On the Order of Magnitude of the Double Fourier Transform

Mathematical Notes -     

Nowhere monotone functions and functions of nonmonotonic type

We investigate the relationships between the notions of a continuous function being monotone on no interval, monotone at no point, of monotonic type on no interval, and of monotonic type at no point. In particular, we characterize the set of all points at which a function that has one of the weaker properties fails to have one of the stronger properties. A theorem of Garg about level sets of continuous, nowhere monotone functions is strengthened by placing control on the location in the domain where the level sets are large. It is shown that every continuous function that is of monotonic type on no interval has large intersection with every function in some second category set in each of the spaces , and .

     

Functions not first-return integrable

Benedetto Bongiorno constructed a certain class of improperly Riemann integrable functions on [0,1] which are not first-return integrable. He asked if all improper Riemann integrable functions which are not Lebesgue integrable are not first-return integrable. Recently David Fremlin provided a clever example to show that this is not the case. It remains open as to which functions are first-return integrable. We prove two general theorems which imply the existence of a large class of improperly Riemann integrable functions which are not first-return integrable. As a corollary we obtain that there is an improperly Riemann integrable function which is C^∞ on (0,1] yet fails to be first-return integrable.     

Rotational analysis of 3664 Å band of NiCl

Nickel chloride was excited in a high frequency discharge source and the band at 3664 Å was photographed at an inverse dispersion of 0.59 Å/mm on a two metre plane grating spectrograph. The rotational analysis was carried out and the molecular constants of the upper state are reported. The rotational isotopic shifts due to³⁷Cl support the rotational analysis. The electronic transition involved is identified to beβ ²Δ_5/2 →X ₁ ²Δ_5/2.     

Pseudoarcs, pseudocircles, Lakes of Wada and generic maps on

We prove a Bruckner–Garg type theorem for the fiber structure of a generic map from a continuum X into the unit interval I. We also study the specific case of X=S². We show that each nondegenerate component of each fiber of a generic map in C(S²,I) is figure-eight-like. This together with a result by Krasinkiewicz and Levin gives that each nondegenerate component of each fiber of a generic map in C(S²,I) is hereditarily indecomposable and figure-eight-like. We also show that pseudoarcs, pseudocircles and Lakes of Wada appear in abundance in fibers of a generic map in C(S²,I). We also exhibit a general method for proving when a P-like hereditarily indecomposable continuum is Q-like when Q is a certain graph containing P.     

On weighted absolute convergence of multiple Fourier series of a function of $${(p_{1} ,\ldots,p_{N})}$$–$${(\Lambda^1 ,\ldots ,\Lambda^N)}$$(Λ1,…,ΛN)– BV

We obtain a sufficient condition for the weighted β-absolute convergence of the multiple Fourier series

$$\sum_{k_{1} \in\mathbb{Z}} \cdots \sum_{k_{N} \in\mathbb{Z}} \gamma_{k_{1} ,\ldots ,k_{N}} |\hat{f}(k_1 ,\ldots ,k_N)|^\beta,$$

where \({0 < \beta < 2}\) and \({\gamma_{k_{1} ,\ldots ,k_{N}}}\) is a weight sequence, of a function f of the class \({(\Lambda^1 ,\ldots ,\Lambda^N)BV^{(p_{1} ,\ldots ,p_{N})}([0, 2\pi]^N)}\).

     

Irrational Rotations and Differentiability

In this paper we investigate first return differentiation when the trajectories are restricted to irrational rigid rotations.     

Combined computational and experimental study of substituent effects on the thermodynamics of H(2), CO,arene, and alkane addition to iridium

The thermodynamics of small-molecule (H(2), arene, alkane, and CO) addition to pincer-ligated iridium complexes of several different configurations (three-coordinate d(8), four-coordinate d(8), and five-coordinate d(6)) have been investigated by computational and experimental means. The substituent para to the iridium (Y) has been varied in complexes containing the (Y-PCP)Ir unit (Y-PCP = eta(3)-1,3,5-C(6)H(2)[CH(2)PR(2)](2)Y; R = methyl for computations; R = tert-butyl for experiments); substituent effects have been studied for the addition of H(2), C-H, and CO to the complexes (Y-PCP)Ir, (Y-PCP)Ir(CO), and (Y-PCP)Ir(H)(2). Para substituents on arenes undergoing C-H bond addition to (PCP)Ir or to (PCP)Ir(CO) have also been varied computationally and experimentally. In general, increasing electron donation by the substituent Y in the 16-electron complexes, (Y-PCP)Ir(CO) or (Y-PCP)Ir(H)(2), disfavors addition of H-H or C-H bonds, in contradiction to the idea of such additions being oxidative. Addition of CO to the same 16-electron complexes is also disfavored by increased electron donation from Y. By contrast, addition of H-H and C-H bonds or CO to the three-coordinate parent species (Y-PCP)Ir is favored by increased electron donation. In general, the effects of varying Y are markedly similar for H(2), C-H, and CO addition. The trends can be fully rationalized in terms of simple molecular orbital interactions but not in terms of concepts related to oxidation, such as charge-transfer or electronegativity differences.