Scalar Basis Research Articles

Near-linear algebra

In this paper, we demonstrate that the realm of near-vector spaces enables us to address nonlinear problems while also providing access to most of the tools that linear algebra offers. We establish fundamental results for near-vector spaces, which serve to extend classical linear algebra into the realm of near-linear algebra. Within this paper, we finalize the algebraic proof that for a given scalar group [Formula: see text], any nonempty [Formula: see text]-subspace that remains stable under addition and scalar multiplication constitutes an [Formula: see text]-subspace. We prove that any quotient of a near-vector space by an [Formula: see text]-subspace is itself a near-vector space, along with presenting the First Isomorphism Theorem for near-vector spaces. In doing so, we obtain comprehensive descriptions of the span. By defining linear independence outside the quasi-kernel, we introduce a new concept of basis. We also establish that near-vector spaces are characterized based on the presence of a scalar basis.

Complex S3-symmetric 3HDM

CP violation plays a very important role in nature with implications both for Particle Physics and for Cosmology. Accounting for the observed matter-antimatter asymmetry of the Universe requires the existence of new sources of CP violation beyond the Standard Model. In models with an extended scalar sector CP violation can emerge either explicitly, i.e., at the Lagrangian level, or spontaneously. Spontaneous CP violation occurs in the framework of the electroweak symmetry breaking whenever the Lagrangian conserves CP and the vacuum breaks it. This requires that not all vacuum expectation values be real. In the context of multi-Higgs extensions of the Standard Model imposing the existence of a scalar basis where all couplings are real is a sufficient condition for CP to be explicitly conserved. We discuss a three-Higgs-doublet model with an underlying S3 symmetry, allowing in principle for complex couplings. In this framework it is possible to have either spontaneous or explicit CP violation in the scalar sector, depending on the regions of parameter space corresponding to the different possible vacua of the S3 symmetric potential. We list all possible vacuum structures allowing for CP violation in the scalar sector specifying whether it can be explicit or spontaneous. It is by now established that CP is violated in the flavour sector and that the Cabibbo-Kobayashi-Maskawa matrix is complex. In order to understand what are the possible sources of CP violation in the Yukawa sector we analyse the implications of the different available choices of representations for the quarks under the S3 group. This classification is based strictly on the exact S3-symmetric scalar potential with no soft symmetry breaking terms. The scalar sector of one such model was explored numerically. After applying the theoretical and the most important experimental constraints the available parameter space is shown to be able to give rise to light neutral scalars at the mathcal{O} (MeV) scale.

Explicit Conversion Formulas Between Spherical Wave Expansions With Scalar and Vector Expansion Coefficients

Explicit formulas for converting spherical wave expansions with vector basis functions and scalar expansion coefficients into spherical wave expansions with scalar basis functions and vector expansion coefficients and vice versa are presented. The formulas are given in terms of Wigner-3- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$j$ </tex-math></inline-formula> -symbols. The conversion formulas are derived by spherical harmonics expansions of products of two spherical harmonics and by levering on recurrence relations of the associated Legendre functions. The expansions with vector coefficients are redundant and explicit formulas are given for the linear combination of vector coefficients of which the expanded fields cancel identically to zero. The redundancy in the expansion can be used to find compact expansions with a minimum expansion order. For every vector spherical wave function, two expansions using scalar spherical harmonics are presented: on the one hand, a radial-component free expansion and on the other hand, a minimum-order expansion. The correctness of all expansions is verified with a simple computer code up to a mode order of 65. The agreement of the expanded near and far fields in both types of expansions is better than 14 digits of accuracy for most cases.

New recursion relations for tree-level correlators in anti–de Sitter spacetime

We present for the first time classical multiparticle solutions in Anti de Sitter space (AdS) involving scalars, gluons, and gravitons. They are recursively defined through multiparticle currents which reduce to Berends-Giele currents in the flat space limit. This construction exposes a compact definition of tree-level boundary correlators using a general prescription that removes unphysical boundary contributions. Similarly to the flat space perturbiner, a convenient gauge choice leads to a scalar basis for all degrees of freedom, while the tensor structure is exclusively captured by field theory vertices. This provides a fully automated way to compute AdS boundary correlators to any multiplicity and cosmological wavefunction coefficients after Wick-rotating to de Sitter space.

Theoretical investigations of electronic spectra of silver atom using all-electron scalar relativistic basis

Exploration of the tractable size basis set that can provide reliable estimates of computed properties for systems containing heavy elements has been the subject of interest in quantum chemistry over many decades. In this connection, the recently developed segmented all-electron relativistically contracted (SARC) basis set proposed by Rolfes et al. [J. Comput. Chem. 41, 1842 (2020)] appears to be worth studying. In the present attempt, ground and excited state properties of the Ag atom is computed on this basis at the Fock-space multi-reference coupled cluster (FSMRCC) level of theory with four-component relativistic spinors. The computed quantities resulting from the SARC basis are subsequently compared with those obtained using an even-tempered basis to assess the efficacy of the SARC basis. Computations have also been performed with the extended SARC basis to improve the quality of the property of interest. The accuracy of the computed quantities such ionization energy, electron affinity, excitation energies, etc., obtained using the SARC basis at the FSMRCC level of theory demonstrates that the SARC basis (particularly the extended one), which is primarily designed for quantum chemical calculations at the two-component scalar relativistic level for systems containing heavy atoms, can be used for fully relativistic calculations. The magnetic dipole hyperfine structure constant A, oscillator strengths f and transition rates Afi calculated using the SARC basis also agree well with the experiment and with the available theoretical estimates.

Parallel Higher Order DGTD and FETD for Transient Electromagnetic-Circuital-Thermal Co-Simulation

A hybrid higher order discontinuous Galerkin time-domain (DGTD) method and finite-element time-domain (FETD) method with parallel technique is proposed for electromagnetic (EM)–circuital–thermal co-simulation in this article. For electromagnetic simulation, DGTD method with higher order hierarchical vector basis functions is used to solve Maxwell equation. Circuit simulation is carried out by modified nodal analysis method. For thermal simulation, FETD method with higher order interpolation scalar basis functions is adopted to solve heat conduction equation. To implement electromagnetic–circuital–thermal co-simulation, the electromagnetic and circuital equations are strongly coupled through voltages, currents, and electric fields at the lumped ports first. Then the electromagnetic and thermal equations are weakly coupled with electromagnetic loss and temperature-dependent medium parameters. Finally, large-scale parallel technique is used to accelerate the process of multiphysics simulation. Numerical results are given to validate the correctness and capability of the proposed electromagnetic–circuital–thermal co-simulation method.

A hybrid spectral element-infinite element approach for 3D controlled-source electromagnetic modeling

In this study, we developed a novel hybrid approach for 3D controlled-source electromagnetic (CSEM) forward modeling. The hybrid approach combines vector spectral element method (SEM) with scalar infinite element method (IFEM) by using infinite element instead of traditional truncated boundary. Based on Maxwell's equations, the curl-curl control equation satisfied by secondary electric field is derived. The boundary value problem in differential form is transformed into integral weak form by using Galerkin weighted residual method. Then the global problem is discretized by element subdivision and high-order orthogonal basis function interpolation. The computational domain is discretized using hexahedral spectral elements and a single layer of infinite elements is added outside the domain. The SEM with high-order Gauss-Lobatto-Legendre (GLL) vector basis functions is used to represent the interior electric field, and the IFEM with high-order Gauss-Lobatto-Legendre (GLL) scalar basis functions is uesd as a boundary condition. Finally, the global numerical solution is obtained by solving large-scale linear equations. The layered model verifies the accuracy and correctness of the approach, and the 3D model verifies the efficiency and numerical stability of the approach. Numerical experiments show that the hybrid SEM-IFEM approach developed in this study can reach similar accuracy to the conventional spectral element method, but the computational domain required can be much smaller, and the corresponding CPU time and memory requirement are greatly reduced, namely, the high precision three-dimensional electromagnetic response can be obtained with less computational resource consumption.

Waves and heaviside propagator in transmission lines

We characterize all plane and modal waves of lossy multiconductor lines. They are obtained by solving matrix dierential equations in closed-form. First-order and second-order telegraph formulations are considered. Wavesare described in terms of a scalar basis coupled with a nite number of matrixiterates. Modal waves are decomposed in forward and backward propagators. This decomposition can be applied in obtaining transmission and reection matrices at junctions. Eects due to initial, boundary or external inputs are described in terms of an integral contribution due to initial data, a convolution due to external disturbances and a bilinear term involving boundary values. For a two conductor transmission line satisfying the Heaviside condition, the fundamental response is obtained for the rst-order system and for the telegraph formulation. Numerical examples are given for lossy and lossless transmission lines and for determination of initial values of steady solutions.

A Data‐Driven Approach to Functional Map Construction and Bases Pursuit

AbstractWe propose a method to simultaneously compute scalar basis functions with an associated functional map for a given pair of triangle meshes. Unlike previous techniques that put emphasis on smoothness with respect to the Laplace–Beltrami operator and thus favor low‐frequency eigenfunctions, we aim for a basis that allows for better feature matching. This change of perspective introduces many degrees of freedom into the problem allowing to better exploit non‐smooth descriptors. To effectively search in this high‐dimensional space of solutions, we incorporate into our minimization state‐of‐the‐art regularizers. We solve the resulting highly non‐linear and non‐convex problem using an iterative scheme via the Alternating Direction Method of Multipliers. At each step, our optimization involves simple to solve linear or Sylvester‐type equations. In practice, our method performs well in terms of convergence, and we additionally show that it is similar to a provably convergent problem. We show the advantages of our approach by extensively testing it on multiple datasets in a few applications including shape matching, consistent quadrangulation and scalar function transfer.

All-electron scalar relativistic basis sets for the elements Rb-Xe.

Segmented all-electron relativistically contracted (SARC) basis sets are presented for the elements 37 Rb-54 Xe, for use with the second-order Douglas-Kroll-Hess approach and the zeroth-order regular approximation. The basis sets have a common set of exponents produced with established heuristic procedures, but have contractions optimized individually for each scalar relativistic Hamiltonian. Their compact size and loose segmented contraction, which is in line with the construction of SARC basis sets for heavier elements, makes them suitable for routine calculations on large systems and when core spectroscopic properties are of interest. The basis sets are of triple-zeta quality and come in singly or doubly polarized versions, which are appropriate for both density functional theory and correlated wave function theory calculations. The quality of the basis sets is assessed against large decontracted reference basis sets for a number of atomic and ionic properties, while their general applicability is demonstrated with selected molecular examples.

Unsteady aerodynamic modeling based on fuzzy scalar radial basis function neural networks

In this paper, a fuzzy scalar radial basis function neural network is proposed, in order to overcome the limitation of traditional aerodynamic reduced-order models having difficulty in adapting to input variables with different orders of magnitude. This network is a combination of fuzzy rules and standard radial basis function neural network, and all the basis functions are defined as scalar basis functions. The use of scalar basis function will increase the flexibility of the model, thus enhancing the generalization capability on complex dynamic behaviors. Particle swarm optimization algorithm is used to find the optimal width of the scalar basis function. The constructed reduced-order models are used to model the unsteady aerodynamics of an airfoil in transonic flow. Results indicate that the proposed reduced-order models can capture the dynamic characteristics of lift coefficients at different reduced frequencies and amplitudes very accurately. Compared with the conventional reduced-order model based on recursive radial basis function neural network, the fuzzy scalar radial basis function neural network shows better generalization capability for different test cases with multiple normalization methods.

A 3-D Dispersive Time-Domain Meshless Formulation for Frequency-Dependent Materials

This communication presents a novel transient meshless formulation for analyzing wave propagation through linear dispersive materials. Many real materials are linear dispersive; thus, their constitutive parameters are not constant and change with frequency. However, the conventional numerical techniques are not proper tools for numerical analysis of dispersive materials, considering that they are formulated based on the assuming constant constitutive parameters for the materials. Therefore, until now, a number of numerical techniques have been proposed for modeling frequency behavior of dispersive materials. On the other hand, meshless methods are new and powerful numerical techniques which their capability in simulating the problem domain without using connection information among nodes makes them efficient techniques for modeling problems with complex geometries. In this communication, we have demonstrated that by incorporating some approaches into the meshless methods, we can turn them into suitable tools for modeling dispersive media. In order to obtain a meshless analysis of a dispersive medium, we have considered the scalar radial basis function meshless method, and for taking the frequency behavior of the medium into account, the auxiliary differential equations method has been used. Hence, the proposed dispersive meshless method not only models the frequency behavior of the dispersive materials but also provides more flexibility in simulating problems with complex geometries. In addition, the efficiency and accuracy of our proposed method are investigated by two numerical examples.

Scalar and electromagnetic nonparaxial bases composed as superpositions of simple vortex fields with complex foci.

We derive bases constructed from simple vortices and complex focus fields and show that they are useful in the description of strongly focused fields. Both scalar and electromagnetic fields are considered, and in each case two types of basis are discussed: bases that use standard polynomials but whose orthogonality condition requires a non-uniform directional weight factor, and bases that are orthogonal with uniform weight but that require new polynomials. Their performance is studied by fitting prescribed fields, where it is seen that the accuracy provided by both types of bases is comparable.

Comparison between staggered grid finite–volume and edge–based finite–element modelling of geophysical electromagnetic data on unstructured grids

This study compares two finite–element (FE) and three finite–volume (FV) schemes which use unstructured tetrahedral grids for the modelling of electromagnetic (EM) data. All these schemes belong to a group of differential methods where the electric field is defined along the edges of the elements. The FE and FV schemes are based on both the EM–field and the potential formulations of Maxwell's equations. The EM–field FE scheme uses edge–based (vector) basis functions while the potential FE scheme uses vector and scalar basis functions. All the FV schemes use staggered tetrahedral–Voronoï grids. Three examples are used for comparisons in terms of accuracy and in terms of the computation resources required by generic iterative and direct solvers for solving the problems. Two of these examples represent survey scenarios with electric and magnetic sources and the results are compared with those from the literature while the third example is a comparison against analytical solutions for an electric dipole source. Exactly the same mesh is used for all examples to allow for direct comparison of the various schemes. The results show that while the FE and FV schemes are comparable in terms of accuracy and computation resources, the FE schemes are slightly more accurate but also more expensive than the FV schemes.

Computationally fast EM field propagation through axi-symmetric media using cylindrical harmonic decomposition.

We describe and provide a systematic procedure for computationally fast propagation of arbitrary vector electromagnetic (EM) fields through an axially symmetric medium. A cylindrical harmonic field propagator is chosen for this purpose and in most cases, this is the best and the obvious choice. Firstly, we describe the cylindrical harmonic decomposition technique in terms of both scalar and vector basis for a given input excitation field. Then we formulate a generalized discrete Fourier-Hankel transform to achieve efficient vector basis decomposition. We allow a slower, pre-computation step, that finds a representation of the axi-symmetric medium as a transfer matrix in a discrete, cylindrical-harmonic basis. We find this matrix from a series of axi-symmetric (2D) finite element simulations (also known as the 2.5D technique). This transfer matrix approach significantly reduces the computational load when the transverse size or range exceeds about 30 wavelengths. This matrix is independent of the input excitation field for a given space-bandwidth product and hence makes it reusable for different excitation fields. We numerically validate the above approaches for different axi-symmetric EM scattering media which include a hemispherical gradient-index Maxwell's fish-eye lens, a transformation optics designed spherical invisibility cloak, a thin aspheric lens, and a cylindrical perfect lens.

Clearing Analysis on Phases: Exact Limiting Probabilities for Skip-Free, Unidirectional, Quasi-Birth-Death Processes

A variety of problems in computing, service, and manufacturing systems can be modeled via infinite repeating Markov chains with an infinite number of levels and a finite number of phases. Many such chains are quasi-birth-death processes with transitions that are skip-free in level, in that one can only transition between consecutive levels, and unidirectional in phase, in that one can only transition from lower-numbered phases to higher-numbered phases. We present a procedure, which we call Clearing Analysis on Phases (CAP), for determining the limiting probabilities of such Markov chains exactly. The CAP method yields the limiting probability of each state in the repeating portion of the chain as a linear combination of scalar bases raised to a power corresponding to the level of the state. The weights in these linear combinations can be determined by solving a finite system of linear equations.

The Efficient Mixed FEM With the Impedance Transmission Boundary Condition for Graphene Plasmonic Waveguides

A mixed finite-element method with an impedance transmission boundary condition (ITBC) is proposed to solve the graphene plasmonic modes. The new variational formulation combines the Gauss’ law with the transverse components of vectorial Helmholtz equation and models the field penetration through a thin graphene sheet by ITBC. The second-order edge-based vector LT/QN basis functions are applied to expand the transverse components of the electric field, and the nodal-based scalar basis functions are employed to discretize its longitudinal component. Numerical results on some designed graphene-based waveguides clearly demonstrate that the proposed method is efficient and accurate for the determination of graphene plasmonic modes.

Potential energy curves and spectroscopic parameters of the 24 Λ-S states and 54 Ω states of the F2 + cation including the spin-orbit coupling effect*

This work calculated the PECs of 24 Λ-S states and 54 Ω states of F2 + cation. The calculations were done with the CASSCF method, which was followed by the internally contracted MRCI approach. Core-valence correlation correction, scalar relativistic correction and basis set extrapolation were taken into account. Of these 24 Λ-S states, the 22Σg −, 22Σu −, 24Σg −, 14Δu, and 24Πg states were found to be repulsive. The X2Πg, A2Πu 14Δg, 14Πg and 24Πg states were found to be inverted with the spin-orbit coupling effect included. The 12Δg, 24Πu, 14Πg, 14Σu +, 22Πu, 14Σg −, 24Σu −, and 12Σg + states were found to be weakly bound. The 24Σu − state had double wells. The avoided crossings of PECs were observed between the A2Πu and 22Πu states, the X2Πg and 22Πg states, the 12Σu − and 22Σu − states, the 14Πu and 24Πu states, and the 14Σ− g and 24Σ− g states. Some spectroscopic parameters were determined and the vibrational properties of several weakly-bound states were predicted. The spin-orbit coupling effect on the spectroscopic parameters was evaluated. Comparison with available experimental data shows that the methodology used in this paper is highly accurate for this system.

On the resilience of scalar and vector vortex modes in turbulence.

Free-space optical communication with spatial modes of light has become topical due to the possibility of dramatically increasing communication bandwidth via Mode Division Multiplexing (MDM). While both scalar and vector vortex modes have been used as transmission bases, it has been suggested that the latter is more robust in turbulence. Using orbital angular momentum as an example, we demonstrate theoretically and experimentally that the crosstalk due to turbulence is the same in the scalar and vector basis sets of such modes. This work brings new insights about the behaviour of vector and scalar modes in turbulence, but more importantly it demonstrates that when considering optimal modes for MDM, the choice should not necessarily be based on their vectorial nature.

The Mixed Finite-Element Method With Mass Lumping for Computing Optical Waveguide Modes

A new mixed finite-element method is proposed for computing the anisotropic, lossy, and open waveguide modes. By incorporating Gauss' law into the vectorial wave equation, the variational formulation is completely free of spurious modes. It utilizes the curl-conforming linear tangential and quadratic normal (LT/QN) edge elements to expand the tangential component of the electric field, and the modified nodal-based scalar basis functions to expand the longitudinal component. A smaller scale generalized eigenvalue problem is obtained by employing the mass lumping technique. Numerical examples verify that the mixed finite-element method with mass lumping is free of any spurious eigenmode and has high efficiency.