We propose two linearly implicit energy-preserving schemes for the complex modified Korteweg–de Vries equation, based on the invariant energy quadratization method. First, a new variable is introduced and a new Hamiltonian system is constructed for this equation. Then the Fourier pseudospectral method is used for the space discretization and the Crank–Nicolson leap-frog schemes for the time discretization. The proposed schemes are linearly implicit, which is only needed to solve a linear system at each time step. The fully discrete schemes can be shown to conserve both mass and energy in the discrete setting. Some numerical examples are also presented to validate the effectiveness of the proposed schemes.

Image registration is the process of finding an alignment between two or more images so that their appearances match. It has been widely studied and applied to several fields, including medical imaging and biology, where it is related to morphometrics. In this paper, we present a construction of conformal diffeomorphisms which is based on constrained optimization. We consider a set of different penalty terms that aim to enforce conformality, based on discretizations of the Cauchy–Riemann equations and geometric principles, and demonstrate them experimentally on a variety of images.

We propose a variation of the pointwise residual method for solving primal and dual ill-posed linear programming with approximate data, sensitive to small perturbations. The method leads to an auxiliary problem, which is also a linear programming problem. Theorems of existence and convergence of approximate solutions are established and optimal estimates of approximation of initial problem solutions are achieved.

Motivated by the manufacture of carbon fibre components, this paper considers the smooth draping of loosely woven fabric over rigid obstacles, both smooth and nonsmooth. The draped fabric is modelled as the continuum limit of a Chebyshev net of two families of short rigid rods that are freely pivoted at their joints. This approach results in a system of nonlinear hyperbolic partial differential equations whose characteristics are the fibres in the fabric. The analysis of this system gives useful information about the drapability of obstacles of many shapes and also poses interesting theoretical questions concerning well-posedness, smoothness and computability of the solutions.

Let $\boldsymbol{f}$ be a square-integrable, zero-mean, random vector with observable realizations in a Hilbert space H, and let $\boldsymbol{g}$ be an associated square-integrable, zero-mean, random vector with realizations which are not observable in a Hilbert space K. We seek an optimal filter in the form of a closed linear operator X acting on the observable realizations of a proximate vector $\boldsymbol{f}_{\epsilon } \approx \boldsymbol{f}$ that provides the best estimate $\widehat{\boldsymbol{g}}_{\epsilon} = X \boldsymbol{f}_{\epsilon}$ of the vector $\boldsymbol{g}$. We assume the required covariance operators are known. The results are illustrated with a typical example.

We assume that human carrying capacity is determined by food availability. We propose three classes of human population dynamical models of logistic type, where the carrying capacity is a function of the food production index. We also employ an integration-based parameter estimation technique to derive explicit formulas for the model parameters. Using actual population and food production index data, numerical simulations of our models suggest that an increase in food availability implies an increase in carrying capacity, but the carrying capacity is “self-limiting” and does not increase indefinitely.