Waddellborup7762

The question of whether a population will persist or go extinct is of key interest throughout ecology and biology. Various mathematical techniques allow us to generate knowledge regarding individual behaviour, which can be analysed to obtain predictions about the ultimate survival or extinction of the population. A common model employed to describe population dynamics is the lattice-based random walk model with crowding (exclusion). This model can incorporate behaviour such as birth, death and movement, while including natural phenomena such as finite size effects. click here Performing sufficiently many realizations of the random walk model to extract representative population behaviour is computationally intensive. Therefore, continuum approximations of random walk models are routinely employed. However, standard continuum approximations are notoriously incapable of making accurate predictions about population extinction. Here, we develop a new continuum approximation, the state-space diffusion approximation, which explicitly accounts for population extinction. Predictions from our approximation faithfully capture the behaviour in the random walk model, and provides additional information compared to standard approximations. We examine the influence of the number of lattice sites and initial number of individuals on the long-term population behaviour, and demonstrate the reduction in computation time between the random walk model and our approximation.Recent reconstructions of total solar irradiance (TSI) postulate that quiet-Sun variations could give significant changes to the solar power input to Earth's climate (radiative climate forcings of 0.7-1.1 W m-2 over 1700-2019) arising from changes in quiet-Sun magnetic fields that have not, as yet, been observed. Reconstructions without such changes yield solar forcings that are smaller by a factor of more than 10. We study the quiet-Sun TSI since 1995 for three reasons (i) this interval shows rapid decay in average solar activity following the grand solar maximum in 1985 (such that activity in 2019 was broadly equivalent to that in 1900); (ii) there is improved consensus between TSI observations; and (iii) it contains the first modelling of TSI that is independent of the observations. Our analysis shows that the most likely upward drift in quiet-Sun radiative forcing since 1700 is between +0.07 and -0.13 W m-2. Hence, we cannot yet discriminate between the quiet-Sun TSI being enhanced or reduced during the Maunder and Dalton sunspot minima, although there is a growing consensus from the combinations of models and observations that it was slightly enhanced. We present reconstructions that add quiet-Sun TSI and its uncertainty to models that reconstruct the effects of sunspots and faculae.We consider versions of the grasshopper problem (Goulko & Kent 2017 Proc. R. Soc. A473, 20170494) on the circle and the sphere, which are relevant to Bell inequalities. For a circle of circumference 2π, we show that for unconstrained lawns of any length and arbitrary jump lengths, the supremum of the probability for the grasshopper's jump to stay on the lawn is one. For antipodal lawns, which by definition contain precisely one of each pair of opposite points and have length π, we show this is true except when the jump length ϕ is of the form π(p/q) with p, q coprime and p odd. For these jump lengths, we show the optimal probability is 1 - 1/q and construct optimal lawns. For a pair of antipodal lawns, we show that the optimal probability of jumping from one onto the other is 1 - 1/q for p, q coprime, p odd and q even, and one in all other cases. For an antipodal lawn on the sphere, it is known (Kent & Pitalúa-García 2014 Phys. Rev. A90, 062124) that if ϕ = π/q, where q ∈ N , then the optimal retention probability of 1 - 1/q for the grasshopper's jump is provided by a hemispherical lawn. We show that in all other cases where 0  less then  ϕ  less then  π/2, hemispherical lawns are not optimal, disproving the hemispherical colouring maximality hypotheses (Kent & Pitalúa-García 2014 Phys. Rev. A90, 062124). We discuss the implications for Bell experiments and related cryptographic tests.As it is known, the existence of the Wiener-Hopf factorization for a given matrix is a well-studied problem. Severe difficulties arise, however, when one needs to compute the factors approximately and obtain the partial indices. This problem is very important in various engineering applications and, therefore, remains to be subject of intensive investigations. In the present paper, we approximate a given matrix function and then explicitly factorize the approximation regardless of whether it has stable partial indices. For this reason, a technique developed in the Janashia-Lagvilava matrix spectral factorization method is applied. Numerical simulations illustrate our ideas in simple situations that demonstrate the potential of the method.In the work, we obtain an effective criterion of the stability of the partial indices for matrix polynomials under an arbitrary sufficiently small perturbation. Verification of the stability is reduced to calculation of the ranks for two explicitly defined Toeplitz matrices. Furthermore, we define a notion of the stability of the partial indices in the given class of matrix functions. This means that we will consider an allowable small perturbation such that a perturbed matrix function belong to the same class as the original one. We prove that in the class of matrix polynomials the Gohberg-Krein-Bojarsky criterion is preserved, i.e. new stability cases do not arise. Our proof of the stability criterion in this class does not use the Gohberg-Krein-Bojarsky theorem.We develop a general framework for the description of instabilities on soap films using the Björling representation of minimal surfaces. The construction is naturally geometric and the instability has the interpretation as being specified by its amplitude and transverse gradient along any curve lying in the minimal surface. When the amplitude vanishes, the curve forms part of the boundary to a critically stable domain, while when the gradient vanishes the Jacobi field is maximal along the curve. In the latter case, we show that the Jacobi field is maximally localized if its amplitude is taken to be the lowest eigenfunction of a one-dimensional Schrödinger operator. We present examples for the helicoid, catenoid, circular helicoids and planar Enneper minimal surfaces, and emphasize that the geometric nature of the Björling representation allows direct connection with instabilities observed in soap films.