Leading-order Research Articles

Parabolic equations with unbounded lower-order coefficients in Sobolev spaces with mixed norms

We prove the \(L_{p,q}\)-solvability of parabolic equations in divergence form with full lower-order terms. The coefficients and non-homogeneous terms belong to mixed Lebesgue spaces with the lowest integrability conditions. In particular, the coefficients for the lower-order terms are not necessarily bounded. We study both the Dirichlet and conormal derivative boundary value problems on irregular domains. We also prove embedding results for parabolic Sobolev spaces, the proof of which is of independent interest.

A method to remove lower order contributions in multi-particle femtoscopic correlation functions

In recent years the femtoscopy technique has been used by the ALICE Collaboration in small colliding systems at the LHC to investigate the strong-interaction of hadron pairs in the low-energy regime. The extension of this technique to the study of many-body correlations aims to deliver in the next years the first experimental measurements of the genuine many-hadron interactions, provided that the contributions due to the lower order terms are properly accounted for. In this paper we present a method that allows to determine the residual lower order contributions to the three-body correlation functions, based on the cumulant decomposition approach and on kinematic transformations. A procedure to simulate genuine three-body correlations in three-baryon correlation functions is also developed. A qualitative study of the produced correlation signal is performed by varying the strength of the adopted three-body interaction model and comparisons with the expectations for the lower order contributions to the correlation function are shown. The method can be also applied to evaluate the combinatorial background in the two-body correlation functions, providing an improved statistical accuracy with respect to the standard techniques. The example of the contribution by the pK^+K^- channel to the recently measured pupphi correlation is discussed.

Nonlinear elliptic equations with a lower-order term depending on the gradient in Orlicz spaces

Nonlinear elliptic equations with a lower-order term depending on the gradient in Orlicz spaces

Scaling dimensions at large charge for cubic ϕ3 theory in six dimensions

The $O(N)$ model with scalar quartic interactions at its ultraviolet fixed point, and the $O(N)$ model with scalar cubic interactions at its infra-red fixed point are conjectured to be equivalent. This has been checked by comparing various features of the two models at their respective fixed points. Recently, the scaling dimensions of a family of operators of fixed charge $Q$ have been shown to match at the FPs up to $\cal{O}\left(\frac{1}{N^2}\right)$at leading order (LO) and next-to-leading order (NLO) in $Q$ using a semiclassical computation which is valid to all orders in the coupling. Here we perform a complementary but overlapping comparison using a perturbative calculation in six dimensions, up to three-loop order in the coupling, to compare these critical scaling dimensions beyond NLO in $Q$, in fact to all relevant orders in $Q$. We also obtain the corresponding results at $\cal{O}\left(\frac{1}{N^3}\right)$ for the cubic theory.

LOWER-ORDER TERMS OF THE ONE-LEVEL DENSITY OF A FAMILY OF QUADRATIC HECKE -FUNCTIONS

AbstractIn this paper, we study lower-order terms of the one-level density of low-lying zeros of quadratic Hecke L-functions in the Gaussian field. Assuming the generalized Riemann hypothesis, our result is valid for even test functions whose Fourier transforms are supported in $(-2, 2)$ . Moreover, we apply the ratios conjecture of L-functions to derive these lower-order terms as well. Up to the first lower-order term, we show that our results are consistent with each other when the Fourier transforms of the test functions are supported in $(-2, 2)$ .

Optimal non-adaptive probabilistic group testing in general sparsity regimes

Abstract In this paper, we consider the problem of noiseless non-adaptive probabilistic group testing, in which the goal is high-probability recovery of the defective set. We show that in the case of $n$ items among which $k$ are defective, the smallest possible number of tests equals $\min \{ C_{k,n} k \log n, n\}$ up to lower-order asymptotic terms, where $C_{k,n}$ is a uniformly bounded constant (varying depending on the scaling of $k$ with respect to $n$) with a simple explicit expression. The algorithmic upper bound follows from a minor adaptation of an existing analysis of the Definite Defectives algorithm, and the algorithm-independent lower bound builds on existing works for the regimes $k \le n^{1-\varOmega (1)}$ and $k = \varTheta (n)$. In sufficiently sparse regimes (including $k = o\big ( \frac{n}{\log n} \big )$), our main result generalizes that of Coja-Oghlan et al. (2020) by avoiding the assumption $k \le n^{1-\varOmega (1)}$, whereas in sufficiently dense regimes (including $k = \omega \big ( \frac{n}{\log n} \big )$), our main result shows that individual testing is asymptotically optimal for any non-zero target success probability, thus strengthening an existing result of Aldridge (2019, IEEE Trans. Inf. Theory, 65, 2058–2061) in terms of both the error probability and the assumed scaling of $k$.

Measurement of GMPT Coefficients for Improved Object Characterisation in Metal Detection

Magnetic polarizability tensors (MPTs) have become popular for characterising conducting permeable objects and assisting with the identification of hidden objects in metal detection for applications in security screening, humanitarian demining and scrap sorting. A rigorous mathematical justification of the complex symmetric rank 2 MPT object characterisation has been established based on the leading order term in an asymptotic expansion of the perturbed field for small objects. However, the accuracy of an MPT object characterisation is limited by the tensor’s small number of independent coefficients. By considering higher order terms in the asymptotic expansion, generalised magnetic polarizabilty tensors (GMPTs) have been introduced and the purpose of this work is to show that GMPT coefficients can, for the first time, be measured in practice. GMPTs offer the possibility to better discriminate between objects and, hence, the potential for better classification and identification, overcoming the limitations of a rank 2 MPT object characterisation. In a metal detector, the low-frequency background fields generated by a set of coils is almost always non-uniform and using GMPTs allow us to make a virtue of this. In this work we include both measurements and simulations to demonstrate the advantages that using GMPTs offer over using an MPT characterisation alone.

J/ψ associated production with a bottom quark pair from the Higgs boson decay in next-to-leading order QCD

In this work, we investigate the next-to-leading order (NLO) QCD correction to $J/\ensuremath{\psi}$ associated production with a bottom quark pair from the Higgs boson decay within the nonrelativistic QCD framework. From numerical results, we find that the decay width of process $H\ensuremath{\rightarrow}b+J/\ensuremath{\psi}+\overline{b}$ at leading order (LO) mainly comes from the contribution of the Fock state ${^{3}S}_{1}^{(8)}$, and the NLO QCD corrections significantly enhance the decay width at LO accuracy by about two times. At NLO accuracy, the Fock states ${^{3}S}_{1}^{(8)}$ and ${^{3}P}_{J}^{(8)}$ channels give the main contribution, accounting for about 68% and 29% of the total decay width of $J/\ensuremath{\psi}$ associated production with a bottom quark pair at NLO accuracy from the Higgs boson decay, respectively. Considering the dominant contribution of color octet (CO) channels at NLO accuracy, the inclusive decay process $H\ensuremath{\rightarrow}b+J/\ensuremath{\psi}+\overline{b}+X$ has the potential to be found in future colliders with high energy/luminosity. The study of $J/\ensuremath{\psi}$ associated production with a bottom quark pair from the Higgs boson decay is not only useful to study the mechanism of color-octet, but also to assist in the investigation of the coupling for the Higgs boson with the bottom quark.

Scope and convergence of the hopping parameter expansion in finite-temperature quantum chromodynamics with heavy quarks around the critical point

Abstract Hopping parameter expansion is a useful tool for investigating heavy dynamical quarks in lattice quantum chromodynamics (QCD), though its range of applicability has sometimes been questioned. We study the convergence and the valid range of the hopping parameter expansion in determining the critical point (critical quark mass) of QCD with heavy quarks at finite temperature and density. On lattices with sufficiently large spatial extent, the terms in the hopping parameter expansion are classified into Wilson loop terms and Polyakov-type loop terms. We first study the case of the worst convergence in which all the gauge link variables are unit matrices and thus the Wilson loops and the Polyakov-type loops get their maximum values. We perform explicit calculation up to more than the 100th order of the hopping parameter expansion. We show that the hopping parameter expansion is convergent up to the chiral limit of free Wilson quarks. We then perform a Monte Carlo simulation to measure correlation among Polyakov-type loop terms up to the 20th order of the hopping parameter expansion. In previous studies, strong correlation between the leading-order Polyakov loop term and the next-to-leading-order bent Polyakov loop terms was reported and used to construct an effective theory to incorporate the next-to-leading-order effect by a shift of the leading-order coupling parameter. We establish that the strong correlation among Polyakov-type loop terms also holds at higher orders of the hopping parameter expansion, and extend the effective theory to incorporate higher-order effects up to high orders. Using the effective theory, we study the truncation error of the hopping parameter expansion. We find that the previous next-to-leading-order result for the critical point for Nt = 4 is reliable. For Nt ≥ 6, we need to incorporate higher-order effects in the effective theory.

Existence and uniqueness of renormalized solution for nonlinear parabolic equations in Musielak Orlicz spaces

This paper is devoted to the study of a class of parabolic equation of type$$ \frac{\partial u}{\partial t} -div(A(x,t,u,\nabla u) +B(x,t,u)) =f \quad\mbox{in}\quad Q_T, $$where $div(A(x,t,u,\nabla u)$ is a Leray-Lions type operator, $B(x,t,u)$ is a nonlinear lower order term and $f\in L^{1}(Q_{T})$.We show the existence and the uniqueness of renormalized solution in the framework of Musielak-Orlicz spaces.

Does Comma Selection Help to Cope with Local Optima?

One hope when using non-elitism in evolutionary computation is that the ability to abandon the current-best solution aids leaving local optima. To improve our understanding of this mechanism, we perform a rigorous runtime analysis of a basic non-elitist evolutionary algorithm (EA), the $(\mu,\lambda)$ EA, on the most basic benchmark function with a local optimum, the jump function. We prove that for all reasonable values of the parameters and the problem, the expected runtime of the $(\mu,\lambda)$~EA is, apart from lower order terms, at least as large as the expected runtime of its elitist counterpart, the $(\mu+\lambda)$~EA (for which we conduct the first runtime analysis on jump functions to allow this comparison). Consequently, the ability of the $(\mu,\lambda)$~EA to leave local optima to inferior solutions does not lead to a runtime advantage. We complement this lower bound with an upper bound that, for broad ranges of the parameters, is identical to our lower bound apart from lower order terms. This is the first runtime result for a non-elitist algorithm on a multi-modal problem that is tight apart from lower order terms.

Omega results for cubic field counts via lower-order terms in the one-level density

Abstract In this paper, we obtain a precise formula for the one-level density of L-functions attached to non-Galois cubic Dedekind zeta functions. We find a secondary term which is unique to this context, in the sense that no lower-order term of this shape has appeared in previously studied families. The presence of this new term allows us to deduce an omega result for cubic field counting functions, under the assumption of the Generalised Riemann Hypothesis. We also investigate the associated L-functions Ratios Conjecture and find that it does not predict this new lower-order term. Taking into account the secondary term in Roberts’s conjecture, we refine the Ratios Conjecture to one which captures this new term. Finally, we show that any improvement in the exponent of the error term of the recent Bhargava–Taniguchi–Thorne cubic field counting estimate would imply that the best possible error term in the refined Ratios Conjecture is $O_\varepsilon (X^{-\frac 13+\varepsilon })$ . This is in opposition with all previously studied families in which the expected error in the Ratios Conjecture prediction for the one-level density is $O_\varepsilon (X^{-\frac 12+\varepsilon })$ .

Generalized $p$-Laplacian systems with lower order terms

This work is devoted to studying the existence of solutions to systems of $p$-Laplacian type. We prove the existence of at least one weak solution, under some assumptions, by applying Galerkin's approximation and the theory of Young measures.

Gradient estimates for the p-Laplacian parabolic equations with a low-order term

Gradient estimates for the p-Laplacian parabolic equations with a low-order term

Electric dipole moment of light nuclei in chiral effective field theory

CP-violating interactions at quark level generate CP-violating nuclear interactions and currents, which could be revealed by looking at the presence of a permanent nuclear electric dipole moment. Within the framework of chiral effective field theory, we discuss the derivation of the CP-violating nuclear potential up to next-to-next-to leading order (N2LO) and the preliminary results for the charge operator up to next-to leading order (NLO). Moreover, we introduce some renormalization argument which indicates that we need to promote the short-distance operator to the leading order (LO) in order to reabsorb the divergences generated by the one pion exchange. Finally, we present some selected numerical results for the electric dipole moments of 2H, 3He and 3H discussing the systematic errors introduced by the truncation of the chiral expansion.

Inverse problems for the fractional Laplace equation with lower order nonlinear perturbations

<p style='text-indent:20px;'>We study the inverse problem for the fractional Laplace equation with multiple nonlinear lower order terms. We show that the direct problem is well-posed and the inverse problem is uniquely solvable. More specifically, the unknown nonlinearities can be uniquely determined from exterior measurements under suitable settings.</p>

The dimensional estimates of exponential growth solutions to uniformly elliptic equations of non-divergence form

<p style='text-indent:20px;'>In this paper, we demonstrate that, for general second order uniformly elliptic equations of non-divergence form in unbounded cylinders with zero boundary condition, the space of solutions with a given exponential growth is of finite dimension. In particular, we generalize a result of Hang and Lin (1999) [<xref ref-type="bibr" rid="b4">4</xref>] in the non-divergence's setting by using a different approach. This allows us to consider the general lower order terms. Furthermore, a sharp estimate of the dimension is established by using the mean value inequality.</p>

Generalized Keller–Osserman Conditions for Fully Nonlinear Degenerate Elliptic Equations

We discuss the existence of entire (i.e. defined on the whole space) subsolutions of fully nonlinear degenerate elliptic equations, giving necessary and sufficient conditions on the coefficients of the lower order terms which extend the classical Keller–Osserman conditions for semilinear elliptic equations. Our analysis shows that, when the conditions of existence of entire subsolutions fail, a priori upper bounds for local subsolutions can be obtained.

Improved Coding Over Sets for DNA-Based Data Storage

Error-correcting codes over sets, with applications to DNA storage, are studied. The DNA-storage channel receives a set of sequences, and produces a corrupted version of the set, including sequence loss, symbol substitution, symbol insertion/deletion, and limited-magnitude errors in symbols. Various parameter regimes are studied. New bounds on code parameters are provided, which improve upon known bounds. New codes are constructed, at times matching the bounds up to lower-order terms or small constant factors.

Partitions of normalised multiple regression equations for datum transformations

Multiple regression equations (MREs) provide an empirical direct method of transforming coordinates between geodetic datums. Since they offer a means of modelling distortions, they are capable of a more accurate fit to datum-shift datasets than more basic direct methods. MRE models of datum shifts traditionally consist of polynomials based on relative latitude and longitude. However, the limited availability of low-power terms often leads to high-power terms being included, and these are a potential cause of instability. This paper introduces three variations based on simple partitions and 2 or 4 smoothly conjoined polynomials. The new types are North/South, East/West and Four-Quadrant. They increase the availability of low-order terms, enabling distortions to be modelled with fewer side effects. Case studies in Great Britain, Slovenia and Western Australia provide examples of partitioned MREs that are more accurate than conventional MREs with the same number of terms.