The spectral analysis and exponential stability of a bi‐directional coupled wave‐ODE system

In this paper, an unstable linear time invariant (LTI) ODE system is stabilized exponentially by the PDE compensato—a wave equation with Kelvin‐Voigt (K‐V) damping. Direct feedback connections between the ODE system and wave equation are established: The velocity of the wave equation enters the ODE through the variable v_t(1,t); meanwhile, the output of the ODE is fluxed into the wave equation. It is found that the spectrum of the system operator is composed of two parts: point spectrum and continuous spectrum. The continuous spectrum consists of an isolated point , and there are two branches of asymptotic eigenvalues: the first branch approaches to , and the other branch tends to ?∞. It is shown that there is a sequence of generalized eigenfunctions, which forms a Riesz basis for the Hilbert state space. As a consequence, the spectrum‐determined growth condition and exponential stability of the system are concluded.     

Relaxation oscillations in a slow–fast modified Leslie–Gower model

In this paper, we prove the existence and uniqueness of relaxation oscillation cycle of a slow–fast modified Leslie–Gower model via the entry–exit function and geometric singular perturbation theory. Numerical simulations are also carried out to illustrate our theoretical result.     

A note on submodular function minimization by Chubanov’s LP algorithm

Recently Dadush et al. (2017) have devised a polynomial submodular function minimization (SFM) algorithm based on their LP algorithm. In the present note we also show a weakly polynomial algorithm for SFM based on the recently developed linear programming feasibility algorithm of Chubanov (2017) to stimulate further research on SFM.     

Network topology of deeply supercooled water

Empirical potential structure refinements have been made to recent high-energy x-ray diffraction data, providing molecular models of deeply supercooled water. The average O-O coordination number is found to drop from 5.13 at 293?K to 4.85 at 244?K, within 3.5?Å. Triplet O-O-O bond angle distributions reveal a broad peak centred at 96.4° at 293?K which shifts to 100.0° at 244?K, indicative of the local geometry becoming increasingly tetrahedral with decreasing temperature. However, although the number of non-bonded interstitial molecules between the first and second shells is depleted upon cooling, the number of interstitial molecules forming triplets that are embedded within the hydrogen bonded tetrahedral network at θ_OOO?=?53°, remains constant. This is consistent with previous observations of an invariant O-O coordination number with temperature (4.24 out to 3.3?Å) and corresponds to non-bonded molecules positioned at close to half the ideal tetrahedral angle. Both -O-O-O- and hydrogen-bonded -O-H-O- ring length distributions show increases in 6 and 7-membered rings upon supercooling. This is concomitant with a shift and increase in intensity of peaks at r₄ ～8.7?Å and r₅ ～10.8?Å in the oxygen-oxygen pair distribution function, which in the models correspond to correlations between adjacent and next-nearest-neighbour hydrogen-bonded rings.     

Well‐posedness of fractional integro‐differential equations in vector‐valued functional spaces

We study the well‐posedness of the fractional differential equations with infinite delay on Lebesgue–Bochner spaces and Besov spaces , where A and B are closed linear operators on a Banach space X satisfying ,  and . Under suitable assumptions on the kernels a and b, we completely characterize the well‐posedness of in the above vector‐valued function spaces on by using known operator‐valued Fourier multiplier theorems. We also give concrete examples where our abstract results may be applied.     

Brownian motion with drift on spaces with varying dimension

Many properties of Brownian motion on spaces with varying dimension (BMVD in abbreviation) have been explored in Chen and Lou (2018). In this paper, we study Brownian motion with drift on spaces with varying dimension (BMVD with drift in abbreviation). Such a process can be conveniently defined by a regular Dirichlet form that is not necessarily symmetric. Through the method of Duhamel’s principle, it is established in this paper that the transition density of BMVD with drift has the same type of two-sided Gaussian bounds as that for BMVD (without drift). As a corollary, we derive Green function estimate for BMVD with drift.