Studies in fluorinated 1,3-diketones and related compounds Part X. Synthesis of some new polyfluorinated 1,3-diketones,their copper 1,3-diketonates and spectral studies

The reaction of tetrafluoro-1,2-diiodobenzene with tellurium yields an oil which, on treatment with bromine yields perfluorotelluranthrene- 9,9,10,10-tetrabromide. The latter is reduced by sodium sulfide to perfluorotelluranthrene. The crystal structure of perfluorotelluranthrene is reported.     

On some nonself mappings

Let X be a Banach space, let K be a non–empty closed subset of X and let T : K → X be a non–self mapping. The main result of this paper is that if T satisfies the contractive–type condition (1.1) below and maps ?K (?K the boundary of K) into K then T has a unique fixed point in K.     

The decay of Ho and Ho

The disintegration characteristics of ^164mHo and ^164gHo have been studied using a high-resolution Ge(Li) spectrometer. Radioactive samples of ^164mHo and ^164gHo were obtained through the (n, 2n) reaction on spectroscopically pure holmium oxide with 14 MeV neutrons. The isomeric cross-section ratio for the yields of the ground and metastable states has been estimated to be 1.0±0.25. Gamma rays of 37.7 and 56.1 keV energy have been assigned to the decay of ^164mHo and 73.4 and 91.5 keV γ-rays are attributed to the decay of ^164gHo by electron capture and negaton emission, respectively. The branching ratio for the electron capture decay of ^164gHo to the levels in ¹⁶⁴Dy has been estimated accurately from the analysis of X-ray intensities. The results have been incorporated into a decay scheme.     

Poisson-Jacobi transforms

The solutions of the Jacobi difference heat equation ▽ _n v(n, t)=(δ/δt)v (n, t) are considered. The theory of Poisson-Jacobi transform is developed to solve the equation. Convergence and inversion theorems are established.

     

Some problems and conjectures in number theory

In this article, we shall discuss some interesting, viable, meaningful, applicable and productive conjectures and methods to deal with some fundamental results in the theory of numbers.     

Geometrical solution of some cubic equations with non-real roots

A general cubic equation ax ³ + bx ² + cx + d = 0 where a, b, c, d ∈R, a ≠ 0 has three roots with two possibilities—either all three roots are real or one root is real and the remaining two roots are imaginary. Dealing with the second possibility this paper attempts to give the geometrical locations of the imaginary roots of the equation under three different sets of conditions. These sets of conditions include: (i) the real root of the given cubic equation is given, (ii) the real part of an imaginary root is given, and (iii) the imaginary part of an imaginary root is given.     

A short Proof of Bezout's Theorem in P2

We will give a new short proof of Bezout's Theorem for complex algebraic curves in P ² which is local.