Stationary Phase Points Research Articles

Non-sterile cultivation of Yarrowia lipolytica in fed-batch mode for the production of lipids and biomass.

A reduction in the cost of production and energy requirement is necessary for developing sustainable commercial bioprocesses. Bypassing sterilization, which is an energy and cost-intensive part of bioprocesses could be a way to achieve this. In this study, nonsterile cultivation of Yarrowia lipolytica was done on a synthetic medium containing acetic acid as the sole carbon source using two different strategies in the fed-batch mode. The contamination percentages throughout the process were measured using flow cytometry and complemented using brightfield microscopy. Maximum biomass and lipid yields of 0.57 (g biomass/g substrate) and 0.17 (g lipids/g substrate), respectively, and maximum biomass and lipid productivities of 0.085 and 0.023 g/L/h, respectively, were obtained in different fed-batch strategies. Feeding at the point of stationary phase resulted in better biomass yield and productivity with less than 2% contamination till 48 h. Feeding to maintain a minimum acetic level resulted in better lipid yield and productivity with less than 2% contamination during the complete process. The results of this study demonstrate the potential for cultivating Y. lipolytica in nonsterile conditions and monitoring the contamination throughout the process using flow cytometry.

Geometric approach for determining stationary phase points in radar imaging

This paper proposes a geometric approach to find out the stationary phase points in radar imaging, namely the coordinates of the transmitting and the receiving antennas that have the greatest contribution to a specified point in the wavenumber spectrum of the scattered field. The foundation of the proposed approach is that, the Green’s function can be approximated by a locally plane wave when the distance between the individual antennas and the object is much larger than the wavelength. In this way, the stationary phase points can be easily found through geometric calculations, much more convenient than the prevailing algebraic approach of letting the first derivative of the phase term to be zero. Six examples for different propagation backgrounds and antenna setups are provided in this paper. It is demonstrated that, the proposed method is beneficial in eliminating extraneous roots of the stationary phase points, which could be generated by letting the first derivative of the phase term to be zero in the stationary phase method.

Long-time asymptotics for the integrable nonlocal Lakshmanan–Porsezian–Daniel equation with decaying initial value data

In this work, we study the Cauchy problem of integrable nonlocal Lakshmanan-Porsezian-Daniel equation with rapid attenuation of initial data. The basic Riemann–Hilbert problem of integrable nonlocal Lakshmanan-Porsezian-Daniel equation is constructed from Lax pair. Using Deift-Zhou nonlinear steepest descent method, the explicit long-time asymptotic formula of integrable nonlocal Lakshmanan-Porsezian-Daniel equation is derived, which is different from the local model. Besides, compared to the nonlocal nonlinear Schrödinger equation, since the increase of real stationary phase points, the long-time asymptotic formula for nonlocal Lakshmanan-Porsezian-Daniel equation becomes more complex.

The Fast Physical Optics Method With the Linear Amplitude and Quadratic Phase Terms

The physical optics (PO) scattered fields from the electrically large scatterers contain slowly varying amplitude and highly oscillatory phase terms. In this work, first, we consider quadratic approximations for the phase term of the PO scattered fields. Second, the linear amplitude of the PO scattered fields [linear-fast PO (L-FPO)] is proposed. The simplification of the linear amplitude term is adopted for two reasons. One is that the contributions of the PO scattered fields mainly come from the high-frequency critical points, that is, the stationary phase points, the boundary phase points, and the vertex points. When the wave frequency goes large, the linearization of the amplitude term could guarantee the accuracy of the PO scattered fields from the high-frequency critical points. The other reason is that the L-FPO method proposed in this work avoids the time-consuming and complicated calculation of Fresnel integrals. The computational cost of the Fresnel integrals is shrunk drastically. For the aircraft model, the CPU time of the proposed method is reduced by one order of magnitude compared with Gordon’s method and nearly half compared with the Q-FPO method. In the process of the PO method of determining the lit region of the scatterer, the new shadowing technique pairing a quadratic quadrilateral patch into two subtriangles achieves high accuracy in the determination of the shadow boundary.

Long Time Asymptotic Behavior for the Nonlocal mKdV Equation in Solitonic Space–Time Regions

We study the long time asymptotic behavior for the Cauchy problem of an integrable real nonlocal mKdV equation with nonzero initial data in the solitonic regions \begin{align*} &q_t(x,t)-6\sigma q(x,t)q(-x,-t)q_{x}(x,t)+q_{xxx}(x,t)=0, &q(x,0)=q_{0}(x),\ \ \lim_{x\to \pm\infty} q_{0}(x)=q_{\pm}, \end{align*} where $|q_{\pm}|=1$ and $q_{+}=\delta q_{-}$, $\sigma\delta=-1$. In our previous article, we have obtained long time asymptotics for the nonlocal mKdV equation in the solitonic region $-6<\xi<6$ with $\xi=\frac{x}{t}$. In this paper, we calculate the asymptotic expansion of the solution $q(x,t)$ for other solitonic regions $\xi<-6$ and $\xi>6$. Based on the Riemann-Hilbert problem of the the Cauchy problem, further using the $\bar{\partial}$ steepest descent method, we derive different long time asymptotic expansions of the solution $q(x,t)$ in above two different space-time solitonic regions. In the region $\xi<-6$, phase function $\theta(z)$ has four stationary phase points on the $\mathbb{R}$. Correspondingly, $q(x,t)$ can be characterized with an $\mathcal{N}(\Lambda)$-soliton on discrete spectrum, the leading order term on continuous spectrum and an residual error term, which are affected by a function ${\rm Im}\nu(\zeta_i)$. In the region $\xi>6$, phase function $\theta(z)$ has four stationary phase points on $i\mathbb{R}$, the corresponding asymptotic approximations can be characterized with an $\mathcal{N}(\Lambda)$-soliton with diverse residual error order $\mathcal{O}(t^{-1})$.

Long-time asymptotic behavior of the coupled dispersive AB system in low regularity spaces

In this paper, we mainly investigate the long-time asymptotic behavior of the solution for coupled dispersive AB systems with weighted Sobolev initial data, which allows soliton solutions via the Dbar steepest descent method. Based on the spectral analysis of Lax pairs, the Cauchy problem of coupled dispersive AB systems is transformed into a Riemann–Hilbert problem, and the existence and uniqueness of its solution is proved by the vanishing lemma. The stationary phase points play an important role in determining the long-time asymptotic behavior of these solutions. We demonstrate that in any fixed time cone Cx1,x2,v1,v2=(x,t)∈R2∣x=x0+vt,x0∈x1,x2,v∈v1,v2, the long-time asymptotic behavior of the solution for coupled dispersive AB systems can be expressed by N(I) solitons on the discrete spectrum, the leading order term O(t−1/2) on the continuous spectrum, and the allowable residual O(t−3/4).

Defocusing NLS equation with nonzero background: Large-time asymptotics in a solitonless region

We consider the Cauchy problem for the defocusing Schrödinger (NLS) equation with a nonzero backgroundiqt+qxx−2(|q|2−1)q=0,q(x,0)=q0(x),limx→±∞⁡q0(x)=±1. Recently, for the space-time region |x/(2t)|<1 which is a solitonic region without stationary phase points on the jump contour, Cuccagna and Jenkins presented the asymptotic stability of the N-soliton solutions for the NLS equation by using the ∂¯ generalization of the Deift-Zhou nonlinear steepest descent method. Their large-time asymptotic expansion takes the form(0.1)q(x,t)=T(∞)−2qsol,N(x,t)+O(t−1), whose leading term is N-soliton and the second term O(t−1) is a residual error from a ∂‾-equation. In this paper, we are interested in the large-time asymptotics in the space-time region |x/(2t)|>1 which is outside the soliton region, but there will be two stationary points appearing on the jump contour R. We found an asymptotic expansion that is different from (0.1)(0.2)q(x,t)=e−iα(∞)(1+t−1/2h(x,t))+O(t−3/4), whose leading term is a nonzero background, the second t−1/2 order term is from the continuous spectrum and the third term O(t−3/4) is a residual error from a ∂‾-equation. The above two asymptotic results (0.1) and (0.2) imply that the region |x/(2t)|<1 considered by Cuccagna and Jenkins is a fast decaying soliton solution region, while the region |x/(2t)|>1 considered by us is a slow decaying nonzero background region.

Efficient ArcSAR Focusing in the Wavenumber Domain

Arc synthetic aperture radar (ArcSAR) is a ground-based remote imaging technology with the ability to cover wide fields of view. However, because its azimuth is formed by scanning angle rotation, the focusing of ArcSAR images is different from classical SAR focusing. Since the existing imaging methods cannot give a good balance between computational efficiency and accuracy, the wavenumber domain algorithm (WMA) could become an interesting alternative. Due to the fact that in ArcSAR imaging, the slant range measurement depends on a sine term of the scanning angle, no dedicated WMA-based approach for ArcSAR imaging has been formulated yet. The main challenge in this context is that the solution of the stationary phase point cannot be resolved explicitly via Fourier transform (FT). This article proposes a new wavenumber domain imaging method, which exploits the sine law in the process of solving the stationary phase point during FT along the direction of the received echo using the triangular relationship formed by the target, the radar, and the rotation center of radar and then obtains the exact phase error expression in range and angular wavenumber domain without any approximation of the slant range or scanning angle. Using this formulation, we develop the corresponding phase error compensation method and complete image focusing. Through point target simulation and experiments on real ArcSAR data, the effectiveness of this method is verified in terms of imaging accuracy and computational efficiency.

A General Method of Series Reversion for Synthetic Aperture Radar Imaging

Synthetic aperture radar (SAR) imaging usually needs to be converted between the time domain and the frequency domain, in which the solution of stationary phase point determines the accuracy of time–frequency conversion and the final image quality. Theoretically, the stationary phase point can be accurately solved by the method of series reversion (MSR) in an arbitrary configuration (e.g., bistatic and/or nonlinear trajectory). However, the existing methods based on MSR are proposed based on the assumption of specific signal form. In other words, it is necessary to reuse MSR to derive the time–frequency conversion for different signal forms, which brings great inconvenience to engineering applications. In this article, we aim to propose a general method based on MSR. Based on this method, the accurate time–frequency conversion can be derived by simply arranging any signal to the standard form specified in this article first and then simply replacing the variables.

Seismic Interferometry in Linear Radon Domain Applying to Noise Passive Data

We explored a new method for noise passive data to retrieve the virtual-source gathers in linear Radon domain. Seismic interferometry based on the cross correlation algorithm can be viewed as cancellation of a common part of a raypath from a physical source to two different receivers, and a stacking operator is then applied to the crosscorrelagram to take advantage of the contributions from all the sources. As the contributions of retrieving the virtual-shot gathers mainly come from the stationary-phase point, limited passive data with the same ray-parameter plays the major role. That is, the interference effects of most of passive data are counteracted through stacking with each other. Seismic interferometry in linear-Radon domain is proposed for selecting traces with the same ray parameters to retrieve the virtual-shot gathers. As the common-receiver gathers are needed by linear-Radon interferometry, a rearranged operator is used to rearrange noise passive data according to the periodicity. In this way, the common-receiver gathers of noise passive data can be obtained. Then, the common-receiver gathers are transformed into linear-Radon domain and the selected-ray range is used for cross correlation, which can save the computation time significantly. That has important practical significance in processing noise passive data.

Long-time asymptotics for the focusing Fokas-Lenells equation in the solitonic region of space-time

We study the long-time asymptotic behavior of the focusing Fokas-Lenells (FL) equationuxt+αβ2u−2iαβux−αuxx−iαβ2|u|2ux=0 with generic initial data in a Sobolev space which supports bright soliton solutions. The FL equation is an integrable generalization of the well-known Schrodinger equation, and also linked to the derivative Schrodinger model, but it exhibits several different characteristics from theirs. (i) The Lax pair of the FL equation involves an additional spectral singularity at k=0. (ii) Four stationary phase points will appear during asymptotic analysis, which require a more detailed necessary description to obtain the long-time asymptotics of the focusing FL equation. Based on the Riemann-Hilbert problem for the initial value problem of the focusing FL equation, we show that inside any fixed time-spatial coneC(x1,x2,v1,v2)={(x,t)∈R2|x=x0+vt,x0∈[x1,x2],v∈[v1,v2]}, the long-time asymptotic behavior of the solution u(x,t) for the focusing FL equation can be characterized with an N(I)-soliton on discrete spectrums and a leading order term O(t−1/2) on continuous spectrum up to a residual error order O(t−3/4). The main tool is a ∂‾-generalization of the Deift-Zhou nonlinear steepest descent method.

The Polar Coordinate Transformation Numerical Steepest Descent Path Method for Calculating Physical Optical Scattering Field of Quadric Concave or Convex Surfaces

In this letter, the polar coordinate transformation numerical steepest descent path (PCT-NSDP) method is proposed to calculate the physical optics (PO) scattered field from concave or convex quadratic surfaces. In the proposed method, only three triangles integral domains are calculated, which avoids the complicated decomposition in the former NSDP methods. The PCT-NSDP method has a closed-form formula of the contribution of PO stationary phase point. Therefore, the calculation time can be much smaller. Moreover, the workload is frequency-independent. The numerical examples of the radar cross section of a quadratic convex surface demonstrate the high accuracy and efficiency of the PCT-NSDP method.

Fast and Accurate Solution for the Surface Fields Scattering of a Dipole Antenna Placed on Planar Multi‐Layered Earth

AbstractFast and accurate solution has been derived to the electromagnetic (EM) scattering problem from a dipole antenna placed on a planar multi‐layered earth. The derivation consists of three steps. First, reducing the field integrals to combinations of recognized Sommerfeld integrals (SIs) and expressed it as an infinite series. Next, in the complex ‐ plane, we replaced the reflection coefficient by using an exponential function which tends to the summation of algebraic series. Finally, by using the Shoeib technique, which divided the integrand into two factors and helped to calculate the stationary phase point, closed‐form expressions for the far‐field had been derived speedy and appropriately. This method allows to reduce the integral representations for the field components generated by the dipole source to a well‐known standard integral, whose assessment is easy. Numerical simulations are executed to show the validity of the advanced method, which can be valid irrespective of the operating frequency, over the preceding solutions, in addition to its benefits in saving time over others techniques. The results obtained are compared with those mentioned elsewhere.

Focus Improvement of Airborne High-Squint Bistatic SAR Data Using Modified Azimuth NLCS Algorithm Based on Lagrange Inversion Theorem

In this paper, a modified azimuth nonlinear chirp scaling (NLCS) algorithm is derived for high-squint bistatic synthetic aperture radar (BiSAR) imaging to solve its inherent difficult issues, including the large range cell migration (RCM), azimuth-dependent Doppler parameters, and the sensibility of the higher order terms. First, using the Lagrange inversion theorem, an accurate spectrum suitable for processing airborne high-squint BiSAR data is introduced. Different from the spectrum that is based on the method of series reversion (MSR), it is allowed to derive the bistatic stationary phase point while retaining the double square root (DSR) of the slant range history. Based the spectrum, a linear RCM correction is used to remove the most of the linear RCM components and mitigate the range-azimuth coupling, and, then, bulk secondary range compression is implemented to compensate the residual RCM and cross-coupling terms. Following this, a modified azimuth NLCS operation is applied to eliminate the azimuth-dependence of Doppler parameters and equalize the azimuth frequency modulation for azimuth compression. The experimental results, with better focusing performance, prove the high accuracy and effectiveness of the proposed algorithm.

Power law decay of local density of states oscillations near a line defect in a system with semi-Dirac points

We theoretically study the power-law decay behavior of the local density of states (LDOS) oscillations near a line defect in system with semi-Dirac points by using a low-energy k.p Hamiltonian. We find that the LDOS oscillations are strongly anisotropic and sensitively depend on the orientation of the line defect. We analytically obtain the decay indexes of the LDOS oscillations near a line defect running along different directions by using the stationary phase approximation. Specifically, when the line defect is perpendicular to the linear dispersion direction, the decay index is -5/4 whereas it becomes -1/4 if the system is gapped, both of which are different from the decay index -3/2 in isotropic Dirac systems. In contrast, when the line defect is perpendicular to the parabolic dispersion direction, the decay index is always -1/2 regardless of whether the system is gapped or not, which is the same as that in a conventional semimetal. In general, when the defect runs along an arbitrary direction, the decay index sensitively depends on the incident energy for a certain orientation of the line defect. It varies from -5/4 to -1/2 due to the absence of strict stationary phase point. Our results indicate that the decay index -5/4 provides a fingerprint to identify semi-Dirac points in 2D electron systems.

Numerical analysis of spin foam dynamics and the flatness problem

In this paper, we apply a recently proposed numerical algorithm for finding stationary phase points in spin foam amplitudes. We study a spin foam amplitude with three vertices and a bulk face in 4d BF theory. We fix the boundary coherent states to three possible triangulations, one with zero deficit angle on the bulk face and two with non-zero deficit angle. We compute the amplitude numerically and we find a stationary phase point already at low spins in all the three cases. We comment on how this result contrasts with the claims of flatness problem in spin foam theories. We point out where these arguments may be misleading and we propose further computations to definitively answer the question.

Normal modes separation by use of ambient noise cross-correlation function in shallow sea

The possibility to extract the time-domain Green's function from ambient noise cross-correlation function was demonstrated by a number of authors. This approach has a weak point: the accumulation time of the noise signal. The accumulation time is significantly reduced when using mode signals and separating normal modes. Both numerical study of the spectrogram of the noise cross-correlation function and experiment in Arctic shelf demonstrated that dispersion curves observed in the spectrogram in the case of large distances between hydrophones, with a decrease in this distance are transformed into separate maxima localized both in frequency and in time delay. These maxima arise due to the presence of a stationary phase point near the minimum of the group velocities of the modes, and also due to modes interference, which is more efficient at small distances between hydrophones. At small distances, working with a signal in the frequency band where only the lowest hydroacoustic mode is excited has a number of advantages, the main of which is a decrease in the accumulation time. We demonstrate the change in the regime of modes separation with a change in the distance between hydrophones and discuss the optimal conditions for the applicability of the method.

Classical instanton solutions in quantum field theory

Instantons are non-trivial solutions of classical Euclidean equations of motion with a finite action. They provide stationary phase points in the path integral for tunnel amplitude between two topologically distinct vacua. It make them useful in many applications of quantum theory, especially for describing the wave function of systems with a degenerate vacua in the framework of the path integrals formalism. Our goal is to introduce the current situation about research on instantons and prepare for experiments. In this paper we give a review of instanton effects in quantum theory. We find in stanton solutions in some quantum mechanical problems, namely, in the problems of the one-dimensional motion of a particle in two-well and periodic potentials. We describe known instantons in quantum field theory that arise, in particular, in the two-dimensional Abelian Higgs model and in SU(2) Yang – Mills gauge fields. We find instanton solutions of two-dimensional scalar field models with sine-Gordon and double-well potentials in a limited spatial volume. We show that accounting of instantons significantly changes the form of the Yukawa potential for the sine-Gordon model in two dimensions.

Searching for classical geometries in spin foam amplitudes: a numerical method

We develop a numerical method to investigate the semiclassical limit of spin foam amplitudes with many vertices. We test it using the Ponzano–Regge model, a spin foam model for three-dimensional euclidean gravity, and a transition amplitude with three vertices. We study the summation over bulk spins, and we identify the stationary phase points that dominate it and that correspond to classical geometries. We complement with the numerical analysis of a four vertex transition amplitude and with a modification of the model that includes local curvature. We discuss the generalization of our results to the four-dimensional EPRL spin foam model, and we provide suggestions for new computations.

Quality of Chaetoceros calcitrans Cultured with Different Concentrations of Potassium Nitrate (KNO3)

Addition of different fertilizer concentrations into cultivation media affects the cell density of microalgae. Potassium nitrate (KNO3), is one of the components in Guillard fertilizer composition commonly used for microalgae culture. This study aims to determine the quality of Chaetoceros calcitrans when cultured with different concentration of KNO3. This research was conducted from November 2019 to January 2020 at Balai Perikanan Budidaya Air Payau (BPBAP) Situbondo and Biosain Laboratory, Jember State Politeknik, East Java. This study consists of four treatments and three replicates. The object in this study was C. calcitrans culture with initial density 105 cells/ml. The main parameters observed were cell density, biomass, protein and amino acids contents and medium parameters such as temperature, pH, DO, salinity, nitrate and phosphate. There were four concentrations of KNO3 used in this study, namely control (75 g/l), treatment Group A (100 g/l), treatment B (125 g/l) and treatment C (150 g/l). The results showed that the cells density of C. calcitrans at control, treatment Group A, B and C were 110.4; 105.2; 108.3; and 100.8 (×104 cells/ml), respectively. This study indicated that different concentration of KNO3 affect the starting point of stationary phase, but One Way ANOVA test showed that those treatments had no significant effect (P?0,05) on the growth rate and dry biomass of C. calcitrans. Finally, we found that the protein content in addition of 75, 100, 125 and 150 g/l KNO3 were 9.748; 8.802; 6.812; and 3.776%, respectively.