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This article focuses on analyzing the finite-time ruin probability within a specific class of discrete risk models. These models incorporate dependent claims, an interest rate component, and stationary noise terms exhibiting semi-heavy-tailed behavior. In this framework, the claim amount follows a unilateral linear dependent process with independent and identically distributed noise terms, while the discount factor is determined by both the interest rate and time. The finite-time ruin probability has been derived under insurance risk conditions resembling the gamma distribution. [ABSTRACT FROM AUTHOR]

Interkingdom signalling within a holobiont allows host and symbionts to communicate and to regulate each other's physiological and developmental states. Here we show that a suite of signalling molecules that function as neurotransmitters and neuromodulators in most animals with nervous systems, specifically dopamine and trace amines, are produced exclusively by the bacterial symbionts of the demosponge Amphimedon queenslandica. Although sponges do not possess a nervous system, A. queenslandica expresses rhodopsin class G-protein-coupled receptors that are structurally similar to dopamine and trace amine receptors. When sponge larvae, which express these receptors, are exposed to agonists and antagonists of bilaterian dopamine and trace amine receptors, we observe marked changes in larval phototactic swimming behaviour, consistent with the sponge being competent to recognise and respond to symbiont-derived trace amine signals. These results indicate that monoamines synthesised by bacterial symbionts may be able to influence the physiology of the host sponge. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fast spectral-Galerkin method for solving PDEs involving an integral fractional Laplacian in Rd, which is built upon two essential components: (i) the Dunford-- Taylor formulation of the fractional Laplacian; and (ii) Fourier-like biorthogonal mapped Chebyshev functions (MCFs) as basis functions. As a result, the fractional Laplacian can be fully diagonalized, and the complexity of solving an elliptic fractional PDE is quasi-optimal, i.e., O((N log2 N)d) with N being the number of modes in each spatial direction. Ample numerical tests for various decaying exact solutions show that the convergence of the fast solver perfectly matches the order of theoretical error estimates. With a suitable time discretization, the fast solver can be directly applied to a large class of nonlinear fractional PDEs. As an example, we solve the fractional nonlinear Schrödinger equation by using the fourth-order time-splitting method together with the proposed MCF--spectral-Galerkin method. [ABSTRACT FROM AUTHOR]

Many PDEs involving fractional Laplacian are naturally set in unbounded domains with underlying solutions decaying slowly and subject to certain power law. Their numerical solutions are underexplored. This paper aims at developing accurate spectral methods using rational basis (or modified mapped Gegenbauer functions) for such models in unbounded domains. The main building block of the spectral algorithms is the explicit representations for the Fourier transform and fractional Laplacian of the rational basis, derived from some useful integral identities related to modified Bessel functions. With these at our disposal, we can construct rational spectral-Galerkin and direct collocation schemes by precomputing the associated fractional differentiation matrices. We obtain optimal error estimates of rational spectral approximation in the fractional Sobolev spaces and analyze the optimal convergence of the proposed Galerkin scheme. We also provide ample numerical results to show that the rational method outperforms the Hermite function approach. [ABSTRACT FROM AUTHOR]