We rigorously prove results on spiky patterns for the Gierer-Meinhardt system [5] with a large number of jump discontinuities in the diffusion coefficient of the inhibitor. Using numerical computations in combination with a Turing-type instability analysis, this system has been investigated by Benson, Maini and Sherratt [1], [3], [9]. We first review results on the case of two segments given in...

Aperiodic substitution tilings provide popular models for quasicrystals, materials exhibiting aperiodic order. We study the graph Laplacian associated with four from mutual local derivability class of Penrose tiling, as well Ammann–Beenker tiling. In each case we exhibit locally-supported eigenfunctions, which necessarily cause jump discontinuities in integrated density states these models. By ...

Empirical evidence of asset price discontinuities or ‘‘jumps’’ in financial markets has been well documented in the literature. Recently, Aït-Sahalia and Jacod (2009b) defined a general ‘‘jump activity index’’ to describe the degree of jump activities for asset price semimartingales, and provided a consistent estimatorwhen the underlying process contains both a continuous and a jump component. ...

We study ballistic transport in integrable quantum lattice models, i.e., the spin XXZ and Hubbard chains, close to noninteracting limit. It is by now well established that stiffnesses of charge currents reveal, at high temperatures, a discontinuous reduction (jump) when interaction introduced. show jumps are related large degeneracy parent models and, more generally, can appear other with macro...

A unified theory for aerodynamic and condensation shock waves in vapor-droplet flows in the presence of an inert carrier gas is presented. Same conservation equations apply across discontinuous models for both types of wave. Exact (as well as approximate), explicit analytical jump conditions across such discontinuities are derived subject to several boundary conditions. Collectively they may be...

We study the Hankel determinant generated by Laguerre weight with jump discontinuities at $t_k, k=1,\cdots,m$. By employing ladder operator approach to establish Riccati equations, we show that $\sigma_n(t_1,\cdots,t_m)$, logarithmic derivative of $n$-dimensional determinant, satisfies a generalization $\sigma$-from Painlev\'{e} V equation. Through investigating Riemann-Hilbert problem for asso...

We develop the process of discounting when underlying rates follow a jump-diffusion process, that is, when, in addition to diffusive behavior, suffer series finite discontinuities located at random Poissonian times. Jump amplitudes are also and governed by an arbitrary density. Such model may describe economic evolution, specially extreme situations occur (pandemics, global wars, etc.). When, b...