Error bounds and exponential improvement for the asymptotic expansion of the Barnes GG-function

In this paper we establish new integral representations for the remainder term of the known asymptotic expansion of the logarithm of the Barnes $G$-function. Using these representations, we obtain explicit and numerically computable error bounds for the asymptotic series, which are much simpler than the ones obtained earlier by other authors. We find that along the imaginary axis, suddenly infinitely many exponentially small terms appear in the asymptotic expansion of the Barnes $G$-function. Employing one of our representations for the remainder term, we derive an exponentially improved asymptotic expansion for the logarithm of the Barnes $G$-function, which shows that the appearance of these exponentially small terms is in fact smooth, thereby proving the Berry transition property of the asymptotic series of the $G$-function.Comment: 14 pages, accepted for publication in Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science

The conception of inter-communal structures in the urban-rural relations of Bihor County, Romania

Urbanisation or urban drift often results in imbalanced regional development where the urban thrives and the rural becomes deprived. Efforts to correct this imbalance in the form of growth pole models rarely met with success because of many factors, not least, political. There are nevertheless growth pole models of polycentric development which can foster harmonious relationships between urban and rural areas. This paper illustrates this point with special reference to the conception of eleven Territorial Planning Units (TPU) in the Romanian county of Bihor. The hallmark of the TPU is the establishment of inter-communal structures which manage to circumvent the unhealthy polarisation of conventional urbanisation, reduce regional disparities and strengthen rural- urban relationships because they pay due attention to the needs and interests of local communities

Error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal

In (Boyd, Proc. R. Soc. Lond. A 447 (1994) 609--630), W. G. C. Boyd derived a resurgence representation for the gamma function, exploiting the reformulation of the method of steepest descents by M. Berry and C. Howls (Berry and Howls, Proc. R. Soc. Lond. A 434 (1991) 657--675). Using this representation, he was able to derive a number of properties of the asymptotic expansion for the gamma function, including explicit and realistic error bounds, the smooth transition of the Stokes discontinuities, and asymptotics for the late coefficients. The main aim of this paper is to modify the resurgence formula of Boyd making it suitable for deriving better error estimates for the asymptotic expansions of the gamma function and its reciprocal. We also prove the exponentially improved versions of these expansions complete with error terms. Finally, we provide new (formal) asymptotic expansions for the coefficients appearing in the asymptotic series and compare their numerical efficacy with the results of earlier authors.Comment: 22 pages, accepted for publication in Proceedings of the Royal Society of Edinburgh, Section A: Mathematical and Physical Science

LHC optics and elastic scattering measured by the TOTEM experiment

The TOTEM experiment at the LHC has measured proton-proton elastic scattering in dedicated runs at $\sqrt{s}=7$ and 8 TeV centre-of-mass LHC energies. The proton-proton total cross-section $\sigma_{\rm tot}$ has been derived for both energies using a luminosity independent method. TOTEM has excluded a purely exponential differential cross-section for elastic proton-proton scattering with significance greater than 7$\sigma$ in the $|t|$ range from 0.027 to 0.2 GeV$^{2}$ at $\sqrt{s}=8$ TeV.Comment: Proceedings, 17th Lomonosov Conferenc

The resurgence properties of the incomplete gamma function I

In this paper we derive new representations for the incomplete gamma function, exploiting the reformulation of the method of steepest descents by C. J. Howls (Howls, Proc. R. Soc. Lond. A 439 (1992) 373--396). Using these representations, we obtain a number of properties of the asymptotic expansions of the incomplete gamma function with large arguments, including explicit and realistic error bounds, asymptotics for the late coefficients, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities.Comment: 36 pages, 4 figures. arXiv admin note: text overlap with arXiv:1311.2522, arXiv:1309.2209, arXiv:1312.276