On the generalized Fréchet functional equation with constant coefficients and its stability

We study a generalization of the Fréchet functional equation, stemming from a characterization of inner product spaces. We show, in particular, that under some weak additional assumptions each solution of such an equation is additive. We also obtain a theorem on the Ulam type stability of the equation. In its proof we use a fixed point result to show the existence of an exact solution of the equation that is close to a given approximate solution.     

Natural extensions of holomorphic motions

We consider an arbitrary real analytic family X_z, , over the closed unit disc , of real analytic plane Jordan curves X_z. Ifj _e ^iθ ,e ^iθ ∋ ∂D, is an arbitrary real-analytic family of orientation-reversing homeomorphisms of fixingX _e ^iθ pointwise, we show that there is a unique holomorphic motion of extending the given motion of Jordan curves and consistent with the given family of involutions. If these generalized reflections are defined using the barycentric extension construction of Douady-Earle-Nag, then the resulting extension method for holomorphic motions of X is natural, that is Moebius-invariant and continuous with respect to variation of the given motion of X₀.     

Cardinality of the set of extreme extensions¶of a quasi-measure

Cardinals that arise as the number of extreme quasi-measure extensions of a quasi-measure [resp., measure] μ defined on an algebra [resp., σ-algebra] of sets to a larger algebra [resp., σ-algebra] of sets are characterized in the general case as well as under some natural assumptions on μ. Received: 19 September 2000     

Conformal geometry of foliations

In this article, we begin a systematic study of conformal properties of codimension-1 foliations. We first define and study local conformal invariants. A case of particular interest is that of harmonic foliations of the plane. Then we study existence of totally umbilical and “Dupin” foliations on compact 3-manifolds of constant curvature.      

Polynomial symplectomorphisms

Let be the field of real or complex numbers. Let (X ²ⁿ, )be a symplectic affine space. We study the group of polynomialsymplectomorphisms of X. We show that for an arbitrary k thegroup of polynomial symplectomorphisms acts k-transitively onX. Moreover, if 2 l 2n – 2 then elements of this groupcan be characterized by polynomial automorphisms which preservethe symplectic type of all algebraic l-dimensional subvarietiesof X.     

Syntheses and reactions of terpene β-hydroxyselenides and β-hydroxydiselenides

A convenient method for the synthesis of optically active trans-hydroxyselenides and trans-hydroxydiselenides from the bicyclic terpene group based on the reactions of sodium selenide or sodium diselenide with cis- and trans-(+)-3-carane, trans-(+)-2-carane and (?)-β-pinane epoxides is described. The corresponding cis-hydroxy and cis-methoxydiselenides were obtained in the reaction of sodium diselenide with β-hydroxy- and β-methoxytosylates. The influence of a hydroxy group at the β-position on the diastereomeric ratio of the products of the asymmetric methoxyselenenylation of styrene has been established by composition of the products, crystal structure analyses, and theoretical calculations using a DFT method on the B3LYP level (6-311G(d)).     

Significance of comproportionation reaction in multi-step electrochemical reduction of fullerene C60

Voltammetry at platinum ultramicroelectrode in 1,2-dichlorobenzene shows that for a high concentration of supporting electrolyte, THAPF₆, reduction waves 1, 2 and 3 of C₆₀ are of identical height. When supporting electrolyte concentration is lowered, the second and the third wave become lower than the first one. This feature does not agree with simple reduction mechanism under diffusion-migration transport, but supports the model, by Amatore and coworkers, that assumes very fast comproportionation reaction. According to this model, higher-charged fullerene anions are present only in the proximity of the electrode surface, because they react with lower-charged intermediates and are eventually converted to single-charged fullerene anions. Only single-charged anions are transported into the bulk of the solution.     

Immersions of spheres and algebraically constructible functions

Let Λ be an algebraic set and let (n is even) be a polynomial mapping such that for each there is r(λ) > 0 such that the mapping g _λ  =  g(· , λ) restricted to the sphere S ⁿ (r) is an immersion for every 0  <  r  <  r (λ), so that the intersection number I(g _λ|S ⁿ (r)) is defined. Then is an algebraically constructible function. I. Karolkiewicz and A. Nowel supported by the grant BW/5100-5-0286-7.     

Undirected simple connected graphs with minimum number of spanning trees

We show that for positive integers n, m with n(n−1)/2≥m≥n−1, the graph L_n,m having n vertices and m edges that consists of an (n−k)-clique and k−1 vertices of degree 1 has the fewest spanning trees among all connected graphs on n vertices and m edges. This proves Boesch’s conjecture [F.T. Boesch, A. Satyanarayana, C.L. Suffel, Least reliable networks and reliability domination, IEEE Trans. Commun. 38 (1990) 2004-2009].