Leading-order Research Articles

The Boussinesq approximation for buoyant flows

The aim of this paper is to present a simplified, yet rigorous, deduction of the Boussinesq approximated governing equations for buoyant flows. In order to carry out the core deduction procedure, a simplified version of the manifold asymptotic analyses available in the literature is discussed. The method adopted in this study is focussed on the local balance equations valid for a general, not necessarily Newtonian, fluid. The analysis is carried out by demonstrating the leading order terms in the governing equations for the asymptotic limit which characterises the approximation. The role played by the effect of viscous dissipation is also taken into account.

Quantum Chaos, Random Matrices, and Irreversibility in Interacting Many-Body Quantum Systems.

The Pauli master equation describes the statistical equilibration of a closed quantum system. Simplifying and generalizing an approach developed in two previous papers, we present a derivation of that equation using concepts developed in quantum chaos and random-matrix theory. We assume that the system consists of subsystems with strong internal mixing. We can then model the system as an ensemble of random matrices. Equilibration results from averaging over the ensemble. The direction of the arrow of time is determined by an (ever-so-small) coupling to the outside world. The master equation holds for sufficiently large times if the average level densities in all subsystems are sufficiently smooth. These conditions are quantified in the text, and leading-order correction terms are given.

Complete Ionization for a Non-autonomous Point Interaction Model in d = 2

We consider the two dimensional Schr\"odinger equation with time dependent delta potential, which represents a model for the dynamics of a quantum particle subject to a point interaction whose strength varies in time. First, we prove global well-posedness of the associated Cauchy problem under general assumptions on the potential and on the initial datum. Then, for a monochromatic periodic potential (which also satisfies a suitable no-resonance condition) we investigate the asymptotic behavior of the survival probability of a bound state of the time-independent problem. Such probability is shown to have a time decay of order $\mathcal{O}(\log t/t)^2$, up to lower order terms.

Spectral analysis of Euler–Bernoulli beam model with distributed damping and fully non-conservative boundary feedback matrix

The distribution of natural frequencies of the Euler–Bernoulli beam resting on elastic foundation and subject to an axial force in the presence of several damping mechanisms is investigated. The damping mechanisms are: ( i ) an external or viscous damping with damping coefficient ( − a 0 ( x )), ( ii ) a damping proportional to the bending rate with the damping coefficient a 1 ( x ). The beam is clamped at the left end and equipped with a four-parameter (α, β, κ 1 , κ 2 ) linear boundary feedback law at the right end. The 2 × 2 boundary feedback matrix relates the control input (a vector of velocity and its spacial derivative at the right end) to the output (a vector of shear and moment at the right end). The initial boundary value problem describing the dynamics of the beam has been reduced to the first order in time evolution equation in the state Hilbert space of the system. The dynamics generator has a purely discrete spectrum (the vibrational modes). Explicit asymptotic formula for the eigenvalues as the number of an eigenvalue tends to infinity have been obtained. It is shown that the boundary control parameters and the distributed damping play different roles in the asymptotical formulas for the eigenvalues of the dynamics generator. Namely, the damping coefficient a 1 and the boundary controls κ 1 and κ 2 enter the leading asymptotical term explicitly, while damping coefficient a 0 appears in the lower order terms.

Fixed-Time Stabilization for Nonlinear Systems With Low-Order and High-Order Nonlinearities via Event-Triggered Control

This paper investigates the fixed-time stabilization problem for a class of nonlinear systems via event-triggered control. The event-triggered mechanism can be applied to the nonlinear system with the coexistence of low-order and high-order nonlinearities. In order to deal with these low-order and high-order terms as well as achieve the objective of fixed-time stabilization, the initial value of the system is divided into two cases, and then the event-triggered controller is designed, respectively. By switching control theory, it is proved that the nonlinear system is globally fixed-time stable under the designed controller and the Zeno behavior can be excluded. Finally, two simulations show that the proposed technology is effective.

Asymptotic Evaluation of a Lattice Sum Associated with the Laplacian Matrix

The Laplacian matrix is of fundamental importance in the study of graphs, networks, random walks on lattices, and arithmetic of curves. In certain cases, the trace of its pseudoinverse appears as the only non-trivial term in computing some of the intrinsic graph invariants. Here we study a double sum Fn which is associated with the trace of the pseudo inverse of the Laplacian matrix for certain graphs. We investigate the asymptotic behavior of this sum as n → ∞. Our approach is based on classical analysis combined with asymptotic and numerical analysis, and utilizes special functions. We determine the leading-order term, which is of size n2 log n, and develop general methods to obtain the secondary main terms in the asymptotic expansion of Fn up to errors of $$\mathcal {O}({\rm{log}} \; n)$$ and $$\mathcal {O}(1)$$ as n → ∞. We provide some examples to demonstrate our methods.

An Experimental Design Approach for Regret Minimization in Logistic Bandits

In this work we consider the problem of regret minimization for logistic bandits. The main challenge of logistic bandits is reducing the dependence on a potentially large problem dependent constant that can at worst scale exponentially with the norm of the unknown parameter vector. Previous works have applied self-concordance of the logistic function to remove this worst-case dependence providing regret guarantees that move the reduce the dependence on this worst case parameter to lower order terms with only polylogarithmic dependence on the main term and as well as linear dependence on the dimension of the unknown parameter. This work improves upon the prior art by 1) removing all scaling of the worst case term on the main term and 2) reducing the dependence on the dependence to scale with the square root of dimension in the fixed arm setting by employing an experimental design procedure. Our regret bound in fact takes a tighter instance (i.e., gap) dependent regret bound for the first time in logistic bandits. We also propose a new warmup sampling algorithm that can dramatically reduce the lower order term in the regret in general and prove that it can exponentially reduce the lower order term's dependency on the worst case parameter in some instances. Finally, we discuss the impact of the bias of the MLE on the logistic bandit problem in d dimensions, providing an example where d^2 lower order regret (cf., it is d for linear bandits) may not be improved as long as the MLE is used and how bias-corrected estimators may be used to make it closer to d.

Spatiotemporal dynamics in a twisted, circular waveguide array

AbstractWe consider the existence and spectral stability of nonlinear discrete localized solutions representing light pulses propagating in a twisted multicore optical fiber. By considering an even number, N, of waveguides, we derive asymptotic expressions for solutions in which the bulk of the light intensity is concentrated as soliton‐like pulses confined to a single waveguide. The leading order terms obtained are in very good agreement with results of numerical computations. Furthermore, as in the model without temporal dispersion, when the twist parameter, ϕ, is given by , these standing waves exhibit optical suppression, in which a single waveguide remains unexcited, to leading order. Spectral computations and numerical evolution experiments suggest that these standing wave solutions are stable for values of the coupling parameter less than a critical value, at which point a spectral instability results from the collision of an internal eigenvalue with the eigenvalues at the origin. This critical value has a maximum when .

Existence and regularity of solutions to elliptic equation with singular convection term and lower order term

In this paper, we consider the effects of singular convection term and lower order term on the existence and regularity of solutions to the following elliptic Dirichlet problem: where is a bounded smooth domain with with is a non‐negative function. The main results of this paper show the combined effects of singular convection term and lower order term on the existence and regularity of solution to this problem.

EW radiative corrections to theory predictions of charge asymmetry for W-boson hadroproduction

The higher-order predictions of the lepton charge asymmetry (the W-boson charge asymmetry ) for the leptonic final state of the W-boson hadroproduction in proton–proton (pp) collisions are presented. The predictions from the state-of-the-art computations are reported for achieving adequate description of the by including next-to-leading order (NLO) electroweak (EW) radiative corrections in combination with next-to-NLO (NNLO) quantum chromodynamics (QCD) radiative corrections. The combined predictions NNLO QCD+NLO EW and NNLO QCD × NLO EW, based on standard additive and factorised combination prescriptions in turn, are provided in the fiducial phase space of the pseudorapidity of the decay lepton (of the rapidity of the W-boson), comprising both central and forward detector acceptance regions as η l ≤ 4.5 (y W ≤ 4.5). The inclusion of the NLO EW effects for the () constitutes additional input for the relative u- and d-quark densities in the proton, which is also of high importance in the domain of the high-precision studies. The predicted distributions are compared with the actual measurements by CERN Large Hadron Collider (LHC) experiments at 8 TeV pp collisions energies. The combined predictions for the () distributions are also provided in comparisons with the NNLO QCD predictions at both 13 TeV and 14 TeV energies. The impact of the NLO EW corrections for the () distributions is extensively assessed by means of relative correction factor analysis with respect to the NNLO QCD predictions, in addition to a detailed K-factor analysis with respect to the leading order (LO) accuracy. The predicted results show that the NLO EW effects have larger impact in the forward η l region of the contrary to the central η l region, and is sizable in some of the y W ranges of the . The paper suggests inclusion of the presented EW corrections at NLO to have explicit accounting for the EW effects for the () in phenomenological studies.

A Priori Direct Numerical Simulation Analysis of the Closure of Cross-Scalar Dissipation Rate of Reaction Progress Variable and Mixture Fraction in Turbulent Stratified Flames

The cross-scalar dissipation rate of reaction progress variable and mixture fraction widetilde{{varepsilon_{cxi } }} plays an important role in the modelling of stratified combustion. The evolution and statistical behaviour of widetilde{{varepsilon_{cxi } }} have been analysed using a direct numerical simulation (DNS) database of statistically planar turbulent stratified flames with a globally stochiometric mixture. A parametric analysis has been conducted by considering a number of DNS cases with a varying initial root-mean-square velocity fluctuation u^{prime } and initial scalar integral length scale ell_{phi }. The explicitly Reynolds averaged DNS data suggests that the linear relaxation model for widetilde{{varepsilon_{cxi } }} is inadequate for most cases, but its performance appears to improve with increasing initial ell_{phi } and u^{prime } values. An exact transport equation for widetilde{{varepsilon_{cxi } }} has been derived from the first principle, and the budget of the unclosed terms of the widetilde{{varepsilon_{cxi } }} transport equation has been analysed in detail. It has been found that the terms arising from the density variation, scalar-turbulence interaction, chemical reaction rate and molecular dissipation rate play leading order roles in the widetilde{{varepsilon_{cxi } }} transport. These observations have been justified by a scaling analysis, which has been utilised to identify the dominant components of the leading order terms to aid model development for the unclosed terms of the widetilde{{varepsilon_{cxi } }} transport equation. The performances of newly proposed models for the unclosed terms have been assessed with respect to the corresponding terms extracted from DNS data, and the newly proposed closures yield satisfactory predictions of the unclosed terms in the widetilde{{varepsilon_{cxi } }} transport equation.

The Nonsmooth Dynamics of Combined Slip and Spin Motion Under Dry Friction

We investigate the motion of rigid bodies subject to combined slipping and spinning over a rough flat surface in the presence of dry friction. Integration of Coulomb friction forces over the contact area gives rise to a dynamical system with an isolated discontinuity of codimension 3. Recent results about such vector fields are applied to the motion of flat bodies under the assumption of known, time-independent distributions of normal contact forces and to general bodies where kinematic constraints enforce a state-dependent normal contact force distribution with a discontinuity at the sticking state. In both cases, the equations of motion are transformed into a smooth slow–fast dynamical system. The fixed points of the fast flow indicate the possible directions of combined slip–spin motion immediately before a body stops. We also introduce an approximation of the frictional forces and moments by the leading-order term of a spherical harmonic expansion, which allows for an explicit formulation of the equations of motion. The approximate model captures important empirically observed features of the motion. It is proven analytically and illustrated by examples that the number of fixed points in the approximate model is 2, 4, or 6.

A systematic analysis of the memory term in coarse-grained models: The case of the Markovian approximation

The systematic development of coarse-grained (CG) models via the Mori–Zwanzig projector operator formalism requires the explicit description of a deterministic drift term, a dissipative memory term and a random fluctuation term. The memory and fluctuating terms are related by the fluctuation–dissipation relation and are more challenging to sample and describe than the drift term due to complex dependence on space and time. This work proposes a rational basis for a Markovian data-driven approach to approximating the memory and fluctuating terms. We assumed a functional form for the memory kernel and under broad regularity hypothesis, we derived bounds for the error committed in replacing the original term with an approximation obtained by its asymptotic expansions. These error bounds depend on the characteristic time scale of the atomistic model, representing the decay of the autocorrelation function of the fluctuating force; and the characteristic time scale of the CG model, representing the decay of the autocorrelation function of the momenta of the beads. Using appropriate parameters to describe these time scales, we provide a quantitative meaning to the observation that the Markovian approximation improves as they separate. We then proceed to show how the leading-order term of such expansion can be identified with the Markovian approximation usually considered in the CG theory. We also show that, while the error of the approximation involving time can be controlled, the Markovian term usually considered in CG simulations may exhibit significant spatial variation. It follows that assuming a spatially constant memory term is an uncontrolled approximation which should be carefully checked. We complement our analysis with an application to the estimation of the memory in the CG model of a one-dimensional Lennard–Jones chain with different masses and interactions, showing that even for such a simple case, a non-negligible spatial dependence for the memory term exists.

The chiral Gross-Neveu model on the lattice via a Landau-forbidden phase transition

We study the phase diagram of the (1 + 1)-dimensional Gross-Neveu model with both {g}_x^2{left(overline{psi}psi right)}^2 and {g}_y^2{left(overline{psi}i{gamma}_5psi right)}^2 interaction terms on a spatial lattice. The continuous chiral symmetry, which is present in the continuum model when {g}_x^2={g}_y^2 , has a mixed ’t Hooft anomaly with the charge conservation symmetry, which guarantees the existence of a massless mode. However, the same ’t Hooft anomaly also implies that the continuous chiral symmetry is broken explicitly in our lattice model. Nevertheless, from numerical matrix product state simulations we find that for certain parameters of the lattice model, the continuous chiral symmetry reemerges in the infrared fixed point theory, even at strong coupling. We argue that, in order to understand this phenomenon, it is crucial to go beyond mean-field theory (or, equivalently, beyond the leading order term in a 1/N expansion). Interestingly, on the lattice, the chiral Gross-Neveu model appears at a Landau-forbidden second order phase transition separating two distinct and unrelated symmetry-breaking orders. We point out the crucial role of two different ’t Hooft anomalies or Lieb-Schultz-Mattis obstructions for this Landau-forbidden phase transition to occur.

Multiplicity dependence of charged-particle jet production in pp collisions at mathbf {sqrt{s}}=mathbf {13~TeV}

The multiplicity dependence of jet production in pp collisions at the centre-of-mass energy of sqrt{s} = 13 {mathrm {TeV}} is studied for the first time. Jets are reconstructed from charged particles using the anti-k_mathrm {T} algorithm with resolution parameters R varying from 0.2 to 0.7. The jets are measured in the pseudorapidity range |eta _{mathrm{jet}}|< 0.9-R and in the transverse momentum range 5<p_mathrm {T,jet}^{mathrm{ch}}<140 {mathrm {GeV}}/c. The multiplicity intervals are categorised by the ALICE forward detector V0. The p_{mathrm T} differential cross section of charged-particle jets are compared to leading order (LO) and next-to-leading order (NLO) perturbative quantum chromodynamics (pQCD) calculations. It is found that the data are better described by the NLO calculation, although the NLO prediction overestimates the jet cross section below 20 {mathrm {GeV}}/c. The cross section ratios for different R are also measured and compared to model calculations. These measurements provide insights into the angular dependence of jet fragmentation. The jet yield increases with increasing self-normalised charged-particle multiplicity. This increase shows only a weak dependence on jet transverse momentum and resolution parameter at the highest multiplicity. While such behaviour is qualitatively described by the present version of PYTHIA, quantitative description may require implementing new mechanisms for multi-particle production in hadronic collisions.

Nonlinear corrections to the single differential cross section for neutral current e−p scattering in the NLO approximation

We present the effects of nonlinear corrections to the single differential cross section d\sigma/dQ^2 and the reduced cross section \sigma_r (x,Q^2) for the neutral current (NC) e-p scattering at the leading order (LO) and the next-to-leading order (NLO) approximations in perturbative quantum chromo dynamics (QCD). Technically, based on the double Laplace transform method, we first derive the effects of the nonlinear corrections to the proton structure functions F2(x,Q^2) and FL(x,Q^2) and consequently obtain the corresponding single differential and reduced cross sections. Our results clearly indicate the consistency of the nonlinear behavior of the quark and gluon distributions at low x values. Our numerical results ( obtained in a range of the virtuality 8.5 < Q2 < 5000 GeV ^2 and the Bjorken scale 10^(-5)< x < 1) show that the effects of these nonlinear corrections to the proton structure functions are more noticeable at x < 0.001 and, to some extent, control the incremental trend of these functions at low x values. Moreover, a comparison of our numerical results of the single differential and reduced cross sections at the NLO approximations with those of H1 collaboration data shows that the nonlinear corrections increase the accuracy of calculations rather than the linear calculations at low to moderate Q^2 values for low x values.

Thermodynamics for a Rotating Chiral Fermion System in the Uniform Magnetic Field

We study the thermodynamics for a uniformly rotating system of chiral fermions under the uniform magnetic field. Then, we obtain the mathematical expressions of some thermodynamic quantities in terms of the series with respect to the external magnetic field B, the angular velocity Ω and the chemical potential μ, expanded around B=0, Ω=0 and μ=0. Our results given by such series are a generalization of the expressions available in the references simply corresponding to the lower-order terms of our findings. The zero-temperature limit of our results is also discussed.

Twist-3 gluon fragmentation contribution to hyperon polarization in semi-inclusive deep inelastic scattering

We derive the twist-3 gluon fragmentation function (FF) contribution to the transversely polarized hyperon production in semi-inclusive deep inelastic scattering, $ep\ensuremath{\rightarrow}e{\mathrm{\ensuremath{\Lambda}}}^{\ensuremath{\uparrow}}X$, in the leading order (LO) with respect to the QCD coupling in the framework of the collinear twist-3 factorization. Together with the known result for the contribution from the twist-3 distribution in the proton and the twist-3 quark FFs for the hyperon, this completes the LO cross section for this process. The constraint relations among the twist-3 FFs are taken into account. The formula is relevant to large-${P}_{T}$ hyperon production in the future electron-ion-collider experiment.

Asymptotic stability of small ground states for NLS under random perturbations

We consider the cubic Schrödinger (NLS) equation on the Euclidean space perturbed by a short-range potential V . The presence of a negative simple eigenvalue for -\Delta+V gives rise to a curve of small and localized nonlinear ground states that yield some time-periodic solutions known to be asymptotically stable in the energy space. We study the persistence of these coherent states under rough perturbations. We shall construct a large measure set of small scaling-supercritical solutions below the energy space that display some asymptotic stability behaviors. The main difficulty is the need to handle the interactions of localized and dispersive terms in the modulation equations. To do so, we use a critical-weighted strategy to combine probabilistic nonlinear estimates in critical spaces based on U^p , V^q (controlling higher-order terms) with some local energy decay estimates (controlling lower-order terms). We also revisit in the perturbed setting the analysis of Bényi, Oh, and Pocovnicu (2015) on the probabilistic global well-posedness and scattering for small supercritical initial data. We use a distorted Fourier transform and semiclassical functional calculus to generalize probabilistic and bilinear Strichartz estimates.

Discrete weighted Hardy inequality in 1-D

In this paper we consider weighted versions of one dimensional discrete Hardy's inequality on the half-line with power weights of the form nα; namely, we consider:(0.1)∑n=1∞|u(n)−u(n−1)|2nα≥c(α)∑n=1∞|u(n)|2n2nα.We prove the above inequality when α∈[0,1)∪[5,∞) with the sharp constant c(α). Furthermore, when α∈[1/3,1)∪{0}, we prove an improved version of (0.1) by adding infinitely many positive lower order terms in the RHS. More precisely, we prove(0.2)∑n=1∞|u(n)−u(n−1)|2nα≥c(α)∑n=1∞|u(n)|2n2nα+∑k=3∞bk(α)∑n=2∞|u(n)|2nknα, for non-negative constants bk(α).