Algebraic Treatment Research Articles

Discrete Method of Images for 3D Radio Propagation Modeling

Discretization by rasterization is introduced into the method of images (MI) in the context of 3D deterministic radio propagation modeling as a way to exploit spatial coherence of electromagnetic propagation for fine-grained parallelism. Traditional algebraic treatment of bounding regions and surfaces is replaced by computer graphics rendering of 3D reflections and double refractions while building the image tree. The visibility of reception points and surfaces is also resolved by shader programs. The proposed rasterization is shown to be of comparable run time to that of the fundamentally parallel shooting and bouncing rays. The rasterization does not affect the signal evaluation backtracking step, thus preserving its advantage over the brute force ray-tracing methods in terms of accuracy. Moreover, the rendering resolution may be scaled back for a given level of scenario detail with only marginal impact on the image tree size. This allows selection of scene optimized execution parameters for faster execution, giving the method a competitive edge. The proposed variant of MI can be run on any GPU that supports real-time 3D graphics.

Anharmonicity effects in the frictionlike mode of graphite

Graphite is a prototypical solid lubricant demanding a thorough understanding of its low-friction behavior. The ${E}_{2g}$(1) Raman active vibrational mode of graphite is associated with the rigid-layer relative movement of its graphene sheets. Thus, this mode can provide a good means of exploring the low resistance of graphene layers to slip with respect to each other. To take advantage of this fact, the anharmonicity of the ${E}_{2g}$(1) mode has to be carefully characterized and evaluated since the atomic arrangement of carbon atoms in the ambient condition ABA stacking of graphite evidences potential asymmetry. The calculated one-dimensional energetic profile of the ${E}_{2g}$(1) mode reveals this local anisotropy around the energy minima and can be microscopically interpreted in terms of electron density interactions. Morse-type potentials accurately fit the energetic profiles at different interlayer separations, and provide simple analytical expressions for evaluating harmonic and anharmonic contributions to the $\mathrm{\ensuremath{\Gamma}}$-point ${E}_{2g}$(1) frequency ${\ensuremath{\omega}}_{{E}_{2g}(1)}$ under a perturbative algebraic treatment. We quantify how the anharmonic contribution increases with the available energy ($E$) at zero pressure, and how this contribution decreases as hydrostatic pressure ($p$) or uniaxial stress is applied for a given available energy. The calculated ${\ensuremath{\omega}}_{{E}_{2g}(1)}\ensuremath{-}p$ and ${\ensuremath{\omega}}_{{E}_{2g}(1)}\ensuremath{-}E$ trends indicate an increasing (decreasing) of frictional forces in graphite with pressure (temperature). Our conclusions are supported by the good agreement of the calculated frequencies with existing Raman experiments under hydrostatic pressure conditions.

An algebraic study of BLUPs under two linear random-effects models with correlated covariance matrices

Assume that a pair of general Linear Random-effects Models (LRMs) are given with a correlated covariance matrix for their error terms. This paper presents an algebraic approach to the statistical analysis and inference of the two correlated LRMs using some state-of-the-art formulas in linear algebra and matrix theory. It is shown first that the best linear unbiased predictors (BLUPs) of all unknown parameters under LRMs can be determined by certain linear matrix equations, and thus the BLUPs under the two LRMs can be obtained in exact algebraic expressions. We also discuss algebraical and statistical properties of the BLUPs, as well as some additive decompositions of the BLUPs. In particular, we present necessary and sufficient conditions for the separated and simultaneous BLUPs to be equivalent. The whole work provides direct access to a very simple algebraic treatment of predictors/estimators under two LRMs with correlated covariance matrices.

Topological features in crystal structures: a quotient graph assisted analysis of underlying nets and their embeddings

Topological properties of crystal structures may be analysed at different levels, depending on the representation and the topology that has been assigned to the crystal. Considered here is the combinatorial or bond topology of the structure, which is independent of its realization in space. Periodic nets representing one-dimensional complexes, or the associated graphs, characterize the skeleton of chemical bonds within the crystal. Since periodic nets can be faithfully represented by their labelled quotient graphs, it may be inferred that their topological features can be recovered by a direct analysis of the labelled quotient graph. Evidence is given for ring analysis and structure decomposition into building units and building networks. An algebraic treatment is developed for ring analysis and thoroughly applied to a description of coesite. Building units can be finite or infinite, corresponding to 1-, 2- or even 3-periodic subnets. The list of infinite units includes linear chains or sheets of corner- or edge-sharing polyhedra. Decomposing periodic nets into their building units relies on graph-theoretical methods classified as surgery techniques. The most relevant operations are edge subdivision, vertex identification, edge contraction and decoration. Instead, these operations can be performed on labelled quotient graphs, evidencing in almost a mechanical way the nature and connection mode of building units in the derived net. Various examples are discussed, ranging from finite building blocks to 3-periodic subnets. Among others, the structures of strontium oxychloride, spinel, lithiophilite and garnet are addressed.

Momentum space orthogonal polynomial projection quantization

The orthogonal polynomial projection quantization (OPPQ) is an algebraic method for solving Schrödinger’s equation by representing the wave function as an expansion in terms of polynomials orthogonal with respect to a suitable reference function R(x), which decays asymptotically not faster than the bound state wave function. The expansion coefficients are obtained as linear combinations of power moments . In turn, the 's are generated by a linear recursion relation derived from Schrödinger’s equation from an initial set of low order moments. It can be readily argued that for square integrable wave functions representing physical states . Rapidly converging discrete energies are obtained by setting Ω coefficients to zero at arbitrarily high order. This paper introduces an extention of OPPQ in momentum space by using the representation , where Qn(k) are polynomials orthogonal with respect to a suitable reference function T(k). The advantage of this new representation is that it can help solving problems for which there is no coordinate space moment equation. This is because the power moments in momentum space are the Taylor expansion coefficients, which are recursively calculated via Schrödinger’s equation. We show the convergence of this new method for the sextic anharmonic oscillator and an algebraic treatment of Gross–Pitaevskii nonlinear equation.

The Decision Problem of Modal Product Logics with a Diagonal, and Faulty Counter Machines

In the propositional modal (and algebraic) treatment of two-variable first-order logic equality is modelled by a `diagonal' constant, interpreted in square products of universal frames as the identity (also known as the `diagonal') relation. Here we study the decision problem of products of two arbitrary modal logics equipped with such a diagonal. As the presence or absence of equality in two-variable first-order logic does not influence the complexity of its satisfiability problem, one might expect that adding a diagonal to product logics in general is similarly harmless. We show that this is far from being the case, and there can be quite a big jump in complexity, even from decidable to the highly undecidable. Our undecidable logics can also be viewed as new fragments of first- order logic where adding equality changes a decidable fragment to undecidable. We prove our results by a novel application of counter machine problems. While our formalism apparently cannot force reliable counter machine computations directly, the presence of a unique diagonal in the models makes it possible to encode both lossy and insertion-error computations, for the same sequence of instructions. We show that, given such a pair of faulty computations, it is then possible to reconstruct a reliable run from them.

Algebraic Reynolds stress modeling of turbulence subject to rapid homogeneous and non-homogeneous compression or expansion

A recently developed explicit algebraic Reynolds stress model (EARSM) by Grigoriev et al. [“A realizable explicit algebraic Reynolds stress model for compressible turbulent flow with significant mean dilatation,” Phys. Fluids 25(10), 105112 (2013)] and the related differential Reynolds stress model (DRSM) are used to investigate the influence of homogeneous shear and compression on the evolution of turbulence in the limit of rapid distortion theory (RDT). The DRSM predictions of the turbulence kinetic energy evolution are in reasonable agreement with RDT while the evolution of diagonal components of anisotropy correctly captures the essential features, which is not the case for standard compressible extensions of DRSMs. The EARSM is shown to give a realizable anisotropy tensor and a correct trend of the growth of turbulence kinetic energy K, which saturates at a power law growth versus compression ratio, as well as retaining a normalized strain in the RDT regime. In contrast, an eddy-viscosity model results in a rapid exponential growth of K and excludes both realizability and high magnitude of the strain rate. We illustrate the importance of using a proper algebraic treatment of EARSM in systems with high values of dilatation and vorticity but low shear. A homogeneously compressed and rotating gas cloud with cylindrical symmetry, related to astrophysical flows and swirling supercritical flows, was investigated too. We also outline the extension of DRSM and EARSM to include the effect of non-homogeneous density coupled with “local mean acceleration” which can be important for, e.g., stratified flows or flows with heat release. A fixed-point analysis of direct numerical simulation data of combustion in a wall-jet flow demonstrates that our model gives quantitatively correct predictions of both streamwise and cross-stream components of turbulent density flux as well as their influence on the anisotropies. In summary, we believe that our approach, based on a proper formulation of the rapid pressure-strain correlation and accounting for the coupling with turbulent density flux, can be an important element in CFD tools for compressible flows.

Solutions of a constrained Hermitian matrix-valued function optimization problem with applications

Abstract. Let f(X) = (XC + D )M(XC + D) − G be a given nonlinear Hermitian matrix-valued function with M = M∗ and G = G∗, and assume that the variable matrix X satisfies the consistent linear matrix equation XA = B. This paper shows how to characterize the semi-definiteness of f(X) subject to all solutions of XA = B. As applications, a standard method is obtained for finding analytical solutions X0 of X0A = B such that the matrix inequality f(X) < f(X0) or f(X) 4 f(X0) holds for all solutions of XA = B. The whole work provides direct access, as a standard example, to a very simple algebraic treatment of the constrained Hermitian matrix-valued function and the corresponding semi-definiteness and optimization problems.

Simple Algebraic Treatment of Unsteady Heat Conduction in Solid Spheres with Heat Convection Exchange to Fluids Covering the Entire Time Domain

Simple Algebraic Treatment of Unsteady Heat Conduction in Solid Spheres with Heat Convection Exchange to Fluids Covering the Entire Time Domain

Simple Algebraic Treatment of Unsteady Heat Conduction in Solid Spheres with Heat Convection Exchange to Fluids Covering the Entire Time Domain

Accurate estimates of temperature histories in solid bodies of regular shape (slabs, cylinders and spheres) exposed to cold fluids have been traditionally done by the method of separation of variables evaluating infinite Fourier series. The infinite Fourier series correspond to the exact, analytic solution of the unidirectional heat conduction equation in various coordinate systems. The central goal of this technical paper is to bypass this traditional procedure involving infinite Fourier series. The idea is to predict the three most important temperatures (mean, surface and center) in a solid sphere by implementing a new 1-D composite lumped analysis, which constitutes a natural extension of the standard 1-D lumped analysis. The computational methodology to be proposed is effortless and brings to the table a handful of compact algebraic equations for the mean, surface and center temperatures that cover the complete gamma of the controlling Biot numbers (0 < Bi < 100) in the entire time domain, i.e., at all dimensionless times or Fourier numbers (0 < τ < ∞).

Ground-state stabilization of quantum finite-level systems by dissipation

Control by dissipation, or environment engineering, constitutes an important methodology within quantum coherent control which was proposed to improve the robustness and scalability of quantum control systems. The system–environment coupling, often considered to be detrimental to quantum coherence, also provides the means to steer the system to desired states. This paper aims to develop the theory for engineering of the dissipation, based on a ground-state Lyapunov stability analysis of open quantum systems via a Heisenberg-picture approach. In particular, Lyapunov stability conditions expressed as operator inequalities allow a purely algebraic treatment of the environment engineering problem, which facilitates the integration of quantum components into a large-scale quantum system and draws an explicit connection to the classical theory of vector Lyapunov functions and decomposition–aggregation methods for control of complex systems. This leads to tractable algebraic conditions concerning the ground-state stability and scalability of quantum systems. The implications of the results in relation to dissipative quantum computing and state engineering are also discussed in this paper.

The Lie Group Structure of the Butcher Group

The Butcher group is a powerful tool to analyse integration methods for ordinary differential equations, in particular Runge--Kutta methods. In the present paper, we complement the algebraic treatment of the Butcher group with a natural infinite-dimensional Lie group structure. This structure turns the Butcher group into a real analytic Baker--Campbell--Hausdorff Lie group modelled on a Fr\'echet space. In addition, the Butcher group is a regular Lie group in the sense of Milnor and contains the subgroup of symplectic tree maps as a closed Lie subgroup. Finally, we also compute the Lie algebra of the Butcher group and discuss its relation to the Lie algebra associated to the Butcher group by Connes and Kreimer.

Residual Weyl symmetry out of conformal geometry and its BRST structure

The conformal structure of second order in $m$-dimensions together with the so-called (normal) conformal Cartan connection, is considered as a framework for gauge theories. The dressing field scheme presented in a previous work amounts to a decoupling of both the inversion and the Lorentz symmetries such that the residual gauge symmetry is the Weyl symmetry. On the one hand, it provides straightforwardly the Riemannian parametrization of the normal conformal Cartan connection and its curvature. On the other hand, it also provides the finite transformation laws under the Weyl rescaling of the various geometric objects involved. Subsequently, the dressing field method is shown to fit the BRS differential algebra treatment of infinitesimal gauge symmetry. The dressed ghost field encoding the residual Weyl symmetry is presented. The related so-called algebraic connection supplies relevant combinations found in the literature in the algebraic study of the Weyl anomaly.

Ladder operators and associated algebra for position-dependent effective mass systems

An algebraic treatment of shape-invariant quantum-mechanical position-dependent effective mass systems is discussed. Using shape invariance, a general recipe for construction of ladder operators and associated algebraic structure of the pertaining system, is obtained. These operators are used to find exact solutions of general one-dimensional systems with spatially varying mass. We apply our formalism to specific translationally shape-invariant potentials having position-dependent effective mass.

Organisation of the Analytical, Stoichiometric, and Thermodynamic Information for water Chemistry Calculations

Abstract A common feature of the chemical processes of the hydrosphere and water treatment plants is that essentially the same types of chemical equilibrium reactions occur in both fields. These equilibria could be acid/base, complexation, redox, precipitation, and interfacial processes. Since these reactions may also occur in combination, the aqueous environments are unavoidably multispecies systems. Due to multiple equilibria, the state of aggregation, the state of oxidation, as well as the electric charge of the species may change dramatically. Calculation of the equilibrium concentration of the species is facilitated by the availability of analytical, stoichiometric, and thermodynamic information that are consistently organised into an ASTI matrix. The matrix makes it possible to apply a uniform algebraic treatment for all occurring equilibria even, if later on, further reactions have to be included in the chemical model. The use of the ASTI matrix enables us to set up the necessary mass balance equations and equilibrium relationships, which together form a non-linear system of equations (NLSE). The goal of our paper is to show that the use of the ASTI matrix approach in cooperation with the powerful engineering calculation software, MATHCAD14, results in fast and easy handling of the NLSE-s and, consequently, the calculations of speciation in aqueous systems. The paper demonstrates the method of application in three examples: the calculation of the pH dependence of the solubility of calcite in closed and open systems, the calculation of the pH and pε in a system where acid/base reactions, complexation equilibria, and redox equilibria occur, and a study of adsorption of lead ions on aluminium oxide.

Algebraic and group treatments to nonlinear displaced number states and their nonclassicality features: A new approach

Recently, nonlinear displaced number states (NDNSs) have been manually introduced, in which the deformation function f(n) has been artificially added to the previously well-known displaced number states (DNSs). Indeed, just a simple comparison has been performed between the standard coherent state and nonlinear coherent state for the formation of NDNSs. In the present paper, after expressing enough physical motivation of our procedure, four distinct classes of NDNSs are presented by applying algebraic and group treatments. To achieve this purpose, by considering the DNSs and recalling the nonlinear coherent states formalism, the NDNSs are logically defined through an algebraic consideration. In addition, by using a particular class of Gilmore–Perelomov-type of SU(1, 1) and a class of SU(2) coherent states, the NDNSs are introduced via group-theoretical approach. Then, in order to examine the nonclassical behavior of these states, sub-Poissonian statistics by evaluating Mandel parameter and Wigner quasi-probability distribution function associated with the obtained NDNSs are discussed, in detail.

Duality relations for hypergeometric series

We explicitly give the relations between the hypergeometric solutions of the general hypergeometric equation and their duals, as well as similar relations for ‐hypergeometric equations. They form a family of very general identities for hypergeometric series. Although they were foreseen already by Bailey in the 1930s on analytic grounds, we give a purely algebraic treatment based on general principles in general differential and difference modules.

Quantum physics and signal processing in rigged Hilbert spaces by means of special functions, Lie algebras and Fourier and Fourier-like transforms

Quantum physics and signal processing in the line R are strictly related to Fourier transform and Weyl-Heisenberg algebra. We discuss here the addition of a new discrete variable that measures the degree of the Hermite functions and allows to obtain the projective algebra io(2). A rigged Hilbert space is found and a new discrete basis in R obtained. All operators defined on R are shown to belong to the universal enveloping algebra of io(2) allowing, in this way, their algebraic treatment. Introducing in the half-line a Fourier-like transform, the procedure is extended to R+ and can be easily generalized to Rn and to spherical cohordinate systems.

The structure of rotational bands in alpha-cluster nuclei

In this contribution, I discuss an algebraic treatment of alpha-cluster nuclei based on the introduction of a spectrum generating algebra for the relative motion of the alpha-clusters. Particular attention is paid to the discrete symmetry of the geometric arrangement of the α -particles, and the consequences for the structure of the rotational bands in the 12 C and 16 O nuclei.

Bond angles around a tetravalent atom.

Relationships among the six bond angles about a central tetravalent atom depend on symmetry, ranging from the most symmetrical Td point group to the least symmetrical C1 point group having only the identity element. Exact relationships are derived here in two ways: (1) a purely algebraic treatment of the general mathematical conditions among the bond angles, followed by factorizations that arise from various symmetry constraints and (2) a reverse approach based on geometric analysis, starting with the most symmetrical Td case and relaxing constraints stepwise to lower point groups. The mathematical formulas show systematically how the degrees of freedom among the bond angles increase from zero to a maximum of five as the symmetry is relaxed from the Td symmetry.