Leading-order Research Articles

One-level density of zeros of Dirichlet L-functions over function fields

TextWe compute the one-level density of zeros of order ℓ Dirichlet L-functions over function fields Fq[t] for ℓ=3,4 in the Kummer setting (q≡1(modℓ)) and for ℓ=3,4,6 in the non-Kummer setting (q≢1(modℓ)). In each case, we obtain a main term predicted by Random Matrix Theory (RMT) and lower order terms not predicted by RMT. We also confirm the symmetry type of the families is unitary, supporting Katz and Sarnak's philosophy. VideoFor a video summary of this paper, please visit https://youtu.be/0tGmXHvVKF0.

A regularity result for nonuniformly elliptic equations with lower order terms

A regularity result for nonuniformly elliptic equations with lower order terms

Review of the hyperfine structure theory of hydrogen molecular ions

The study of high-precision spectroscopy for hydrogen molecular ions enables the determination of fundamental constants, such as the proton-to-electron mass ratio, the deuteron-to-electron mass ratio, the Rydberg constant, and the charge radii of proton and deuteron. This can be accomplished through a combination of high precision experimental measurements and theoretical calculations. The spectroscopy of hydrogen molecular ions reveals abundant hyperfine splittings, necessitating not only an understanding of rovibrational transition frequencies but also a thorough grasp of hyperfine structure theory to extract meaningful physical information from the spectra. This article reviews the history of experiments and theories related to the spectroscopy of hydrogen molecular ions, with a particular focus on the theory of hyperfine structure. As far back as the second half of the last century, the hyperfine structure of hydrogen molecular ions was described by a comprehensive theory based on its leading-order term, known as the Breit-Pauli Hamiltonian. Thanks to the advancements in non-relativistic quantum electrodynamics (NRQED) at the beginning of this century, a systematic development of next-to-leading-order theory for hyperfine structure has been achieved and applied to <inline-formula><tex-math id="M8">\begin{document}$\text{H}_2^+$\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$\text{HD}^+$\end{document}</tex-math></inline-formula> in recent years, including the establishment of the <inline-formula><tex-math id="M10">\begin{document}$m\alpha^7\ln(\alpha)$\end{document}</tex-math></inline-formula> order correction. For the hyperfine structure of <inline-formula><tex-math id="M11">\begin{document}$\text{H}_2^+$\end{document}</tex-math></inline-formula>, theoretical calculations show good agreement with experimental measurements after decades of work. However, for HD<sup>+</sup>, discrepancies have been observed between measurements and theoretical predictions that cannot be accounted for by the theoretical uncertainty in the non-logarithmic term of the <inline-formula><tex-math id="M13">\begin{document}$m\alpha^7$\end{document}</tex-math></inline-formula> order correction. To address this issue, additional experimental measurements are needed for mutual validation, as well as independent tests of the theory, particularly regarding the non-logarithmic term of the <inline-formula><tex-math id="M14">\begin{document}$m\alpha^7$\end{document}</tex-math></inline-formula> order correction.

Fractional Schrödinger equation involving a nonlinearity dependent on a lower-order fractional term

Fractional Schrödinger equation involving a nonlinearity dependent on a lower-order fractional term

Degenerate singular parabolic problems with natural growth

In this paper, we study the existence and regularity results for nonlinear singular parabolic problems with a natural growth gradient term \[\begin{cases}\frac{\partial u}{\partial t}-\operatorname{div}((a(x,t)+u^{q})|\nabla u|^{p-2}\nabla u)+d(x,t)\frac{|\nabla u|^{p}}{u^{\gamma}}=f & \text{ in } Q,\\ u(x,t)=0 & \text{ on } \Gamma, \\ u(x,t=0)=u_{0}(x) & \text{ in } \Omega, \end{cases}\] where \(\Omega\) is a bounded open subset of \(\mathbb{R}^{N}\), \(N\gt 2\), \(Q\) is the cylinder \(\Omega \times (0,T)\), \(T\gt 0\), \(\Gamma\) the lateral surface \(\partial \Omega \times (0,T)\), \(2\leq p\lt N\), \(a(x,t)\) and \(b(x,t)\) are positive measurable bounded functions, \(q\geq 0\), \(0\leq\gamma\lt 1\), and \(f\) non-negative function belongs to the Lebesgue space \(L^{m}(Q)\) with \(m\gt 1\), and \(u_{0}\in L^{\infty}(\Omega)\) such that \[\forall\omega\subset\subset\Omega\, \exists D_{\omega}\gt 0:\, u_{0}\geq D_{\omega}\text{ in }\omega.\] More precisely, we study the interaction between the term \(u^{q}\) (\(q>0\)) and the singular lower order term \(d(x,t)|\nabla u|^{p}u^{-\gamma}\) (\(0\lt\gamma\lt 1\)) in order to get a solution to the above problem. The regularizing effect of the term \(u^q\) on the regularity of the solution and its gradient is also analyzed.

A direct proof of existence of weak solutions to elliptic problems

We provide a direct proof of existence and uniqueness of weak solutions to a broad family of strongly nonlinear elliptic equations with lower-order terms. The leading part of the operator satisfies general growth conditions settling the problem in the framework of fully anisotropic and inhomogeneous Musielak-Orlicz spaces generated by an $N$-function \linebreak $M\colon \Omega\times\mathbb R^d\to [0,\infty)$. Neither $\nabla_2$ nor $\Delta_2$ conditions are imposed on $M$. Our results cover among others problems with anisotropic polynomial, Orlicz, variable exponent, and double phase growth.

ΔS = 0 hadronic parity violation in next-to-leading order QCD: Anomalous dimension matrices and their implications

We construct the effective Hamiltonian for hadronic parity violation in strangeness-nonchanging (ΔS=0) processes in next-to-leading order (NLO) in QCD, for all isosectors, and at a renormalization scale of 2GeV, thus extending our earlier leading-order (LO) analysis [1,2]. Hadronic parity violation, studied in the context of the low-energy interactions of nucleons and nuclei, exposes the complex interplay of weak and strong interactions in these systems, and thus supports our extension to NLO. Here we exploit the flavor-blind nature of QCD interactions to construct the needed anomalous dimension matrices from those computed in flavor physics, which we then use to refine our effective Hamiltonian and finally our predicted parity-violating meson-nucleon coupling constants, to find improved agreement with few-body experiments.

On Ultradifferentiable Regularity of Perturbations by Lower Order Terms of Globally $$C^\infty$$ Hypoelliptic Ultradifferentiable Pseudodifferential Operators

On Ultradifferentiable Regularity of Perturbations by Lower Order Terms of Globally $$C^\infty$$ Hypoelliptic Ultradifferentiable Pseudodifferential Operators

Regularity for solutions of elliptic $$p(x)-$$Laplacian type equations with lower order terms and hardy potential

Regularity for solutions of elliptic $$p(x)-$$Laplacian type equations with lower order terms and hardy potential

Theoretical Analyses of Multiobjective Evolutionary Algorithms on Multimodal Objectives.

Multiobjective evolutionary algorithms are successfully applied in many real-world multiobjective optimization problems. As for many other AI methods, the theoretical understanding of these algorithms is lagging far behind their success in practice. In particular, previous theory work considers mostly easy problems that are composed of unimodal objectives. As a first step towards a deeper understanding of how evolutionary algorithms solve multimodal multiobjective problems, we propose the OneJumpZeroJump problem, a bi-objective problem composed of two objectives isomorphic to the classic jump function benchmark. We prove that the simple evolutionary multiobjective optimizer (SEMO) with probability one does not compute the full Pareto front, regardless of the runtime. In contrast, for all problem sizes n and all jump sizes k∈[4..n2-1], the global SEMO (GSEMO) covers the Pareto front in an expected number of Θ((n-2k)nk) iterations. For k=o(n), we also show the tighter bound 32enk+1±o(nk+1), which might be the first runtime bound for an MOEA that is tight apart from lower-order terms. We also combine the GSEMO with two approaches that showed advantages in single-objective multimodal problems. When using the GSEMO with a heavy-tailed mutation operator, the expected runtime improves by a factor of at least kΩ(k). When adapting the recent stagnation-detection strategy of Rajabi and Witt (2022) to the GSEMO, the expected runtime also improves by a factor of at least kΩ(k) and surpasses the heavy-tailed GSEMO by a small polynomial factor in k. Via an experimental analysis, we show that these asymptotic differences are visible already for small problem sizes: A factor-5 speed-up from heavy-tailed mutation and a factor-10 speed-up from stagnation detection can be observed already for jump size 4 and problem sizes between 10 and 50. Overall, our results show that the ideas recently developed to aid single-objective evolutionary algorithms to cope with local optima can be effectively employed also in multiobjective optimization.

Evaluation of a high-order central-difference solver for highly compressible flows out of thermochemical equilibrium

A high-order shock-capturing central finite-difference scheme is evaluated for numerical simulations of hypersonic high-enthalpy flows out of thermochemical equilibrium. The scheme is an extension to thermochemical out-of-equilibrium flows of the technique presented in Sciacovelli et al. (2021) for high-speed flows in chemical nonequilibrium. It relies on a standard tenth-order accurate central-difference approximation of the inviscid fluxes, supplemented with a high-order accurate nonlinear artificial dissipation term of ninth-order accuracy in smooth flow regions. Close to flow discontinuities, a shock-capturing low-order term is activated based on a highly selective shock sensor. To enable robust and non oscillatory solutions in regions of strong discontinuities of the thermodynamic variables, including the vibrational temperature, a shock detector consisting of a combination of a pressure-based term and Ducros’ vorticity/dilatation sensor is applied to all conservation equations except that of vibrational energy. For the latter, a shock sensor based on the vibrational temperature itself is adopted instead, to account for the loose coupling between vibrational energy and pressure. The accuracy and robustness of the proposed approach is demonstrated for a variety of thermochemical non-equilibrium configurations, ranging from one-dimensional benchmarks to three-dimensional turbulent flows, for which the ILES capabilities of the selective high-order numerical dissipation are also showcased.

Sound attenuation in high mach number oscillating bubble media

We study theoretically and numerically sound attenuation in bubble-containing media when the bubbles are freely oscillating at high Mach numbers. This paper expands one of the main forms of bubble-related acoustic damping factors by extending the previous theories to higher Mach numbers, further improves the theories of nonlinear sound propagation in bubble-containing media. A nonlinear sound propagation model incorporating second-order liquid compression terms is developed, expressing the sound velocity and density in the medium as a function of the driving pressure, and taking into account the higher-order liquid compression effects on sound propagation. The correctness of the proposed model is verified by comparing with a linear model and a nonlinear model containing only low-order Mach number terms. When the bubble oscillates at a high Mach number, radiation damping, which is directly related to Mach number, becomes the main damping component affecting sound attenuation. The higher the driving amplitude, the stronger the nonlinear effect, and the greater the impact of high-order liquid compression effects on the sound attenuation, the more necessary it is to use the proposed model to calculate the sound attenuation. For high Mach numbers, varying the bubble radius and bubble number density, respectively, the difference between the proposed model and the model containing only low-order Mach number terms in capturing the pressure-dependent attenuation is calculated. Due to stronger radiation damping in smaller bubbles, the effect of compressibility becomes more important. The smaller the bubble radius, the greater the half-quality factor of the curve related to the difference in attenuation calculated by the two models, the more necessary it is to calculate the pressure-dependent attenuation using the proposed model. Here, the half-quality factor is defined as the corresponding frequency bandwidth when the curve falls from the maximum value to 22 times. Without considering the coupling effect between bubbles, the half-quality factor of the curve is not affected by the bubble number density.

Electroweak corrections to Higgs boson decays in a Complex Singlet extension of the SM and their phenomenological impact

The complex singlet extension CxSM of the Standard Model (SM) is a simple extension of the SM with two visible Higgs bosons in the spectrum and a Dark Matter (DM) candidate. In this paper we complete the computation of the next-to-leading (NLO) electroweak (EW) corrections to on-shell and non-loop-induced Higgs decays. Our calculations are implemented in the code EWsHDECAY which also includes the relevant QCD corrections. Performing an extensive parameter scan in the model and including all relevant theoretical and experimental single- and di-Higgs as well as DM constraints, we obtain a viable parameter sample. We find that current DM constraints are able to test the model in DM mass regions where collider searches are not sensitive. The relative EW corrections turn out to be large for scenarios with relatively large couplings, threshold effects or small leading-order (LO) widths. Otherwise, they are of typical EW size and can amount up to about 20–25%. The theory uncertainty derived from the change of the renormalization scheme dependence then is of a few per cent. While the NLO corrections applied in the constraints due to single- and di-Higgs searches impact the validity of specific parameter points, the overall shape of the allowed parameter region is not yet sensitive to the EW corrections. This picture will change with further increased experimental precision in the future and necessitates precise predictions on the theory side as presented in this paper.

Singular orthotropic functionals with nonstandard growth conditions

We pursue the study of a model convex functional with orthotropic structure and nonstandard growth conditions, this time focusing on the sub-quadratic case. We prove that bounded local minimizers are locally Lipschitz. No restrictions on the ratio between the highest and the lowest growth rates are needed. The result holds also in presence of a non-autonomous lower order term, under sharp integrability assumptions. Finally, we prove higher differentiability of bounded local minimizers as well.

Regularity results for local solutions to some anisotropic elliptic equations

In this paper we study the higher integrability of local solutions for a class of anisotropic equations with lower order terms whose growth coefficients lay in Marcinkiewicz spaces. A condition for the boundedness of such solutions is also given.

Regularity theory and Green’s function for elliptic equations with lower order terms in unbounded domains

We consider elliptic operators in divergence form with lower order terms of the form Lu = -{{textrm{div}}}(A cdot nabla u + b u ) - c cdot nabla u - du, in an open set Omega subset mathbb {R}^n, n ge 3, with possibly infinite Lebesgue measure. We assume that the n times n matrix A is uniformly elliptic with real, merely bounded and possibly non-symmetric coefficients, and either b, c in L^{n,infty }_{text {loc}}({Omega }) and d in L_{text {loc}}^{frac{n}{2}, infty }(Omega ), or |b|^2, |c|^2, |d| in mathcal {K}_{text {loc}}(Omega ), where mathcal {K}_{text {loc}}(Omega ) stands for the local Stummel–Kato class. Let {mathcal {K}_{text {Dini}}}(Omega ) be a variant of mathcal {K}(Omega ) satisfying a Carleson-Dini-type condition. We develop a De Giorgi/Nash/Moser theory for solutions of Lu = f - {{textrm{div}}}g, where f and |g|^2 in {mathcal {K}_{text {Dini}}}(Omega ) if, for q in [n, infty ), any of the following assumptions holds: (i) |b|^2, |d| in {mathcal {K}_{text {Dini}}}(Omega ) and either c in L^{n,q}_{text {loc}}(Omega ) or |c|^2 in mathcal {K}_{text {loc}}(Omega ); (ii) {{textrm{div}}}b +d le 0 and either b+c in L^{n,q}_{text {loc}}(Omega ) or |b+c|^2 in mathcal {K}_{text {loc}}(Omega ); (iii) -{{textrm{div}}}c + d le 0 and |b+c|^2 in {mathcal {K}_{text {Dini}}}(Omega ). We also prove a Wiener-type criterion for boundary regularity. Assuming global conditions on the coefficients, we show that the variational Dirichlet problem is well-posed and, assuming -{{textrm{div}}}c +d le 0, we construct the Green’s function associated with L satisfying quantitative estimates. Under the additional hypothesis |b+c|^2 in mathcal {K}'(Omega ), we show that it satisfies global pointwise bounds and also construct the Green’s function associated with the formal adjoint operator of L. An important feature of our results is that all the estimates are scale invariant and independent of Omega , while we do not assume smallness of the norms of the coefficients or coercivity of the associated bilinear form.

A priori and a posteriori error analysis for semilinear problems in liquid crystals

In this paper, we develop a unified framework for the a priori and a posteriori error control of different lowest-order finite element methods for approximating the regular solutions of systems of partial differential equations under a set of hypotheses. The systems involve cubic nonlinearities in lower order terms, non-homogeneous Dirichlet boundary conditions, and the results are established under minimal regularity assumptions on the exact solution. The key contributions include (i) results for existence and local uniqueness of the discrete solutions using Newton–Kantorovich theorem, (ii) a priori error estimates in the energy norm, and (iii) a posteriori error estimates that steer the adaptive refinement process. The results are applied to conforming, Nitsche, discontinuous Galerkin, and weakly over penalized symmetric interior penalty schemes for variational models of ferronematics and nematic liquid crystals. The theoretical estimates are corroborated by substantive numerical results.

Training a deep operator network as a surrogate solver for two-dimensional parabolic-equation models.

Parabolic equations (PEs) are useful for modeling sound propagation in a range-dependent environment. However, this approach entails approximating a leading-order cross-derivative term in the PE square-root operators. Deep operator networks (DeepONets) are designed to approximate operators. In this paper, we train DeepONets to take complex sound pressure and speed of sound at any depth location of interest as inputs and approximate the PE square operator in modeling two-dimensional sound propagation. Once trained, a network can predict the far field for a wide variety of environmental conditions, without needing to approximate the operator or calculate the whole mode trajectory and at a lower computational cost. The original DeepONet learns the operator of a single function. By contrast, the modified version presented here learns multiple-input operators with Fourier features. Using computational and theoretical examples, we demonstrate that DeepONet is efficient for learning complex ocean acoustic physics with good accuracy.

Existence of renormalized solutions to fully anisotropic and inhomogeneous elliptic problems

We will present the proof of existence and uniqueness of renormalized solutions to a broad family of strongly non-linear elliptic equations with lower order terms and data of low integrability. The leading part of the operator satisfies general growth conditions settling the problem in the framework of fully anisotropic and inhomogeneous Musielak–Orlicz spaces. The setting considered in this paper generalized known results in the variable exponents, anisotropic polynomial, double phase and classical Orlicz setting.

A priori bounds for GIETs, affine shadows and rigidity of foliations in genus two

We prove a rigidity result for foliations on surfaces of genus two, which can be seen as a generalization to higher genus of Herman’s theorem on circle diffeomorphisms and, correspondingly, flows on the torus. We prove in particular that, if a smooth, orientable foliation with non-degenerate (Morse) singularities on a closed surface of genus two is minimal, then, under a full measure condition for the rotation number, it is differentiably conjugate to a linear foliation.The corresponding result at the level of Poincaré sections is that, for a full measure set of (standard) interval exchange transformations (IETs for short) with d=4 or d=5 continuity intervals and irreducible combinatorics, any generalized interval exchange transformation (GIET for short) which is topologically conjugate to a standard IET from this set and satisfies an obstruction expressed in terms of boundary operator (which is automatically satisfied when the GIET arises as a Poincaré map of a smooth foliation) is mathcal{C}^{1}-conjugate to it. This in particular settles a conjecture by Marmi, Moussa and Yoccoz in genus two. Our results also show that this conjecture on the rigidity of GIETs can be reduced to the study of affine IETs, or more precisely of Birkhoff sums of piecewise constant observables over standard IETs, in genus g geq 3.Our approach is via renormalization, namely we exploit a suitable acceleration of the Rauzy-Veech induction (an acceleration which makes Oseledets generic effective) on the space of GIETs. For in ly renormalizable, irrational GIETs of any number of intervals dgeq 2 we prove a dynamical dichotomy on the behaviour of the orbits under renormalization, by proving that either an orbit is recurrent to certain bounded sets in the space of GIETs, or it diverges and it is approximated (up to lower order terms) by the orbit of an affine IET (a case that we refer to as affine shadowing). This result can in particular be used, in conjunction with previous work by Marmi-Moussa and Yoccoz on the existence of wandering intervals for affine IETs, to prove, a priori bounds in genus two and is therefore at the base of the rigidity result.