Level sets of multiple ergodic averages

We propose to study multiple ergodic averages from multifractal analysis point of view. In some special cases in the symbolic dynamics, Hausdorff dimensions of the level sets of multiple ergodic average limit are determined by using Riesz products.Comment: This note was refused by Proceedings of AMS although the referee said "In my opinion this is a nice application of the Riesz product technique to solve, in principle, a hard problem when considered in its full generality. Nevertheless, I think it needs some extra work to see how this example seats in a more general context and explore how far this technique can go." We should say that Riesz product works perfectly in the situation described in this note, but Riesz product has its limit--we don't think that Riesz product technique can solve the problem in its generalit

On the frequency of partial quotients of regular continued fractions

We consider sets of real numbers in $[0,1)$ with prescribed frequencies of partial quotients in their regular continued fraction expansions. It is shown that the Hausdorff dimensions of these sets, always bounded from below by $1/2$, are given by a modified variational principle.Comment: Accepted by Mathematical Proceedings of the Cambridge Philosophical Societ

Parameters to characterize the internal recirculation of an oxidation ditch

Mixed liquor circulates ceaselessly in the closed-loop corridor in an oxidation ditch (OD), which is significantly different from other wastewater treatment processes. The internal recirculation ratio (IRR), i.e., the ratio between circulation flow rate (QCC) and influent flow rate (QIn), and the circulatory period (T), i.e. the time consumed for the mixed liquor to complete one lap in the circular corridor, was used to quantify the internal recirculation characteristics of the OD system. In order to elucidate the characteristics and  applicability of IRR and T, this study obtained the numerical relationship between IRR and T by formula derivation. It also discusses the factors influencing IRR and analyses the applications of IRR and T. The results showed that IRR = QCC/QIn = HRT/T = HRT • IRF (HRT = hydraulic retention time of the mixed liquor in the circular corridor; IRF = internal recirculation frequency). Moreover, three kinds of parameters had an effect on IRR: QIn; reactor dimensions, i.e., length (Lmid), width (B), and height (H) of the circular corridor; and horizontal velocity of the mixed liquor in the circular corridor (v). QIn changed IRR by altering HRT. However, B, H, Lmid, and v changed IRR by altering IRF and T. Furthermore, the same IRR corresponded to many different HRT and IRF. Therefore, when QIn and QCC varied in the OD system, using HRT and IRF to evaluate the variation of QIn and QCC, respectively, was better than using IRR to evaluate their synthetical variation. IRF and T were useful for directly and precisely characterizing the circulation speed and circulation flow rate in the circular corridor, while IRR was more useful for evaluating the dilution effect of reflux on influent

2-[(4-Bromo­phenyl­imino)­meth­yl]-4,6-di­iodo­phenol

The title compound, C13H8BrI2NO, was prepared by the reaction of 3,5-diiodo­salicyl­aldehyde with 4-bromo­phenyl­amine in ethanol. There is an intra­molecular O—H⋯N hydrogen bond in the mol­ecule, which generates an S(6) ring. The dihedral angle between the benzene rings is 2.6 (3)°

2-[(2-Chloro­phen­yl)imino­meth­yl]-4,6-di­iodo­phenol

The asymmetric unit of the title compound, C13H8ClI2NO, contains half of the mol­ecule situated on a mirror plane. The hy­droxy group is involved in the formation of an intra­molecular O—H⋯N hydrogen bond. π–π inter­actions between the benzene rings of neighbouring mol­ecules [centroid–centroid distance = 3.629 (3) Å] form stacks along the b axis. In the crystal, weak C—H⋯O and C—H⋯Cl inter­actions are observed