Scalar Case Research Articles

Holomorphic functional calculus and vector-valued Littlewood–Paley–Stein theory for semigroups

We study vector-valued Littlewood–Paley–Stein theory for semigroups \{T_{t}\}_{t>0} of regular contractions on L_{p}(\Omega) for a fixed 1<p<\infty . We prove that if a Banach space X is of martingale cotype q , then there is a constant C such that \biggl\|\biggl(\int_{0}^{\infty}\biggl\|t\frac{\partial}{\partial t}P_{t} (f)\biggr\|_{X}^{q}\,\frac{dt}{t}\biggr)^{1/q}\biggr\|_{L_p(\Omega)}\le C\|f\|_{L_p(\Omega; X)}, \quad\forall f\in L_{p}(\Omega; X), where \{P_{t}\}_{t>0} is the Poisson semigroup subordinated to \{T_{t}\}_{t>0} . Let \mathsf{L}^{P}_{\mathsf{c}, q, p}(X) be the least constant C , and let \mathsf{M}_{\mathsf{c}, q}(X) be the martingale cotype q constant of X . We show \mathsf{L}^{P}_{\mathsf{c},q, p}(X)\lesssim \max(p^{1/q}, p') \mathsf{M}_{\mathsf{c},q}(X). Moreover, the order \max(p^{1/q}, p') is optimal as p\to1 and p\to\infty . If X is of martingale type q , the reverse inequality holds. If additionally \{T_{t}\}_{t>0} is analytic on L_{p}(\Omega; X) , the semigroup \{P_{t}\}_{t>0} in these results can be replaced by \{T_{t}\}_{t>0} itself. Our new approach is built on holomorphic functional calculus. Compared with the previous approaches, ours is more powerful in several aspects: (a) it permits us to go much further beyond the setting of symmetric submarkovian semigroups; (b) it yields the optimal orders of growth on p for most of the relevant constants; (c) it gives new insights into the scalar case for which our orders of the best constants in the classical Littlewood–Paley–Stein inequalities for symmetric submarkovian semigroups are better than those of Stein. In particular, we resolve a problem of Naor and Young on the optimal order of the best constant in the above inequality when X is of martingale cotype q and \{P_{t}\}_{t>0} is the classical Poisson or heat semigroup on \mathbb{R}^{d} .

Influence of Primary Coma on the Tightly Focusing Characteristics of Circular Basis Hybrid Order Poincaré Sphere Beams

Analogous to the Poincaré sphere, a hybrid order Poincaré sphere is used to represent the ellipse field singularities (C-points). We study the tight focusing properties of generic bright and dark hybrid order Poincaré sphere beams in the presence of primary coma. The role of the polarization singularity index and handedness of the polarization of the hybrid order Poincaré sphere beams on the focused structure has been discussed. Results have been presented for the total intensity, component intensities, and component phase distributions for left- and right-handed bright and dark star and lemon types singularities. The presence of primary coma distorted the focal plane intensity distributions for both positive and negative index generic C-points. Coma is known to disturb the circular symmetry of the focal plane intensity distribution. Similarly in tight focusing polarization is known to disturb the symmetry. Therefore, a beam with structured and inhomogeneous polarization distribution tightly focused under the influence of coma is a fit case to study. It is found that the presence of primary coma aberration in the focusing system produces a positional shift of the high-intensity peaks and a reduction of the intensity on one side of the center. As the strength of the primary coma increases, the focal plane intensity distributions shift more and more toward the right from the initial position. Unlike the scalar vortex case, in the case of hybrid order Poincaré sphere beams, the focal plane intensity distribution undergoes rotation, as the helicity of the hybrid order Poincaré sphere beams is inverted, in addition to shifting. All the component phase distributions are found to be embedded with phase vortices of charge ±1.

Gravitational action for a massive Majorana fermion in 2d quantum gravity

We compute the gravitational action of a free massive Majorana fermion coupled to two-dimensional gravity on compact Riemann surfaces of arbitrary genus. The structure is similar to the case of the massive scalar. The small-mass expansion of the gravitational yields the Liouville action at zeroth order, and we can identify the Mabuchi action at first order. While the massive Majorana action is a conformal deformation of the massless Majorana CFT, we find an action different from the one given by the David-Distler-Kawai (DDK) ansatz.

Modular Hamiltonian in the semi infinite line. Part II. Dimensional reduction of Dirac fermions in spherically symmetric regions

In this article, we extend our study on a new class of modular Hamiltonians on an interval attached to the origin on the semi-infinite line, introduced in a recent work dedicated to scalar fields. Here, we shift our attention to fermions and similarly to the scalar case, we investigate the modular Hamiltonians of theories which are obtained through dimensional reduction, this time, of a free massless Dirac field in d dimensions. By following the same methodology, we perform dimensional reduction on both the physical and modular Hamiltonians. This process enables us to establish a correspondence: we identify the modular Hamiltonian in an interval connected to the origin to the one obtained from the reduction of the modular Hamiltonian pertaining to the conformal parent theory on a sphere. Intriguingly, although the resulting one-dimensional theories lack conformal symmetry due to the presence of a term proportional to 1/r, the corresponding modular Hamiltonians are local functions in the energy density. This phenomenon mirrors the well known behaviour observed in the conformal case and indicates the existence of a residual symmetry, characterized by a subset of the original conformal group.Furthermore, through an analysis of the spectrum of the modular Hamiltonians, we derive an analytic expression for the associated entanglement entropy. Our findings also enable us to successfully recover the conformal anomaly coefficient from the universal piece of the entropy in even dimensions, as well as the universal constant F term in d = 3, by extending the radial regularization scheme originally introduced by Srednicki to perform the sum over angular modes.

Detectability of dense-environment effects on black-hole mergers: The scalar field case, higher-order ringdown modes, and parameter biases

Detectability of dense-environment effects on black-hole mergers: The scalar field case, higher-order ringdown modes, and parameter biases

Positive radial solutions for Dirichlet problems in the ball

We are concerned with existence, localization and multiplicity of positive radial solutions to Dirichlet problems with ϕ-Laplacians in a ball, in both scalar and system cases. Our approach essentially relies on fixed point index computations and a main feature is that it avoids any Harnack type inequality. Applications to some problems involving operators with Uhlenbeck structure are discussed.

Scalar QNM spectra of Kerr and Reissner-Nordström revealed by eigenvalue repulsions in Kerr-Newman

Recent studies of the gravito-electromagnetic frequency spectra of Kerr-Newman (KN) black holes have revealed two families of quasinormal modes (QNMs), namely photon sphere modes and near-horizon modes. However, they can only be unambiguously distinguished in the Reissner-Nordström (RN) limit, due to a phenomenon called eigenvalue repulsion (also known as level repulsion, avoided crossing or the Wigner-Teller effect), whereby the two families can interact strongly near extremality. We find that these features are also present in the QNM spectra of a scalar field in KN, where the perturbation modes are described by ODEs and thus easier to explore. Starting from the RN limit, we study how the scalar QNM spectra of KN dramatically changes as we vary the ratio of charge to angular momentum, all the way until the Kerr limit, while staying at a fixed distance from extremality. This scalar field case clarifies the (so far puzzling) relationship between the QNM spectra of RN and Kerr black holes and the nature of the eigenvalue repulsions in KN, that ultimately settle the fate of the QNM spectra in Kerr. We study not just the slowest-decaying QNMs (both for ℓ = m = 0 and ℓ = m = 2), but several sub-dominant overtones as well, as these turn out to play a crucial role understanding the KN QNM spectra. We also give a new high-order WKB expansion of KN QNMs that typically describes the photon sphere modes beyond the eikonal limit, and use a matched asymptotic expansion to get a very good approximation of the near-horizon modes near extremality.

Symmetry and monotonicity of singular solutions to p-Laplacian systems involving a first order term

Abstract We consider positive singular solutions (i.e. with a non-removable singularity) of a system of PDEs driven by p-Laplacian operators and with the additional presence of a nonlinear first order term. By a careful use of a rather new version of the moving plane method, we prove the symmetry of the solutions. The result is already new in the scalar case.

Hilfer proportional nonlocal fractional integro-multipoint boundary value problems

Abstract In this article, we introduce and study a boundary value problem for ( k , χ ¯ * ) \left(k,{\bar{\chi }}_{* }) -Hilfer generalized proportional fractional differential equation of order in an interval (1, 2], equipped with integro-multipoint nonlocal boundary conditions. In the scalar case setting, the existence results are proved via Leray-Schauder nonlinear alternative and Krasnosel’skiĭ’s fixed point theorem, while the existence of a unique solution is established by applying Banach’s contraction mapping principle. In Banach’s space setting, an existence result is proved via Mönch’s fixed point theorem and the measure of noncompactness. Finally, the obtained theoretical results are well illustrated by constructed examples.

Orientation-selective elliptic higher-order Poincaré sphere beam arrays

In this study, a new class of light-beam arrays, called orientation-selective elliptic higher-order Poincaré sphere (OS-EHOPS) beam arrays, is introduced as an extension of the scalar case reported in a previous study (Wang et al., 2020). For each beamlet of such beam arrays, the state of polarization represented by higher-/hybrid-order Poincaré spheres is spatially inhomogeneous, whereas the spatial profile takes on an elliptical shape with controllable orientation and ellipticity. The design procedure, which is rooted in multi-coordinate transformations, is comprehensively described and analyzed. Furthermore, we established an experimental setup involving a phase-only spatial light modulator that served as a Dammann grating and a common-path interferometer to effectively generate OS-EHOPS beam arrays. Two types of OS-EHOPS beam arrays, with hexagonal and double-ring-arranged structures, were synthesized to verify the feasibility of our method. Our design and optical system have potential applications in beam shaping and orientation-selective microfabrication.

Coupled-mode separation of multiply scattered wavefield components in two-dimensional waveguides.

For a waveguide that is invariant in one of the horizontal directions, this paper presents a mathematically exact partial-wave decomposition of the wavefield for assessment of multiple scattering by horizontally displaced medium anomalies. The decomposition is based on discrete coupled-mode theory and combination of reflection/transmission matrices. In particular, there is no high-frequency ray approximation. Full details are presented for the scalar case with plane-wave incidence from below. An application of interest concerns ground motion induced by seismic waves, which may be severely amplified by local medium anomalies such as alluvial valleys. Global optimization techniques are used to design an artificial medium termination at depth for a normal-mode representation of the field. A derivation of a horizontal source array to produce an incident plane wave gives, as a by-product, an extension of a previous Fourier-transform relation involving Bessel functions. Like purely numerical methods, such as finite differences and finite elements, the method can handle all kinds of (two-dimensional) anomaly shapes.

Convergence of Derivative-Free Iterative Methods with or without Memory in Banach Space

A method without memory as well as a method with memory are developed free of derivatives for solving equations in Banach spaces. The convergence order of these methods is established in the scalar case using Taylor expansions and hypotheses on higher-order derivatives which do not appear in these methods. But this way, their applicability is limited. That is why, in this paper, their local and semi-local convergence analyses (which have not been given previously) are provided using only the divided differences of order one, which actually appears in these methods. Moreover, we provide computable error distances and uniqueness of the solution results, which have not been given before. Since our technique is very general, it can be used to extend the applicability of other methods using linear operators with inverses along the same lines. Numerical experiments are also provided in this article to illustrate the theoretical results.

A Large Time Step Wave-Adding Simple Scheme for Unsteady Compressible Flows

In this paper, we construct a large time step wave-adding scheme (LTS-WA) for unsteady Euler equations in compressible flows with a much simpler strategy in the wave-adding procedure, named LTS-WAS. The new method does not need to calculate coefficients of contribution and also add the waves according to their types and families, which makes the scheme easy for coding and optimisation. The new scheme has a high speedup ratio, and its formulation is the same in scalar and system cases. The scheme is extended to 3D compressible flows using the dimension split method. Numerical experiments show that the new scheme maintains the advantages of the original LTS-WA scheme at large CFL numbers, and has a higher efficiency.

Inverse scattering problem for the matrix modified Korteweg–de Vries equation with finite density type initial data

The theory of inverse scattering is developed to study the initial-value problem for the matrix modified Korteweg–de Vries (mKdV) equation with the 2m×2m(m≥1) Lax pairs and finite density type initial data. In direct scattering problem, by introducing a suitable uniform transformation we establish the proper complex z-plane in order to discuss the Jost eigenfunctions, scattering matrix and their analyticity and symmetry. Moreover, the asymptotic behavior of the Jost eigenfunctions and scattering matrix are analyzed needed in inverse problem. Then, the generalized Riemann–Hilbert problem of the matrix mKdV equation is established by using the analyticity of the modified eigenfunctions and scattering coefficients, from which the reconstruction formula of potential function with reflection-less case is derived successfully. Finally, some interesting phenomena in scalar and matrix cases are presented, as well as theoretical analysis of asymptotic behavior of solutions in a appendix.

Asymptotic States and S-Matrix Operator in de Sitter Ambient Space Formalism

Within the de Sitter ambient space framework, the two different bases of the one-particle Hilbert space of the de Sitter group algebra are presented for the scalar case. Using field operator algebra and its Fock space construction in this formalism, we discuss the existence of asymptotic states in de Sitter QFT under an extension of the adiabatic hypothesis and prove the Fock space completeness theorem for the massive scalar field. We define the quantum state in the limit of future and past infinity on the de Sitter hyperboloid in an observer-independent way. These results allow us to examine the existence of the S-matrix operator for de Sitter QFT in ambient space formalism, a question which is usually obscure in spacetime with a cosmological event horizon for a specific observer. Some similarities and differences between QFT in Minkowski and de Sitter spaces are discussed.

On incorporation of heavy-quark mass into soft-wall holographic models

In this paper, we consider the soft-wall holographic model with the linear dilaton background. The model leads to a Hydrogen-like meson spectrum which can be interpreted as the static limit with very large quark masses when the Coulomb interaction dominates. The mass scale introduced by the linear dilaton is matched to the quark mass. The resulting model is analyzed for the scalar, vector and tensor cases. The electromagnetic coupling constants predicted by the model are decreasing with the radial number in contrast to the soft-wall model with quadratic dilaton where these couplings represent a universal constant. The given prediction is qualitatively consistent with the corresponding experimental data in vector quarkonia. The proposed model can thus be used as a constituent part of more elaborated holographic models for heavy quarkonia. A particular example of such a model is put forward.

Evaluation of the Resolution in Inverse Scattering of Dielectric Cylinders for Medical Applications.

The inverse scattering problem has numerous significant applications, including in geophysical explorations, medical imaging, and radar imaging. To achieve better performance of the imaging system, theoretical knowledge of the resolution of the algorithm is required for most of these applications. However, analytical investigations about the resolution presently feel inadequate. In order to estimate the achievable resolution, we address the point spread function (PSF) evaluation of the scattered field for a single frequency and the multi-view case both for the near and the far fields and the scalar case when the angular domain of the incident field and observation ranges is a round angle. Instead of the common free space condition, an inhomogeneous background medium, consisting of a homogeneous dielectric cylinder with a circular cross-section in free space, is assumed. In addition, since the exact evaluation of the PSF can only be accomplished numerically, an analytical approximation of the resolution is also considered. For the sake of its comparison, the truncated singular value decomposition (TSVD) algorithm can be used to implement the exact PSF. We show how the behavior of the singular values and the resolution change by varying the permittivity of the background medium. The usefulness of the theoretical discussion is demonstrated in localizing point-like scatterers within a dielectric cylinder, so mimicking a scenario that may occur in breast cancer imaging. Numerical results are provided to validate the analytical investigations.

Two-Component GW Calculations: Cubic Scaling Implementation and Comparison of Vertex-Corrected and Partially Self-Consistent GW Variants.

We report an all-electron, atomic orbital (AO)-based, two-component (2C) implementation of the GW approximation (GWA) for closed-shell molecules. Our algorithm is based on the space-time formulation of the GWA and uses analytical continuation (AC) of the self-energy, and pair-atomic density fitting (PADF) to switch between AO and auxiliary basis. By calculating the dynamical contribution to the GW self-energy at a quasi-one-component level, our 2C-GW algorithm is only about a factor of 2-3 slower than in the scalar relativistic case. Additionally, we present a 2C implementation of the simplest vertex correction to the self-energy, the statically screened G3W2 correction. Comparison of first ionization potentials (IPs) of a set of 67 molecules with heavy elements (a subset of the SOC81 set) calculated with our implementation against results from the WEST code reveals mean absolute deviations (MAD) of around 70 meV for G0W0@PBE and G0W0@PBE0. We check the accuracy of our AC treatment by comparison to full-frequency GW calculations, which shows that in the absence of multisolution cases, the errors due to AC are only minor. This implies that the main sources of the observed deviations between both implementations are the different single-particle bases and the pseudopotential approximation in the WEST code. Finally, we assess the performance of some (partially self-consistent) variants of the GWA for the calculation of first IPs by comparison to vertical experimental reference values. G0W0@PBE0 (25% exact exchange) and G0W0@BHLYP (50% exact exchange) perform best with mean absolute deviations (MAD) of about 200 meV. Explicit treatment of spin-orbit effects at the 2C level is crucial for systematic agreement with experiment. On the other hand, eigenvalue-only self-consistent GW (evGW) and quasi-particle self-consistent GW (qsGW) significantly overestimate the IPs. Perturbative G3W2 corrections increase the IPs and therefore improve the agreement with experiment in cases where G0W0 alone underestimates the IPs. With a MAD of only 140 meV, 2C-G0W0@PBE0 + G3W2 is in best agreement with the experimental reference values.

WEIGHTED \(S^p\)-PSEUDO \(S\)-ASYMPTOTICALLY PERIODIC SOLUTIONS FOR SOME SYSTEMS OF NONLINEAR DELAY INTEGRAL EQUATIONS WITH SUPERLINEAR PERTURBATION

This work is concerned with the existence of positive weighted pseudo \(S\)-asymptotically periodic solution in Stepanov-like sense for some systems of nonlinear delay integral equations. In this context, we will first be interested in establishing a suitable composition theorem, and then some existing results concerning the \(S\)-asymptotic periodicity in the scalar case are developed here for the vector case. We point out that, in this paper, we adopt some changes in the definitions, which, although slight, are necessary to accomplish the work.

Likelihood Based Inference and Bias Reduction in the Modified Skew-t-Normal Distribution

In this paper, likelihood-based inference and bias correction based on Firth’s approach are developed in the modified skew-t-normal (MStN) distribution. The latter model exhibits a greater flexibility than the modified skew-normal (MSN) distribution since it is able to model heavily skewed data and thick tails. In addition, the tails are controlled by the shape parameter and the degrees of freedom. We provide the density of this new distribution and present some of its more important properties including a general expression for the moments. The Fisher’s information matrix together with the observed matrix associated with the log-likelihood are also given. Furthermore, the non-singularity of the Fisher’s information matrix for the MStN model is demonstrated when the shape parameter is zero. As the MStN model presents an inferential problem in the shape parameter, Firth’s method for bias reduction was applied for the scalar case and for the location and scale case.