scholarly journals Dielectric Saturation in Ionic Crystals. I. Rigid Ion Model

1967 ◽  
Vol 20 (1) ◽  
pp. 11 ◽  
Author(s):  
J Oitmaa
Keyword(s):  
Dielectric Constant ◽  
Explicit Expression ◽  
Ionic Crystal ◽  
Applied Field ◽  
Order Term ◽  
Higher Order ◽  
Ionic Crystals ◽  
Dielectric Saturation ◽  
Higher Order Term ◽  
Rigid Ion Model

The higher-order term in the static dielectric constant of an ionic crystal, which is proportional to the square of the applied field, is calculated using a rigid ion model. An explicit expression is obtained for this term, involving anharmonic coefficients, and is evaluated for NaCI.

scholarly journals Dielectric Saturation in Ionic Crystals. II. Deformable Ions

1968 ◽  
Vol 21 (4) ◽  
pp. 439
Author(s):  
J Oitmaa
Keyword(s):  
Dielectric Constant ◽  
Shell Model ◽  
Ionic Crystal ◽  
Higher Order ◽  
Static Dielectric Constant ◽  
Ionic Crystals ◽  
Dependent Term ◽  
Field Dependent ◽  
Dielectric Saturation

The lowest order field-dependent term in ~he static dielectric constant is calculated for an ionic crystal with deformable ions, and is evaluated numerically for NaI using a simple shell model. Some terms of higher order are also calculated.

Derivation of Skyrme Lagrangian and higher-order term

1990 ◽  
Vol 41 (9) ◽  
pp. 2930-2932
Author(s):  
Akihiro Ito
Keyword(s):  
Order Term ◽  
Higher Order ◽  
Higher Order Term

scholarly journals A rewriting calculus for cyclic higher-order term graphs

2007 ◽  
Vol 17 (3) ◽  
pp. 363-406 ◽  
Author(s):  
PAOLO BALDAN ◽  
CLARA BERTOLISSI ◽  
HORATIU CIRSTEA ◽  
CLAUDE KIRCHNER
Keyword(s):  
Order Term ◽  
Equational Theory ◽  
Term Rewriting ◽  
Higher Order ◽  
Graph Representation ◽  
First Order ◽  
Rewrite Rules ◽  
Evaluation Mechanism ◽  
Matching Constraints ◽  
Higher Order Term

The Rewriting Calculus (ρ-calculus, for short) was introduced at the end of the 1990s and fully integrates term-rewriting and λ-calculus. The rewrite rules, acting as elaborated abstractions, their application and the structured results obtained are first class objects of the calculus. The evaluation mechanism, which is a generalisation of beta-reduction, relies strongly on term matching in various theories.In this paper we propose an extension of the ρ-calculus, called ρg-calculus, that handles structures with cycles and sharing rather than simple terms. This is obtained by using recursion constraints in addition to the standard ρ-calculus matching constraints, which leads to a term-graph representation in an equational style. Like in the ρ-calculus, the transformations are performed by explicit application of rewrite rules as first-class entities. The possibility of expressing sharing and cycles allows one to represent and compute over regular infinite entities.We show that the ρg-calculus, under suitable linearity conditions, is confluent. The proof of this result is quite elaborate, due to the non-termination of the system and the fact that ρg-calculus-terms are considered modulo an equational theory. We also show that the ρg-calculus is expressive enough to simulate first-order (equational) left-linear term-graph rewriting and α-calculus with explicit recursion (modelled using a letrec-like construct).

Asymptotic expansions for laminar forced-convection heat and mass transfer Part 2. Boundary-layer flows

1966 ◽  
Vol 24 (2) ◽  
pp. 339-366 ◽  
Author(s):  
J. D. Goddard ◽  
Andreas Acrivos
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Mass Transfer ◽  
Prandtl Number ◽  
Order Term ◽  
Higher Order ◽  
Convection Heat ◽  
Boundary Layer Flows ◽  
Prandtl Numbers ◽  
Higher Order Term

This is the second of two articles by the authors dealing with asymptotic expansions for forced-convection heat or mass transfer to laminar flows. It is shown here how the method of the first paper (Acrivos & Goddard 1965), which was used to derive a higher-order term in the large Péclet number expansion for heat or mass transfer to small Reynolds number flows, can yield equally well higher-order terms in both the large and the small Prandtl number expansions for heat transfer to laminar boundary-layer flows. By means of this method an exact expression for the first-order correction to Lighthill's (1950) asymptotic formula for heat transfer at large Prandtl numbers, as well as an additional higher-order term for the small Prandtl number expansion of Morgan, Pipkin & Warner (1958), are derived. The results thus obtained are applicable to systems with non-isothermal surfaces and arbitrary planar or axisymmetric flow geometries. For the latter geometries a derivation is given of a higher-order term in the Péclet number expansion which arises from the curvature of the thermal layer for small Prandtl numbers. Finally, some applications of the results to ‘similarity’ flows are also presented.

Accurate Description of Near-Crack-Tip Fields for the Estimation of Inelastic Zone Extent in Quasi-Brittle Materials

2012 ◽  
Vol 525-526 ◽  
pp. 529-532 ◽  
Author(s):  
Václav Veselý ◽  
Jakub Sobek ◽  
Lucie Šestáková ◽  
Stanislav Seitl
Keyword(s):  
Finite Element ◽  
Order Term ◽  
Higher Order ◽  
Fe Analysis ◽  
Displacement Fields ◽  
Deterministic Method ◽  
Stress And Displacement Fields ◽  
Higher Order Terms ◽  
Higher Order Term ◽  
Selection Of

A description of stress and displacement fields by means of the Williams power series using also higher-order terms is the focus of this paper. Coefficients of this series are determined via the over-deterministic method from the results of conventional finite element (FE) analysis. A study is conducted into the selection of the FE node set whose results are processed in this regression technique. Coefficients up to the twelfth term were determined with high precision. The effect of the position of the FE node set on the accuracy of the values of the higher-order term coefficients is reported.

Mikroskopisches Modell eines Ionenkristalles mit einem Brown’schen Untergitter. I. Allgemeine Theorie

1978 ◽  
Vol 33 (7) ◽  
pp. 808-814 ◽  
Author(s):  
C. H. Tillmanns ◽  
L. Merten ◽  
G. Börstel
Keyword(s):  
Complex System ◽  
Dipole Moment ◽  
Simple Model ◽  
Relaxation Process ◽  
Ionic Crystal ◽  
Equation Of Motion ◽  
System Of Equations ◽  
Ionic Crystals ◽  
Debye Relaxation ◽  
Rigid Ion Model

The rigid ion model is extended to ionic crystals with a Brownian sublattice. This sublattice is formed by particles which have the possibility to assume at least two positions along a direction overcoming a potential barrier. Such a Debye relaxation process is connected with a spontaneous dipole moment. Taking into account this additional dipole moment in the equation of motion for ionic crystals we get the corresponding equation for ionic crystals with a Brownian sublattice. In part II of this paper this complex system of equations, which depends on temperature, is solved for a simple model of a diatomic ionic crystal.

LANDAU-DE GENNES MODEL WITH THE INCLUSION OF DENSITY CHANGE FOR THE NEMATIC-ISOTROPIC PHASE TRANSITION

1996 ◽  
Vol 10 (16) ◽  
pp. 777-778 ◽  
Author(s):  
B. NANDI ◽  
P.K. MUKHERJEE ◽  
M. SAHA
Keyword(s):  
Phase Transition ◽  
Temperature Difference ◽  
Order Term ◽  
Higher Order ◽  
Density Change ◽  
Sixth Order ◽  
Isotropic Phase ◽  
Definite Improvement ◽  
Higher Order Term ◽  
Good Agreement

The Landau-de Gennes model of the nematic-isotropic phase transition with the inclusion of the density change and a higher order term (sixth-order term) in a simple way is examined. The temperature difference T NI −T* obtained is in good agreement with the experimental one. We also obtained the value of the ratio (Q*–Q NI )/Q NI . The value of (Q*–Q NI )/Q NI is not improved over the earlier attempt, but there is a definite improvement in the value of T NI –T* which is obtained as 1.045 K.

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