On sharp Burkholder–Rosenthal-type inequalities for infinite-degree U-statistics

In this paper, we present a method that allows one to obtain a number of sharp inequalities for expectations of functions of infinite-degree U-statistics. Using the approach, we prove, in particular, the following result: Let D be the class of functions f :R₊→R₊ such that the function f(x+z)−f(x) is concave in xR₊ for all zR₊. Then the following estimate holds: for all fD and all U-statistics ∑_1i1<<ilnY_i1,…,il(X_i1,…,X_il) with nonnegative kernels Y_i1,…,il :R^l→R₊, 1i_kn; i_r≠i_s, r≠s; k,r,s=1,…,l; l=0,…,m, in independent r.v.'s X₁,…,X_n. Similar inequality holds for sums of decoupled U-statistics. The class D is quite wide and includes all nonnegative twice differentiable functions f such that the function f″(x) is nonincreasing in x>0, and, in particular, the power functions f(x)=x^t, 1<t2; the power functions multiplied by logarithm f(x)= (x+x₀)^t ln(x+x₀), 1<t<2, x₀max(e^{(3t2−6t+2)/(t(t−1)(2−t))},1); and the entropy-type functions f(x)=(x+x₀)ln(x+x₀), x₀1. As an application of the results, we determine the best constants in Burkholder–Rosenthal-type inequalities for sums of U-statistics and prove new decoupling inequalities for those objects. The results obtained in the paper are, to our knowledge, the first known results on the best constants in sharp moment estimates for U-statistics of a general type.     

Analyses of the complex proton NMR spectra: determination of anisotropic proton chemical shifts of oriented molecules by a two dimensional experiment

NMR spectra of molecules oriented in the liquid crystalline media provide information on the molecular structure and order parameter. However, the numerical iterative analysis of the proton spectra of strongly coupled spins is difficult and time consuming. Such analysis is simplified if nearly accurate starting parameters are available. One such parameter is the chemical shift which in the oriented media is very different from the isotropic values due to anisotropic contributions. In this study, we have explored the possibility of obtaining chemical shifts in the oriented phase to aid the analysis of the spectra. A two dimensional experiment in which FSLG decoupling employed during the t1 period eliminates the homonuclear dipolar couplings and retains only the chemical shifts has been implemented. Experiments on the molecule cis,cis-mucononitrile demonstrate that the chemical shifts obtained by this procedure are nearly the same as the chemical shifts derived by iterative analysis of the one dimensional spectrum of the molecule following the standard procedure. The method has also been used to analyse the spectrum of 1-iodopropane using the chemical shifts obtained from the proposed experiment as the starting parameters.     

A new approach to solve a set of nonlinear split boundary value problems

In this paper, we propose a new approach based on conjunction of the orthogonal collocation on finite elements method with decoupling and quasi-linearization technique to approximate solutions of a set of nonlinear split boundary value problems. The numerical stability, the convergence and the accuracy of the results are checked by this algorithm. The approach developed in this study is illustrated by some numerical examples. These examples are solved using a special software package which implements the proposed algorithms.     

Decoupled Artificial Photosynthesis

Natural photosynthesis (NP) generates oxygen and carbohydrates from water and CO₂ utilizing solar energy to nourish lives and balance CO₂ levels. Following nature, artificial photosynthesis (AP), typically, overall water or CO₂ splitting, produces fuels and chemicals from renewable energy. However, hydrogen evolution or CO₂ reduction is inherently coupled with kinetically sluggish water oxidation, lowering efficiencies and raising safety concerns. Decoupled systems have thus emerged. In this review, we elaborate how decoupled artificial photosynthesis (DAP) evolves from NP and AP and unveil their distinct photoelectrochemical mechanisms in energy capture, transduction and conversion. Advances of AP and DAP are summarized in terms of photochemical (PC), photoelectrochemical (PEC), and photovoltaic-electrochemical (PV-EC) catalysis based on material and device design. The energy transduction process of DAP is emphasized. Challenges and perspectives on future researches are also presented.