A noteworthy improvement of the 3-diazo-4-oxo-3,4-dihydroquinoline photosynthesis of indole-3-carboxamides

The use of benzene, in place of methylene chloride as solvent, allows the synthesis of amides of indole-3-carboxylic acid via the photolysis of 3-diazo-4-oxo-3,4-dihydroquinoline to occur in significantly higher yield and requiring shorter reaction times. This modification makes unnecessary the time consuming and costly chromatographic purification step, and allows the synthesis of novel amides which could not he isolated using the previous method.     

Syntheses of some azolopyridopyrimidines

Syntheses of isomeric azolopyridopyrimidines are described. As starting material the corresponding pyridopyrimidines were used. It could be established that in many cases a transformation of a functional group with the purpose to form an annelated five-membered ring proceeded with ring opening at the pyrimidine part. Subsequent ring closure with one carbon synthons gave then the desired tricyclic heterocycles.     

Shortness exponents of families of graphs

Known estimates of the maximal length of simple circuits in certain 3-connected planar graphs are surveyed and improved in several directions.     

Cauchy’s residue theorem for a class of real valued functions

Let [a, b] be an interval in ℝ Rand let F be a real valued function defined at the endpoints of [a, b] and with a certain number of discontinuities within [a, b]. Assuming F to be differentiable on a set [a, b] | E to the derivative f, where E is a subset of [a, b] at whose points F can take values ±∞ or not be defined at all, we adopt the convention that F and f are equal to 0 at all points of E and show that KH-vt ∝ _a ^b f = F(b) − F(a), where KH-vt denotes the total value of the Kurzweil-Henstock integral. The paper ends with a few examples that illustrate the theory.     

Stability of Roots of Polynomials Under Linear Combinations of Derivatives

Let T=α ₀ I+α ₁ D+⋅⋅⋅+α _n D ⁿ , where D is the differentiation operator and a₀ 1 0\alpha_{0}\not=0 , and let f be a square-free polynomial with large minimum root separation. We prove that the roots of Tf are close to the roots of f translated by −α ₁/α ₀.