Search for Tracker Potentials in Quintessence Theory

We report a significant finding in Quintessence theory that the the scalar fields with tracker potentials have a model-independent scaling behaviour in the expanding universe. So far widely discussed exponential,power law or hyperbolic potentials can simply mimic the tracking behaviour over a limited range of redshift. In the small redshift range where the variation of the tracking parameter $\epsilon$ may be taken to be negligible, the differential equation of generic potentials leads to hyperbolic sine and hyperbolic cosine potentials which may approximate tracker field in the present day universe. We have plotted the variation of tracker potential and the equation of state of the tracker field as function of the redshift $z$ for the model-independent relation derived from tracker field theory; we have also plotted the variation of $V(\Phi)$ in terms of the scalar field $\Phi$ for the chosen hyperbolic cosine function and have compared with the curves obtained by reconstruction of $V(\phi)$ from the real observational data from the supernovae.Comment: 11 pages, 3 figures, late

"Rare" Fluctuation Effects in the Anderson Model of Localization

We discuss the role of rare fluctuation effects in quantum condensed matter systems. In particular, we present recent numerical results of the effect of resonant states in Anderson's original model of electron localization. We find that such resonances give rise to anomalous behavior of eigenstates not just far in the Lifshitz tail, but rather for a substantial fraction of eigenstates, especially for intermediate disorder. The anomalous behavior includes non-analyticity in various properties as a characteristic. The effect of dimensionality on the singularity, which is present in all dimensions, is described, and the behavior for bounded and unbounded disorder is contrasted

Singular Behavior of Eigenstates in Anderson's Model of Localization

We observe a singularity in the electronic properties of the Anderson Model of Localization with bounded diagonal disorder, which is clearly distinct from the well-established mobility edge (localization-delocalization transition) that occurs in dimensions $d>2$. We present results of numerical calculations for Anderson's original (box) distribution of onsite disorder in dimensions $d$ = 1, 2 and 3. To establish this hitherto unreported behavior, and to understand its evolution with disorder, we contrast the behavior of two different measures of the localization length of the electronic wavefunctions - the averaged inverse participation ratio and the Lyapunov exponent. Our data suggest that Anderson's model exhibits richer behavior than has been established so far.Comment: Correction to v1: Fig.3 caption now displaye

Singular Behavior of Anderson Localized Wavefunctions for a Two-Site Model

We show analytically that the apparent non-analyticity discovered recently in the inverse participation ratio (IPR) of the eigenstates in Anderson's model of localization is also present in a simple two-site model, along with a concurrent non-analyticity in the density of states (DOS) at the same energy. We demonstrate its evolution from two sites to the thermodynamic limit by numerical methods. For the two site model, non-analyticity in higher derivatives of the DOS and IPR is also proven to exist for all bounded distributions of disorder

Formulation of Mathematical Mode! of Picketing of Liquor Shops and Warehouses

Before getting independence of INDIA from British regime, large number of Leaders of India was required to take strong agitation against British Government for getting freedom. One of the prominent leaders was Mr. M. K. Gandhi. During the period 1920 to 1942 in Central Provinces and Berar specifically pertaining to the period June 1930 to September 1930 [1], strong agitations took place towards reducing income to Government by way of reducing liquor consumption. Several events took place towards this objective. Based on the facts, the attempt is made in this paper to present the entire agitation as one social phenomena in the form of a Mathematical Model co-relating the fall in liquor revenue in terms of various causes responsible for this fall in revenue. It is only through the Mathematical Model that it is possible to get quantitative idea of intensity of interaction of causes on effects of any phenomena may be it be scientific or socio-economic or of any other type. Particularly the approach of Field Data Based Model [2] is applicable in such a situation as this is a Field Phenomena. Such models serve as most reliable tools to plan future such activities. This could be known as a process of PROGNOSIS