Professional correspondence

This letter discusses interpolation methods; it is prompted by Akima`s note on local methods of univariate interpolation (April 1996 issue of SIGNUM Newsletter). That note compared various univariate methods, concluding that Algorithm 697 is probably the best. Purpose of this letter is provide information about PCHIC comparable to that given in Akima`s note. The 4 methods (Alg. 514, Alg. 697, PCHIC0, PCHIC1) are applied to the 5 analytic functions in Ellis and McLain`s paper (x{sup 3}, x{sup 4}, exp(-x{sup 2}/2), tanh x, sin x)

LIP: The Livermore Interpolation Package, Version 1.3

This report describes LIP, the Livermore Interpolation Package. Because LIP is a stand-alone version of the interpolation package in the Livermore Equation of State (LEOS) access library, the initials LIP alternatively stand for the ''LEOS Interpolation Package''. LIP was totally rewritten from the package described in [1]. In particular, the independent variables are now referred to as x and y, since the package need not be restricted to equation of state data, which uses variables {rho} (density) and T (temperature). LIP is primarily concerned with the interpolation of two-dimensional data on a rectangular mesh. The interpolation methods provided include piecewise bilinear, reduced (12-term) bicubic, and bicubic Hermite (biherm). There is a monotonicity-preserving variant of the latter, known as bimond. For historical reasons, there is also a biquadratic interpolator, but this option is not recommended for general use. A birational method was added at version 1.3. In addition to direct interpolation of two-dimensional data, LIP includes a facility for inverse interpolation (at present, only in the second independent variable). For completeness, however, the package also supports a compatible one-dimensional interpolation capability. Parametric interpolation of points on a two-dimensional curve can be accomplished by treating the components as a pair of one-dimensional functions with a common independent variable. LIP has an object-oriented design, but it is implemented in ANSI Standard C for efficiency and compatibility with existing applications. First, a ''LIP interpolation object'' is created and initialized with the data to be interpolated. Then the interpolation coefficients for the selected method are computed and added to the object. Since version 1.1, LIP has options to instead estimate derivative values or merely store data in the object. (These are referred to as ''partial setup'' options.) It is then possible to pass the object to functions that interpolate or invert the interpolant at an arbitrary number of points. The first section of this report describes the overall design of the package, including both forward and inverse interpolation. Sections 2-6 describe each interpolation method in detail. The software that implements this design is summarized function-by-function in Section 7. For a complete example of package usage, refer to Section 8. The report concludes with a few brief notes on possible software enhancements. For guidance on adding other functional forms to LIP, refer to Appendix B. The reader who is primarily interested in using LIP to solve a problem should skim Section 1, then skip to Sections 7.1-4. Finally, jump ahead to Section 8 and study the example. The remaining sections can be referred to in case more details are desired. Changes since version 1.1 of this document include the new Section 3.2.1 that discusses derivative estimation and new Section 6 that discusses the birational interpolation method. Section numbers following the latter have been modified accordingly

LIP: The Livermore Interpolation Package, Version 1.4

This report describes LIP, the Livermore Interpolation Package. Because LIP is a stand-alone version of the interpolation package in the Livermore Equation of State (LEOS) access library, the initials LIP alternatively stand for the 'LEOS Interpolation Package'. LIP was totally rewritten from the package described in [1]. In particular, the independent variables are now referred to as x and y, since the package need not be restricted to equation of state data, which uses variables {rho} (density) and T (temperature). LIP is primarily concerned with the interpolation of two-dimensional data on a rectangular mesh. The interpolation methods provided include piecewise bilinear, reduced (12-term) bicubic, and bicubic Hermite (biherm). There is a monotonicity-preserving variant of the latter, known as bimond. For historical reasons, there is also a biquadratic interpolator, but this option is not recommended for general use. A birational method was added at version 1.3. In addition to direct interpolation of two-dimensional data, LIP includes a facility for inverse interpolation (at present, only in the second independent variable). For completeness, however, the package also supports a compatible one-dimensional interpolation capability. Parametric interpolation of points on a two-dimensional curve can be accomplished by treating the components as a pair of one-dimensional functions with a common independent variable. LIP has an object-oriented design, but it is implemented in ANSI Standard C for efficiency and compatibility with existing applications. First, a 'LIP interpolation object' is created and initialized with the data to be interpolated. Then the interpolation coefficients for the selected method are computed and added to the object. Since version 1.1, LIP has options to instead estimate derivative values or merely store data in the object. (These are referred to as 'partial setup' options.) It is then possible to pass the object to functions that interpolate or invert the interpolant at an arbitrary number of points. The first section of this report describes the overall design of the package, including both forward and inverse interpolation. Sections 2-6 describe each interpolation method in detail. The software that implements this design is summarized function-by-function in Section 7. For a complete example of package usage, refer to Section 8. The report concludes with a few brief notes on possible software enhancements. For guidance on adding other functional forms to LIP, refer to Appendix B. The reader who is primarily interested in using LIP to solve a problem should skim Section 1, then skip to Sections 7.1-4. Finally, jump ahead to Section 8 and study the example. The remaining sections can be referred to in case more details are desired. Changes since version 1.1 of this document include the new Section 3.2.1 that discusses derivative estimation and new Section 6 that discusses the birational interpolation method. Section numbers following the latter have been modified accordingly

Piecewise Cubic Interpolation Methods

Interpolation of one-dimensional data using piecewise cubic interpolants is considered. Methods are presented for modifying the derivative values in the Hermite representation in order to eliminate the ''bumps'' and ''wiggles'' that frequently plague the more common cubic spline or Akima interpolants. The resulting interpolant is C/sup 1/, but generally not C/sup 2/. The report consists of a reproduction of a poster prepared for a meeting. 27 figures

Nuclear energy density functional from chiral pion-nucleon dynamics: Isovector spin-orbit terms

We extend a recent calculation of the nuclear energy density functional in the systematic framework of chiral perturbation theory by computing the isovector spin-orbit terms: $(\vec \nabla \rho_p- \vec \nabla \rho_n)\cdot(\vec J_p-\vec J_n) G_{so}(k_f)+ (\vec J_p-\vec J_n)^2 G_J(k_f)$. The calculation includes the one-pion exchange Fock diagram and the iterated one-pion exchange Hartree and Fock diagrams. From these few leading order contributions in the small momentum expansion one obtains already a good equation of state of isospin-symmetric nuclear matter. We find that the parameterfree results for the (density-dependent) strength functions $G_{so}(k_f)$ and $G_J(k_f)$ agree fairly well with that of phenomenological Skyrme forces for densities $\rho > \rho_0/10$. At very low densities a strong variation of the strength functions $G_{so}(k_f)$ and $G_J(k_f)$ with density sets in. This has to do with chiral singularities $m_\pi^{-1}$ and the presence of two competing small mass scales $k_f$ and $m_\pi$. The novel density dependencies of $G_{so}(k_f)$ and $G_J(k_f)$ as predicted by our parameterfree (leading order) calculation should be examined in nuclear structure calculations.Comment: 9 pages, 3 figure, published in: Physical Review C68, 014323 (2003

Exact results for the optical absorption of strongly correlated electrons in a half-filled Peierls-distorted chain

In this second of three articles on the optical absorption of electrons in a half-filled Peierls-distorted chain we present exact results for strongly correlated tight-binding electrons. In the limit of a strong on-site interaction $U$ we map the Hubbard model onto the Harris-Lange model which can be solved exactly in one dimension in terms of spinless fermions for the charge excitations. The exact solution allows for an interpretation of the charge dynamics in terms of parallel Hubbard bands with a free-electron dispersion of band-width $W$, separated by the Hubbard interaction $U$. The spin degrees of freedom enter the expressions for the optical absorption only via a momentum dependent but static ground state expectation value. The remaining spin problem can be traced out exactly since the eigenstates of the Harris-Lange model are spin-degenerate. This corresponds to the Hubbard model at temperatures large compared to the spin exchange energy. Explicit results are given for the optical absorption in the presence of a lattice distortion $\delta$ and a nearest-neighbor interaction $V$. We find that the optical absorption for $V=0$ is dominated by a peak at $\omega=U$ and broad but weak absorption bands for $| \omega -U | \leq W$. For an appreciable nearest-neighbor interaction, $V>W/2$, almost all spectral weight is transferred to Simpson's exciton band which is eventually Peierls-split.Comment: 50 pages REVTEX 3.0, 6 postscript figures; hardcopy versions before May 96 are obsolete; accepted for publication in The Philosophical Magazine

Ceramic Substrates for High-temperature Electronic Integration

One of the most attractive ways to increase power handling capacity in power modules is to increase the operating temperature using wide-band-gap semiconductors. Ceramics are ideal candidates for use as substrates in high-power high-temperature electronic devices. The present article aims to determine the most suitable ceramic material for this application

Dielectric response of charge induced correlated state in the quasi-one-dimensional conductor (TMTTF)2PF6

Conductivity and permittivity of the quasi-one-dimensionsional organic transfer salt (TMTTF)2PF6 have been measured at low frequencies (10^3-10^7 Hz) between room temperature down to below the temperature of transition into the spin-Peierls state. We interpret the huge real part of the dielectric permittivity (up to 10^6) in the localized state as the realization in this compound of a charge ordered state of Wigner crystal type due to long range Coulomb interaction.Comment: 11 pages, 3 .eps figure