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.     

Selective Determination of Total Capsaicinoids in Plant Material Using Poly(Gallic Acid)‐modified Electrode

Novel voltammetric approach for the selective determination of total capsaicinoids has been developed using glassy carbon electrode modified with multi‐walled carbon nanotubes and poly(gallic acid) (PGA/MWNT/GCE). The modified electrode provides significant improvements in the capsaicinoids voltammetric characteristics in comparison to GCE and MWNT/GCE. The electrooxidation of capsaicinoids is irreversible adsorption‐controlled process with the anodic transfer coefficient of 0.49–0.53 and heterogeneous electron transfer rate constant of 1300–2400 s^?1. The analytical ranges of 0.010–1.0 and 1.0–50 μM for capsaicin, 0.025–0.75 and 0.75–75 μM for dihydrocapsaicin and 0.025–5.0 and 5.0–75 μM for nonivamide with the detection limits of 2.9, 5.9 and 6.1 nM, respectively, have been obtained using differential pulse voltammetry (DPV). The selectivity of the capsaicinoids quantification in the presence of ascorbic acid, α‐tocopherol and carotenoids is shown. The method has been tested on the samples of red hot pepper spices and Capsicum annuum L. tinctures. The results correspond to the chromatographic data.     

Goldstone mode of a magnon Bose−Einstein condensate in MnCO<Subscript>3</Subscript>

The dependence of the enhancement of the Raman scattering on the size of a dielectric column is measured in structures with the spatial modulation of the height and lateral sizes of the dielectric coated with a thick metal layer (10–80 nm). It is established that, in the case of a thick metal coating (silver, gold, and copper coatings are used) at dimensions of the dielectric column close to the laser pump wavelength, considerable enhancement of the Raman signal oscillating upon the variation of the geometrical dimensions of the structure is observed. It is shown that the observed resonance enhancement of the Raman signal is associated with the transformation of the electromagnetic radiation into localized plasmon–polariton modes, and the efficiency of such transformation is determined by the commensurability of the wavelength of the plasmon–polariton mode and the planar size of the metal film. For different metal coatings, the dependence of the enhancement of the Raman scattering on the laser wavelength is measured.     

Nuclear quadrupole resonance in organosilicon nitrogen compounds

Conclusions Based on the¹⁴N NQR data, the p – d interaction in dimethylaminosilanes is weaker than in the amides of phosphorus acids.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 10, pp. 2331–2333, October, 1975.     

Voltammetric Determination of Hesperidin on the Electrode Modified with SnO2 Nanoparticles and Surfactants

Highly sensitive voltammetric method for hesperidin determination is developed using glassy carbon electrode modified with SnO₂ nanoparticles (SnO₂ NPs) and surfactants. The highest oxidation currents of hesperidin are observed in the case of cationic 0.50 mM cetylpyridinium bromide (CPB). Parameters of hesperidin electrooxidation have been calculated. Adsorptive anodic differential pulse voltammetry (AdADPV) in phosphate buffer pH 7.0 after 120 s accumulation provides linear dynamic ranges of 0.10–10 and 10–75 μM with the detection limit of 0.077 μM. The selectivity of hesperidin response is proved in the presence of inorganic ions, saccharides, ascorbic acid and other natural phenolics.