D Hele-Shaw moving boundary problem with surface tension and single-point suction. These solutions are domains that bifurcate from the trivial spherical solution. The moving domains have a geometric centre located at the suction point and they are axially symmetric. We show stability with respect to perturbations that preserve these properties.

We consider a model for the evolution of damage in elastic materials originally proposed by Michel Frémond. For the corresponding PDE system, we prove existence and uniqueness of a local in time strong solution. The main novelty of our result stands in the fact that, differently from previous contributions, we assume no occurrence of any type of regularising terms.

We examine the game theoretic properties of a model of crime first introduced by Short et al. (2010 Phys. Rev. E82, 066114) as the SBD Adversarial Game. We identify the rationalizable strategies and one-shot equilibria under multiple equilibrium refinements. We further show that SBD's main result about the effectiveness of defecting-punishers (“Informants”) in driving the system to evolve to the cooperative equilibrium under an imitation dynamic generalizes to a best response dynamic, though only under certain parameter regimes. The nature of this strategy's role, however, differs significantly between the two dynamics: in the SBD imitation dynamic, Informants are sufficient but not necessary to achieve the cooperative equilibrium, while under the best response dynamic, Informants are necessary but not sufficient for convergence to cooperation. Since a policy of simply converting citizens to Informants will not guarantee success under best response dynamics, we identify alternative strategies that may help the system reach cooperation in this case, e.g., the use of moderate but not too severe punishments on criminals.

The aim of this article is to study the existence and uniqueness of solution for a quasistatic fully coupled thermoelastic problem arising from some metallurgical processes. We consider mixed boundary conditions for both submodels, and a Robin boundary condition for the thermal one. Furthermore, the reference temperature, the thermal conductivity and the Lamé's parameters are assumed to depend on the material point.

In this paper, we study the initial-boundary value problem of a repulsion Keller–Segel system with a logarithmic sensitivity modelling the reinforced random walk. By establishing an energy–dissipation identity, we prove the existence of classical solutions in two dimensions as well as existence of weak solutions in the three-dimensional setting. Moreover, it is shown that the weak solutions enjoy an eventual regularity property, i.e., it becomes regular after certain time T > 0. An exponential convergence rate towards the spatially homogeneous steady states is obtained as well. We adopt a new approach developed recently by the author to study the eventual regularity. The argument is based on observation of the exponential stability of constant solutions in scaling-invariant spaces together with certain dissipative property of the global solutions in the same spaces.

We study a mathematical model proposed in the literature with the aim of describing the interactions between tumor cells and the immune system, when a periodic treatment of immunotherapy is applied. Combining some techniques from non-linear analysis (degree theory, lower and upper solutions, and theory of free-homeomorphisms in the plane), we give a detailed global analysis of the model. We also observe that for certain therapies, the maximum level of aggressiveness of a cancer, for which the treatment works (or does not work), can be computed explicitly. We discuss some strategies for designing therapies. The mathematical analysis is completed with numerical results and conclusions.

Loss functions with a large number of saddle points are one of the major obstacles for training modern machine learning (ML) models efficiently. First-order methods such as gradient descent (GD) are usually the methods of choice for training ML models. However, these methods converge to saddle points for certain choices of initial guesses. In this paper, we propose a modification of the recently proposed Laplacian smoothing gradient descent (LSGD) [Osher et al., arXiv:1806.06317], called modified LSGD (mLSGD), and demonstrate its potential to avoid saddle points without sacrificing the convergence rate. Our analysis is based on the attraction region, formed by all starting points for which the considered numerical scheme converges to a saddle point. We investigate the attraction region’s dimension both analytically and numerically. For a canonical class of quadratic functions, we show that the dimension of the attraction region for mLSGD is $\lfloor (n-1)/2\rfloor$, and hence it is significantly smaller than that of GD whose dimension is $n-1$.

In this paper, we consider a general single population model with delay and patch structure, which could model the population loss during the dispersal. It is shown that the model admits a unique positive equilibrium when the dispersal rate is smaller than a critical value. The stability of the positive equilibrium and associated Hopf bifurcation are investigated when the dispersal rate is small or near the critical value. Moreover, we show the effect of network topology on Hopf bifurcation values for a delayed logistic population model.

A credit default swap (CDS) is an exchange of premium payments for a compensation for the occurrence of a credit event. Counterparty risks refer to defaults of parties holding CDS contracts. In this paper we develop a valuation/pricing model for a CDS subject to counterparty risks. Using the Cox–Ingersoll–Ross (CIR) model for interest rate and first arrival times of Poisson processes with variable intensities for the occurrences of credit default and counterparty defaults, we derive a mathematical formulation and make a full theoretical investigation. In addition, we develop a full theory for the corresponding infinite horizon problem and establish its connection with the asymptotic long expiry behaviour of finite horizon problem. Furthermore, we establish a connection between two major frameworks for default times: the structure model approach and the intensity model approach. We show that a solution of the structure model can be obtained as the limit of a sequence of solutions of intensity models. Regarded as an important theoretical development, we remove a constraint typically imposed on the parameters of the CIR model; that is, the well-posedness (existence, uniqueness and continuous dependence of parameters) of the mathematical model holds for any empirically calibrated parameters for the CIR model.

The aim of this paper is to develop a theory for the growth of multiple crystals in a polymer melt. This leads to nonlinear moving boundary problems for the temperature, with normal growth speed of the crystal boundaries determined by a nonlinear Gibbs-Thompson relation. Particular attention is paid to the effect of impingement, i.e., the event of two crystals hitting each other, which stops the growth on the contact interface. In one space dimension, the well-posedness of the growth model coupled to the heat equation is shown for an arbitrary number of crystals, both in a quasi-stationary and in an instationary situation. The resulting evolution of a fixed crystal in presence of other crystals is compared to the pure single crystal case. Finally, some basic features of the model in higher spatial dimensions and the main problems encountered in the attempt to prove a general well-posedness result are discussed.

We investigate a reaction–diffusion problem in a two-component porous medium with a nonlinear interface condition between the different components. One component is connected and the other one is disconnected. The ratio between the microscopic pore scale and the size of the whole domain is described by the small parameter $\epsilon$. On the interface between the components, we consider a dynamic Wentzell-boundary condition, where the normal fluxes from the bulk domains are given by a reaction–diffusion equation for the traces of the bulk solutions, including nonlinear reaction kinetics depending on the solutions on both sides of the interface. Using two-scale techniques, we pass to the limit $\epsilon \to 0$ and derive macroscopic models, where we need homogenisation results for surface diffusion. To cope with the nonlinear terms, we derive strong two-scale convergence results.

The Helmholtz equation $-\nabla\cdot (a\nabla u) - \omega^2 u = f$ is considered in an unbounded wave guide $\Omega := \mathbb{R} \times S \subset \mathbb{R}^d$ , $S\subset \mathbb{R}^{d-1}$ a bounded domain. The coefficient a is strictly elliptic and either periodic in the unbounded direction $x_1 \in \mathbb{R}$ or periodic outside a compact subset; in the latter case, two different periodic media can be used in the two unbounded directions. For non-singular frequencies $\omega$ , we show the existence of a solution u. While previous proofs of such results were based on analyticity arguments within operator theory, here, only energy methods are used.

We introduce an approach and a software tool for solving coupled energy networks composed of gas and electric power networks. Those networks are coupled to stochastic fluctuations to address possibly fluctuating demand due to fluctuating demands and supplies. Through computational results, the presented approach is tested on networks of realistic size.

We consider the GI/M/1 – K queue which has a capacity of K customers. Using singular perturbation methods, we construct asymptotic approximations to the stationary queue length distribution. We assume that K is large and treat several different parameter regimes. Extensive numerical comparisons are used to show the quality of the proposed approximations.

The linearized Ginzburg–Landau equations in both semi-infinite strips and rectangles are transformed into equivalent one-dimensional integral equations. Then, the properties of the integral equations are utilized to prove that the onset field for a semi-infinite strip is isolated. We solve the integral equations numerically to obtain the onset field for both rectangles and semi-infinite strips. A formal asymptotic expansion of the onset field in the long rectangle limit is also obtained. Using this formal expansion, we show that the onset field converges in this limit faster than any finite exponential rate, and as a byproduct, that the onset mode in a semi-infinite strip must be asymptotically symmetric.