Principal part of resolvent and factorization of an increasing nonanalytic operator-function

Let a selfadjoint operator-valued functionL() be given on the interval [a,b] such thatL(a)0,L(b)0,L()0 (ab), andL() has a certain smoothness (for instance, it satisfies Hölder's condition). It turns out that the spectral theory of the operator-valued functionL() can be reduced to the spectral theory of one operatorZ, the spectrum of which lies on (a, b) and which is similar to a selfadjoint operator. In particular, the factorization takes place:L()=M()(I–Z), where the operator-valued functionM() is invertible on [a, b]. Earlier similar results were known only for analytic operator-valued functions. The authors had to use new methods for the proof of the described theorem. The key moment is the decomposition ofL ^–1() into the sume of its principal and regular parts.     

New necessary conditions on the existence of abelian difference sets

In this paper we develop a new method to obtain identities in a group algebraGF(p)G if an abelian difference set of ordern0 (modp) exists inG. We give an explicit formula ifp ² orp ³ is the exactp-power dividingn. This generalizes the approach of Wilbrink, Arasu and the author. The proof presented here uses some knowledge about field extensions of thep-adic numbers.     

On abelian difference set codes

In this paper we determine the ranks of the incidence matrices that belong to the following types of difference sets: Twin prime power difference sets, biquadratic residues and biquadratic residues with 0. We also prove a conjecture of Assmus and Key on the code generated by the hyperovals of PG(2, q).     

Optimally scaled matrices,necessary and sufficient conditions

Summary We shall in this paper consider the problem of determination a row or column scaling of a matrixA, which minimizes the condition number ofA. This problem was studied by several authors. For the cases of the maximum norm and of the sum norm the scale problem was completely solved by Bauer [1] and Sluis [5]. The condition ofA subordinate to the pair of euclidean norms is the ratio /, where and are the maximal and minimal eigenvalue of (A ^H A)^1/2 respectively. The euclidean case was considered by Forsythe and Strauss [3]. Shapiro [6] proposed some approaches to a numerical solution in this case. The main result of this paper is the presentation of necessary and sufficient conditions for optimal scaling in terms of maximizing and minimizing vectors. A uniqueness proof for the solution is offered provided some normality assumption is satisfied.     

A Theoretical Study of Complex Formation between Formaldehyde and Lithium

Ab initio SCF and CI calculations on the cationic and neutral complexes of formaldehyde and lithium are reported. For the cationic complex CH₂O/Li⁺, the stabilization energy of 41.7 kcal/mol obtained from the SCF calculation increases to 51.6 kcal/mol if a configuration interaction is introduced. For the neutral complex CH₂O^?/Li⁺, the C_2v-conformer of the ²A₁-state with the equilibrium bond distances of d(C? O) = 1.23 Å and d (O? Li) = 1.90 Å is calculated to be more stable than the ₂B₁-state with d (C? O) = 1.34 Å, and d (O? Li) = 1.65 Å. Charge transfer and polarization effects upon complex formation are discussed.     

NMR studies of interionic interactions in N-methylformamide

The NMR chemical shifts of alkali and thallium(I) salts with various monovalent anions have been measured in N-methylformamide solution. Lithium-7 chemical shifts are virtually concentration and counter-ion independent, presumably due to an absence of direct cation-anion interactions. The sodium-23, potassium-39 and cesium-133 chemical shifts of the salts studied depend on the anion and vary linearly with the concentration. The observed behavior can be accounted for by the formation of collisional ion pairs. On the other hand, the thallium-205 chemical shifts of thallium(I) nitrate and perchlorate were anion-dependent and varied non-linearly with the salt concentration. These results are indicative of contact ion pair formation; formation constants were calculated to be 2.6±0.4 M ^–1 for TlNO ₃ and 1.7±0.5 M ^–1 for TlClO ₄ . The cesium-133 NMR spectra of several mixed electrolyte systems also have been measured in N-methylformamide solution. The¹³³Cs chemical shifts also change linearly with the concentrations of the salts added to 0.10 M CsI/NMF solutions. The influence of the anions on the chemical shifts is the same as that observed for cesium salts alone.