High-Order Harmonic Generation and Molecular Orbital Tomography: Characteristics of Molecular Recollision Electronic Wave Packets

We investigate the orientation dependence of molecular high-order harmonic generation (HHG) both numerically and analytically. We show that the molecular recollision electronic wave packets (REWPs) in the HHG are closely related to the ionization potential as well as the particular orbital from which it ionized. As a result, the spectral amplitude of the molecular REWP can be significantly different from its reference atom (i.e., with the same ionization potential as the molecule under study) in some energy regions due to the interference between the atomic cores of the molecules. This finding is important for molecular orbital tomography using HHG[Nature \textbf{432}, 867(2004)].Comment: 4 pages, 4 figure

Place and informal care in an ageing society:reviewing the state of the art in geographical gerontology

Who cares for our frail older populations and where is fast becoming a critical issue for policy-makers and practitioners in many high income countries as they grapple with the economic and welfare implications of increasing longevity. This demographic shift is, of course, a major success story. However, increased life expectancy is also bringing with it a growth in those numbers of older people, particularly the oldest old, who are experiencing multiple morbidities and a declining ability to undertake those instrumental activities of daily life (IADLs) that are so important to maintaining independence and dignity in later life. At the same time, policy and practice has shifted away from residential or institutional care for our older population to focus on ‘ageing in place’. Here, older people are to be supported to remain within their own homes for as long as possible. Conceptually, this has meant that services and care previously delivered within a single institutional environment, have been redesigned for delivery within domestic settings where frail older people would also benefit from the informal care support from family, friends and neighbours. On the one hand, this has meant that many older people have benefited from the familiarity, sense of safety and support that care provided within the domestic setting has engendered; on the other, changing family structures, a decline in community and sweeping health and welfare cuts in an era of economic austerity have left growing numbers of older people increasingly lonely, isolated and at risk. Understanding who cares, where, the form that care takes and how this is being differentially experienced by our older populations have been issues of growing concern for geographers interested in health and ageing. In this paper I review the current ‘state of the art’ of geographical gerontology around informal care and the home and illustrate how those working in this field are making an important contribution to multidisciplinary debates around care of our older populations

Singular Trudinger--Moser inequality involving LpL^{p} norm in bounded domain

In this paper, we use the method of blow-up analysis and capacity estimate to derive the singular Trudinger--Moser inequality involving $N$-Finsler--Laplacian and $L^{p}$ norm, precisely, for any $p>1$, $0\leq\gamma<\gamma_{1}:= \inf\limits_{u\in W^{1, N}_{0}(\Omega)\backslash \{0\}}\frac{\int_{\Omega}F^{N}(\nabla u)dx}{\| u\|_p^N}$ and $0\leq\beta<N$, we have \begin{align} \sup_{u\in W_{0}^{1,N}(\Omega),\;\int_{\Omega}F^{N}(\nabla u)dx-\gamma\| u\|_p^N\leq1}\int_{\Omega}\frac{e^{\lambda_{N}(1-\frac{\beta}{N})\lvert u\rvert^{\frac{N}{N-1}}}}{F^{o}(x)^{\beta}}\;\mathrm{d}x<+\infty\notag, \end{align} where $\lambda_{N}=N^{\frac{N}{N-1}} \kappa_{N}^{\frac{1}{N-1}}$ and $\kappa_{N}$ is the volume of a unit Wulff ball in $\mathbb{R}^N$, moreover, extremal functions for the inequality are also obtained. When $F=\lvert\cdot\rvert$ and $p=N$, we can obtain the singular version of Tintarev type inequality by the obove inequality, namely, for any $0\leq\alpha<\alpha_{1}(\Omega):=\inf\limits_{u\in W^{1, N}_{0}(\Omega)\backslash \{0\}}\frac{\int_{\Omega}|\nabla u|^Ndx}{\| u\|_N^N}$ and $0\leq\beta<N$, it holds $$\sup_{u\in W_{0}^{1,N}(\Omega),\;\int_{\Omega}\lvert\nabla u\rvert^{N}\;\mathrm{d}x-\alpha\|u\|_{N}^{N}\leq1}\int_{\Omega}\frac{e^{\alpha_{N}(1-\frac{\beta}{N})\lvert u\rvert^{\frac{N}{N-1}}}}{\lvert x\rvert^{\beta}}\;\mathrm{d}x<+\infty,$$ where $\alpha_{N}:=N^{\frac{N}{N-1}}\omega_{N}^{\frac{1}{N-1}}$ and $\omega_{N}$ is the volume of unit ball in $\mathbb{R}^{N}$. Our results extend many well-known Trudinger--Moser type inequalities to more general setting