Abstract: We are interested in solving Liouville-type problems to explore constancy properties for maps or differential forms on Riemannian manifolds. Geometric structures on manifolds, the existence of constancy properties for maps or differential forms, and energy growth for maps or differential forms are intertwined. In this article, we concentrate on discovery of solutions to Liouville-type problems where manifolds are Euclidean spaces (i.e. flat Riemannian manifolds) and maps become real-valued functions. Liouville-type results of vanishing properties for functions are obtained. The original work in our research findings is to extend the q-energy for a function from finite in Lq space to infinite in non-Lq space by applying p-balanced technique where q = p = 2. Calculation skills such as Hölder's Inequality and Tests for Series have been used to evaluate limits and integrations for function energy. Calculation ideas and computational techniques for solving Liouville-type problems shown in this article, which are utilized in Euclidean spaces, can be universalized as a successful algorithm, which works for both maps and differential forms on Riemannian manifolds. This innovative algorithm has a far-reaching impact on research work of solving Liouville-type problems in the general settings involved with infinite energy. The p-balanced technique in this algorithm provides a clue to success on the road of q-energy extension from finite to infinite.
Abstract: This exploratory study gives an overview of the
evolution of the main financial and performance indicators of the
Academic Spin-Off’s and High Growth Academic Spin-Off’s in year
3 and year 6 after its creation in the region of Catalonia in Spain. The
study compares and evaluates results of these different measures of
performance and the degree of success of these companies for each
University.
We found that the average Catalonian Academic Spin-Off is small
and have not achieved the sustainability stage at year 6. On the
contrary, a small group of High Growth Academic Spin-Off’s
exhibits robust performance with high profits in year 6. Our results
support the need to increase selectivity and support for these
companies especially near year 3, because are the ones that will bring
wealth and employment. University role as an investor has rigid
norms and habits that impede an efficient economic return from their
ASO investment.
Universities with high performance on sales and employment in
year 3 not always could sustain this growth in year 6 because their
ASO’s are not profitable. On the contrary, profitable ASO exhibit
superior performance in all measurement indicators in year 6. We
advocate the need of a balanced growth (with profits) as a way to
obtain subsequent continuous growth.
Abstract: Disparity in India has been persisting since independence causing many socioeconomic problems and its removal has become the most prime objective of the planned development in India. Hence the paper attempts to study the disparity at State and Regional level and gives inclusive planning guidelines to achieve balanced regional development. At State level, the relative socioeconomic backwardness of Vidarbha Region based on Interregional analysis using selected indicators like Foreign Direct Investment, Human Development Index, Per Capita District Domestic Product has been assessed and broad guidelines have been proposed. In the later part at Regional level, the relative backwardness of districts based on Intraregional analysis using socioeconomic indicators has been assessed within Nagpur sub region and factors responsible for backwardness & disparity have been indicated. The policy guidelines for Identified sub region have been proposed based on the most significant factor and their extent of relationship explaining backwardness Nagpur sub region.