Arithmetic level raising on triple product of Shimura curves and Gross-Schoen diagonal cycles II: Bipartite Euler system

In this article, we study the Gross-Schoen diagonal cycle on the triple product of Shimura curves at a place of good reduction. We prove the unramified arithmetic level raising theorem for triple product of Shimura curves and we deduce from it the second reciprocity law which relates the image of the diagonal cycle under the Abel-Jacobi map to certain period integral of Gross-Kudla type. Along with the first reciprocity law we proved in a previous work, we show that the Gross-Schoen diagonal cycles form a Bipartite Euler system for the symmetric cube motive of modular forms. As an application we provide some evidence for the rank $1$ case of the Bloch-Kato conjecture for the symmetric cube motive of a modular form

The nonextensive parameter for the rotating astrophysical systems with power-law distributions

We study the nonextensive parameter for the rotating astrophysical systems with power-law distributions, including both the rotating self-gravitating system and the rotating space plasma. We extend the equation of nonextensive parameter to complex system with arbitrary force field, and derive a general equation of the q-parameter, most generally including both the rotating self-gravitating systems and the rotating space plasmas. At the same time, we reproduce the kappa-distribution in space plasmas and obtain equations of the kappa-parameter. We show that the q-parameter is related not only to the temperature gradient, the gravitational force and the electromagnetic force, but also to the inertial centrifugal force and Coriolis force. Thus the rotation introduces significant effect on nonextensivity in the systems. Several examples are given to illustrate the nonextensive effect introduced by the rotation.Comment: 11 pages, 56 reference

Exploring offence statistics in stockholm city using spatial analysis tools

ABSTRACT The objective of this paper is to investigate changes in offence patterns in Stockholm City using methods from spatial statistics. The paper has two parts. The first is a brief description of methodological procedures to obtain robust geographical units for spatial statistical analysis. The second part focuses on a discussion of the results of different types of spatial statistical analyses of offence patterns for Stockholm City. Standardised offence rates (SOR) are calculated and mapped using GIS for three offences: residential burglary, theft of and from cars and vandalism. The Getis-Ord statistic is used to identify crime clusters or hot spots and finally offence patterns are analysed as a function of socio-economic variables using the linear regression model. The findings of previous Swedish studies on crime patterns, mostly by Wikström (1991), and the insights provided by North American and British theories on crime patterns provide a background for this study. Results suggest that whilst there have been no dramatic changes in the geographies of these offences in Stockholm City during the last decade, there have been some shifts both in terms of geographical patterns and in their association with underlying socio-economic conditions.

Providing scientific visualisation for spatial data analysis: criteria and an assessment of SAGE

A consistent theme in recent work on developing exploratory spatial data analysis (ESDA) has been the importance attached to visualization techniques, particularly following the pioneering development of packages such as REGARD by Haslett et al (1990). The focus on visual techniques is often justified in two ways: (a) the power of modern graphical interfaces means that graphics is no longer a way of simply presenting results in the form of maps or graphs, but a tool for the extraction of information from data; (b)graphical, exploratory methods are felt to be more intuitive for non-specialists to use than methods of numerical spatial statistics enabling wider participation in the process of getting data insights. Despite the importance attached to visualisation techniques, very little work has been done to assess the effectiveness of techniques, either in the wider scientific visualisation community, or among those working with spatial data. This paper will describe a theoretical framework for developing visualisation tools for ESDA that incorporates a data model of what the analyst is looking for based on the concepts of "rough" and "smooth" elements of a data set and a theoretical scheme for assessing visual tools. The paper will include examples of appropriate tools and a commentary on the effectiveness of some existing packages