Asymptotic Expansion Research Articles

Logarithmic terms in discrete heat kernel expansions in the quadrant

In the context of lattice walk enumeration in cones, we consider the number of walks in the quarter plane with fixed starting and ending points, prescribed step-set, and given length. After renormalisation, this number may be interpreted as a discrete heat kernel in the quadrant. We propose a new method to compute complete asymptotic expansions of these numbers of walks as their length tends to infinity, based on two main ingredients: explicit expressions for the underlying generating functions in terms of elliptic Jacobi theta functions along with a duality known as Jacobi transformation. This duality allows us to pass from a classical Taylor expansion of the series to an expansion at the critical point of the model. We work through two examples. First, we present our approach on the well-known Kreweras model, which is algebraic, and show how to obtain a complete asymptotic expansion in this case. We then consider a more generic (so-called infinite group) model and find the associated complete asymptotic expansion. In this second case, we prove the existence of logarithmic terms in the asymptotic expansion, and we relate the coefficients appearing in the expansion to polyharmonic functions. To our knowledge, this is the first time that logarithmic terms have been observed in the asymptotics of a class of lattice walks confined to a quadrant.

Ensemble Generalization of the Perdew-Zunger Self-Interaction Correction: A Way Out of Multiple Minima and Symmetry Breaking.

The Perdew-Zunger (PZ) self-interaction correction (SIC) is an established tool to correct unphysical behavior in density functional approximations. Yet, the PZ-SIC is well-known to sometimes break molecular symmetries. An example of this is the benzene molecule, for which the PZ-SIC predicts a symmetry-broken electron density and molecular geometry, since the method does not describe the two possible Kekulé structures on an even footing, leading to local minima [Lehtola et al. J. Chem. Theory Comput. 2016, 12, 3195]. The PZ-SIC is often implemented with Fermi-Löwdin orbitals (FLOs), yielding the FLO-SIC method, which likewise has issues with symmetry breaking and local minima [Trepte et al. J. Chem. Phys. 2021, 155, 224109]. In this work, we propose a generalization of the PZ-SIC─the ensemble PZ-SIC (E-PZ-SIC) method─which shares the asymptotic computational scaling of the PZ-SIC (albeit with an additional prefactor). The E-PZ-SIC is straightforwardly applicable to various molecules, merely requiring one to average the self-interaction correction over all possible Kekulé structures, in line with chemical intuition. We showcase the implementation of the E-PZ-SIC with FLOs, as the resulting E-FLO-SIC method is easy to realize on top of an existing implementation of the FLO-SIC. We show that the E-FLO-SIC indeed eliminates symmetry breaking, reproducing a symmetric electron density and molecular geometry for benzene. The ensemble approach suggested herein could also be employed within approximate or locally scaled variants of the PZ-SIC and its FLO-SIC versions.

Asymptotic profiles of zero points of solutions to the heat equation

In this paper, we consider the asymptotic profiles of zero points for the spatial variable of the solutions to the heat equation. By giving suitable conditions for the initial data, we prove the existence of zero points by extending the high-order asymptotic expansion theory for the heat equation. This reveals a previously unknown asymptotic profile of zero points diverging at O ( t ) O(t) . In a one-dimensional spatial case, we show the zero point’s second and third-order asymptotic profiles in a general situation. We also analyze a zero level set in high-dimensional spaces and obtain results that extend the results for the one-dimensional spatial case.

Instantons in ϕ4 theories: Transseries, virial theorems, and numerical aspects

We discuss numerical aspects of instantons in two- and three-dimensional ϕ4 theories with an internal O(N) symmetry group, the so-called N-vector model. By combining asymptotic transseries expansions for large arguments with convergence acceleration techniques, we obtain high-precision values for certain integrals of the instanton that naturally occur in loop corrections around instanton configurations. Knowledge of these numerical properties is necessary in order to evaluate corrections to the large-order factorial growth of perturbation theory in ϕ4 theories. The results contribute to the understanding of the mathematical structures underlying the instanton configurations. Published by the American Physical Society 2024

Finiteness theorems and counting conjectures for the flux landscape

In this paper, we explore the string theory landscape obtained from type IIB and F-theory flux compactifications. We first give a comprehensive introduction to a number of mathematical finiteness theorems, indicate how they have been obtained, and clarify their implications for the structure of the locus of flux vacua. Subsequently, in order to address finer details of the locus of flux vacua, we propose three mathematically precise conjectures on the expected number of connected components, geometric complexity, and dimensionality of the vacuum locus. With the recent breakthroughs on the tameness of Hodge theory, we believe that they are attainable to rigorous mathematical tools and can be successfully addressed in the near future. The remainder of the paper is concerned with more technical aspects of the finiteness theorems. In particular, we investigate their local implications and explain how infinite tails of disconnected vacua approaching the boundaries of the moduli space are forbidden. To make this precise, we present new results on asymptotic expansions of Hodge inner products near arbitrary boundaries of the complex structure moduli space.

Microstructure evolution and fluid transport in porous media: a formal asymptotic expansions approach

Microstructure evolution and fluid transport in porous media: a formal asymptotic expansions approach

Renormalization-group improved resummation of super-leading logarithms

A new strategy is presented for systematically treating super-leading logarithmic contributions including higher-order Glauber exchanges for non-global LHC observables in renormalization-group (RG) improved perturbation theory. This represents an important improvement over previous approaches, as it allows for the consistent inclusion of the scale dependence of the strong coupling, thereby providing more reliable estimates of the scale uncertainties in theoretical predictions. The key idea is to rearrange the relevant RG evolution operator in such a way that all double-logarithmic corrections are exponentiated from the outset. This forms the starting point for the first resummation of super-leading logarithms at leading order in RG-improved perturbation theory for arbitrary 2 → M scattering processes. Moreover, the asymptotic scaling of subleading logarithmic corrections from higher-order Glauber exchanges is determined, demonstrating their parametric suppression.

Rigorous Holographic Bound on AdS Scale Separation.

We give an elementary proof of the following property of unitary, interacting four-dimensional N=2 superconformal field theories: At large central charge c, there exist at least sqrt[c] single-trace, scalar superconformal primary operators with dimensions Δ≲sqrt[c] (suppressing multiplicative logarithmic corrections). This follows from a stronger, more refined bound on the spectral density in terms of the asymptotic growth rate of the central charge. The proof employs known results on the structure of Coulomb branch operators. Interpreted holographically, this bounds the possible degree of scale separation in semiclassical AdS_{5} half-maximal supergravity. In particular, the bulk must contain an infinite tower of charged scalar states of energies parametrically below the large black hole threshold E_{BH}∼c. We address the extreme case of AdS_{5} pure supergravity, ruling it out as the asymptotic limit of certain sequences in theory space, though the general question remains open.

Soliton resolution for the modified short pulse equation with the weighted Sobolev initial data

In this paper, the Cauchy problem of the modified short pulse (mSP) equation with initial conditions in weighted Sobolev space is studied by using [Formula: see text]-steepest descent method. Based on the spectral analysis of Lax pair and the transformations of field variables and independent variables, a basic Riemann–Hilbert problem (RHP) is established, and the solution of Cauchy problem for the mSP equation is transformed into the corresponding RHP. The asymptotic expansion of the solution of mSP equation is derived in a fixed space-time cone [Formula: see text] According to this, the soliton resolution conjecture of the mSP equation is given, in which the leading term is characterized by an [Formula: see text]-soliton on the discrete spectrum, the second term comes from the soliton–radiation interactions on the continuum spectrum, and the error term [Formula: see text] is generated by the corresponding [Formula: see text]-problem.

Gevrey Versus q-Gevrey Asymptotic Expansions for Some Linear q-Difference–Differential Cauchy Problem

Gevrey Versus q-Gevrey Asymptotic Expansions for Some Linear q-Difference–Differential Cauchy Problem

Derivation and numerical resolution of 2D shallow water equations for multi-regime flows of Herschel–Bulkley fluids

This paper presents mathematical modelling and simulation of thin free-surface flows of viscoplastic fluids with a Herschel–Bulkley rheology over complex topographies with basal perturbations. Using the asymptotic expansion method, depth-averaged models (lubrication and shallow water type models) are derived for 3D (three-dimensional) multi-regime flows on non-flat inclined topographies with varying basal slipperiness. Starting from the Navier–Stokes equations, two flow regimes corresponding to different balances between shear and pressure forces are presented. Flow models corresponding to these regimes are calculated as perturbations of the zeroth-order solutions. The classical reference models in the literature are recovered by considering their respective cases on a flat-inclined surface. In the second regime case, a pressure term is non-negligible. Mathematically, it leads to a corrective term to the classical regime equations. Flow solutions of the two regimes are compared; the difference appears in particular in the vicinity of sharp changes of slopes. Nonetheless, both regime models are compared with experiments and are found to be in good agreement. Furthermore, numerical examples are shown to illustrate the robustness of the present shallow water models to simulate viscoplastic flows in 3D and over an inclined topography with local perturbations in basal elevation and basal slipperiness. The derived models are adequate for direct (engineering and geophysical) applications to real-world flow problems presenting Herschel–Bulkley rheology like lava and mud flows.

Federated dual averaging learning algorithm with delayed gradients for composite optimization

Federated learning offers a promising paradigm for training machine learning models in distributed edge networks. This paper focuses on a critical aspect of federated learning, specifically examining the influence of communication delays between the server and edge devices. We study the problem of federated composite optimization (FCO) in the scenario where communication delays are introduced due to client-sides. To tackle this issue, we propose a federated dual averaging learning algorithm that incorporates delayed gradients. We conduct a thorough convergence analysis of our algorithm taking into account the impact of the delayed gradient information. Our analysis demonstrates that under the assumption of bounded expected random delays, the algorithm exhibits the asymptotic convergence towards the optimal value. Furthermore, we present explicit convergence rates of the proposed algorithm for fixed and random delays. Our results show that the impact of delays on FCO problems with smooth loss functions gradually becomes negligible. Finally, to demonstrate the efficacy of the proposed algorithm, we conduct numerical simulations on the FEMNIST dataset utilizing ℓ1-regularized logistic regression. These experimental results confirm our findings that the algorithm converges effectively even in the presence of delays.

Incompressible boundary layer flow past a hump with temperature-dependent viscosity

The problem of boundary layer flow past a hump placed on a flat surface, where viscosity is coupled with temperature, is investigated using triple-deck theory. The incompressible Navier–Stokes equations supplemented by the heat and viscosity equations are considered in the limit that the Reynolds number is large. The triple-deck scalings are adopted, and asymptotic expansions are used to obtain some novel triple-deck equations coupled with the thermal boundary layer. We have used the same viscosity–temperature dependence as in Jasmine and Gajjar [Int. J. Heat Mass Transfer 48, 1022–1037 (2005)], in which viscosity varies inversely proportional to the temperature. The linear analysis is performed, and the nonlinear equations are solved numerically. Our results show that increasing the characteristic constant of the viscosity leads to enhanced peaks and troughs in the pressure and in the skin friction over the hump, as compared to the constant viscosity case. This suggests that temperature-dependent viscosity is more likely to provoke separation.

Wigner Function Method for Describing Electromagnetic Field in Plasma-Like Media with Spatial Dispersion and Resonant Dissipation

The paper gives a systematic presentation of the Wigner function method (Weyl formalism) for modeling the propagation and absorption of electromagnetic waves in anisotropic and gyrotropic dissipative media with spatial dispersion. A general kinetic equation for the Wigner function (tensor) is formulated, and its asymptotic expansion up to the second order for smoothly inhomogeneous and weakly dissipative media is constructed. As a result, a modification of the method of the kinetic equation for rays is proposed, based on the stochastic description of rays, making it possible to increase the accuracy of numerical modeling of wave problems with strong transverse inhomogeneity of the absorption coefficient without increasing the amount of calculations. The technique developed can be used to describe the propagation, absorption, and scattering of electron-cyclotron waves in high-temperature plasma of magnetic traps for controlled fusion in cases where standard modeling methods do not provide the necessary accuracy.

Symmetry breaking of rotating convection due to non-Oberbeck–Boussinesq effects

The non-Oberbeck–Boussinesq (NOB) effects arising from variations in thermal expansivity are theoretically and numerically studied in the context of rotating Rayleigh–Bénard convection in forms of two-dimensional rolls. The thermal expansivity increases with pressure (depth), and its variation is measured by a dimensionless factor ε. Utilizing an asymptotic expansion with weak nonlinearity, we derive an amplitude equation, revealing that NOB effects amplify the magnitude of convection. An ε2-order NOB correction leads to a symmetry breaking about the horizontal mid-plane, manifested in the strengthening of convection near the bottom and its weakening near the top, forming bottom-heavy profiles. At ε3-order, the conjunction of NOB effects and nonlinear advection leads to a horizontal symmetry breaking. The values of Taylor number and Prandtl number determine whether upward or downward plumes are stronger. Numerical calculations validate the theoretical analyses in weakly nonlinear regime. This work advances our understanding of hydrothermal plumes in some winter lakes on Earth and in the subglacial oceans on icy moons as well as tracer transport from the seafloor to the ice shell.

Frequency Response of the Diffuse-Charge Dynamics in Electrochemical Systems with a Graphene Electrode

We investigate the frequency response of diffuse-charge dynamics related to a 1:1 symmetric electrolyte containing a graphene electrode by solving the governing Poisson-Nernst-Planck equations subjected to appropriate boundary conditions in the asymptotic limit ε = λ D /H → 0, where λ D is the Debye screening length and H is the half-thickness of the electrolyte. Using the method of matched asymptotic expansion, we first solve the leading order non-linear problem for equilibrium state at a nonzero applied DC voltage in the presence of Stern layer(s). Then, we extend the leading order asymptotic analysis to derive an analytic expression for the impedance of the graphene-based electrochemical cell when a small AC voltage perturbation is added to the applied DC voltage. Finally, we use a suitable scaling of the impedance parameters to expose the impacts of the ion concentration and the DC bias voltage on the frequency response for possible applications involving the graphene-electrolyte interface.

Hydrodynamic instability of shear imposed falling film over a uniformly heated inclined undulated substrate

Linear and weakly nonlinear stability analyses of an externally shear-imposed, gravity-driven falling film over a uniformly heated wavy substrate are studied. The longwave asymptotic expansion technique is utilized to formulate a single nonlinear free surface deflection equation. The linear stability criteria for the onset of instability are derived using the normal mode form in the linearized portion of the surface deformation equation. Linear stability theory reveals that the flow-directed sturdy external shear grows the surface wave instability by increasing the net driving force. On the contrary, the upstream-directed imposed shear may reduce the surface mode instability by restricting the gravity-driving force, which has the consequence of weakening the bulk velocity of the liquid film. However, the surface mode can be stabilized/destabilized by increasing the temperature-dependent density/surface-tension variation. Furthermore, the bottom steepness shows dual behavior on the surface instability depending upon the wavy wall's portion (uphill/downhill). At the downhill portion, the surface wave becomes more unstable than at the bottom substrate's uphill portion. Moreover, the multi-scale method is incorporated to obtain the complex Ginzburg–Landau equation in order to study the weakly nonlinear stability, confirming the existence of various flow regions of the liquid film. At any bottom portion (uphill/downhill), the flow-directed external shear expands the supercritical stable zones, which causes an amplification in the nonlinear wave amplitude, and the backflow-directed shear plays a counterproductive role. On the other hand, the supercritical stable region decreases or increases as long as the linear variation of density or surface tension increases with respect to the temperature, whereas the sub-critical unstable region exhibits an inverse trend.

Theoretical evaluation of high-permeability wellbore skin effect on aquifer response under pumping

Skin zones with distinct hydraulic properties often form around wellbores during drilling operations. Conventional skin-factor models, developed for low-permeability (positive) skin, assume discontinuous hydraulic heads between the wellbore and aquifer. However, these models may be inadequate for high-permeability (negative) skin scenarios resulting from intensive well development. This study derives a new skin-factor approach for negative skin approaches based on the two-zone model approach. We derive a new skin-factor approach that maintains continuous heads between the wellbore and aquifer while allowing for discontinuous fluxes. Using Laplace and finite cos-Fourier transformations, we develop a groundwater flow model for aquifer pumping under a partially penetrating well, incorporating this new approach. An asymptotic expansion method is employed to obtain a large-time solution. The proposed model is also compared to the existing two-zone model. Sensitivity analysis reveals that high-permeability skin significantly affects drawdown, particularly when there is a sudden increase in porosity within the skin zone. This revised approach provides a more accurate framework for modeling various skin effects, offering valuable insights for groundwater management and well hydraulic modeling in both hydrogeology and petroleum engineering applications.

Stellarator equilibrium axis-expansion to all orders in distance from the axis for arbitrary plasma beta

A systematic theory of the asymptotic expansion of the magnetohydrostatics (MHS) equilibrium in the distance from the magnetic axis is developed to include arbitrary smooth currents near the magnetic axis. Compared with the vacuum and the force-free system, an additional magnetic differential equation must be solved to obtain the pressure-driven currents. It is shown that there exist variables in which the rest of the MHS system closely mimics the vacuum system. Thus, a unified treatment of MHS fields is possible. The mathematical structure of the near-axis expansions to arbitrary order is examined carefully to show that the double-periodicity of physical quantities in a toroidal domain can be satisfied order by order. The essential role played by the leading-order Birkhoff–Gustavson normal form in solving the magnetic differential equations is highlighted. Several explicit examples of vacuum, force-free and MHS equilibrium in different geometries are presented.

The maximum norm error estimate and Richardson extrapolation methods of a second-order box scheme for a hyperbolic-difference equation with shifts

The study aims at the development and theoretical analyses of a box method used to solve a kind of hyperbolic equations with shifts. By using the discrete energy method, it is shown that numerical solutions converge to exact solutions with an order of O(τ 2 + h 2) in L ∞-norm as τ = O(h). Here, τ and h are temporal and spatial meshsizes, respectively. According to local truncation error, the asymptotic expansion formula of the numerical solutions is derived by introducing two auxiliary problems. By applying this asymptotic expansion formula, a class of Richardson extrapolations methods have been derived to achieve extrapolation solutions with an order of O(τ 4 + h 4) in L ∞-norm. Numerical results show the high performance of the proposed methods and the exactness of theoretical findings.