Cosmic rays from remnants of quasars?

Considerations of the collision losses for protons traversing the 2.7 K black body microwave radiation field have led to the conclusion that the highest energy cosmic rays, those observed at $\geq 10^{20}$ eV, must come from sources within the present epoch. In light of this constraint, it is here suggested that these particles may be accelerated near the event horizons of spinning supermassive black holes associated with presently inactive quasar remnants. The required emf is generated by the black hole induced rotation of externally supplied magnetic field lines threading the horizon. Producing the observed flux of the highest energy cosmic rays would constitute a negligible drain on the black hole dynamo. Observations with upcoming air shower arrays and space missions may lead to the identification of candidate dormant galaxies which harbor such black holes. Although the highest energy events observed so far are accounted for within the context of this scenario, a spectral upper bound at $\sim 10^{21}$ eV is expected since the acceleration to higher energies appears to be precluded, on general grounds.Comment: Mon. Not. Roy. Astr. Soc., in press. 5 pages, 1 tabl

On homogeneous skewness of unimodal distributions

We introduce a new concept of skewness for unimodal continuous distributions which is built on the asymmetry of the density function around its mode. The asymmetry is captured through a skewness function. We call a distribution homogeneously skewed if this skewness function is consistently positive or negative throughout its domain, and partially homogeneously skewed if the skewness function changes its sign at most once. This type of skewness is shown to exist in many popular continuous distributions such as Triangular, Gamma, Beta, Lognormal and Weibull. Two alternative ways of partial ordering among the partially homogeneously skewed distributions are described. Extensions of the notion to broader classes of distributions including discrete distributions have also been discussed

Modal and Relevance Logics for Qualitative Spatial Reasoning

Qualitative Spatial Reasoning (QSR) is an alternative technique to represent spatial relations without using numbers. Regions and their relationships are used as qualitative terms. Mostly peer qualitative spatial reasonings has two aspect: (a) the first aspect is based on inclusion and it focuses on the ”part-of” relationship. This aspect is mathematically covered by mereology. (b) the second aspect focuses on topological nature, i.e., whether they are in ”contact” without having a common part. Mereotopology is a mathematical theory that covers these two aspects. The theoretical aspect of this thesis is to use classical propositional logic with non-classical relevance logic to obtain a logic capable of reasoning about Boolean algebras i.e., the mereological aspect of QSR. Then, we extended the logic further by adding modal logic operators in order to reason about topological contact i.e., the topological aspect of QSR. Thus, we name this logic Modal Relevance Logic (MRL). We have provided a natural deduction system for this logic by defining inference rules for the operators and constants used in our (MRL) logic and shown that our system is correct. Furthermore, we have used the functional programming language and interactive theorem prover Coq to implement the definitions and natural deduction rules in order to provide an interactive system for reasoning in the logic

Design of adaptive multi-arm multi-stage clinical trials

Two-arm group sequential designs have been widely used for over forty years, especially for studies with mortality endpoints. The natural generalization of such designs to trials with multiple treatment arms and a common control (MAMS designs) has, however, been implemented rarely. While the statistical methodology for this extension is clear, the main limitation has been an efficient way to perform the computations. Past efforts were hampered by algorithms that were computationally explosive. With the increasing interest in adaptive designs, platform designs, and other innovative designs that involve multiple comparisons over multiple stages, the importance of MAMS designs is growing rapidly. This dissertation proposes a group sequential approach to design MAMS trial where the test statistic is the maximum of the cumulative score statistics for each pair-wise comparison, and is evaluated at each analysis time point with respect to efficacy and futility stopping boundaries while maintaining strong control of the family wise error rate (FWER). In this dissertation we start with a break-through algorithm that will enable us to compute MAMS boundaries rapidly. This algorithm will make MAMS design a practical reality. For designs with efficacy-only boundaries, the computational effort increases linearly with number of arms and number of stages. For designs with both efficacy and futility boundaries the computational effort doubles with successive increases in number of stages. Previous attempts to obtain MAMS boundaries were confined to smaller problems because their computational effort grew exponentially with number of arms and number of stages. We will next extend our proposed group sequential MAMS design to permit adaptive changes such as dropping treatment arms and increasing the sample size at each interim analysis time point. In order to control the FWER in the presence of these adaptations the early stopping boundaries must be re-computed by invoking the conditional error rate principle and the closed testing principle. This adaptive MAMS design is immensely useful in phase~2 and phase~3 settings. An alternative to the group sequential approach for MAMS design is the p-value combination approach. This approach has been in place for the last fifteen years.This alternative MAMS approach is based on combining independent p-values from the incremental data of each stage. Strong control of the FWER for this alternative approach is achieved by closed testing. We will compare the operating characteristics of the two approaches both analytically and empirically via simulation. In this dissertation we will demonstrate that the MAMS group sequential approach dominates the traditional p-value combination approach in terms of statistical power

Dynamical effects of an extended cloud of Dark Matter on dwarf spheroidals

If the dwarf spheroidals are embedded in an extended cloud of dark matter then their density profiles can be reproduced by assuming a Maxwellian distribution of velocities for the constituent stars. The observed luminosity profiles of dwarf spheroidals imply densities for the dark matter in the range 10-26 to 10-25 g cm-3, and mass-to-luminosity ratios which are typically an order of magnitude greater than those of globular clusters. Neutrinos of mass ~10 eV and <v> ~1000 km s-1 can provide this requisite density for the background

Towards a physical model for galactic rotation curves

Extensive and meticulous observations of the rotation curves of galaxies show that they are either flat or gently going up, but rarely decreasing, at large galactocentric distances. Here we show that the gravitational potential which would lead to such rotation curves arises naturally when the visible matter modelled as a collisionless Maxwellian gas is embedded in a dark halo of collisionless particles with a much higher dispersion in velocities