Blowing up with infinite energy of conformal metrics on Sn

1999 ◽  
Vol 24 (5-6) ◽  
pp. 785-799 ◽  
Author(s):  
Chiun-Chuan Chen ◽  
Chiun-Chuan Chen ◽  
Chang-Shou Lin ◽  
Chang-Shou Lin
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon

This chapter examines various ways to construct symplectic manifolds and submanifolds. It begins by studying blowing up and down in both the complex and the symplectic contexts. The next section is devoted to a discussion of fibre connected sums and describes Gompf’s construction of symplectic four-manifolds with arbitrary fundamental group. The chapter also contains an exposition of Gromov’s telescope construction, which shows that for open manifolds the h-principle rules and the inclusion of the space of symplectic forms into the space of nondegenerate 2-forms is a homotopy equivalence. The final section outlines Donaldson’s construction of codimension two symplectic submanifolds and explains the associated decompositions of the ambient manifold.


2021 ◽  
Vol 10 (1) ◽  
pp. 952-971
Author(s):  
Ahmed Alsaedi ◽  
Bashir Ahmad ◽  
Mokhtar Kirane ◽  
Berikbol T. Torebek

Abstract This paper is devoted to the study of initial-boundary value problems for time-fractional analogues of Korteweg-de Vries, Benjamin-Bona-Mahony, Burgers, Rosenau, Camassa-Holm, Degasperis-Procesi, Ostrovsky and time-fractional modified Korteweg-de Vries-Burgers equations on a bounded domain. Sufficient conditions for the blowing-up of solutions in finite time of aforementioned equations are presented. We also discuss the maximum principle and influence of gradient non-linearity on the global solvability of initial-boundary value problems for the time-fractional Burgers equation. The main tool of our study is the Pohozhaev nonlinear capacity method. We also provide some illustrative examples.


2021 ◽  
Vol 5 (1) ◽  
pp. 191-222
Author(s):  
Karima Kourtit

AbstractThe contemporary ‘digital age’ prompts the need for a re-assessment of urban planning principles and practices. Against the background of current data-rich urban planning, this study seeks to address the question whether an appropriate methodological underpinning can be provided for smart city governance based on a data-driven planning perspective. It posits that the current digital technology age has a drastic impact on city strategies and calls for a multi-faceted perspective on future urban development, termed here the ‘XXQ-principle’ (which seeks to attain the highest possible level of quality for urban life). Heterogeneity in urban objectives and data embodied in the XXQ-principle can be systematically addressed by a process of data decomposition (based on a ‘cascade principle’), so that first, higher-level urban policy domains are equipped with the necessary (‘big’) data provisions, followed by lower-ranking urban governance levels. The conceptual decomposition principle can then be translated into a comprehensive hierarchical model architecture for urban intelligence based on the ‘flying disc’ model, including key performance indicators (KPIs). This new model maps out the socio-economic arena of a complex urban system according to the above cascade system. The design of this urban system architecture and the complex mutual connections between its subsystems is based on the ‘blowing-up’ principle that originates from a methodological deconstruction-reconstruction paradigm in the social sciences. The paper advocates the systematic application of this principle to enhance the performance of smart cities, called the XXQ performance value. This study is not empirical, although it is inspired by a wealth of previous empirical research. It aims to advance conceptual and methodological thinking on principles of smart urban planning.


2014 ◽  
Vol 12 (12) ◽  
Author(s):  
Dina Abuzaid ◽  
Randa Ben Mahmoud ◽  
Hichem Chtioui ◽  
Afef Rigane

AbstractIn this paper, we consider the problem of the existence of conformal metrics with prescribed scalar curvature on the standard sphere S n, n ≥ 3. We give new existence and multiplicity results based on a new Euler-Hopf formula type. Our argument also has the advantage of extending well known results due to Y. Li [16].


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