Isometric multipliers of a vector valued Beurling algebra on a discrete semigroup

Let (S,ω) be a weighted abelian semigroup, let M _ω (S) be the semigroup of ω-bounded multipliers of S, and let \(\mathcal {A}\) be a strictly convex commutative Banach algebra with identity. It is shown that T is an onto isometric multiplier of \(\ell ^{1}(S,\omega , \mathcal {A})\) if and only if there exists an invertible σ ∈ M _ω (S), a unitary point \(a \in \mathcal {A}\), and a k>0 such that \(T(f)= ka{\sum }_{x \in S} f(x)\delta _{\sigma (x)}\) for each \(f={\sum }_{x \in S}f(x)\delta _{x} \in \ell ^{1}(S,\omega ,\mathcal {A})\). It is also shown that an isomorphism from \(\ell ^{1}(S_{1},\omega _{1},\mathcal {A})\) onto \(\ell ^{1}(S_{2},\omega _{2}, \mathcal {B})\) induces an isomorphism from \(M(\ell ^{1}(S_{1},\omega _{1},\mathcal {A}))\), the set of all multipliers of \(\ell ^{1}(S_{1},\omega _{1},\mathcal {A})\), onto \(M(\ell ^{1}(S_{2},\omega _{2},\mathcal {B}))\).     

Approximation of quadrilaterals by rational quadrilaterals in the plane

Many questions about triangles and quadrilaterals with rational sides, diagonals and areas can be reduced to solving certain diophantine equations. We look at a number of such questions including the question of approximating arbitrary triangles and quadrilaterals by those with rational sides, diagonals and areas. We transform these problems into questions on the existence of infinitely many rational solutions on a two parameter family of quartic curves. This is further transformed to a two parameter family of elliptic curves to deduce our main result concerning density of points on a line which are at a rational distance from three collinear points (Theorem 4). We deduce from this a new proof of density of rational quadrilaterals in the space of all quadrilaterals (Theorem 39). The other main result (Theorem 3) of this article is on the density of rational triangles which is related to analyzing rational points on the unit circle. Interestingly, this enables us to deduce that parallelograms with rational sides and area are dense in the class of all parallelograms. We also give a criterion for density of certain sets in topological spaces using local product structure and prove the density Theorem 6 in the appendix section. An application of this proves the density of rational points as stated in Theorem 31.     

Synthesis of phenanthro[a] phenazine derivatives: a novel ring structure forming discotic liquid crystals

Phenanthro[a] phenazine, a new heterocyclic ring structure of both biological and physical interest, has been prepared by coupling 1,2-phenylenediamine with triphenylene-1,2-diquinone derivatives in high yields. 2,3,6,7,10-Penta-alkoxyphenanthro[a] phenazines are mesogenic over a broad temperature range. These novel unsymmetrical, coloured, polar, heteroaromatic discotic mesogens are potential candidates for the study of conducting, photoconducting, electroluminescence, photorefrective, etc. properties.     

Generation of tunable 16 μm radiation from CO2 by cascade lasing

In this paper we propose a scheme to generate tunable 16 μm radiation from CO₂ molecules by cascade lasing. The stimulating 9.5 μm radiation is generated internally by the fast rotating mirror Q-switching technique. The optical scheme proposed by us uses an intracavity prism to separate the 9.5 μm and the 16 μm beams. This facilitates independent tuning of the two beams if required. In the present configuration, only the 16 μm cavity is dispersive. The 9.5 μm beam grows spontaneously in a stable semiconfocal resonator. We have developed a theoretical model to simulate the proposed scheme. The model predicts the energy and power of 16 μm radiation. The calculated values are much higher than the previously obtained experimental values. The results point out the feasibility of developing a laser system based on the theoretical design parameters presented in this paper. Such laser systems can find application in uranium isotope separation studies.     

Precision two-photon spectroscopy of alkali elements

In this paper, we have briefly reviewed the work on two-photon spectroscopy of alkali elements and its applications. The technique of Doppler-free two-photon spectroscopy is briefly summarized. A review of various techniques adopted for measuring absolute frequencies of the atomic transitions and precision measurements of isotope shifts and hyperfine structures (HFS) is presented. Some of the recent works on precision measurements of HFS constants of 6s ² S _1/2 level of ³⁹K and ⁴¹K, 9s ² S _1/2level and 7d ² D _3/2 level of ¹³³Cs are also discussed.     

Higher-Order $(F,\alpha, \beta, \rho, d, E) $-Convexity in Fractional Programming

In this paper we define higher order $(F,\alpha, \beta, \rho,d, E)$-convex function with respect to $E$-differentiable function $K$ and obtain optimality conditions for nonlinear programming problem (NP) from the concept of higher order $(F,\alpha, \beta, \rho,d)$-convexity. Here, we establish Mond-Weir and Wolfe duality for (NP) and utilize these duality in nonlinear fractional programming problem.     

Main injector particle production experiment at Fermilab

The main injector particle production (MIPP) experiment at Fermilab uses particle beams of charged pions, kaons, proton and antiproton with beam momenta of 5?C90 GeV/c to measure particle production cross-sections of various nuclei including liquid hydrogen, MINOS target and thin targets of beryllium, carbon, bismuth and uranium. The physics motivation to perform such cross-section measurements is described here. Recent results on the analysis of NuMI target and forward neutron cross-sections are presented here. Preliminary cross-section measurements for 58 GeV/c proton on liquid hydrogen target are also presented. A new method is described to correct for low multiplicity inefficiencies in the trigger using KNO scaling.     

Effect of isospin-dependent cross-section on fragment production in the collision of charge asymmetric nuclei

To understand the role of isospin effects on fragmentation due to the collisions of charge asymmetric nuclei, we have performed a complete systematical study using isospin-dependent quantum molecular dynamics model. Here simulations have been carried out for $^{124}{\rm X}_{n}+^{124}{\rm X}_{n}$ , where n varies from 47 to 59 and for $^{40}{\rm Y}_{m}+^{40}{\rm Y}_{m}$ , where m varies from 14 to 23. Our study shows that isospin-dependent cross-section shows its influence on fragmentation in the collision of neutron-rich nuclei.