Leading-order Research Articles

General approach on shadow radius and photon spheres in asymptotically flat spacetimes and the impact of mass-dependent variations

Recent observations of black hole shadows have revolutionized our ability to probe gravity in extreme environments. This manuscript presents a novel analytic method to calculate, in leading-order terms, the key parameters of photon sphere and shadow radius. This method offers advantages for complex metrics where traditional approaches are cumbersome. We further explore the impact of black hole mass on the photon sphere radius, providing insights into black hole interactions with their surroundings. Our findings contribute significantly to black hole physics and gravity under extreme conditions. By leveraging future advancements in observations, such as the next-generation Event Horizon Telescope (ngEHT), this work paves the way for even more precise tests of gravity near black holes.

Multivalued Variational Inequalities with Generalized Fractional Φ-Laplacians

In this article, we examine variational inequalities of the form ⟨A(u),v−u⟩+⟨F(u),v−u⟩≥0,∀v∈Ku∈K,, where A is a generalized fractional Φ-Laplace operator, K is a closed convex set in a fractional Musielak–Orlicz–Sobolev space, and F is a multivalued integral operator. We consider a functional analytic framework for the above problem, including conditions on the multivalued lower order term F such that the problem can be properly formulated in a fractional Musielak–Orlicz–Sobolev space, and the involved mappings have certain useful monotonicity–continuity properties. Furthermore, we investigate the existence of solutions contingent upon certain coercivity conditions.

ℏ 4 quantum corrections to semiclassical transmission probabilities

The combination of vibrational perturbation theory with the replacement of the harmonic oscillator quantization condition along the reaction coordinate with an imaginary action to be used in the uniform semiclassical approximation for the transmission probability has been shown in recent years to be a practical method for obtaining thermal reaction rates. To date, this theory has been developed systematically only up to second order in perturbation theory. Although it gives the correct leading order term in an ℏ2 expansion, its accuracy at lower temperatures, where tunneling becomes important, is not clear. In this paper, we develop the theory to fourth order in the action. This demands developing the quantum perturbation theory up to sixth order. Remarkably, we find that the fourth order theory gives the correct ℏ4 term in the expansion of the exact thermal rate. The relative magnitude of the fourth order correction as compared to the second order term objectively indicates the accuracy of the second order theory. We also extend the previous modified second order theory to the fourth order case, creating an ℏ2 modified potential for this purpose. The resulting theory is tested on the standard examples—symmetric and asymmetric Eckart potentials and a Gaussian potential. The modified fourth order theory is remarkably accurate for the asymmetric Eckart potential.

Asymptotic analysis of high‐order solitons of an equivalent Kundu–Eckhaus equation

In this work, we study the asymptotic characteristics of high‐order solitons for the focusing Kundu–Eckhaus (KE) equation. Based on the loop group theory, we construct the general Darboux transformation within the framework of Riemann–Hilbert problems to derive the general high‐order soliton solution. Using high‐order Bäcklund transformation, we derive the leading order term of the determinant solution to obtain the asymptotic representation for the high‐order soliton solution. Furthermore, this method is also extended to the construction of more general high‐order cases with multiple poles. We further find that if a soliton propagates along the logarithm characteristic curve, the high‐order soliton can be decomposed into individual solitons with the same amplitude and velocity. Finally, these solutions are theoretically and graphically analyzed in detail.

Solitary solutions to the steady Euler equations with piecewise constant vorticity in a channel

We consider a two-dimensional, two-layer, incompressible, steady flow, with vorticity which is constant in each layer, in an infinite channel with rigid walls. The velocity is continuous across the interface, there is no surface tension or difference in density between the two layers, and the flow is inviscid. Unlike in previous studies, we consider solutions which are localised perturbations rather than periodic or quasi-periodic perturbations of a background shear flow. We rigorously construct a curve of exact solutions and give the leading order terms in an asymptotic expansion. We also give a thorough qualitative description of the fluid particle paths, which can include stagnation points, critical layers, and streamlines which meet the boundary.

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In this paper we prove existence and regularity results for a class of parabolic problems with irregular initial data and lower order terms singular with respect to the solution. We prove that, even if the initial datum is not bounded but only in L1(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1(\\Omega )$$\\end{document}, there exists a solution that “instantly” becomes bounded. Moreover we study the behavior in time of these solutions showing that this class of problems admits global solutions which all have the same behavior in time independently of the size of the initial data.

A personal perspective of the present status and future challenges facing thermal reaction rate theory.

Reaction rate theory has been at the center of physical chemistry for well over one hundred years. The evolution of the theory is not only of historical interest. Reliable and accurate computation of reaction rates remains a challenge to this very day, especially in view of the development of quantum chemistry methods, which predict the relevant force fields. It is still not possible to compute the numerically exact rate on the fly when the system has more than at most a few dozen anharmonic degrees of freedom, so one must consider various approximate methods, not only from the practical point of view of constructing numerical algorithms but also on conceptual and formal levels. In this Perspective, I present some of the recent analytical results concerning leading order terms in an ℏ2m series expansion of the exact rate and their implications on various approximate theories. A second aspect has to do with the crossover temperature between tunneling and thermal activation. Using a uniform semiclassical transmission probability rather than the "primitive" semiclassical theory leads to the conclusion that there is no divergence problem associated with a "crossover temperature." If one defines a semiclassical crossover temperature as the point at which the tunneling energy of the instanton equals the barrier height, then it is a factor of two higher than its previous estimate based on the "primitive" semiclassical approximation. In the low temperature tunneling regime, the uniform semiclassical theory as well as the "primitive" semiclassical theory were based on the classical Euclidean action of a periodic orbit on the inverted potential. The uniform semiclassical theory wrongly predicts that the "half-point," which is the energy at which the transmission probability equals 1/2, for any barrier potential, is always the barrier energy. We describe here how augmenting the Euclidean action with constant terms of order ℏ2 can significantly improve the accuracy of the semiclassical theory and correct this deficiency. This also leads to a deep connection with and improvement of vibrational perturbation theory. The uniform semiclassical theory also enables an extension of the quantum version of Kramers' turnover theory to temperatures below the "crossover temperature." The implications of these recent advances on various approximate methods used to date are discussed at length, leading to the conclusion that reaction rate theory will continue to challenge us both on conceptual and practical levels for years to come.

ℏ2 Corrections to Semiclassical Transmission Coefficients.

The uniform semiclassical expression for the energy-dependent transmission probability through a barrier has been a staple of reaction rate theory for almost 90 years. Yet, when using the classical Euclidean action, the transmission probability is identical to 1/2 when the energy equals the barrier height since the Euclidean action vanishes at this energy. This result is generally incorrect. It also leads to an inaccurate estimate of the leading order term in an ℏ2n expansion of the thermal transmission coefficient. The central result of this paper is that adding an ℏ2 dependent correction to the uniform semiclassical expression, whether as a constant action or as a shift in the energy scale, not only corrects this inaccuracy but also leads to a theory that is more accurate than the previous one for almost any energy. Shifting the energy scale is a generalization of the vibrational perturbation theory 2 (VPT2) and is much more accurate than the "standard" VPT2 theory, especially when the potential is asymmetric. Shifting the action by a constant is a generalization of a result obtained by Yasumori and Fueki (YF) only for the Eckart barrier. The resulting modified VPT2 and YF semiclassical theories are applied to the symmetric and asymmetric Eckart barrier, a Gaussian barrier, and a tanh barrier. The one-dimensional theories are also generalized to many-dimensional systems. Their effect on the thermal instanton theory is discussed.

Applying Lin's method to constructing heteroclinic orbits near the heteroclinic chain

In this paper, we apply Lin's method to study the existence of heteroclinic orbits near the degenerate heteroclinic chain under ‐dimensional periodic perturbations. The heteroclinic chain consists of two degenerate heteroclinic orbits and connected by three hyperbolic saddle points . Assume that the degeneracy of the unperturbed heteroclinic orbit is , the splitting index is . By applying Lin's method, we construct heteroclinic orbits connected and near the unperturbed heteroclinic chain. The existence of these orbits is equivalent to finding zeros of the corresponding bifurcation function. The lower order terms of the bifurcation function is the map from to . Using the contraction mapping principle, we provide a detailed analysis on how zeros can exist based on different cases of splitting indices , and then obtain the existence of the heteroclinic orbits which backward asymptotic to and forward asymptotic to .

A Singular Tempered Sub-Diffusion Fractional Equation with Changing-Sign Perturbation

In this paper, we establish some new results on the existence of positive solutions for a singular tempered sub-diffusion fractional equation involving a changing-sign perturbation and a lower-order sub-diffusion term of the unknown function. By employing multiple transformations, we transform the changing-sign singular perturbation problem to a positive problem, then establish some sufficient conditions for the existence of positive solutions of the problem. The asymptotic properties of solutions are also derived. In deriving the results, we only require that the singular perturbation term satisfies the Carathéodory condition, which means that the disturbance influence is significant and may even achieve negative infinity near some time singular points.

Dynamical de Sitter black holes in a quasi-stationary expansion

We revisit and improve the analytic study (Gregory et al 2018 Class. Quantum. Grav. 35 155008) of spherically symmetric but dynamical black holes in Einstein’s gravity coupled to a real scalar field. We introduce a series expansion in a small parameter ε that implements slow time dependence. At the leading order (LO), the generic solution is a quasi-stationary Schwarzschild–de Sitter (SdS) metric, i.e. one where time-dependence enters only through the mass and cosmological constant parameters of SdS. The two coupled ODEs describing the LO time dependence are solved up to quadrature for an arbitrary scalar potential. Higher order corrections can be consistently computed, as we show by explicitly solving the Einstein equations at the next to LO as well. We comment on how the quasi-stationary expansion we introduce here is equivalent to the non-relativistic 1/c expansion.

Z -boson decays into S -wave quarkonium plus a photon up to O(αsv2) corrections

In this paper, we calculate the decay widths and branching fractions for the decays Z→H+γ up to O(αsv2) accuracy within the framework of nonrelativistic QCD, where H stands for the S-wave quarkonium ηc, J/ψ, ηb, or ϒ, respectively. To compare with the leading-order terms, those corrections show good perturbative behavior as expected. It is found that contributions from the next-to-leading order QCD correction O(αsv0), the relativistic correction O(αs0v2) and their joint correction O(αsv2) are sizable and comparable to each other, especially for the charmonium case. Thus we need to take all of them into consideration for a sound estimation. For a high luminosity electron-positron collider running around the Z pole, due to Z-boson resonance effect, sizable events could be produced from those rare decay channels. Published by the American Physical Society 2024

Quadratic gravity in analogy to quantum chromodynamics: Light fermions in its landscape

We investigate a nonperturbative approach to quantum gravity built in terms of analogies between quadratic gravity and quantum chromodynamics. This approach is based on a conjectured phase transition between quadratic gravity in the trans-Planckian regime and an effective field theory, with general relativity as the leading-order term, below the Planck scale. We point out that not all aspects of the analogy between quadratic gravity and quantum chromodynamics are desired. A possible mechanism of chiral symmetry breaking driven by quantum gravity fluctuations could make this setup incompatible with our observed Universe. Here, we put forward a first investigation of chiral symmetry breaking in the context of quadratic gravity. We find indication that gravity, despite being an attractive force, does not trigger chiral symmetry breaking in a nonperturbative regime. This result is based on a (functional) renormalization group analysis of four-fermion interactions coupled to quadratic gravity. We also comment on the particularities associated with the single-flavor case. In summary, the analogy between quadratic gravity and chromodynamics passes this first phenomenological viability test. Published by the American Physical Society 2024

Existence and uniqueness results for an elliptic equation with blowing-up coefficient and singular lower order term

Existence and uniqueness results for an elliptic equation with blowing-up coefficient and singular lower order term

Applications of moments of Dirichlet coefficients in elliptic curve families

The moments of the coefficients of elliptic curve [Formula: see text]-functions are related to numerous important arithmetic problems. Rosen and Silverman proved a conjecture of Nagao relating the first moment of one-parameter families satisfying Tate’s conjecture to the rank of the corresponding elliptic surface over [Formula: see text]; one can also construct families of moderate rank by finding families with large first moments. Michel proved that if [Formula: see text] is not constant, then the second moment of the family is of size [Formula: see text]; these two moments show that for suitably small support the behavior of zeros near the central point agree with that of eigenvalues from random matrix ensembles, with the higher moments impacting the rate of convergence. In his thesis, Miller noticed a negative bias in the second moment of every one-parameter family of elliptic curves over [Formula: see text] whose second moment had a (by him) calculable closed-form expression, specifically the first lower-order term which does not average to zero is on average negative. This Bias Conjecture has now been confirmed for many families; however, these are highly non-generic families as they are specially chosen so that the resulting Legendre sums can be determined. For cohomological reasons, each subsequent term in the second moment expansion is smaller than the previous by a factor on the order of [Formula: see text], and thus numerically, it is hard to see a term of size [Formula: see text] with a small negative average as it can be masked by a term of size [Formula: see text] which averages to zero. Inspired by the recent successes by Yang-Hui He, Kyu-Hwan Lee, Thomas Oliver, Alexey Pozdnyakov and others in investigations of murmurations of elliptic curve coefficients with machine learning techniques, we pose a similar problem for trying to understand the Bias Conjecture. As a start to this program, we numerically investigate the bias conjecture and provide a visual representation of the bias for the second moment. We find a one-parameter family of elliptic curves whose bias is positive for half the primes. However, the numerics do not offer conclusive evidence that negative bias for the other half is enough to overwhelm the positive bias. Without an explicit expansion for the second moment, we are not able to extract potential negative bias of the order [Formula: see text] term.

Diagonalizing the Born-Oppenheimer Hamiltonian via Moyal perturbation theory, nonadiabatic corrections, and translational degrees of freedom.

This article describes a method for calculating higher order or nonadiabatic corrections in Born-Oppenheimer theory and its interaction with the translational degrees of freedom. The method uses the Wigner-Weyl correspondence to map nuclear operators into functions on the classical phase space and the Moyal star product to represent operator multiplication on those functions. These are explained in the body of the paper. The result is a power series in κ2, where κ = (m/M)1/4 is the usual Born-Oppenheimer parameter. The lowest order term is the usual Born-Oppenheimer approximation, while higher order terms are nonadiabatic corrections. These are needed in calculations of electronic currents, momenta, and densities. The separation of nuclear and electronic degrees of freedom takes place in the context of the exact symmetries (for an isolated molecule) of translations and rotations, and these, especially translations, are explicitly incorporated into our discussion. This article presents an independent derivation of the Moyal expansion in molecular Born-Oppenheimer theory. We show how electronic currents and momenta can be calculated within the framework of Moyal perturbation theory; we derive the transformation laws of the electronic Hamiltonian, the electronic eigenstates, and the derivative couplings under translations; we discuss in detail the rectilinear motion of the molecular center of mass in the Born-Oppenheimer representation; and we show how the elimination of the translational components of the derivative couplings leads to a unitary transformation that has the effect of exactly separating the translational degrees of freedom.

Surface wrinkling of a film coated to a graded substrate

We study the surface wrinkling of a stiff thin elastic film bonded to a compliant graded elastic substrate subject to compressive stress generated either by compression or growth of the bilayer. Our aim is to clarify the influence of the modulus gradient on the onset and surface pattern in this bilayer. Within the framework of finite elasticity, an exact bifurcation condition is obtained using the Stroh formulation and the surface impedance matrix method. Further analytical progress is made by focusing on the case of short wavelength limit for which the Wentzel–Kramers–Brillouin method can be used to resolve the eigenvalue problem of ordinary differential equations with variable coefficients. An explicit bifurcation condition is obtained from which the critical buckling load and the critical wavelength are derived asymptotically. In particular, we consider two distinct situations depending on the ratio β of the shear modulus at the substrate surface to that at infinity. If β is of O(1) or small, the parameters related to modulus gradient all appear in the higher-order terms and play an insignificant role in the bifurcation. In that case, it is the modulus ratio between the film and substrate surface that governs the onset of surface wrinkling. If, however, β≫1, the modulus gradient affects the critical condition through leading-order terms. Through our analysis we unravel the influence of different material and geometric parameters, including the modulus gradient, on the bifurcation threshold and the associated wavelength which can be of importance in many biological and technological settings.

Induced Isotensor Interactions in Heavy-Ion Double-Charge-Exchange Reactions and the Role of Initial and Final State Interactions

The role of initial state (ISI) and final state (FSI) ion–ion interactions in heavy-ion double-charge-exchange (DCE) reactions A(Z,N)→A(Z±2,N∓2) are studied for double single-charge-exchange (DSCE) reactions given by sequential actions of the isovector nucleon–nucleon (NN) T-matrix. In momentum representation, the second-order DSCE reaction amplitude is shown to be given in factorized form by projectile and target nuclear matrix elements and a reaction kernel containing ISI and FSI. Expanding the intermediate propagator in a Taylor series with respect to auxiliary energy allows us to perform the summation in the leading-order term over intermediate nuclear states in closure approximation. The nuclear matrix element attains a form given by the products of two-body interactions directly exciting the n2p−2 and p2n−2 DCE transitions in the projectile and the target nucleus, respectively. A surprising result is that the intermediate propagation induces correlations between the transition vertices, showing that DSCE reactions are a two-nucleon process that resembles a system of interacting spin–isospin dipoles. Transformation of the DSCE NN T-matrix interactions from the reaction theoretical t-channel form to the s-channel operator structure required for spectroscopic purposes is elaborated in detail, showing that, in general, a rich spectrum of spin scalar, spin vector and higher-rank spin tensor multipole transitions will contribute to a DSCE reaction. Similarities (and differences) to two-neutrino double-beta decay (DBD) are discussed. ISI/FSI distortion and absorption effects are illustrated in black sphere approximation and in an illustrative application to data.

On the some equations inequalities in Musielak-Orlicz spaces with measure data

Our aim in this present work is to prove the existence of a solution for the nonlinear unilateral parabolic problems associated to the equation ∂u - div a(x, t, u, ∇u) - div Φ(x, t, u) = μ in QT = Ω × (0, T), ∂t where the lower order term Φ satisfies a generalized natural growth condition and the datum μ belongs to L1(Q) + E (Q).

Sharp-Interface Limit of a Multi-phase Spectral Shape Optimization Problem for Elastic Structures

We consider an optimization problem for the eigenvalues of a multi-material elastic structure that was previously introduced by Garcke et al. (Adv. Nonlinear Anal. 11:159–197, 2022). There, the elastic structure is represented by a vector-valued phase-field variable, and a corresponding optimality system consisting of a state equation and a gradient inequality was derived. In the present paper, we pass to the sharp-interface limit in this optimality system by the technique of formally matched asymptotics. Therefore, we derive suitable Lagrange multipliers to formulate the gradient inequality as a pointwise equality. Afterwards, we introduce inner and outer expansions, relate them by suitable matching conditions and formally pass to the sharp-interface limit by comparing the leading order terms in the state equation and in the gradient equality. Furthermore, the relation between these formally derived first-order conditions and results of Allaire and Jouve (Comput. Methods Appl. Mech. Eng. 194:3269–3290, 2005) obtained in the framework of classical shape calculus is discussed. Eventually, we provide numerical simulations for a variety of examples. In particular, we illustrate the sharp-interface limit and also consider a joint optimization problem of simultaneous compliance and eigenvalue optimization.