Scalar Case Research Articles

Statistical Distribution Theory and Fractional Calculus

This is an overview paper. This paper is an attempt to show that fractional calculus can be reached through statistical distribution theory. This paper brings together results on fractional integrals and fractional derivatives of the first and second kinds in the real and complex domains in the scalar, vector, and matrix-variate cases, and shows that all these results can be reached through statistical distribution theory. It is shown that the whole area of fractional integrals can be reached through distributions of products and ratios in the scalar variable case and distributions of symmetric products and symmetric ratios in the matrix-variate cases. While summarizing the materials, the real domain results are also listed side by side with the complex domain results so that a comparative study is possible. Fractional integrals and derivatives in the real domain mean that the parameters involved could be real or complex with appropriate conditions, the arbitrary function is real-valued, and the variables involved are all real. These in the complex domain mean that the parameters could be real or complex and the arbitrary function is still real-valued but the variables involved are in the complex domain. Fully complex domain means the variables as well as the arbitrary function are in the complex domain. Most of the materials on fractional integrals and fractional derivatives involving a single matrix or a number of matrices in the real or complex domain are of this author. Slight modifications of the results, compared with the published works in various papers, are there in various sections. In the paragraph on notations, the lemmas that are taken from this author’s own book on Jacobians are common with published works and hence the similarity index with this author’s works will be high. Section Matrix-Variate Joint Distributions and Fractional Integrals in Many Matrix-Variate Cases material on a statistical approach to Kiryakova’s multi-index fractional integral and its extension to the real scalar case of second kind integrals as well as extensions of first and second kind integrals to real and complex matrix-variate cases are believed to be new. Matrix differential operators are introduced in Section Fractional Derivatives and, with the help of these operators, fractional derivatives are constructed from the corresponding fractional integrals. These operators are applicable in a large variety of functions. Applicability is shown through identities created from scale transformed gamma random variables. Some concluding remarks are given and some open problems are pointed out in Section Concluding Remarks.

Ultralight dark matter explanation of NANOGrav observations

The angular correlation of pulsar residuals observed by NANOGrav and other pulsar timing array collaborations show evidence in support of the Hellings-Downs correlation expected from stochastic gravitational wave background (SGWB). In this paper, we offer a nongravitational wave explanation of the observed pulsar timing correlations as caused by an ultralight Lμ−Lτ gauge boson dark matter (ULDM). ULDM can affect the pulsar correlations in two ways. The gravitational potential of vector ULDM gives rise to a Shapiro time delay of the pulsar signals and a nontrivial angular correlation (as compared to the scalar ULDM case). In addition, if the pulsars have a nonzero charge of the dark matter gauge group, then the electric field of the local dark matter causes an oscillation of the pulsar and a corresponding Doppler shift of the pulsar signal. We point out that pulsars carry a significant charge of muons, and thus the Lμ−Lτ vector dark matter contributes to both the Doppler oscillations and the time delay of the pulsar signals. The synergy between these two effects provides a better fit to the shape of the angular correlation function, as observed by the NANOGrav collaboration, compared to the standard SGWB explanation or the SGWB combined with time delay explanations. Our analysis shows that, in addition to the SGWB signal, there may potentially be excess timing residuals attributable to the Lμ−Lτ ULDM. Published by the American Physical Society 2024

On arbitrary matrix coefficient assignment for the characteristic matrix polynomial of block matrix linear control systems

For block matrix linear control systems, we study the property of arbitrary matrix coefficient assignability for the characteristic matrix polynomial. This property is a generalization of the property of eigenvalue spectrum assignability or arbitrary coefficient assignability for the characteristic polynomial from system with scalar $(s=1)$ block matrices to systems with block matrices of higher dimensions $(s>1)$. Compared to the scalar case $(s=1)$, new features appear in the block cases of higher dimensions $(s>1)$ that are absent in the scalar case. New properties of arbitrary (upper triangular, lower triangular, diagonal) matrix coefficient assignability for the characteristic matrix polynomial are introduced. In the scalar case, all the described properties are equivalent to each other, but in block matrix cases of higher dimensions this is not the case. Implications between these properties are established.

Douglas-Rudin approximation theorem for operator-valued functions on the unit ball of [formula omitted]

Douglas and Rudin proved that any unimodular function on the unit circle T can be uniformly approximated by quotients of inner functions. We extend this result to the operator-valued unimodular functions defined on the boundary of the open unit ball of Cd. Our proof technique combines the spectral theorem for unitary operators with the Douglas-Rudin theorem in the scalar case to bootstrap the result to the operator-valued case. This yields a new proof and a significant generalization of Barclay's result (2009) [4] on the approximation of matrix-valued unimodular functions on T.

A bound preserving cut discontinuous Galerkin method for one dimensional hyperbolic conservation laws

In this paper we present a family of high order cut finite element methods with bound preserving properties for hyperbolic conservation laws in one space dimension. The methods are based on the discontinuous Galerkin framework and use a regular background mesh, where interior boundaries are allowed to cut through the mesh arbitrarily. Our methods include ghost penalty stabilization to handle small cut elements and a new reconstruction of the approximation on macro-elements, which are local patches consisting of cut and un-cut neighboring elements that are connected by stabilization. We show that the reconstructed solution retains conservation and order of convergence. Our lowest order scheme results in a piecewise constant solution that satisfies a maximum principle for scalar hyperbolic conservation laws. When the lowest order scheme is applied to the Euler equations, the scheme is positivity preserving in the sense that positivity of pressure and density are retained. For the high order schemes, suitable bound preserving limiters are applied to the reconstructed solution on macro-elements. In the scalar case, a maximum principle limiter is applied, which ensures that the limited approximation satisfies the maximum principle. Correspondingly, we use a positivity preserving limiter for the Euler equations, and show that our scheme is positivity preserving. In the presence of shocks additional limiting is needed to avoid oscillations, hence we apply a standard TVB limiter to the reconstructed solution. The time step restrictions are of the same order as for the corresponding discontinuous Galerkin methods on the background mesh. Numerical computations illustrate accuracy, bound preservation, and shock capturing capabilities of the proposed schemes.

Pathway to Fractional Integrals, Fractional Differential Equations, and Role of the H-Function

In this paper, the pathway model for the real scalar variable case is re-explored and its connections to fractional integrals, solutions of fractional differential equations, Tsallis statistics and superstatistics in statistical mechanics, the reaction-rate probability integral, Krätzel transform, pathway transform, etc., are explored. It is shown that the common thread in these connections is their H-function representations. The pathway parameter is shown to be connected to the fractional order in fractional integrals and fractional differential equations.

The Dai–Liao-type conjugate gradient methods for solving vector optimization problems

This paper attempts to propose Dai–Liao (DL)-type nonlinear conjugate gradient (CG) methods for solving vector optimization problems. Four variants of the DL method are extended and analysed from the scalar case to the vector setting. We first give a direct vector extended version of the modified Dai–Liao (DL+) method. Then the second algorithm is presented by combining a new extension form of the well-known Hager–Zhang (HZ) parameter. This new scheme equips a good descent property and its parameter also reduces to the classical one in the scalar case. The last two algorithms extend two sufficient descent DL-type methods. Particularly, two different vector forms of their parameters are considered and compared. As a result, we reveal some differences between scalar and vector cases to a certain extent. Without any convex assumption, the sufficient descent property and global convergence of all algorithms are established under inexact line searches. Finally, preliminary numerical experiments illustrate the practical behaviour of our algorithms by using Dolan and Moré performance profile.

ALP contribution to the strong CP problem

We compute the one-loop contribution to the θ¯ parameter of an faxionlike particle (ALP) with CP-odd derivative couplings. Its contribution to the neutron electric dipole moment is shown to be orders of magnitude larger than that stemming from the one-loop ALP contributions to the up- and down-quark electric and chromoelectric dipole moments. This strongly improves existing bounds on ALP-fermion CP-odd interactions and also sets limits on previously unconstrained couplings. The case of a general singlet scalar is analyzed as well. In addition, we explore how the bounds are modified in the presence of a Peccei-Quinn symmetry. Published by the American Physical Society 2024

On spectral inclusion sets and computing the spectra and pseudospectra of bounded linear operators

In this paper, we derive novel families of inclusion sets for the spectra and pseudospectra of large classes of bounded linear operators, and establish convergence of particular sequences of these inclusion sets to the spectrum or pseudospectrum, as appropriate. Our results apply, in particular, to bounded linear operators on a separable Hilbert space that, with respect to some orthonormal basis, have a representation as a bi-infinite matrix that is banded or band-dominated. More generally, our results apply in cases where the matrix entries themselves are bounded linear operators on some Banach space. In the scalar matrix entry case, we show that our methods, given the input information we assume, lead to a sequence of approximations to the spectrum, each element of which can be computed in finitely many arithmetic operations, so that, with our assumed inputs, the problem of determining the spectrum of a band-dominated operator has solvability complexity index one in the sense of Ben-Artzi et al. (2020). As a concrete and substantial application, we apply our methods to the determination of the spectra of non-self-adjoint bi-infinite tridiagonal matrices that are pseudoergodic in the sense of Davies [Commun. Math. Phys. 216 (2001), 687–704].

Decoding the B→Kνν excess at Belle II: Kinematics, operators, and masses

An excess in the branching fraction for B+→K+νν recently measured at Belle II may be a hint of new physics. We perform thorough likelihood analyses for different new physics scenarios such as B→KX with a new invisible particle X, or B→Kχχ through a scalar, vector, or tensor current with χ being a new invisible particle or a neutrino. We find that vector-current three-body decay with mX≃0.6 GeV—which may be dark matter—is most favored, while two-body decay with mX≃2 GeV is also competitive. The best-fit branching fractions for the scalar and tensor cases are a few times larger than for the two-body and vector cases. Past measurements provide further discrimination, although the best-fit parameters stay similar. Published by the American Physical Society 2024

On the truncated matricial moment problem. I

This paper is about the general truncated matrix-valued moment problem. Let Hq denote the complex Hermitian q×q-matrices, q∈N. Suppose that (X,X) is a measurable space and E is a finite-dimensional vector space of measurable mappings of X into Hq. A linear functional Λ on E is called a moment functional if there exists a positive Hq-valued measure μ on (X,X) such that Λ(F)=∫X〈F,dμ〉 for F∈E.We prove a matricial version of the Richter–Tchakaloff theorem which states that each moment functional on E has a finitely atomic representing measure. It is shown that strictly positive linear functionals on E are moment functionals. For a moment functional Λ, we study the set of atoms W(Λ) and define two Carathéodory numbers of Λ. Further, we define and investigate the core set V(Λ) which generalizes the core variety from the scalar case to the matricial setting. A main result of the paper is the equality W(Λ)=V(Λ).

ℓ -Proca stars

Initially applied to the scalar case, we extend the applicability of the multi-field generalization with angular momentum of bosonic stars to the vector case, in order to obtain new configurations that generalize the one-field spherical Proca stars. These new objects, which we call ℓ -Proca stars, arise as stationary and spherically symmetric bosonic stars solutions of the Einstein-(multi)Proca system, whose matter content is formed by an arbitrary odd number of 2ℓ+1 of complex Proca fields with the same mass, time-frequency, radial profile and angular momentum number ℓ . We analyze the system of constraint and evolution radial equations for the matter content to show the consistency of our proposal, and obtain numerically the ground states of these new solutions for the first few values of ℓ using spectral methods.

Light thermal dark matter beyond p-wave annihilation in minimal Higgs portal model

This study explores a minimal renormalizable dark matter (DM) model, incorporating a sub-GeV Majorana DM and a singlet scalar particle ϕ. Using scalar and pseudo-scalar interactions (couplings cs and cp), we investigate implications for DM detection, considering s-wave, p-wave, and combined (s+p wave) contributions in DM annihilation cross-section, as well as loop-correction contributions to DM-nucleon elastic scattering. Identifying a broad parameter space (10 MeV < mχ ≲ mϕ) within the 2σ allowed region, we explore scenarios (|cs| ≫ |cp|, |cs| ≪ |cp|, and |cs| ≈ |cp|). We find that (i) a non-zero pseudo-scalar coupling alleviates direct detection constraints as a comparison with the previous pure scalar coupling case; (ii) CMB observations set stringent limits on pseudo-scalar interaction dominant cases, making s-wave annihilation viable only for mχ > 1 GeV; (iii) the preferred ϕ-resonance region can be tested in the future indirect detection experiments, such as e-ASTROGAM.

Multivariate and Matrix-Variate Logistic Models in the Real and Complex Domains

Several extensions of the basic scalar variable logistic density to the multivariate and matrix-variate cases, in the real and complex domains, are given where the extended forms end up in extended zeta functions. Several cases of multivariate and matrix-variate Bayesian procedures, in the real and complex domains, are also given. It is pointed out that there are a range of applications of Gaussian and Wishart-based matrix-variate distributions in the complex domain in multi-look data from radar and sonar. It is hoped that the distributions derived in this paper will be highly useful in such applications in physics, engineering, statistics and communication problems, because, in the real scalar case, a logistic model is seen to be more appropriate compared to a Gaussian model in many industrial applications. Hence, logistic-based multivariate and matrix-variate distributions, especially in the complex domain, are expected to perform better where Gaussian and Wishart-based distributions are currently used.

Numerical Simulation of the Density Effect on the Macroscopic Transport Process of Tracer in the Ruhrstahl–Heraeus (RH) Vacuum Degasser

Silicon steel (electrical steel) has been used in electric motors that are important components in sustainable new energy Electrical Vehicles (EVs). The Ruhrstahl–Heraeus process is commonly used in the refining process of silicon steel. The refining effect inside the RH degasser is closely related to the flow and mixing of molten steel. In this study, a 260 t RH was used as the prototype, and the transport process of the passive scalar tracer (virtual tracer) and salt tracer (considering density effect) was studied using numerical simulation and water model research methods. The results indicate that the tracer transports from the up snorkel of the down snorkel to the bottom of the ladle, and then upwards from the bottom of the ladle to the top of the ladle. Density and gravity, respectively, play a promoting and hindering role in these two stages. In different areas of the ladle, density and gravity play a different degree of promotion and obstruction. Moreover, in different regions of the ladle, the different circulation strength leads to the different promotion degrees and obstruction degrees of the density. This results in the difference between the concentration growth rate of the salt tracer and the passive scalar in different regions of the ladle top. From the perspective of mixing time, density and gravity have no effect on the mixing time at the bottom of the ladle, and the difference between the passive scalar and NaCl solution tracer is within the range of 1–5%. For a larger dosage of tracer case, the difference range is reduced. However, at the top of the ladle, the average mixing time for the NaCl solution case is significantly longer than that of the passive scalar case, within the range of 3–14.7%. For a larger dosage of tracer case, the difference range is increased to 17.4–41.1%. It indicates that density and gravity delay the mixing of substances at the top area of the ladle, and this should be paid more attention when adding denser alloys in RH degasser.

The dynamics of three-forms in thick branes

In this work, we investigate thick brane models with a single three-form field. We find novel solutions for thick braneworlds where only three-forms exist and interact gravitationally in the bulk, both with and without matter fields. We use an additional scalar field as proxy for the matter fields. As an initial study, we consider the results here in contrast to the single scalar field thick braneworld case. The properties of the specific three-form parameterisation limits the freedom we have to choose the form of the warp factor, leading to a closed system of equations with nontrivial yet unstable solutions. The stability of the gravitational sector for thick brane three-forms is investigated and the models are shown to be unstable against small perturbations of the metric, further indicating that three-forms cannot exist stably in thick braneworld settings.

Turnaround Radius for charged particles in the Reissner–Nordström deSitter spacetime

We investigate the turnaround radius of the Reissner–Nordström deSitter Spacetime and how the turnaround radius changes if a test particle carries charge. We also consider the Martínez–Troncoso–Zanelli (MTZ) solution of conformally coupled gravity and investigate how the turnaround radius changes for a scalar test charge. In both scalar and electric interaction cases we find that the Turnaround Radius depends on the particle’s energy.

The pluriclosed flow for T2-invariant Vaisman metrics on the Kodaira-Thurston surface

In this note we study T2-invariant pluriclosed metrics on the Kodaira-Thurston surface. We obtain a characterization of T2-invariant Vaisman metrics, and notice that the Kodaira-Thurston surface admits Vaisman metrics with non-constant scalar curvature. Then we study the behaviour of the Vaisman condition in relation to the pluriclosed flow. As a consequence, we show that if the initial metric on the Kodaira-Thurston surface is a T2-invariant Vaisman metric, then the pluriclosed flow preserves the Vaisman condition, extending to the non-constant scalar curvature case the previous result in [6].

A novel approach to cosmological particle production

In this work we present a novel approach to the study of cosmological particle production in asymptotically Minkowski spacetimes. We emphasize that it is possible to determine the amount of particle production by focusing on the mathematical properties of the mode function equations, i.e. their singularities and monodromies, sidestepping the need to solve those equations. We consider in detail creation of scalar and spin 1/2 particles in four dimensional asymptotically Minkowski flat FLRW spacetimes. We explain that when the mode function equation for scalar fields has only regular singular points, the corresponding scale factors are asymptotically Minkowski. For Dirac spin 1/2 fields, the requirement of mode function equations with only regular points is more restrictive, and picks up a subset of the aforementioned scale factors. For the scalar case, we argue that there are two different regimes of particle production; while most of the literature has focused on only one of these regimes, the other regime presents enhanced particle production. On the other hand, for Dirac fermions we find a single regime of particle production. Finally, we very briefly comment on the possibility of studying particle production in spacetimes that don't asymptote to Minkowski, by considering mode function equations with irregular singular points.

The Lavrentiev phenomenon in calculus of variations with differential forms

In this article we study convex non-autonomous variational problems with differential forms and corresponding function spaces. We introduce a general framework for constructing counterexamples to the Lavrentiev gap, which we apply to several models, including the double phase, borderline case of double phase potential, and variable exponent. The results for the borderline case of double phase potential provide new insights even for the scalar case, i.e., variational problems with 0-forms.