On the H\'enon-Lane-Emden conjecture

We consider Liouville-type theorems for the following H\'{e}non-Lane-Emden system \hfill -\Delta u&=& |x|^{a}v^p \text{in} \mathbb{R}^N, \hfill -\Delta v&=& |x|^{b}u^q \text{in} \mathbb{R}^N, when $pq>1$, $p,q,a,b\ge0$. The main conjecture states that there is no non-trivial non-negative solution whenever $(p,q)$ is under the critical Sobolev hyperbola, i.e. $\frac{N+a}{p+1}+\frac{N+b}{q+1}>{N-2}$. We show that this is indeed the case in dimension N=3 provided the solution is also assumed to be bounded, extending a result established recently by Phan-Souplet in the scalar case. Assuming stability of the solutions, we could then prove Liouville-type theorems in higher dimensions. For the scalar cases, albeit of second order ($a=b$ and $p=q$) or of fourth order ($a\ge 0=b$ and $p>1=q$), we show that for all dimensions $N\ge 3$ in the first case (resp., $N\ge 5$ in the second case), there is no positive solution with a finite Morse index, whenever $p$ is below the corresponding critical exponent, i.e $1<p<\frac{N+2+2a}{N-2}$ (resp., $1<p<\frac{N+4+2a}{N-4}$). Finally, we show that non-negative stable solutions of the full H\'{e}non-Lane-Emden system are trivial provided \label{sysdim00} N<2+2(\frac{p(b+2)+a+2}{pq-1}) (\sqrt{\frac{pq(q+1)}{p+1}}+ \sqrt{\frac{pq(q+1)}{p+1}-\sqrt\frac{pq(q+1)}{p+1}}).Comment: Theorem 4 has been added in the new version. 23 pages, Comments are welcome. Updated version - if any - can be downloaded at http://www.birs.ca/~nassif/ or http://www.math.ubc.ca/~fazly/research.htm

Glasgow defined : a business perspective

Commerce, trade and business together with education have been the heart of Glasgow over the centuries. Glasgow played a central role in developing Scotland's trans-Atlantic commerce and trade- based economy with the development of the tobacco trade. As the 'second city of the empire' in the late 1800s it was, as Findlay (2011) notes, a central player in the first industrial revolution with its textile, mining, iron and shipping industries. Glasgow's leadership in the manufacture of ships, locomotives and heavy engineering reflected not only a highly skilled workforce, but equally a strong tradition of technological innovation and invention together allied to a strong financial and business services base. Through much of the 20th century Glasgow, along with many other industrial cities, had to confront the problems of industrial change and rising social deprivation and experienced several cycles of decline, renewal and regeneration. However, the traditions of innovation, together with a strong higher education sector, and a vibrant culture and dynamism have enabled Glasgow to change and to renew its economy. Since the 1980s, Glasgow has been rebuilding itself through a series regeneration programmes including: the 'Glasgow Miles Better' campaign, the 1988 Glasgow Garden Festival, being European City of Culture in 1990 and, looking forward, hosting the Commonwealth Games in 2014. These, together with a series of development strategies, most notably the private-sector led Glasgow Action which led the implementation of a services-led McKinsey strategy from the mid-1980’s to the Joint Economic Strategy of Glasgow City Council and Scottish Enterprise set out in 2006 'Step Change' strategy programme and – even more recently - the creation of the Glasgow Economic Commission and the private sector-led Glasgow Economic Leadership all illustrate the innovation and strength of commitment of civic, business and academic partners to continuing and strengthening Glasgow's economic growth and renewal

On Gauge Invariant Cosmological Perturbations in UV-modified Horava Gravity: A Brief Introduction

We revisit gauge invariant cosmological perturbations in UV-modified, z = 3 Horava gravity with one scalar matter field, which has been proposed as a renormalizable gravity theory without the ghost problem in four dimensions. We confirm that there is no extra graviton modes and general relativity is recovered in IR, which achieves the consistency of the model. From the UV-modification terms which break the detailed balance condition in UV, we obtain scale-invariant power spectrums for non-inflationary backgrounds, like the power-law expansions, without knowing the details of early expansion history of Universe. This could provide a new framework for the Big Bang cosmology.Comment: 5 pages, Prepared for Proceeding of International Joint Conference of ICGAC-XIII and IK15 (July 3-7, 2017, Seoul, Republic of Korea