Asymptotic Terms Research Articles

Analysis of 2-D elastic solid with multiple V-notches by a fast multipole BEM with a novel singular element with multi-order asymptotic terms

A novel fast multipole boundary element method (BEM) is developed to analyze the stress fields of plane structures with multiple V-notches. To accurately evaluate the singular stresses in the vicinity of the V-notch tip, a novel singular element with multi-order asymptotic terms (SEMAT) is first proposed. The tractions and displacements on the SEMAT are expressed as the asymptotic expansions, where both the first singular term and other higher-order terms are considered. The SEMAT is then introduced into the fast multipole BEM, and several singularity problems of boundary integrals on the SEMAT are treated with caution. Finally, the fast multipole BEM with the SEMAT is used to obtain the whole stress and displacement fields for the plane V-notch/crack structures. Numerical examples show that the present method is accurate for analyzing the V-notch with opening angle of arbitrary size in both single and bonded materials. It is not necessary to use a dense mesh in modeling, and the computational efficiency of the present method is high for large-scale problems.

Low Frequency Cointegrating Regression with Local to Unity Regressors and Unknown Form of Serial Dependence

This article develops new t and F tests in a low-frequency transformed triangular cointegrating regression when one may not be certain that the economic variables are exact unit root processes. We first show that the low-frequency transformed and augmented OLS (TA-OLS) method exhibits an asymptotic bias term in its limiting distribution. As a result, the test for the cointegration vector can have substantially large size distortion, even with minor deviations from the unit root regressors. To correct the asymptotic bias of the TA-OLS statistics for the cointegration vector, we develop modified TA-OLS statistics that adjust the bias and take account of the estimation uncertainty of the long-run endogeneity arising from the bias correction. Based on the modified test statistics, we provide Bonferroni-based tests of the cointegration vector using standard t and F critical values. Monte Carlo results show that our approach has the correct size and reasonable power for a wide range of local-to-unity parameters. Additionally, our method has advantages over the IVX approach when the serial dependence and the long-run endogeneity in the cointegration system are important.

Projected transverse momentum resummation in top-antitop pair production at LHC

The transverse momentum distribution of the toverline{t} system is of both experimental and theoretical interest. In the presence of azimuthally asymmetric divergences, pursuing resummation at high logarithmic precision is rather demanding in general. In this paper, we propose the projected transverse momentum spectrum textrm{d}{sigma}_{toverline{t}}/textrm{d}{q}_{tau } , which is derived from the classical {overrightarrow{q}}_{textrm{T}} spectrum by integrating out the rejection component {q}_{tau_{perp }} with respect to a reference unit vector overrightarrow{tau} , to serve as an alternative solution to remove these asymmetric divergences, in addition to the azimuthally averaged case textrm{d}{sigma}_{toverline{t}}/textrm{d}mid {overrightarrow{q}}_{textrm{T}}mid . In the context of the effective field theories, SCETII and HQET, we will demonstrate that in spite of the {q}_{tau_{perp }} integrations, the leading asymptotic terms of textrm{d}{sigma}_{toverline{t}}/textrm{d}{q}_{tau } still observe the factorisation pattern in terms of the hard, beam, and soft functions in the vicinity of qτ = 0 GeV. Then, with the help of the renormalisation group equation techniques, we carry out the resummation at NLL+NLO, N2LL+N2LO, and approximate N2LL′+N2LO accuracy on three observables of interest, textrm{d}{sigma}_{toverline{t}}/d{q}_{textrm{T},textrm{in}},textrm{d}{sigma}_{toverline{t}}/textrm{d}{q}_{textrm{T},textrm{out}} , and textrm{d}{sigma}_{toverline{t}}/dDelta {phi}_{toverline{t}} , within the domain {M}_{toverline{t}} ≥ 400 GeV. The first two cases are obtained by choosing overrightarrow{tau} parallel and perpendicular to the top quark transverse momentum, respectively. The azimuthal de-correlation Delta {phi}_{toverline{t}} of the toverline{t} pair is evaluated through its kinematical connection to qT,out. This is the first time the azimuthal spectrum Delta {phi}_{toverline{t}} is appraised at or beyond the N2LL level including a consistent treatment of both beam collinear and soft radiation.

Valuation of barrier and lookback options under hybrid CEV and stochastic volatility

In this paper, we evaluate down-and-out put option and floating strike lookback option prices when the underlying asset is driven by a hybrid model with constant elasticity of variance and stochastic volatility (SVCEV). Usually, it is difficult to get closed-form solutions for those exotic options under stochastic volatility models. Here, we use an asymptotic expansion approach and the Mellin transform method to obtain explicit closed-form formulae for the zero-order and first-order correction terms. In addition, we perform a sensitivity analysis numerically on the asymptotic terms and compare the option prices corresponding to the Black–Scholes, CEV and SVCEV models with those calculated by Monte-Carlo simulations and the binomial tree method to illustrate the accuracy of our pricing formulae.

Asymptotics of the Solution to the Cauchy Problem for a Singularly Perturbed Operator Differential Transport Equation with Weak Diffusion

Formal asymptotic expansions of the solution to the Cauchy problem for a singularly perturbed operator differential transport equation with weak diffusion and small nonlinearity are constructed in the critical case. Under certain conditions imposed on the data of the problem, an asymptotic expansion of the solution is constructed in the form of series in powers of a small parameter with coefficients depending on stretched variables. Problems for determining all terms of the asymptotic expansion are obtained. It is shown that the leading term of the solution asymptotics is determined by solving Cauchy problems for a parabolic Burgers-type equation and, under certain conditions, for a Korteweg–de Vries–Burgers type equation. The remainder terms are estimated with respect to the residual.

Robust Boundary Controller Design with Proportional Integral Observer for a Linear 1D Heat Equation*

The problem of state estimation with boundary disturbance reconstruction and output regulation for linear 1D parabolic systems with a boundary measurement and unknown constant disturbance input on the opposite boundary is addressed. As the disturbance is unmatched it cannot be directly compensated by the measurement. The problem is addressed with an approach that consists of a backstepping observer for the nominal part with an asymptotic compensation term for the disturbance reconstruction and control offset compensation. The main difference to similar studies relies in the fact that the state space does not need to be extended, leading to a reduced-order observer setup. The effectiveness of the approach is illustrated using numerical simulations.

‘Far interaction’ of small spectral perturbations of the Neumann boundary conditions for an elliptic system of differential equations in a three-dimensional domain

A formally selfadjoint system of second-order differential equations is considered in a three-dimensional domain on small parts of whose boundary an analogue of Steklov spectral conditions is set, while the Neumann boundary conditions are set on the rest of the boundary. Under certain algebraic and geometric conditions an asymptotic expression for the eigenvalues of this problem is presented and a limiting problem is put together, which produces the leading asymptotic terms and involves systems of integro-differential equations in half-spaces, interconnected by means of certain integral characteristics of vector-valued eigenfunctions. One example of a concrete problem in mathematical physics describes surface waves in several ice holes made in the ice cover of a water basin, and the asymptotic formula for eigenfrequencies shows that the local wave processes interact independently of the distance between the holes. Another series of applied problems relates to elastic fixings of bodies along small pieces of their surfaces. Possible generalizations are discussed; a number of related open questions are stated. Bibliography: 41 titles.

Impact of the tangential traction for radial hydraulic fracture

The radial (penny-shaped) model of hydraulic fracture is considered. The tangential traction on the fracture walls is incorporated, including an updated evaluation of the energy release rate (fracture criterion), system asymptotics and the need to account for stagnant zone formation near the injection point. The impact of incorporating the shear stress on the construction of solvers, and the effectiveness of approximating system parameters using the first term of the crack tip asymptotics, is discussed. A full quantitative investigation of the impact of tangential traction on solution is undertaken, utilising an extremely effective adaptive time-space solver.

Perturbative connection formulas for Heun equations

Connection formulas relating Frobenius solutions of linear ODEs at different Fuchsian singular points can be expressed in terms of the large order asymptotics of the corresponding power series. We demonstrate that for the usual, confluent and reduced confluent Heun equation, the series expansion of the relevant asymptotic amplitude in a suitable parameter can be systematically computed to arbitrary order. This allows to check a recent conjecture of Bonelli-Iossa-Panea Lichtig-Tanzini expressing the Heun connection matrix in terms of quasiclassical Virasoro conformal blocks.

Spectral asymptotics at thresholds for a Dirac-type operator on [formula omitted

In this article, we provide the spectral analysis of a Dirac-type operator on Z2 by describing the behavior of the spectral shift function associated with a sign–definite trace–class perturbation by a multiplication operator. We prove that it remains bounded outside a single threshold and obtain its main asymptotic term in the unbounded case. Interestingly, we show that the constant in the main asymptotic term encodes the interaction between a flat band and whole non–constant bands. The strategy used is the reduction of the spectral shift function to the eigenvalue counting function of some compact operator which can be studied as a toroidal pseudo–differential operator.

On Gegenbauer Point Processes on the Unit Interval

In this paper we compute the logarithmic energy of points in the unit interval [-1,1] chosen from different Gegenbauer Determinantal Point Processes. We check that all the different families of Gegenbauer polynomials yield the same asymptotic result to third order, we compute exactly the value for Chebyshev polynomials and we give a closed expression for the minimal possible logarithmic energy. The comparison suggests that DPPs cannot match the value of the minimum beyond the third asymptotic term.

Boundary lubrication and super lubricity, textured surfaces

Over last couple of decades, various regimes of lubrication attracted lots of attention. Among these regimes of lubrication, the least understood and studied are the regimes or boundary lubrication and lubricity. Better understanding these regimes of lubrication is necessary due to the fact that many moving joints such as, for example, bearings and gears at least part of their operating time work in these regimes. It may occur due to slow motion of counter parts or their relative overload. This paper proposes a model for analysis of these regimes of lubrication. Specifically, the model takes into account roughness of one of the contact surfaces. It is shown that for sufficiently high applied load it can be expected that contact surfaces are completely separated by a lubricant layer in spite of the presence of surface asperities. The definitions of boundary lubrication regimes and super lubricity are proposed. A method based on the method of matched asymptotic expansions is applied for analyzing such cases and an approximate analytical solution of the problem away from the contact region boundary is found. In the narrow inlet and exit zones adjacent to contact boundaries equations describing the main solution terms are derived. For the case of super lubricity, a resemblance of a superposition between the lubrication and surface roughness effects is determined and the effect of surface roughness on the lubrication film thickness is considered. Specifically, certain roughness conditions are identified for which the lubrication film thickness can be higher than for smooth surfaces while in other cases it can be lower. For the super lubricity regimes, the equations for the main pressure asymptotic terms in the inlet and exit zones are reduced to the corresponding equations for the smooth (not rough) surfaces for which a series of numerical solutions is obtained earlier. A general analysis of the effect of texturing on lubrication conditions in case of super lubricity is conducted based on a simple geometric criterion. Several models of textured surfaces and their influence on the lubrication film thickness in case of super lubricity regimes are considered.

Asymptotic analysis of spectral problems in thick junctions with the branched fractal structure

A spectral problem is considered in a domain that is the union of a domain and a lot of thin trees situated ‐periodically along some manifold on the boundary of The trees have finite number of branching levels. The perturbed Robin boundary condition is given on the th branching layer; are real parameters. The asymptotic analysis of this problem is made as , that is, when the number of the thin trees infinitely increases and their thickness vanishes. In particular, the Hausdorff convergence of the spectrum to the spectrum of the corresponding nonstandard homogenized spectral problem is proved, the leading terms of asymptotics are constructed, and the corresponding asymptotic estimates are justified for the eigenvalues and eigenfunctions.

Autoregressive Models with Time-Dependent Coefficients—A Comparison between Several Approaches

Autoregressive-moving average (ARMA) models with time-dependent (td) coefficients and marginally heteroscedastic innovations provide a natural alternative to stationary ARMA models. Several theories have been developed in the last 25 years for parametric estimations in that context. In this paper, we focus on time-dependent autoregressive (tdAR) models and consider one of the estimation theories in that case. We also provide an alternative theory for tdAR processes that relies on a ρ-mixing property. We compare these theories to the Dahlhaus theory for locally stationary processes and the Bibi and Francq theory, made essentially for cyclically time-dependent models, with our own theory. Regarding existing theories, there are differences in the basic assumptions (e.g., on derivability with respect to time or with respect to parameters) that are better seen in specific cases such as the tdAR(1) process. There are also differences in terms of asymptotics, as shown by an example. Our opinion is that the field of application can play a role in choosing one of the theories. This paper is completed by simulation results that show that the asymptotic theory can be used even for short series (less than 50 observations).

A CLT for second difference estimators with an application to volatility and intensity

In this paper, we introduce a general method for estimating the quadratic covariation of one or more spot parameter processes associated with continuous time semimartingales, and present a central limit theorem that has this class of estimators as one of its applications. The class of estimators we introduce, that we call Two-Scales Quadratic Covariation (TSQC) estimators, is based on sums of increments of second differences of the observed processes, and the intervals over which the differences are computed are rolling and overlapping. This latter feature lets us take full advantage of the data, and, by sufficiency considerations, ought to outperform estimators that are based on only one partition of the observational window. Moreover, a two-scales approach is employed to deal with asymptotic bias terms in a systematic manner, thus automatically giving consistent estimators without having to work out the form of the bias term on a case-to-case basis. We highlight the versatility of our central limit theorem by applying it to a novel leverage effect estimator that does not belong to the class of TSQC estimators. The principal empirical motivation for the present study is that the discrete times at which a continuous time semimartingale is observed might depend on features of the observable process other than its level, such as its spot-volatility process. As an application of the TSQC estimators, we therefore show how it may be used to estimate the quadratic covariation between the spot-volatility process and the intensity process of the observation times, when both of these are taken to be semimartingales. The finite sample properties of this estimator are studied by way of a simulation experiment, and we also apply this estimator in an empirical analysis of the Apple stock. Our analysis of the Apple stock indicates a rather strong correlation between the spot volatility process of the log-prices process and the times at which this stock is traded and hence observed.

Asymptotic series solution of variational stokes problems in planar domain with crack-like singularity

ABSTRACT A class of nonlinear variational problems describing incompressible fluids and solids by stationary Stokes equations given in a planar domain with a crack (infinitely thin flat plate in fluids) is considered. Based on the Fourier asymptotic analysis, general analytical solutions are obtained in polar coordinates as the power series with respect to the distance to the crack tip. The logarithm terms and angular functions are accounted in the asymptotic expansion using recurrence relations. Then boundary conditions imposed between the opposite crack faces in the sector of angle determine admissible exponents and parameters in the power series. For the specific conditions of Dirichlet, Neumann, impermeability, non-penetration and shear crack, the principal asymptotic terms are derived, which verify the singular behaviour. In particular, the analytical solution answers the questions of a square-root singularity at the crack tip and the presence of log-oscillations of variational solutions for the Stokes problems.

Inverse Spectral Problem for Integro-Differential Sturm–Liouville Operators with Discontinuity Conditions

We consider the Sturm–Liouville operator perturbed by a convolution integral operator on a finite interval with Dirichlet boundary conditions and discontinuity conditions in the middle of the interval. We study the inverse problem of recovering the convolution term from the spectrum. The problem is reduced to solving the so-called main nonlinear integral equation with a singularity. To derive and study this equation, we do detailed analysis of kernels of transformation operators for the integro-differential expression considered. We prove global solvability of the main equation, which enables us to prove uniqueness of solution of the inverse problem and to obtain necessary and sufficient conditions for its solvability in terms of asymptotics of the spectrum. The proof is constructive and gives an algorithm for solving the inverse problem.

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 $k$.

Statistics of the first Galois cohomology group : A refinement of Malle’s conjecture

Malle proposed a conjecture for counting the number of $G$-extensions $L/K$ with discriminant bounded above by $X$, denoted $N(K,G;X)$, where $G$ is a fixed transitive subgroup $G\subset S_n$ and $X$ tends towards infinity. We introduce a refinement of Malle's conjecture, if $G$ is a group with a nontrivial Galois action then we consider the set of crossed homomorphisms in $Z^1(K,G)$ (or equivalently $1$-coclasses in $H^1(K,G)$) with bounded discriminant. This has a natural interpretation given by counting $G$-extensions $F/L$ for some fixed $L$ and prescribed extension class $F/L/K$. If $T$ is an abelian group with any Galois action, we compute the asymptotic growth rate of this refined counting function for $Z^1(K,T)$ (and equivalently for $H^1(K,T)$) and show that it is a natural generalization of Malle's conjecture. The proof technique is in essence an application of a theorem of Wiles on generalized Selmer groups, and additionally gives the asymptotic main term when restricted to certain local behaviors. As a consequence, whenever the inverse Galois problem is solved for $G\subset S_n$ over $K$ and $G$ has an abelian normal subgroup $T\trianglelefteq G$ we prove a nontrivial lower bound for $N(K,G;X)$ given by a nonzero power of $X$ times a power of $\log X$. For many groups, including many solvable groups, these are the first known nontrivial lower bounds. These bounds prove Malle's predicted lower bounds for a large family of groups, and for an infinite subfamily they generalize Kl\"uners' counter example to Malle's conjecture and verify the corrected lower bounds predicted by T\"urkelli.

Semiclassical Approach to the Nonlocal Kinetic Model of Metal Vapor Active Media

A semiclassical approach based on the WKB–Maslov method is developed for the kinetic ionization equation in dense plasma with approximations characteristic of metal vapor active media excited by a contracted discharge. We develop the technique for constructing the leading term of the semiclassical asymptotics of the Cauchy problem solution for the kinetic equation under the supposition of weak diffusion. In terms of the approach developed, the local cubic nonlinear term in the original kinetic equation is considered in a nonlocal form. This allows one to transform the nonlinear nonlocal kinetic equation to an associated linear partial differential equation with a given accuracy of the asymptotic parameter using the dynamical system of moments of the desired solution of the equation. The Cauchy problem solution for the nonlinear nonlocal kinetic equation can be obtained from the solution of the associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation. Within the developed approach, the plasma relaxation in metal vapor active media is studied with asymptotic solutions expressed in terms of higher transcendental functions. The qualitative analysis of such the solutions is given.