Continuum Matrix Research Articles

Development of fibre-enriched gluten-free bread: a response surface methodology study

The enrichment of gluten-free (GF) baked products with dietary fibre (DF) seems to be necessary since it has been reported that coeliac patients have generally a low intake of DF due to their GF diet. Response surface methodology was used to optimize a fibre-enriched GF bread formulation based on corn starch, rice flour and hydroxypropylmethyl cellulose. Maize fibre and water were the predictor variables (factors), and loaf specific volume, crumb firmness and crumb L value were the dependent variables (responses) used to assess the product quality. The optimal formulation, determined from the data, contained 6.5% maize fibre and 102.5% water, starch/flour base. The developed mathematical models for the measured responses could be successfully used for their prediction during baking. Shelf-life study of the optimized formulation revealed that bread stored under modified atmosphere packaging exhibited lower crumb firmness and moisture content values, and thus remained softer through storage. Scanning electron microscopy of the crumb showed a continuum matrix between starch and maize fibre, in the optimized formulation, obtaining a more aerated structure.

Elastic properties of sphere-reinforced composites with a mesophase

The elastic properties of a three-phase composite made up of spherical particles coated with a mesophase and dispersed in a continuum matrix were computed through a periodic microfield approach. The composite was idealized as a three-dimensional array of either aligned or staggered hexagonal prisms, each containing one coated particle at the center. While the response of the aligned distribution was too stiff, the results obtained for the staggered arrangement were in good agreement with experimental data and with the predictions of the classical mean-field models for two-phase composites. The model was then used to explore the effect of a mesophase in the elastic modulus of a material with a large mismatch between reinforcement and matrix stiffness (Er/Em=1000). A significant variation in the elastic modulus was found for a sphere radius to mesophase thickness ratio below 10 when the mesophase elastic modulus was comprised between those of the matrix and of the reinforcement. The model predictions of this behavior were in agreement with experimental data found in the literature.

A new method for calculating angular integrals in electron—atom scattering

This paper reports a novel method for calculating certain angular integrals that occur in the matrix elements of the Hamiltonian which describes the scattering of an electron by a complex atom or ion. Specifically the method applies to those matrix elements which involve the quadratically integrable terms, the so-called bound—bound and bound—continuum matrix elements. It considerably reduces the number of integrals which need to be calculated giving an appreciable speed-up in the calculation for low values of the total orbital angular momentum. The application of the method to a data decomposition parallelization scheme for the construction of the Hamiltonian matrix is discussed.

Lattice Calculation of Quarkonium Decay Matrix Elements

We calculate the NRQCD matrix elements for the decays of the lowest-lying S- and P-wave states of charmonium and bottomonium in quenched lattice QCD. We also compute the one-loop relations between the lattice and continuum matrix elements.

Precise analysis of resonance decay law in atomic physics

Straightforward high-precision results for the problem of resonance state decay within the well known Laplace method are obtained. Deviation of the resonance decay law from the conventional exponential time dependence, determined by the pole contributions, is accurately studied for a model system. The asymptote of the background contribution which dominates the decay pattern for large t is shown to be directly related to the threshold behaviour of the discrete state - continuum matrix element. A new scheme for continuum discretization with different sets of the resonance pseudostates is applied and its high accuracy in describing the system evolution over very large time intervals is demonstrated. It is shown that both of the approaches considered do not imply any analytical form for the discrete state - continuum matrix element . A scheme is developed which allows the application of these approaches for arbitrary including the cases when this quantity is obtained within ab initio calculations.

Rigidity and conductivity percolation thresholds in particulate composites

Percolation threshold criteria for particulate composites are derived in terms of the average coordination number in the percolating cluster using the constraint theory of Phillips and Thorpe. Rigidity percolation threshold value depends on the nature of bonding at the contacts between particles. In the absence of any bonding at the contacts, the average coordination number at the rigidity percolation threshold is six. For completely bonded contacts, it is shown that the rigidity percolation threshold is equal to the conductivity percolation threshold and, in three-dimensions, the average coordination number at the threshold is 2.4 (2 in two-dimensions). The threshold criteria are converted to the number and the volume fractions of the percolating phase and the results are compared with available data. Results are also obtained for threshold values in continuum matrix composites.

Structure and conductivity of liquid crystal channel‐like linic complexes of taper‐shaped compounds

AbstractMolecular packing and electrical conductivity were studied in complexes of alkali trifluoromethanesulphonates with low‐molar or polymeric compounds containing both taper‐shaped mesogens were esters of either 3,4,5‐tris [p‐(n‐dodecan‐l‐yloxy) benzyloxy] benzoic acid (I) or 3, 4, 5‐tris (n‐dodecan‐l‐yloxy) benzoic acid (II). In the hexagonal columnar liquid crystal phase the tapered mesogens fan out from the centre of the column, with the ionic receptors forming the central channel and the aliphatic tails constituting the continuum matrix. In the case of side‐chain polymethacrylates the column core also contains the backbone chain. The DC conductivity σ of unoriented samples increases greatly at the crystal‐columnar transition, with only a minor further change upon columnar‐isotropic transition. σ was in the range 10−9 − 10−6 in the columnar phase 40–90 °C, whilst the activation energies for conduction were between 28 kcal mol−1 for the crown ether and only 2 kcal mo−1 for the complex of LiCf3SO3 with the non‐polymeric ester of tri(ethylene oxide) with I.

Integrability in random matrix models

We prove the equivalence between the recent matrix model formulation of 2D gravity and lattice integrable models. For even potentials this system is the Volterra hierarchy, and many properties of the continuum matrix model like the Virasoro conditions on the partition function stem directly from the integrability properties of the lattice model and its hamiltonian properties.

Multiphoton and above-threshold detachment computed for a model negative ion and compared with perturbation theory

Multiphoton excitation of a model negative ion is studied. The model has only a single bound level, and we choose a photon frequency appropriate to five-photon photodetachment (before ponderomotive threshold shifts). Both of these characteristics are compatible with multiphoton detachment studies of the hydrogen negative ion made with CO2 laser radiation. We study the model in a discretized energy basis in order to study independently the influence of bound–continuum (B–C) and continuum-continuum (C–C) dipole matrix elements on the multiphoton detachment process. Because the energy-level structure of the model (and also of negative ions generally) is relatively simple compared with that of atoms, we are able to investigate a number of questions in some detail, using either perturbation theory or exact numerical solutions to Schrödinger’s equation. Interesting consequences are the explicit identification of an interference between B–C-dominated and C–C-dominated detachment channels and the emergence of energy-nonconserving contributions to above-threshold electron spectra.

Multi-channel reaction matrix theory and configuration-interaction in the discrete and in the continuous spectrum. Inclusion of closed channels and derivation of quantum defect theory

We present a derivation of Multichannel Quantum Defect Theory (MQDT) which is based on the configuration-interaction in the continuum (CIC) and reaction matrix (K-matrix) formalisms, generalized so that closed channels are included. The whole development focuses on obtaining and utilizing eigenfunctions composed of smooth parts only. Whereas the theories of Seaton and Fano depend on the use of irregular or unbound functions, ours does not. No separation of configuration space into inner and outer space is necessary. Furthermore, the present theory allows in a practical way the transfer of information from one part of the spectrum to another byinterpolation — as opposed to the standard ab initio MQDT which is limited toextrapolation from the continuous to the discrete spectrum. The formalism is developed for the physically important Coulomb potential. However, its structure is such that it allows the easy determination of those parts which must be altered to accomodate extension to other potentials. Furthermore, it permits the execution of reliable calculations which employ a basis of numerical fixed-core Hartree-Fock (HF) functions for the singly excited configurations, of multiconfigurational HF (MCHF) for the spectroscopic valence states and of analytic virtual orbitals for the correlation vectors.

Gauss quadrature generated by diagonalization ofHin finiteL2bases

Using the $J$ matrix method, diagonalization of ${H}_{0}+V$ in certain finite ${L}^{2}$ basis sets is shown to generate a Gauss quadrature of the continuum. Explicit formulas for the corresponding weight function and orthogonal polynomials are given. This leads to a particularly simple expression for the Fredholm determinant and to the equivalence of the $J$ matrix and Fredholm equivalent-quadrature methods. Consideration of the asymptotic behavior of the polynomials for large degree results in a proof of Heller's derivative rule relating the spacing of pseudostate eigenvalues to the relative normalization of pseudostate and continuum matrix elements, and providing an alternative to Stieltjes imaging for the solution of the classical moment problem. Applications include an accurate quadrature of the sum over a complete set of intermediate states in perturbation expansions.

Continuum calculus and Feynman’s path integrals

An operational calculus is set up with the specific aim of resolving the problem of the integral of functionals in a general complex Banach algebra. The functionals that occur frequently in physics are Feynman’s path integrals for quantum mechanics which usually appear in the form of an exponential of an integral. Through the establishment of two operations, the r differentiation and p integration, we succeed in constructing a formula, in closed form, for the integrals of Feynman type functionals. Applications to known problems (quantum harmonic oscillator, the electron–phonon system) corroborate conventional results. This formula is found to be at once consistent and more general than the method of projection into cylinder functionals of Frederichs type. It does not require an additive Gaussian measure, and it admits integration with finite limit functions. The methodology developed is general and applicable to other branches of mathematics. It is particularly suited to the study of infinitely divisible distributions in probability theory. We rederive, with facility, Lévy’s formulas for continuous sums, and Lévy–Khintchine and Kolmogorov formulas. We also find it applicable to continuum matrix algebra, where the formula for the determinant of matrices of continuous indices is given as a p integral. As to algebraic identities, we give a continuum version of the binary expansion, and retrieve Stirling’s formula of factorials by p integration. The idée-clef lies in the concept of infinitesimal ratio of a function in the same way that differential calculus deals with infinitesimal differences. Then the functional integral appears to be a natural product of the interaction between the conventional integration and the proposed p integration. It also heralds the possibility of a generalized measure theory for integrals where the basic operation between the measure and the integrand is not bilinear.

Antisymmetrization of the Theory of Stripping Reactions

The Butler-Hewitt-McKellar-May theory of stripping reactions is modified to take into account the blocking effect of particle-hole correlations in the core, which populate the final neutron state with probability $1\ensuremath{-}P$. The basic relation of the theory between the direct-reaction matrix element ${M}_{C}$ and the continuum matrix element ${M}_{S}$ is altered to ${M}_{C}={S}^{\frac{1}{2}}{(P\ensuremath{-}S)}^{\ensuremath{-}1}{M}_{S}$.

Analysis of (d,p) reactions

An extension of the method of Butler et al. of (d,p) analysis is noted. The fact that there must be a correlation between spectroscopic factors for discrete states and continuum matrix elements is made clear. Application of this study to the 40Ca(d,p)41Ca reaction confirms a ground state spectroscopic factor in the vicinity of 0.6. It appears clear that there must be an 72 excited state in the 41Ca spectrum to take up the remaining single particle strength. It is possible that this could be a sharp 72 resonance in the low energy continuum.