The effects of a strong quantizing magnetic field on the nucleation of quark matter droplets and on the chemical evolution of a nascent quark phase at the core of a magnetar are investigated. The surface energy of the quark phase diverges logarithmically. As a consequence there cannot be a first order transition to the quark phase. However, a metal-insulator type of second order transition is possible unless the field strength exceeds 10 20 G. The study of the chemical evolution of a newborn quark phase shows that in ␤ equilibrium the system becomes energetically unstable.The theoretical investigation of the effect of a ultrastrong magnetic field on stellar matter in nuclear astrophysics has gotten a new dimension after the discovery of a few magnetars. The observed soft gamma repeaters discovered by BATSE ͓1͔ and KONUS ͓2͔ ͑see also ͓3͔͒ experiments and x-ray sources observed by ASCA, RXTE, and BappoSAX ͓4͔ show the presence of a strong surface magnetic field up to 10 15 G. The discovery of these objects poses a great challenge to the existing models of magnetic field evolution, since they require a very rapid field decay in isolated neutron stars ͓5͔. To investigate the global properties of these strange objects also requires a detailed investigation of stability of dense stellar matter in the presence of ultrastrong magnetic field and knowledge of the exact equation of state of such strongly magnetized matter.The dynamo mechanism, recently proposed by Thompson and Duncan ͓6͔, suggests that the dipole magnetic field of a young neutron star can reach up to 10 15 G. It is generally expected that the internal magnetic field is a few orders of magnitude stronger than surface field strength. Since the strength of internal magnetic field of a neutron star strongly depends on the nature of dense stellar matter present at the core region, it may not necessarily be reflected in its surface magnetic field. However, there is an upper limit of internal magnetic field strength constrained by the scalar virial theorem, which gives B m,max ϳ2ϫ10 8 (M /M ᭪ )(R/R ᭪ ) Ϫ2 G ͓7,8͔. For a typical neutron star of radius Rϭ10 km and mass M ϭM ᭪ , this upper limit is ϳ10 18 G. Beyond this limit, the ultramagnetized neutron stars become unstable. Now there is also a strong belief that a transition to quark phase occurs at the core of a neutron star if the density exceeds a few times normal nuclear density. The transition could be a first order type, with the nucleation of stable quark matter droplets in metastable hadronic matter by fluctuation at the core region. The transition could also be a second order type. Since hadrons ͑nucleons and hyperons͒ do not carry color quantum number, we may call the hadronic matter a color insulator, whereas, in this regard, the quark phase is a color conductor. Therefore, such a second order structural phase transition at high density is analogous to the metal-insulator transition in condensed matter physics, which takes place under high pressure.In this article our aim is to show that a first order ...

In this work, we present sufficient conditions for oscillation of all solutions of a second-order functional differential equation. We consider two special cases when $\gamma >\beta $ γ > β and $\gamma <\beta $ γ < β . This new theorem complements and improves a number of results reported in the literature. Finally, we provide examples illustrating our results and state an open problem.

In this work, we present some new sufficient conditions for the oscillation of a class of second-order neutral delay differential equation. Our oscillation results, complement, simplify and improve recent results on oscillation theory of this type of non-linear neutral differential equations that appear in the literature. An example is provided to illustrate the value of the main results.

The chemical evolution of nascent quark matter core in a newborn compact neutron star is studied in presence of a strong magnetic field.The effective rate of strange quark production in degenerate quark matter core in presence of strong magnetic fields is obtained. The investigations show that in presence of strong magnetic fields a quark matter core becomes energetically unstable and hence a deconfinement transition to quark matter at the centre of a compact neutron star *

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