The maximal solution of a restricted subadditive inequality in numerical analysis

For a fixed non-negative integerp, letU _2p = {U _2p (n)},n ≥ 0, denote the sequence that is defined by the initial conditionsU _2p (0) =U _2p (1) =U _2p (2) = =U _2p (2p) = 1 and the restricted subadditive recursion $$U_{2p} (n + 2p + 1) = \mathop {\min }\limits_{0 \leqslant l \leqslant p} (U_{2p} (n + l) + U_{2p} (n + 2p - l)),n \geqslant 0$$ U _2p is of importance in the theory of sequential search for simple real zeros of real valued continuous 2p-th derivatives In this paper, several closed form expressions forU _2p (n), n > 2p, are determined, thereby providing insight into the structure ofU _2p Two of the properties thus illuminated are (a) the existence of exactlyp + 1 limit points (1 + 1/(p + 1 +i), 0 ≤i ≤p) of the associated sequence {U _2p (n + 1)/U _2p (n)},n ≥ 0, and (b) the relevance toU _2p of the classic number theoretic function ord     

Partitioning Boolean lattices into chains of subsets

We prove that the Boolean lattice of all subsets of an n-set can be partitioned into chains of size four if and only if n9.Research supported in part by N.S.F. grant DMS-8401281.Research supported in part by N.S.F. grant DMS-8406451.     

Soluble models for dynamics driven by a super-diffusive noise

We explicitly discuss scalar Langevin type of equations where the deterministic part is linear, but where the integrated noise source is a non-linear diffusion process exhibiting superdiffusive behavior. We calculate transient and stationary probabilities and study the possibility of noise induced transitions from a unimodal to a bimodal probability shape. Illustrations from finance and dynamical systems are given.     

Anomalous modal structure in a waveguide with a photonic crystal core

We analyze a dielectric waveguide with a photonic crystal core. Using constant frequency contour analysis, we show that the modal behavior of this structure is drastically different from that of a conventional slab waveguide. In particular, at a given frequency the lowest-order guided mode can have an odd symmetry or can have more than one nodal plane in its field distribution. Also, there exist several single-mode regions with a different modal profile in each region. Finally, a single-mode waveguide for the fundamental mode with a large core and strong confinement can be realized. All these behaviors are confirmed by our three-dimensional finite-difference time-domain simulations.     

An AlGaN/GaN HEMT with reduced surface electric field and improved breakdown voltage

A reduced surface electric field in AlGaN/GaN high electron mobility transistor (HEMT) is investigated by employing a localized Mg-doped layer under the two-dimensional electron gas (2-DEG) channel as an electric field shaping layer. The electric field strength around the gate edge is effectively relieved and the surface electric field is distributed evenly as compared with those of HEMTs with conventional source-connected field plate and double field plate structures with the same device physical dimensions. Compared with the HEMTs with conventional source-connected field plate and double field plate, the HEMT with Mg-doped layer also shows that the breakdown location shifts from the surface of the gate edge to the bulk Mg-doped layer edge. By optimizing both the length of Mg-doped layer, L_m, and the doping concentration, a 5.5 times and 3 times the reduction in the peak electric field near the drain side gate edge is observed as compared with those of the HEMTs with source-connected field plate structure and double field plate structure, respectively. In a device with V_GS=-5 V, L_m=1.5 μm, a peak Mg doping concentration of 8× 10¹⁷ cm^-3 and a drift region length of 10 μm, the breakdown voltage is observed to increase from 560 V in a conventional device without field plate structure to over 900 V without any area overhead penalty.