L1L^1 contraction for bounded (non-integrable) solutions of degenerate parabolic equations

We obtain new $L^1$ contraction results for bounded entropy solutions of Cauchy problems for degenerate parabolic equations. The equations we consider have possibly strongly degenerate local or non-local diffusion terms. As opposed to previous results, our results apply without any integrability assumption on the %(the positive part of the difference of) solutions. They take the form of partial Duhamel formulas and can be seen as quantitative extensions of finite speed of propagation local $L^1$ contraction results for scalar conservation laws. A key ingredient in the proofs is a new and non-trivial construction of a subsolution of a fully non-linear (dual) equation. Consequences of our results are maximum and comparison principles, new a priori estimates, and in the non-local case, new existence and uniqueness results

Analysis and assessment of film materials and associated manufacturing processes for a solar sail

Candidate resin manufacturers and film producers were surveyed to determine the availability of key materials and to establish the capabilities of fabricators to prepare ultrathin films of these materials within the capacity/cost/time constraints of the Halley program. Infrared spectra of three candidate samples were obtained by pressing each sample against an internal reflection crystal with the polymer sandwiched between the crystal and the metal backing. The sample size was such that less than one-fourth of the surface of the crystal was covered with the sample. This resulted in weak spectra requiring a six-fold expansion. Internal reflection spectra of the three samples were obtained using both a KRS-5 and a Ge internal reflection crystal. Subtracted infrared spectra of the three samples are presented

Photonic crystal fiber with a hybrid honeycomb cladding

We consider an air-silica honeycomb lattice and demonstrate a new approach to the formation of a core defect. Typically, a high or low-index core is formed by adding a high-index region or an additional air-hole (or other low-index material) to the lattice, but here we discuss how a core defect can be formed by manipulating the cladding region rather than the core region itself. Germanium-doping of the honeycomb lattice has recently been suggested for the formation of a photonic band-gap guiding silica-core and here we experimentally demonstrate how an index-guiding silica-core can be formed by fluorine-doping of the honeycomb lattice.Comment: 5 pages including 3 figures. Accepted for Optics Expres

Continuous dependence estimates for nonlinear fractional convection-diffusion equations

We develop a general framework for finding error estimates for convection-diffusion equations with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators that are generators of pure jump Levy processes (e.g. the fractional Laplacian). As an application, we derive continuous dependence estimates on the nonlinearities and on the Levy measure of the diffusion term. Estimates of the rates of convergence for general nonlinear nonlocal vanishing viscosity approximations of scalar conservation laws then follow as a corollary. Our results both cover, and extend to new equations, a large part of the known error estimates in the literature.Comment: In this version we have corrected Example 3.4 explaining the link with the results in [51,59

Heaps and heapsort on secondary storage

AbstractA heap structure designed for secondary storage is suggested that tries to make the best use of the available buffer space in primary memory. The heap is a complete multi-way tree, with multi-page blocks of records as nodes, satisfying a generalized heap property. A special feature of the tree is that the nodes may be partially filled, as in B-trees. The structure is complemented with priority-queue operations insert and delete-max. When handling a sequence of S operations, the number of page transfers performed is shown to be O(∑i = 1S(1P) log(MP)(NiP)), where P denotes the number of records fitting into a page, M the capacity of the buffer space in records, and Ni, the number of records in the heap prior to the ith operation (assuming P ⩾ 1 and S > M ⩾ c · P, where c is a small positive constant). The number of comparisons required when handling the sequence is O(∑i = 1S log2 Ni). Using the suggested data structure we obtain an optimal external heapsort that performs O((NP) log(MP)(NP)) page transfers and O(N log2 N) comparisons in the worst case when sorting N records

Investigation of the shear-mechanical and dielectric relaxation processes in two mono-alcohols close to the glass transition

Shear-mechanical and dielectric measurements on the two monohydroxy (mono-alcohol) molecular glass formers 2-ethyl-1-hexanol and 2-butanol close to the glass transition temperature are presented. The shear-mechanical data are obtained using the piezoelectric shear-modulus gauge method covering frequencies from 1mHz to 10kHz. The shear-mechanical relaxation spectra show two processes, which follow the typical scenario of a structural (alpha) relaxation and an additional (Johari-Goldstein) beta relaxation. The dielectric relaxation spectra are dominated by a Debye-type peak with an additional non-Debye peak visible. This Debye-type relaxation is a common feature peculiar to mono-alcohols. The time scale of the non-Debye dielectric relaxation process is shown to correspond to the mechanical structural (alpha) relaxation. Glass-transition temperatures and fragilities are reported based on the mechanical alpha relaxation and the dielectric Debye-type process, showing that the two glass-transition temperatures differ by approximately 10K and that the fragility based on the Debye-type process is a factor of two smaller than the structural fragility. If a mechanical signature of the Debye-type relaxation exists in these liquids, its relaxation strength is at most 1% and 3% of the full relaxation strength of 2-butanol and 2-ethyl-1-hexanol respectively. These findings support the notion that it is the non-Debye dielectric relaxation process that corresponds to the structural alpha relaxation in the liquid.Comment: 8 pages, 6 figures. Minor corrections, updated figures, more dielectric data show

Robust numerical methods for nonlocal (and local) equations of porous medium type. Part I: Theory

Abstract. We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations ∂tu − Lσ,μ[φ(u)] = f(x,t) in RN × (0,T), where Lσ,μ is a general symmetric diffusion operator of L ́evy type and φ is merely continuous and non-decreasing. We then use this theory to prove con- vergence for many different numerical schemes. In the nonlocal case most of the results are completely new. Our theory covers strongly degenerate Stefan problems, the full range of porous medium equations, and for the first time for nonlocal problems, also fast diffusion equations. Examples of diffusion op- σ,μ α are the (fractional) Laplacians ∆ and −(−∆)2 for α ∈ (0,2), erators L discrete operators, and combinations. The observation that monotone finite difference operators are nonlocal L ́evy operators, allows us to give a unified and compact nonlocal theory for both local and nonlocal, linear and nonlinear diffusion equations. The theory includes stability, compactness, and conver- gence of the methods under minimal assumptions – including assumptions that lead to very irregular solutions. As a byproduct, we prove the new and general existence result announced in [28]. We also present some numerical tests, but extensive testing is deferred to the companion paper [31] along with a more detailed discussion of the numerical methods included in our theory

Hematite(001)-liquid water interface from hybrid density functional-based molecular dynamics

The atom-scale characterisation of interfaces between transition metal oxides and liquid water is fundamental to our mechanistic understanding of diverse phenomena ranging from crystal growth to biogeochemical transformations to solar fuel production. Here we report on the results of large-scale hybrid density functional theory-based molecular dynamics simulations for the hematite(001)-liquid water interface. A specific focus is placed on understanding how different terminations of the same surface influence surface solvation. We find that the two dominant terminations for the hematite(001) surface exhibit strong differences both in terms of the active species formed on the surface and the strength of surface solvation. According to present simulations, we find that charged oxyanions (-O−) and doubly protonated oxygens (-OH$_{2}^{+}$ ) can be formed on the iron terminated layer via autoionization of neutral -OH groups. No such charged species are found for the oxygen terminated surface. In addition, the missing iron sublayer in the iron terminated surface strongly influences the solvation structure, which becomes less well ordered in the vicinity of the interface. These pronounced differences are likely to affect the reactivity of the two surface terminations, and in particular the energetics of excess charge carriers at the surface

All-optical phase-regenerative multicasting of 40 Gbit/s DPSK signal in a degenerate phase sensitive amplifier

We demonstrate all-optical 1-to-5 differential phase-shift keyed (DPSK) wavelength multicasting at 40 Gbit/s using a degenerate four-wave mixing (FWM) based phase sensitive amplifier (PSA). Phase regenerative properties are reported with a sensitivity improvement of more that 10 dB

Saturation effects in degenerate phase sensitive fiber optic parametric amplifiers

We experimentally study saturation effects in degenerate phase sensitive amplifiers, revealing and explaining a gain regime suitable for all-optical signal processing functions such as phase regeneration