Transverse Mercator with an accuracy of a few nanometers

Implementations of two algorithms for the transverse Mercator projection are described; these achieve accuracies close to machine precision. One is based on the exact equations of Thompson and Lee and the other uses an extension of Krueger's series for the projection to higher order. The exact method provides an accuracy of 9 nm over the entire ellipsoid, while the errors in the series method are less than 5 nm within 3900 km of the central meridian. In each case, the meridian convergence and scale are also computed with similar accuracy. The speed of the series method is competitive with other less accurate algorithms and the exact method is about 5 times slower.Comment: LaTeX, 10 pages, 3 figures. Includes some revisions. Supplementary material is available at http://geographiclib.sourceforge.net/tm.htm

On the relation between adjacent inviscid cell type solutions to the rotating-disk equations

Over a large range of the axial coordinate a typical higher-branch solution of the rotating-disk equations consists of a chain of inviscid cells separated from each other by viscous interlayers. In this paper the leading-order relation between two adjacent cells will be established by matched asymptotic expansions for general values of the parameter appearing in the equations. It is found that the relation between the solutions in the two cells crucially depends on the behaviour of the tangential velocity in the viscous interlayer. The results of the theory are compared with accurate numerical solutions and good agreement is obtained

Two-dimensional symmetric and antisymmetric generalizations of exponential and cosine functions

Properties of the four families of recently introduced special functions of two real variables, denoted here by $E^\pm$, and $\cos^\pm$, are studied. The superscripts $^+$ and $^-$ refer to the symmetric and antisymmetric functions respectively. The functions are considered in all details required for their exploitation in Fourier expansions of digital data, sampled on square grids of any density and for general position of the grid in the real plane relative to the lattice defined by the underlying group theory. Quality of continuous interpolation, resulting from the discrete expansions, is studied, exemplified and compared for some model functions.Comment: 22 pages, 10 figure

Cluster mean-field approximations with the coherent-anomaly-method analysis for the driven pair contact process with diffusion

The cluster mean-field approximations are performed, up to 13 cluster sizes, to study the critical behavior of the driven pair contact process with diffusion (DPCPD) and its precedent, the PCPD in one dimension. Critical points are estimated by extrapolating our data to the infinite cluster size limit, which are in good accordance with recent simulation results. Within the cluster mean-field approximation scheme, the PCPD and the DPCPD share the same mean-field critical behavior. The application of the coherent anomaly method, however, shows that the two models develop different coherent anomalies, which lead to different true critical scaling. The values of the critical exponents for the particle density, the pair density, the correlation length, and the relaxation time are fairly well estimated for the DPCPD. These results support and complement our recent simulation results for the DPCPD

Master Equation Analysis of Thermochemical Nonequilibrium of Nitrogen

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/97092/1/AIAA2012-3305.pd

Finite-Size Scaling Studies of Reaction-Diffusion Systems Part III: Numerical Methods

The scaling exponent and scaling function for the 1D single species coagulation model $(A+A\rightarrow A)$ are shown to be universal, i.e. they are not influenced by the value of the coagulation rate. They are independent of the initial conditions as well. Two different numerical methods are used to compute the scaling properties: Monte Carlo simulations and extrapolations of exact finite lattice data. These methods are tested in a case where analytical results are available. It is shown that Monte Carlo simulations can be used to compute even the correction terms. To obtain reliable results from finite-size extrapolations exact numerical data for lattices up to ten sites are sufficient.Comment: 19 pages, LaTeX, 5 figures uuencoded, BONN HE-94-0

Three variable exponential functions of the alternating group

New class of special functions of three real variables, based on the alternating subgroup of the permutation group $S_3$, is studied. These functions are used for Fourier-like expansion of digital data given on lattice of any density and general position. Such functions have only trivial analogs in one and two variables; a connection to the $E-$functions of $C_3$ is shown. Continuous interpolation of the three dimensional data is studied and exemplified.Comment: 10 pages, 3 figure

Numerical computation of real or complex elliptic integrals

Algorithms for numerical computation of symmetric elliptic integrals of all three kinds are improved in several ways and extended to complex values of the variables (with some restrictions in the case of the integral of the third kind). Numerical check values, consistency checks, and relations to Legendre's integrals and Bulirsch's integrals are included

Glueball spectrum in a (1+1)-dimensional model for QCD

We consider (1+1)-dimensional QCD coupled to scalars in the adjoint representation of the gauge group SU($N$). This model results from dimensional reduction of the (2+1)-dimensional pure glue theory. In the large-N limit we study the spectrum of glueballs numerically, using the discretized \lcq. We find a discrete spectrum of bound states, with the density of levels growing approximately exponentially with the mass. A few low-lying states are very close to being eigenstates of the parton number, and their masses can be accurately calculated by truncated diagonalizations.Comment: 17 pages, uses phyzzx and table.tex, 5 figures available upon request from [email protected]

On surface properties of two-dimensional percolation clusters

The two-dimensional site percolation problem is studied by transfer-matrix methods on finite-width strips with free boundary conditions. The relationship between correlation-length amplitudes and critical indices, predicted by conformal invariance, allows a very precise determination of the surface decay-of-correlations exponent, $\eta_s = 0.6664 \pm 0.0008$, consistent with the analytical value $\eta_s = 2/3$. It is found that a special transition does not occur in the case, corroborating earlier series results. At the ordinary transition, numerical estimates are consistent with the exact value $y_s = -1$ for the irrelevant exponent.Comment: 8 pages, LaTeX with Institute of Physics macros, to appear in Journal of Physics