Lyapunov inequalities for nabla Caputo boundary value problems

ABSTRACT

We will establish uniqueness of solutions to boundary value problems involving the nabla Caputo fractional difference under two-point boundary conditions and give an explicit expression for the Green's functions for these problems. Using the Green's functions for specific cases of these boundary value problems, we will then develop Lyapunov inequalities for certain nabla Caputo BVPs.     

Malliavin and Dirichlet structures for independent random variables

On any denumerable product of probability spaces, we construct a Malliavin gradient and then a divergence and a number operator. This yields a Dirichlet structure which can be shown to approach the usual structures for Poisson and Brownian processes. We obtain versions of almost all the classical functional inequalities in discrete settings which show that the Efron–Stein inequality can be interpreted as a Poincaré inequality or that the Hoeffding decomposition of $U$-statistics can be interpreted as an avatar of the Clark representation formula. Thanks to our framework, we obtain a bound for the distance between the distribution of any functional of independent variables and the Gaussian and Gamma distributions.     

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.     

一类精细化Eulerian多项式的若干性质

最近,孙华定义了一类新的精细化Eulerian多项式,即$$A_n(p,q)=\sum_{\pi\in \mathfrak{S}_n}p^{{\rm odes}(\pi)}q^{{\rm edes}(\pi)},\ \ n\ge 1,$$ 其中$S_n$表示$\{1,2,\ldots,n\}$上全体$n$阶排列的集合, odes$(\pi)$与edes$(\pi)$分别表示$S_n$中排列$\pi$的奇数位与偶数位上降位数的个数.本文利用经典的Eulerian多项式$A_n(q)$ 与Catalan 序列的生成函数$C(q)$,得到精细化Eulerian 多项式$A_n(p,q)$的指数型生成函数及$A_n(p,q)$的显示表达式.在一些特殊情形,本文建立了$A_n(p,q)$与$A_n(0,q)$或$A_n(p,0)$之间的联系,并利用Eulerian数表示多项式$A_n(0,q)$的系数.特别地,这些联系揭示了Euler数$E_n$与Eulerian数$A_{n,k}$之间的一种新的关系.     

Microwave‐associate synthesis of Co3O4 nanoparticles as an effcient nanocatalyst for the synthesis of arylidene barbituric and Meldrum's acid derivatives in green media

In this study, Co₃O₄ nanocatalysts were constructed in environmentally appropriate conditions using controlled, effective, and facile microwave method. The final nanostructures were characterized by SEM, XRD, and TEM analyses. The products had a small size distribution, homogeneous morphology, and crystallographic structures associated with the formation of Co₃O₄ nanostructures. Moreover, EDS mapping analysis confirmed the existence of Co and O elements in the final structure, and the magnetic properties of the samples were investigated by VSM. The application of this nanostructure in a catalytic process was further examined, and the results suggested that it could be used as a novel candidate for the synthesis of arylidene barbituric and Meldrum^,s acid through Knoevenagel condensation of aldehydes by barbituric and Meldrum^,s acid in aqueous media. The high yield of these nanocatalysts would be justified by the nature of the nanostructure as well as the experimental procedure developed in this study, which affected the physicochemical features of the products.     

Synthesis and Characterization of Phenolphthalein‐Containing Sulfonated Poly(arylene ether phosphine oxide)s by Direct Polycondensation for Proton Exchange Membrane

A series of novel phenolphthalein‐containing sulfonated poly(arylene ether phosphine oxide)s (sPAEPP) with various sulfonation degrees were synthesized by direct polycondensation. The structure of sPAEPP was confirmed by ¹H‐NMR, ¹³C‐NMR, and IR spectroscopy. The high‐molecular weight of these polymers was determined by gel permeation chromatography (GPC). The transparent, tough, and flexible membranes could be achieved by solution casting. The macroscopic properties and microstructure of the obtained membranes were investigated in detail. The results showed that these sPAEPP membranes displayed excellent properties in terms of swelling, proton conductivity, and methanol permeability. For example, sPAEPP‐100 membrane exhibited an appropriate water uptake of 33.1%, a swelling ratio of only 11.7% (lower than 20.1% of Nafion 117), a proton conductivity of 0.11 S cm^?1 (similar to that of Nafion 117) at 80 °C, and a methanol permeability of 4.82 × 10^?7 cm² s^?1. Meanwhile, it also presented outstanding oxidative stability. Atomic force microscope (AFM) micrographs showed that the hydrophilic domains of the sPAEPP‐100 membrane formed connected and narrow ionic channels, which contributed to its high proton conductivity and good dimensional stability. As a result, sPAEPP‐100 membrane displays excellent application prospect for fuel cells. © 2019 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2019 , 57, 1097–1104     

A calibrated model of local food system of Hawaii: What are the economic implications of the state's food goals and policies?

We develop a fully calibrated positive mathematical programming model for Hawaii's local food systems—which captures the production and the consumer sides of the market. Then we use the model to assess two proposed policies—a general excise tax (GET) exemption on locally produced foods, and an investment in agricultural infrastructure. For the GET exemption case, our results indicate an economic gain of $118 per $100 cost. On the other hand, an investment in 1,200 acres of land injected to support local production may generate an economic gain of up to $357 per $100 annual cost of the investment. However, these estimates should be considered preliminary, and thus viewed with caution. Although the model is used to capture Hawaii's local food systems, we believe that our model is generalizable and can be adopted by other economies to assess their respective food localization policies. Recommendations for Resource Managers

• Local food policies need to be based on quantitative terms instead of mere armchair speculation because often their potential outcomes may vary significantly.

• The current modeling framework demonstrates the potential of using positive mathematical programming (PMP) in capturing the intricacies of local food systems. However, this exploratory exercise should be viewed as preliminary in nature and the ensuing results were taken with caution because many important factors such as labor availability may have been left out.

• Thus, further model refinements are necessary to better capture the complexities of local food systems such as farm heterogeneity, availability of farm labor, water availability, and interisland transportation of farm products in the case of Hawaii.