Asymptotic Relations Research Articles

High-Frequency Limit of the Inverse Scattering Problem: Asymptotic Convergence from Inverse Helmholtz to Inverse Liouville

We investigate the asymptotic relation between the inverse problems relying on the Helmholtz equation and the radiative transfer equation (RTE) as physical models in the high-frequency limit. In particular, we evaluate the asymptotic convergence of a generalized version of the inverse scattering problem based on the Helmholtz equation, to the inverse scattering problem of the Liouville equation (a simplified version of RTE). The two inverse problems are connected through the Wigner transform that translates the wave-type description on the physical space to the kinetic-type description on the phase space, and the Husimi transform that models data localized both in location and direction. The finding suggests that impinging tightly concentrated monochromatic beams can indeed provide stable reconstruction of the medium, asymptotically in the high-frequency regime. This fact stands in contrast with the unstable reconstruction for the classical inverse scattering problem when the probing signals are plane waves.

Asymptotic relations for the distributional Stockwell and wavelet transforms

Представлены результаты абелева и тауберова типов, характеризующие квазиасимптотическое поведение обобщенных функций из $\mathcal{S}_{0}'(\mathbb{R})$ в терминах их преобразований Стоквелла, а также результат абелева типа, связывающий квазиасимптотическую ограниченность обобщенных функций Лизоркина с асимптотикой их преобразований Стоквелла. Кроме того, получен ряд асимптотических результатов о вейвлет-преобразованиях обобщенных функций.

Sharp Constants of Approximation Theory. IV. Asymptotic Relations in General Settings

In this paper we first introduce the unified definition of the sharp constant that includes constants in three major problems of approximation theory, such as, inequalities for approximating elements, approximation of individual elements, and approximation on classes of elements. Second, we find sufficient conditions that imply limit relations between various sharp constants of approximation theory in general settings. Third, a number of examples from various areas of approximation theory illustrates the general approach.

A multi-level exploration of the relationship between temperature and species diversity: Two cases of marine phytoplankton.

The relationship between temperature (T) and diversity is one of the most important issues in ecology. It provides a key direction not only for exploring the determinants of diversity's patterns, but also for understanding diversity's responses to climate change. Previous studies suggested that T-diversity relationships could be positive, negative, or unimodal. Although these studies accumulated many informative achievements, they might be unsatisfied due to (1) investigating inadequate range of T, (2) selecting incomplete diversity metrics, and (3) making insufficiently detailed analysis of correlation. In this study, species diversity is estimated by four most commonly used diversity metrics and three parameters of species abundance distribution (SAD), and two global datasets of marine phytoplankton (covering a wider range of T) are used to evaluate the T-diversity relationships according to a piecewise model. Results show that all aspects of diversity (except evenness) have the similar relationship with T in the range of lower T, noting that diversity significantly increases as T increases. However, in the range of higher T, diversity may significantly decrease or nearly constant, which indicates that their relationships may be the unimodal or asymptotic. The asymptotic relationship found by this study is assumed that increasing diversity with T will gradually approach the Zipf's law (1:1/2:1/3…). If such assumption can be verified by future investigations, the intrinsic mechanism of the asymptotic relationship is likely to be crucial in understanding the T-diversity relationships.

Spectral and Evolution Analysis of Composite Elastic Plates with High Contrast

We analyse the behaviour of thin composite plates whose material properties vary periodically in-plane and possess a high degree of contrast between the individual components. Starting from the equations of three-dimensional linear elasticity that describe soft inclusions embedded in a relatively stiff thin-plate matrix, we derive the corresponding asymptotically equivalent two-dimensional plate equations. Our approach is based on recent results concerning decomposition of deformations with bounded scaled symmetrised gradients. Using an operator-theoretic approach, we calculate the limit resolvent and analyse the associated limit spectrum and effective evolution equations. We obtain our results under various asymptotic relations between the size of the soft inclusions (equivalently, the period) and the plate thickness as well as under various scaling combinations between the contrast, spectrum, and time. In particular, we demonstrate significant qualitative differences between the asymptotic models obtained in different regimes.

Comparing weighted densities

In this paper we study inequalities between weighted densities of sets of natural numbers corresponding to different weight functions. Depending on the asymptotic relation between the weight functions, we give sharp bounds for possible values of one density when the values of another density are given. In particular, we give a condition for two weight functions to generate equivalent weighted densities.

On Fluxbrane Polynomials for Generalized Melvin-like Solutions Associated with Rank 5 Lie Algebras

We consider generalized Melvin-like solutions corresponding to Lie algebras of rank 5 (A5, B5, C5, D5). The solutions take place in a D-dimensional gravitational model with five Abelian two-forms and five scalar fields. They are governed by five moduli functions Hs(z) (s=1,...,5) of squared radial coordinates z=ρ2, which obey five differential master equations. The moduli functions are polynomials of powers (n1,n2,n3,n4,n5)=(5,8,9,8,5),(10,18,24,28,15),(9,16,21,24,25),(8,14,18,10,10) for Lie algebras A5, B5, C5, D5, respectively. The asymptotic behavior for the polynomials at large distances is governed by some integer-valued 5×5 matrix ν connected in a certain way with the inverse Cartan matrix of the Lie algebra and (in A5 and D5 cases) with the matrix representing a generator of the Z2-group of symmetry of the Dynkin diagram. The symmetry and duality identities for polynomials are obtained, as well as asymptotic relations for solutions at large distances.

Modeling sonar performance using J-divergence

This presentation demonstrates the use of J-divergence as a performance measure for detection in a sonar system. The inherent inaccuracies between system-level performance and the cell-level (PD,PF) detector operating point used in traditional analysis open the door to using approximate performance measures such as J-divergence. The properties of J-divergence making it an appealing choice are covered: summing to accrue J-divergence across multiple independent measurements (e.g., from multiple source signals, waveforms, or arrays), a data-processing inequality dictating that processing cannot improve J-divergence, and an asymptotic relationship to the traditional (PD,PF) operating point. Simple forward models of J-divergence are presented for matched filters and energy detectors when applied to the standard signal models in Gaussian noise. A “design” J-divergence, which is chosen by the desired qualitative performance level, is used in simple equations to obtain the “design” SNR require to achieve it. This provides a direct replacement for the detection threshold (DT) term in the sonar equation that is easier to apply and evaluate. Example applications of J-divergence are presented illustrating its utility in the modeling and adaptation of current systems as well as the design and analysis of new ones.

A Note on Asymptotics Between Singular and Constrained Control Problems of One-Dimensional Diffusions

We study the asymptotic relations between certain singular and constrained control problems for one-dimensional diffusions with both discounted and ergodic objectives. In the constrained control problems the controlling is allowed only at independent Poisson arrival times. We show that when the underlying diffusion is recurrent, the solutions of the discounted problems converge in Abelian sense to those of their ergodic counterparts. Moreover, we show that the solutions of the constrained problems converge to those of their singular counterparts when the Poisson rate tends to infinity. We illustrate the results with drifted Brownian motion and Ornstein-Uhlenbeck process.

Onset of convection in internally heated fluids with strongly temperature-dependent viscosity

We investigate the onset of convection in internally heated fluids with strongly temperature-dependent viscosity by solving numerically a non-linearized system of thermal convection equations in two dimensions for viscosity contrasts up to ∼1035. As the viscosity contrast increases, a high-viscosity stagnant lid develops at the upper surface and convection occurs in a sublayer beneath it. The transition to stagnant-lid convection occurs at about the same viscosity contrast as in Rayleigh–Bénard convection. We obtain asymptotic scaling relationships for the critical Rayleigh number and other parameters in the stagnant-lid regime. We also investigated the possibility of subcritical convection. In contrast to the Rayleigh–Bénard problem, we did not detect a subcritical region for internally heated convection in two-dimensional simulations. The results of this study can help improve our understanding of the conditions under which convection occurs in planetary interiors.

Spectral asymptotic of the Newtonian potential in [formula omitted

We establish the asymptotic relation sn(N)∼ωd4d4π2n−2d, where N is the Newtonian potential defined on the space L2(B). Further, we describe the asymptotic behaviour of the Newtonian potential over the space L2(Rd,dμα), dμα(x)=e−|x|2α(απ)d/2dx, α>0, by giving lower and upper asymptotic estimates of its singular numbers.

Asymptotic relations between interpolation differences and zeta functions

Asymptotic relations between zeta functions (such as, ζ(s),β(s), and other Dirichlet L-functions) and interpolation differences of functions like |y|s and their interpolating entire functions of exponential type 1 are discussed. New criteria for zeros of the zeta functions in the critical strip in terms of integrability of the interpolation differences are obtained as well.

Finite Difference Method on Flat Surfaces with a Flat Unitary Vector Bundle

We establish an asymptotic relation between the spectrum of the discrete Laplacian associated to discretizations of a tileable surface with a flat unitary vector bundle and the spectrum of the Friedrichs extension of the Laplacian with Neumann boundary conditions. Our proof is based on the precise study of the singularities of the eigenvectors near the conical points and corners of a half-translation surface, and on establishing Harnack-type estimates on “almost harmonic” discrete functions, defined on the graphs, which approximate the given surface.

Two-body baryonic Bu,d,s and Bc to charmless final state decays

We study the rates and direct $CP$ violations of two-body baryonic ${\overline{B}}_{u,d,s}\ensuremath{\rightarrow}{\mathbf{\text{B}}\overline{\mathbf{\text{B}}}}^{\ensuremath{'}}$ and ${B}_{c}^{\ensuremath{-}}\ensuremath{\rightarrow}{\mathbf{\text{B}}\overline{\mathbf{\text{B}}}}^{\ensuremath{'}}$ decays, where the final state baryons include low-lying octet and decuplet baryons. We incorporate topological amplitude formalism and the factorization approach. Asymptotic relations at large ${m}_{b}$ are used to simplify decay amplitudes. Using the most up-to-date data on ${\overline{B}}^{0}\ensuremath{\rightarrow}p\overline{p}$ and ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}\mathrm{\ensuremath{\Lambda}}\overline{p}$ decay rates as inputs, rates and direct $CP$ violations of ${\overline{B}}_{u,d,s}\ensuremath{\rightarrow}{\mathbf{\text{B}}\overline{\mathbf{\text{B}}}}^{\ensuremath{'}}$ decays are revised and predicted. It is interesting that the results on rates satisfy all existing experimental bounds and some are close to the bounds. Factorization diagrams contribute to penguin-exchange, exchange, annihilation, and penguin-annihilation amplitudes. Although the resulting penguin-exchange factorization amplitudes are sizable, the rest suffer from severe chiral suppression and are sensitive to nonfactorizable contributions. As the ${\overline{B}}_{s}\ensuremath{\rightarrow}p\overline{p}$ decay is governed by exchange and penguin annihilation amplitudes, the rate predicted in factorization calculation is very rare, but it can be enhanced by including nonfactorizable contributions. The case where the rate is enhanced to saturate the present experimental bound through the enhancement on exchange or penguin-annihilation amplitudes is discussed. As annihilation modes ${B}_{c}^{\ensuremath{-}}\ensuremath{\rightarrow}{\mathbf{\text{B}}\overline{\mathbf{\text{B}}}}^{\ensuremath{'}}$ decays from factorization calculation are found to be very rare, but they can be enhanced by including nonfactorizable contributions as well. Small direct $CP$ violations of pure penguin modes in $\mathrm{\ensuremath{\Delta}}S=\ensuremath{-}1$ ${\overline{B}}_{u,d,s}\ensuremath{\rightarrow}{\mathbf{\text{B}}\overline{\mathbf{\text{B}}}}^{\ensuremath{'}}$ decays are robust predictions of the SM, while vanishing direct $CP$ violations of exchange modes in ${\overline{B}}_{u,d,s}\ensuremath{\rightarrow}{\mathbf{\text{B}}\overline{\mathbf{\text{B}}}}^{\ensuremath{'}}$ decays and in all ${B}_{c}^{\ensuremath{-}}\ensuremath{\rightarrow}{\mathbf{\text{B}}\overline{\mathbf{\text{B}}}}^{\ensuremath{'}}$ decay modes are null tests of the SM.

Optimal singular dividend control with capital injection and affine penalty payment at ruin

In this paper, we extend the optimal dividend and capital injection problem with affine penalty at ruin in (Xu, R. & Woo, J.K. (2020). Insurance: Mathematics and Economics 92: 1–16) to the case with singular dividend payments. The asymptotic relationships between our value function to the one with bounded dividend density are studied, which also help to verify that our value function is a viscosity solution to the associated Hamilton–Jacob–Bellman Quasi-Variational Inequality (HJBQVI). We also show that the value function is the smallest viscosity supersolution within certain functional class. A modified comparison principle is proved to guarantee the uniqueness of the value function as the viscosity solution within the same functional class. Finally, a band-type dividend and capital injection strategy is constructed based on four crucial sets; and the optimality of such band-type strategy is proved by using fixed point argument. Numerical examples of the optimal band-type strategies are provided at the end when the claim size follows exponential and gamma distribution, respectively.

Vibrationally-Resolved X-ray Photoelectron Spectra of Six Polycyclic Aromatic Hydrocarbons from First-Principles Simulations.

Vibrationally resolved C 1s X-ray photoelectron spectra (XPS) of a series of six polycyclic aromatic hydrocarbons (PAHs; phenanthrene, coronene, naphthalene, anthracene, tetracene, and pentacene) were computed by combining the full core hole density functional theory and the Franck-Condon simulations with the inclusion of the Duschinsky rotation effect. Simulated spectra of phenanthrene, coronene, and naphthalene agree well with experiments both in core binding energies (BEs) and profiles, which validate the accuracy of our predictions for the rest molecules with no high-resolution experiments. We found that three types of carbons i (inner C), p (peripheral C bonded to three C atoms), and h (peripheral C bonded to an H atom) show decreasing BEs. In linear PAHs (the latter four), h-type carbons further split into h1 or h2 (on inner or edge benzene ring) subtypes with chemical shifts of ca. 0.2-0.4 eV. All major Franck-Condon-active modes are characterized to be in-plane vibrations: low-frequency (<800 cm-1) C-C ring deformation modes play an essential role in determining the peak asymmetries; and for each h-type carbon a high-frequency (ca. 3600 cm-1) C*-H stretching mode is responsible for the high-energy tail. We found that core ionization leads to reduction of all C*-C and C*-H bond lengths and ring deformation with a definite direction. Based on theoretical spectra of four linear PAHs, we found asymptotic relations and anticipated possible spectral features for even larger linear PAHs. Our calculations provide accurate reference spectra for XPS characterizations of PAHs, which are useful in understanding the vibronic coupling effects in this family.

Dynamical Behaviors of a Translating Liquid Crystal Elastomer Fiber in a Linear Temperature Field.

Liquid crystal elastomer (LCE) fiber with a fixed end in an inhomogeneous temperature field is capable of self-oscillating because of coupling between heat transfer and deformation, and the dynamics of a translating LCE fiber in an inhomogeneous temperature field are worth investigating to widen its applications. In this paper, we propose a theoretic constitutive model and the asymptotic relationship of a LCE fiber translating in a linear temperature field and investigate the dynamical behaviors of a corresponding fiber-mass system. In the three cases of the frame at rest, uniform, and accelerating translation, the fiber-mass system can still self-oscillate, which is determined by the combination of the heat-transfer characteristic time, the temperature gradient, and the thermal expansion coefficient. The self-oscillation is maintained by the energy input from the ambient linear temperature field to compensate for damping dissipation. Meanwhile, the amplitude and frequency of the self-oscillation are not affected by the translating frame for the three cases. Compared with the cases of the frame at rest, the translating frame can change the equilibrium position of the self-oscillation. The results are expected to provide some useful recommendations for the design and motion control in the fields of micro-robots, energy harvesters, and clinical surgical scenarios.

On automatic bias reduction for extreme expectile estimation.

Expectiles induce a law-invariant risk measure that has recently gained popularity in actuarial and financial risk management applications. Unlike quantiles or the quantile-based Expected Shortfall, the expectile risk measure is coherent and elicitable. The estimation of extreme expectiles in the heavy-tailed framework, which is reasonable for extreme financial or actuarial risk management, is not without difficulties; currently available estimators of extreme expectiles are typically biased and hence may show poor finite-sample performance even in fairly large samples. We focus here on the construction of bias-reduced extreme expectile estimators for heavy-tailed distributions. The rationale for our construction hinges on a careful investigation of the asymptotic proportionality relationship between extreme expectiles and their quantile counterparts, as well as of the extrapolation formula motivated by the heavy-tailed context. We accurately quantify and estimate the bias incurred by the use of these relationships when constructing extreme expectile estimators. This motivates the introduction of classes of bias-reduced estimators whose asymptotic properties are rigorously shown, and whose finite-sample properties are assessed on a simulation study and three samples of real data from economics, insurance and finance.Supplementary InformationThe online version contains supplementary material available at 10.1007/s11222-022-10118-x.

The partial transpose and asymptotic free independence for Wishart random matrices, II

Using new combinatorial techniques, we significantly improve the previous results on asymptotic distributions and asymptotic free independence relations of partial transposes of Wishart random matrices. In particular, we give a necessary and sufficient condition for the asymptotic free independence of partial transposes of Wishart matrices with difference block sizes.

Asymptotic relations for semi-classical Laguerre orthogonal polynomials and the associated Hankel determinants

We study recurrence coefficients of semi-classical Laguerre orthogonal polynomials and the associated Hankel determinant generated by a semi-classical Laguerre weight w(x,t)=xαe−x−tx2,x∈(0,∞),α&amp;gt;0,t≥0. If t = 0, it is reduced to the classical Laguerre weight. For t &amp;gt; 0, this weight tends to zero faster than the classical Laguerre weight as x → ∞. In the finite n-dimensional case, we obtain two auxiliary quantities Rn(t) and rn(t) by using the Ladder operator approach. We show that the Hankel determinant has an integral representation in terms of Rn(t), where the quantity Rn(t) is closely related to a second-order nonlinear differential equation. Furthermore, we derive a second-order nonlinear differential equation and also a second-order differential equation for the auxiliary quantity σn(t)=−∑j=0n−1Rj(t), which is also related to the logarithmic derivative of the Hankel determinant. In the infinite n-dimensional case, we consider the asymptotic behaviors of the recurrence coefficients and the scaled Laguerre orthogonal polynomials by using the Coulomb fluid method.