Harnack type estimates and Hölder continuity for non-negative solutions to certain sub-critically singular parabolic partial differential equations

A two-parameter family of Harnack type inequalities for non-negative solutions of a class of singular, quasilinear, homogeneous parabolic equations is established, and it is shown that such estimates imply the Hoelder continuity of solutions. These classes of singular equations include p-Laplacean type equation in the sub-critical range 1<p\le\frac2N/(N+1) and equations of the porous medium type in the sub-critical range 0<m\le(N-2)_+/N

Parallel Exhaustive Search without Coordination

We analyze parallel algorithms in the context of exhaustive search over totally ordered sets. Imagine an infinite list of "boxes", with a "treasure" hidden in one of them, where the boxes' order reflects the importance of finding the treasure in a given box. At each time step, a search protocol executed by a searcher has the ability to peek into one box, and see whether the treasure is present or not. By equally dividing the workload between them, $k$ searchers can find the treasure $k$ times faster than one searcher. However, this straightforward strategy is very sensitive to failures (e.g., crashes of processors), and overcoming this issue seems to require a large amount of communication. We therefore address the question of designing parallel search algorithms maximizing their speed-up and maintaining high levels of robustness, while minimizing the amount of resources for coordination. Based on the observation that algorithms that avoid communication are inherently robust, we analyze the best running time performance of non-coordinating algorithms. Specifically, we devise non-coordinating algorithms that achieve a speed-up of $9/8$ for two searchers, a speed-up of $4/3$ for three searchers, and in general, a speed-up of $\frac{k}{4}(1+1/k)^2$ for any $k\geq 1$ searchers. Thus, asymptotically, the speed-up is only four times worse compared to the case of full-coordination, and our algorithms are surprisingly simple and hence applicable. Moreover, these bounds are tight in a strong sense as no non-coordinating search algorithm can achieve better speed-ups. Overall, we highlight that, in faulty contexts in which coordination between the searchers is technically difficult to implement, intrusive with respect to privacy, and/or costly in term of resources, it might well be worth giving up on coordination, and simply run our non-coordinating exhaustive search algorithms

Local regularity for parabolic nonlocal operators

Weak solutions to parabolic integro-differential operators of order $\alpha \in (\alpha_0, 2)$ are studied. Local a priori estimates of H\"older norms and a weak Harnack inequality are proved. These results are robust with respect to $\alpha \nearrow 2$. In this sense, the presentation is an extension of Moser's result in 1971.Comment: 31 pages, 3 figure

Well-posedness for a class of nonlinear degenerate parabolic equations

In this paper we obtain well-posedness for a class of semilinear weakly degenerate reaction-diffusion systems with Robin boundary conditions. This result is obtained through a Gagliardo-Nirenberg interpolation inequality and some embedding results for weighted Sobolev spaces

Unravelling the role of the group 6 soluble di-iron monooxygenase (SDIMO) SmoABCD in alkane metabolism and chlorinated alkane degradation

Soluble di-iron monooxygenases (SDIMOs) are multi-component enzymes catalysing the oxidation of various substrates. These enzymes are characterized by high sequence and functional diversity that is still not well understood despite their key role in biotechnological processes including contaminant biodegradation. In this study, we analysed a mutant of Rhodoccocus aetherivorans BCP1 (BCP1-2.10) characterized by a transposon insertion in the gene smoA encoding the alpha subunit of the plasmid-located SDIMO SmoABCD. The mutant BCP1-2.10 showed a reduced capacity to grow on propane, lost the ability to grow on butane, pentane and n-hexane and was heavily impaired in the capacity to degrade chloroform and trichloroethane. The expression of the additional SDIMO prmABCD in BCP1-2.10 probably allowed the mutant to partially grow on propane and to degrade it, to some extent, together with the other short-chain n-alkanes. The complementation of the mutant, conducted by introducing smoABCD in the genome as a single copy under a constitutive promoter or within a plasmid under a thiostreptone-inducible promoter, allowed the recovery of the alkanotrophic phenotype as well as the capacity to degrade chlorinated n-alkanes. The heterologous expression of smoABCD allowed a non-alkanotrophic Rhodococcus strain to grow on pentane and n-hexane when the gene cluster was introduced together with the downstream genes encoding alcohol and aldehyde dehydrogenases and a GroEL chaperon. BCP1 smoA gene was shown to belong to the group 6 SDIMOs, which is a rare group of monooxygenases mostly present in Mycobacterium genus and in a few Rhodococcus strains. SmoABCD originally evolved in Mycobacterium and was then acquired by Rhodococcus through horizontal gene transfer events. This work extends the knowledge of the biotechnologically relevant SDIMOs by providing functional and evolutionary insights into a group 6 SDIMO in Rhodococcus and demonstrating its key role in the metabolism of short-chain alkanes and degradation of chlorinated n-alkanes

Riesz potentials and nonlinear parabolic equations

The spatial gradient of solutions to nonlinear degenerate parabolic equations can be pointwise estimated by the caloric Riesz potential of the right hand side datum, exactly as in the case of the heat equation. Heat kernels type estimates persist in the nonlinear cas

Selective Aerobic Oxidation of 5-Hydroxymethylfurfural to 2,5-Diformylfuran or 2-Formyl-5-furancarboxylic Acid in Water by using MgO⋅CeO2 Mixed Oxides as Catalysts

Mixed oxides based on MgO⋅CeO2 were used as efficient catalysts in the aerobic oxidation of 5-hydroxymethylfurfural (5-HMF) to afford, with very high selectivity, either 2,5-diformylfuran (DFF, 99 %) or 2-formyl-5-furancarboxylic acid (FFCA, 90 %), depending on the reaction conditions. 5-Hydroxymethyl-2-furancarboxylic acid (HMFCA, 57–90 %) was formed only at low concentration of 5-HMF (&lt;0.03 m) or in presence of external bases. The conversion of 5-HMF ranged from a few percent to 99 %, according to the reaction conditions. The oxidation was performed in water, with O2 as oxidant, without any additives. The surface characterization of the catalysts gave important information about their acid–base properties, which drive the selectivity of the reaction towards DFF. FFCA was formed from DFF at longer reaction times. Catalysts were studied by XPS and XRD before and after catalytic runs to identify the reason why they undergo reversible deactivation. XRD showed that MgO is hydrated to Mg(OH)2, which, even if not leached out, changes the basic properties of the catalyst that becomes less active after some time. Calcination of the recovered catalyst allows recovery of its initial activity. The catalyst is thus recoverable (&gt;99 %) and reusable. The use of mixed oxides allows tuning of the basicity of the catalysts, avoiding the need for external bases for efficient and selective conversion of 5-HMF and waste formation, resulting in an environmentally friendly, sustainable process