Non-standard Operators Research Articles

Wavelet-based finite element simulation of guided waves containing harmonics

This paper presents a promising numerical scheme for simulation of many harmonics in wave propagation. The wavelet-based adaptive technique eliminates the requirement for a very large number of nodes in finite element method for propagation of such waves. This dynamic adaptive grid selection is based on the fact that very few wavelet coefficients are required to represent a short pulse containing higher harmonics. The method is particularly useful where higher harmonics are ignored due to very high computational cost. In this work, B-spline and Daubechies wavelets-based non-standard (NS) multi-scale operator are applied, and the results are compared with the finite element method.

Explicit Iteration and Unique Positive Solution for a Caputo-Hadamard Fractional Turbulent Flow Model

Hadamard fractional calculus theory has made many scholars enthusiastic and excited because of its special logarithmic function integral kernel. In this paper, we focus on a class of Caputo-Hadamard-type fractional turbulent flow model involving $p(t)$ -Laplacian operator and Erdelyi-Kober fractional integral operator. The $p(t)$ -Laplacian operator involved in our model is the non-standard growth operator which arises in many fields such as elasticity theory, physics, nonlinear electrorheological fluids, ect. It is the first paper that studies a Caputo-Hadamard-type fractional turbulent flow model involving $p(t)$ -Laplacian operator and Erdelyi-Kober fractional integral operator. Different from the constant growth operator, The non-standard growth characteristics of $p(t)$ -Laplacian operator bring great difficulties and challenges. In order to achieve a good survey result, we take advantage of the popular mixed monotonic iterative technique. With the help of this approach, we obtain the uniqueness of positive solution for the new Caputo-Hadamard-type fractional turbulent flow model. In the end, an example is also given to illustrate the main results.

On a nonlocal problem involving a nonstandard nonhomogeneous differential operator

In this article, we are concerned with some class of nonlocal problems involving a nonstandard nonlocal and nonhomogeneous differential operator settled in Musielak–Orlicz–Sobolev spaces. We apply variational approach and look for nontrivial solutions, that is, local minimizers of the corresponding energy functional.

Anisotropic linear triangle finite element approximation for multi-term time-fractional mixed diffusion and diffusion-wave equations with variable coefficient on 2D bounded domain

A fully-discrete approximate scheme is established for the 2D multi-term time-fractional mixed diffusion and diffusion-wave equations with spatial variable coefficient by using linear triangle finite element method in space and classical L1 time-stepping method combined with Crank–Nicolson scheme in time. Then, the unconditionally stable analysis of the fully-discrete scheme is presented by employing some important lemmas. At the same time, both the spatial superclose property in H1-norm and convergence result in L2-norm are derived by skillfully dealing with numerical errors without any restrictions of time step τ and mesh size h. As a necessary way for obtaining the aimed numerical analysis, the relationship between the nonstandard projection operator Rh and the interpolation operator Ih of linear triangle finite element is introduced. Moreover, the global superconvergence is deduced by adopting interpolation postprocessing technique. Finally, numerical tests are provided to illustrate the efficiency and correctness of the theoretical results.

Exothermic dark matter with light mediator after LUX and PandaX-II in 2016

Dark matter (DM) direct detections are investigated for models with the following properties: isospin-violating couplings, exothermic scatterings, and/or a light mediator, with the aim to reduce the tension between the CDMS-Si positive signals and other negative searches. In particular, we focus on the non-standard effective operators which could lead to the spin-independent DM-nucleus scatterings with non-trivial dependences on the transfer momentum or DM velocity. As a result, such effective operator choices have the very mild effects on the final fittings. Furthermore, by including the latest constraints from LUX, PandaX-II, XENON1T and PICO-60, we find that, for almost all the considered models, the predicted CDMS-Si signal regions are either severely constrained or completely excluded by the LUX, PandaX-II, XENON1T and PICO-60 data, including the most promising Xe-phobic exothermic DM models with/without a light mediator. Therefore, we conclude that it is very difficult for the present DM framework to explain the CDMS-Si excess.

Compatible Maxwell solvers with particles I: conforming and non-conforming 2D schemes with a strong Ampere law

This article is the second of a series where we develop and analyze structure-preserving finite element discretizations for the time-dependent 2D Maxwell system with long-time stability properties, and propose a charge-conserving deposition scheme to extend the stability properties in the case where the current source is provided by a particle method. The schemes proposed here derive from a previous study where a generalized commuting diagram was identified as an abstract compatibility criterion in the design of stable schemes for the Maxwell system alone, and applied to build a series of conforming and non-conforming schemes in the 3D case. Here the theory is extended to account for approximate sources, and specific charge-conserving schemes are provided for the 2D case. In this second article we study two schemes which include a strong discretization of the Faraday law. The first one is based on a standard conforming mixed finite element discretization and the long-time stability is ensured by the natural $L^2$ projection for the current, also standard. The second one is a new non-conforming variant where the numerical fields are sought in fully discontinuous spaces. In this 2D setting it is shown that the associated discrete curl operator coincides with that of a classical DG formulation with centered fluxes, and our analysis shows that a non-standard current approximation operator must be used to yield a charge-conserving scheme with long-time stability properties, while retaining the local nature of $L^2$ projections in discontinuous spaces. Numerical experiments involving Maxwell and Maxwell-Vlasov problems are then provided to validate the stability of the proposed methods.

Non-reciprocal wave propagation in modulated elastic metamaterials

Time-reversal symmetry for elastic wave propagation breaks down in a resonant mass-in-mass lattice whose inner-stiffness is weakly modulated in space and in time in a wave-like fashion. Specifically, one-way wave transmission, conversion and amplification as well as unidirectional wave blocking are demonstrated analytically through an asymptotic analysis based on coupled mode theory and numerically thanks to a series of simulations in harmonic and transient regimes. High-amplitude modulations are then explored in the homogenization limit where a non-standard effective mass operator is recovered and shown to take negative values over unusually large frequency bands. These modulated metamaterials, which exhibit either non-reciprocal behaviours or non-standard effective mass operators, offer promise for applications in the field of elastic wave control in general and in one-way conversion/amplification in particular.

Multidimensional effective field theory analysis for direct detection of dark matter

The scattering of dark matter particles off nuclei in direct detection experiments can be described in terms of a multidimensional effective field theory (EFT). A new systematic analysis technique is developed using the EFT approach and Bayesian inference methods to exploit, when possible, the energy-dependent information of the detected events, experimental efficiencies, and backgrounds. Highly dimensional likelihoods are calculated over the mass of the weakly interacting massive particle (WIMP) and multiple EFT coupling coefficients, which can then be used to set limits on these parameters and choose models (EFT operators) that best fit the direct detection data. Expanding the parameter space beyond the standard spin-independent isoscalar cross section and WIMP mass reduces tensions between previously published experiments. Combining these experiments to form a single joint likelihood leads to stronger limits than when each experiment is considered on its own. Simulations using two nonstandard operators (3 and 8) are used to test the proposed analysis technique in up to five dimensions and demonstrate the importance of using multiple likelihood projections when determining constraints on WIMP mass and EFT coupling coefficients. In particular, this shows that an explicit momentum dependence in dark matter scattering can be identified.

Generalised form factor dark matter in the Sun

We study the effects of energy transport in the Sun by asymmetric dark matter with momentum and velocity-dependent interactions, with an eye to solving the decade-old Solar Abundance Problem. We study effective theories where the dark matter-nucleon scattering cross-section goes as vrel2n and q2n with n = −1, 0, 1 or 2, where vrel is the dark matter-nucleon relative velocity and q is the momentum exchanged in the collision. Such cross-sections can arise generically as leading terms from the most basic nonstandard DM-quark operators. We employ a high-precision solar simulation code to study the impact on solar neutrino rates, the sound speed profile, convective zone depth, surface helium abundance and small frequency separations. We find that the majority of models that improve agreement with the observed sound speed profile and depth of the convection zone also reduce neutrino fluxes beyond the level that can be reasonably accommodated by measurement and theory errors. However, a few specific points in parameter space yield a significant overall improvement. A 3–5 GeV DM particle with σSI ∝ q2 is particularly appealing, yielding more than a 6σ improvement with respect to standard solar models, while being allowed by direct detection and collider limits. We provide full analytical capture expressions for q- and vrel-dependent scattering, as well as complete likelihood tables for all models.

A simple and effective axisymmetric convected Helmholtz integral equation

In this paper, we develop an axisymmetric boundary integral equation that derives from a reformulation of the 3D Helmholtz integral formula for the acoustic radiation problems in a subsonic uniform flow. Through the use of a new non-standard derivative operator, the axisymmetric convected Helmholtz integral equation substantially reduces the effects of flow incorporated in the classical convected boundary integral formulations, and involved in the normal derivative and the derivative in the flow direction of the axisymmetric convected Green's function. As for the free term derived from the singular integrals, it is given by a new expression independent of complete elliptic integrals and evaluated analytically as a convected angle in the meridian plane. The numerical treatment of singular integrals requires only the use of standard Gauss quadrature rules. Different test cases are presented.

Transmutations of supersymmetry through soliton scattering and self-consistent condensates

We consider the two most general families of the (1+1)D Dirac systems with transparent scalar potentials, and two related families of the paired reflectionless Schrodinger operators. The ordinary N=2 supersymmetry for such Schrodinger pairs is enlarged up to an exotic N=4 nonlinear centrally extended supersymmetric structure, which involves two bosonic integrals composed from the Lax-Novikov operators for the stationary Korteweg-de Vries hierarchy. Each associated single Dirac system displays a proper N=2 nonlinear supersymmetry with a non-standard grading operator. One of the two families of the first and second order systems exhibits the unbroken supersymmetry, while another is described by the broken exotic supersymmetry. The two families are shown to be mutually transmuted by applying a certain limit procedure to the soliton scattering data. We relate the topologically trivial and nontrivial transparent potentials with self-consistent inhomogeneous condensates in Bogoliubov-de Gennes and Gross-Neveu models, and indicate the exotic N=4 nonlinear supersymmetry of the paired reflectionless Dirac systems.

Elliptic interpolation estimates for non-standard growth operators

We derive a class of potential estimates for elliptic equations with non-standard growth having a measure on the right-hand side; in particular our results allow to interpolate between the pointwise estimates available for solutions to this equation and the ones for the gradient. We allow bounded coefficients in BMO and VMO classes.

Beta decays and non-standard interactions in the LHC era

We consider the role of precision measurements of beta decays and light meson semi-leptonic decays in probing physics beyond the Standard Model in the LHC era. We describe all low-energy charged-current processes within and beyond the Standard Model using an effective field theory framework. We first discuss the theoretical hadronic input which in these precision tests plays a crucial role in setting the baseline for new physics searches. We then review the current and upcoming constraints on the various non-standard operators from the study of decay rates, spectra, and correlations in a broad array of light-quark systems. We finally discuss the interplay with LHC searches, both within models and in an effective theory approach. Our discussion illustrates the independent yet complementary nature of precision beta decay measurements as probes of new physics, showing them to be of continuing importance throughout the LHC era.

Periodic Homogenization of Parabolic Nonstandard Monotone Operators

We study the periodic homogenization for a family of parabolic problems with nonstandard monotone operators leading to Orlicz spaces. After proving the existence theorem based on the classical Galerkin procedure combined with the Stone-Weierstrass theorem, the fundamental in this topic is the determination of the global homogenized problem via the two-scale convergence method adapted to this type of spaces.

Convergence analysis of finite element methods for H(curl; Ω)-elliptic interface problems

In this article we investigate the analysis of a finite element method for solving H(curl; Ω)-elliptic interface problems in general three-dimensional polyhedral domains with smooth interfaces. The continuous problems are discretized by means of the first family of lowest order Nedelec H(curl; Ω)-conforming finite elements on a family of tetrahedral meshes which resolve the smooth interface in the sense of sufficient approximation in terms of a parameter ? that quantifies the mismatch between the smooth interface and the triangulation. Optimal error estimates in the H(curl; Ω)-norm are obtained for the first time. The analysis is based on a ?-strip argument, a new extension theorem for H 1(curl; Ω)-functions across smooth interfaces, a novel non-standard interface-aware interpolation operator, and a perturbation argument for degrees of freedom for H(curl; Ω)-conforming finite elements. Numerical tests are presented to verify the theoretical predictions and confirm the optimal order convergence of the numerical solution.

New physics int→bWdecay at next-to-leading order in QCD

We consider contributions of non-standard tbW effective operators to the decay of an unpolarized top quark into a bottom quark and a W gauge boson at next-to-leading order in QCD. We find that the dipole operator O_{LR} contribution to the transverse-plus W helicity fraction F_+ is significantly enhanced compared to the leading order result at non-vanishing bottom quark mass. Nonetheless, presently the most sensitive observable to direct O_{LR} contributions is the longitudinal W helicity fraction F_L. In particular, the most recent CDF measurement of F_L already provides the most stringent upper bound on O_{LR} contributions, even when compared with indirect bounds from the rare decay B -> X_s gamma.

Convergence analysis of finite element methods for H(div;Ω)-elliptic interface problems

In this article we analyze a finite element method for solving H(div;Ω)-elliptic interface problems in general three-dimensional Lipschitz domains with smooth material interfaces. The continuous problems are discretized by means of lowest order H(div;Ω)-conforming finite elements of the first family (Raviart–Thomas or Nédélec face elements) on a family of unstructured oriented tetrahedral meshes. These resolve the smooth interface in the sense of sufficient approximation in terms of a parameter δ that quantifies the mismatch between the smooth interface and the finite element mesh. Optimal error estimates in the H(div;Ω)-norms are obtained for the first time. The analysis is based on a so-called δ-strip argument, a new extension theorem for H1(div)-functions across smooth interfaces, a novel non-standard interfaceaware interpolation operator, and a perturbation argument for degrees of freedom in H(div;Ω)-conforming finite elements. Numerical tests are presented to verify the theoretical predictions and confirm the optimal order convergence of the numerical solution.

Model-independent constraints on ΔF= 2 operators and the scale of new physics

We update the constraints on new-physics contributions to Delta F=2 processes from the generalized unitarity triangle analysis, including the most recent experimental developments. Based on these constraints, we derive upper bounds on the coefficients of the most general Delta F=2 effective Hamiltonian. These upper bounds can be translated into lower bounds on the scale of new physics that contributes to these low-energy effective interactions. We point out that, due to the enhancement in the renormalization group evolution and in the matrix elements, the coefficients of non-standard operators are much more constrained than the coefficient of the operator present in the Standard Model. Therefore, the scale of new physics in models that generate new Delta F=2 operators, such as next-to-minimal flavour violation, has to be much higher than the scale of minimal flavour violation, and it most probably lies beyond the reach of direct searches at the LHC.

Homogenization of spectral problems in bounded domains with doubly high contrasts

Homogenization of a spectral problem in a bounded domain with a high contrast in both stiffness and density is considered. For a special critical scaling, two-scale asymptotic expansions for eigenvalues and eigenfunctions are constructed. Two-scale limit equations are derived and relate to certain non-standard self-adjoint operators. In particular they explicitly display the first two terms in the asymptotic expansion for the eigenvalues, with a surprising bound for the error of order $\varepsilon^{5/4}$ proved.

Nonstandard operator precedence in Excel

Microsoft's Excel spreadsheet program implements an unusual operator precedence in formulas. This can cause errors in statistical calculations, if it is not properly followed. For example, some errors by Excel in nonlinear regression reported by McCullough and Wilson (2002. On the accuracy of statistical procedures in Microsoft Excel 2000 and Excel XP. Comput. Statist. Data Anal. 40, 713–721) were probably due to programming by the users that did not follow the unusual operator precedence rules. In this note the operator precedence in Excel is explained, the previous nonlinear regression results are corrected, and a simple example of maximum likelihood estimation is given.