On the maximum modulus of complete trigonometric sums

LetQ _k (p) be a set consisting of all polynomials of degreek with integral coefficientsf(x)=a _k x ^k +...+a ₁ x, wherep×a _k . For givenk andp any polynomialf _k,p (x)εQ _k (p) satisfying ‖S(p, f _k,p )‖=sup ‖S(p, f)‖fεQ(p) is called a maximum modular polynomial inQ _k (p), where $$S(p,f) = \sum\limits_{x = 0}^{p - 1} {e^{2\pi if(x)/p} } $$ Moreover, we definec(k, p)=‖S(p, f _k.p (x))‖. The main results are the following theorems.

1. For k=p?1 and p≥3 we have $$c(k,p) = \sqrt {p^2 - 4(p - 1)\sin ^2 \frac{\pi }{p}} $$ Besides, we may take \(f_{k,p} (x) = \prod\limits_{r = 0}^{p - 2} {(x - r)} \)
2. For k=p?s, 2≤s≤(p+1)/2 and p≥5, we have $$c(k,p) \leqslant p - 4(s - 1)\sin ^2 \frac{\pi }{p}$$ .

In Theorems 3 and 4, an interesting connextion between the present question and the famous problem of Prouhet and Tarry is given, some conditions under which the sign of equality in Theorem 2 holds are given and a method used to construct a maximum modular polynomial inQ _k (p) is also given.     

Equivalence of functional-differential equations of neutral type and abstract Cauchy problems

Equivalence of functional-differential equations of neutral type to well-posed abstract Cauchy problems in the state spaces ⁿ }L ^p ,L ^p resp.C is investigated. The assumptions allow non-atomic difference operators.This work was supported by the Fonds zur Förderung der wissenschaftlichen Forschung (Austria) under Grant No. P4534.Work done by this author was done during a stay at the University of Graz from October 1981 till April 1984.     

Low temperature Mössbauer spectra and magnetic ordering of lithiated ferric molybdate: Li2Fe2(MoO4)3

Powder x-ray diffraction, variable temperature magnetic susceptibility, and zero field Mössbauer spectroscopy measurements were used to characterize the new phase Li₂Fe₂ (MoO₄)₃. This material is obtained topochemically simply by the mixing of solutions of lithium iodide in acetonitrile with solid Fe₂(MoO₄)₃ at ambient temperature. Li₂Fe₂(MoO₄)₃ possesses the high temperature orthorhombic ferric molybdate $$\begin{gathered} Fe_2 (MoO_4 )_3 + 2LiI\xrightarrow{{CH_3 CN}}Li_2 Fe_2 (MoO_4 )_3 + I_2 \hfill \\ (solid)(solution)(solid)(solution) \hfill \\ \end{gathered}$$ structure. Guinier photographs were completely indexed in space group Pnca. Magnetic hyperfine splitting of the zero field Mössbauer spectrum below 12.5 K indicates a three-dimensional magnetically ordered state which susceptibility results show to be weakly ferromagnetic owing to probable canting of antiferromagnetically coupled sublattices.     

On the low temperature magnetic properties of some sulfites of divalent iron

Zero field Mössbauer spectra and powder susceptibility measurements show that the condensed, hydrogen bonded network compound FeSO₃ 3H₂O orders antiferromagnetically (T_Néel 9.5 K). The related anhydrous material FeSO₃ also shows antiferromagnetic exchange but no evidence of long range three-dimenstional magnetic order to as low as 1.4 K.Supported by the U.S. National Science Foundation Division of Materials Research—Solid State Chemistry Program Grant No. DMR 8313710.     

Pore Structure Analysis of Porous Particles by Equivalent Model

Starting from Wheeler's composite theory, which takes both the multilayer adsorption and the capillary condensation into consideration, and assuming the equivalent cylindrical model, we have deduced a strict and relatively simpler formula for the calculation of pore size distribution by solving the integral equation directly. The formula is more convenient for application because of its recursion form. Following the same line of reasoning and using the mean value theorem of definite integrals under different conditions, we have unified the equations of Barrett-Joyner-Halenda, Dubinin, Cranston-Inkley, Dollimore-Heal and Roberts. A table of parameters which is required in the calculation is given. Besides the adsorption of nitrogen, the adsorption with water vapor as the adsorbate is investigated and the corresponding empirical equation of adsorption thickness is given.