Implicit Additive Preferences: A Further Generalization of the CES

The CES is generalized by extension of the work of Hanoch (1975) resulting in implicit, direct and indirect relationships between utility and consumption. Expressions for substitution and income elasticities are developed and observed to be variable, rather than constant as in the CES case.Constant elasticity of substitution, implicit functions, preferences, demand

Layout-based substitution tree indexing and retrieval for mathematical expressions

We introduce a new system for layout-based indexing and retrieval of mathematical expressions using substitution trees. Substitution trees can efficiently store and find hierarchically-structured data based on similarity. Previously Kolhase and Sucan applied substitution trees to indexing mathematical expressions in operator tree representation (Content MathML) and query-by-expression retrieval. In this investigation, we use substitution trees to index mathematical expressions in symbol layout tree representation (LaTeX) to group expressions based on the similarity of their symbols, symbol layout, sub-expressions and size. We describe our novel substitution tree indexing and retrieval algorithms and our many significant contributions to the behavior of these algorithms, including: allowing substitution trees to index and retrieve layout-based mathematical expressions instead of predicates; introducing a bias in the insertion function that helps group expressions in the index based on similarity in baseline size; modifying the search function to find expressions that are not identical yet still structurally similar to a search query; and ranking search results based on their similarity in symbols and symbol layout to the search query. We provide an experiment testing our system against the term frequency-inverse document frequency (TF-IDF) keyword-based system of Zanibbi and Yuan and demonstrate that: in many cases, the two systems are comparable; our system excelled at finding expressions identical to the search query and expressions containing relevant sub-expressions; and our system experiences some limitations due to the insertion bias and the presence of LaTeX formatting in expressions. Future work includes: designing a different insertion bias that improves the quality of search results; modifying the behavior of the search and ranking functions; and extending the scope of the system so that it can index websites or non-LaTeX expressions (such as MathML or images). Overall, we present a promising first attempt at layout-based substitution tree indexing and retrieval for mathematical expressions

IMPLICIT ADDITIVE PREFERENCES: A FURTHER GENERALIZATION OF THE CES

The CES is generalized by extension of the work of Hanoch (1975) resulting in implicit, direct and indirect relationships between utility and consumption. Expressions for substitution and income elasticities are developed and observed to be variable, rather than constant as in the CES case.Food Consumption/Nutrition/Food Safety,

Computer simulation of the mathematical modeling involved in constitutive equation development: Via symbolic computations

Development of new material models for describing the high temperature constitutive behavior of real materials represents an important area of research in engineering disciplines. Derivation of mathematical expressions (constitutive equations) which describe this high temperature material behavior can be quite time consuming, involved and error prone; thus intelligent application of symbolic systems to facilitate this tedious process can be of significant benefit. A computerized procedure (SDICE) capable of efficiently deriving potential based constitutive models, in analytical form is presented. This package, running under MACSYMA, has the following features: partial differentiation, tensor computations, automatic grouping and labeling of common factors, expression substitution and simplification, back substitution of invariant and tensorial relations and a relational data base. Also limited aspects of invariant theory were incorporated into SDICE due to the utilization of potentials as a starting point and the desire for these potentials to be frame invariant (objective). Finally not only calculation of flow and/or evolutionary laws were accomplished but also the determination of history independent nonphysical coefficients in terms of physically measurable parameters, e.g., Young's modulus, was achieved. The uniqueness of SDICE resides in its ability to manipulate expressions in a general yet predefined order and simplify expressions so as to limit expression growth. Results are displayed when applicable utilizing index notation

Application of symbolic computations to the constitutive modeling of structural materials

In applications involving elevated temperatures, the derivation of mathematical expressions (constitutive equations) describing the material behavior can be quite time consuming, involved and error-prone. Therefore intelligent application of symbolic systems to faciliate this tedious process can be of significant benefit. Presented here is a problem oriented, self contained symbolic expert system, named SDICE, which is capable of efficiently deriving potential based constitutive models in analytical form. This package, running under DOE MACSYMA, has the following features: (1) potential differentiation (chain rule), (2) tensor computations (utilizing index notation) including both algebraic and calculus; (3) efficient solution of sparse systems of equations; (4) automatic expression substitution and simplification; (5) back substitution of invariant and tensorial relations; (6) the ability to form the Jacobian and Hessian matrix; and (7) a relational data base. Limited aspects of invariant theory were also incorporated into SDICE due to the utilization of potentials as a starting point and the desire for these potentials to be frame invariant (objective). The uniqueness of SDICE resides in its ability to manipulate expressions in a general yet pre-defined order and simplify expressions so as to limit expression growth. Results are displayed, when applicable, utilizing index notation. SDICE was designed to aid and complement the human constitutive model developer. A number of examples are utilized to illustrate the various features contained within SDICE. It is expected that this symbolic package can and will provide a significant incentive to the development of new constitutive theories

Breakdown of the Peierls substitution for the Haldane model with ultracold atoms

We present two independent calculations of the tight-binding parameters for a specific realization of the Haldane model with ultracold atoms. The tunneling coefficients up to next-to-nearest neighbors are computed ab-initio by using the maximally localized Wannier functions, and compared to analytical expressions written in terms of gauge invariant, measurable properties of the spectrum. The two approaches present a remarkable agreement and evidence the breakdown of the Peierls substitution: (i) the phase acquired by the next-to-nearest tunneling amplitude $t_{1}$ presents quantitative and qualitative differences with respect to that obtained by the integral of the vector field A, as considered in the Peierls substitution, even in the regime of low amplitudes of A; (ii) for larger values, also $|t_{1}|$ and the nearest-neighbor tunneling $t_{0}$ have a marked dependence on A. The origin of this behavior and its implications are discussed.Comment: 5 pages, 5 figure