Analytic Coefficients Research Articles

A geometric-analytic study of linear differential equations of order two

<p style='text-indent:20px;'>We study second order linear differential equations with analytic coefficients. One important case is when the equation admits a so called regular singular point. In this case we address some untouched and some new aspects of Frobenius methods. For instance, we address the problem of finding formal solutions and studying their convergence. A characterization of regular singularities is given in terms of the space of solutions. An analytic-geometric classification of such linear polynomial homogeneous ODEs is obtained by the use of techniques from geometric theory of foliations means. This is done by associating to such an ODE a rational Riccati differential equation and therefore a <i>global holonomy group</i>. This group is a computable group of Moebius maps. These techniques apply to classical equations as Bessel and Legendre equations. We also address the problem of deciding which such polynomial equations admit a Liouvillian solution. A normal form for such a solution is then obtained. Our results are concrete and (computationally) constructive and are aimed to shed a new light in this important subject.</p>

Univalent Functions with PositiveCoefficients Involving Touchard Polynomials

The purpose of the present paper is to establish connections between various subclasses of analyticunivalent functions by applying certain convolution operator involving Touchard polynomials. To be more precise, we investigatesuch connections with the classes of analytic univalent functionswith positive coefficients in the openunit disk $\mathbb{U}.$

An Improved Imaging Algorithm for High-Resolution and Highly Squinted One-Stationary Bistatic SAR Using Extended Nonlinear Chirp Scaling Based on Equi-Sum of Bistatic Ranges

The linear range walk correction (LRWC) and the inherent azimuth-variant geometric configuration in the case of one-stationary bistatic synthetic aperture radar (OS-BiSAR) imaging produce 2-D variant range cell migrations (RCMs) and azimuth-dependent Doppler parameters, which makes it more difficult to obtain high-quality image for high-resolution and highly squinted OS-BiSAR. To accommodate these issues, an improved extended nonlinear chirp scaling (ENLCS) algorithm is developed in this letter. An analytic model based on the equi-sum of bistatic ranges (ESBR) after the RCM correction is proposed to reveal the azimuth-variant property of the azimuth distributed targets. Based on this innovative model, analytic coefficients of the ENLCS are derived, and high-quality image formation for OS-BiSAR is accomplished. Simulations are conducted to demonstrate the validity of the proposed algorithm.

Transversity and heavy quark production on hadron colliders

The azimuthal asymmetry of heavy quarks production on double polarized proton–proton and proton–antiproton colliders are studied in this work at next-to-leading order in alpha _s, with some details included. The purpose is to see whether the effect of extracted transversity distribution functions can be seen on present and near future colliders. All analytic one-loop hard coefficients are given. Numerical results for the asymmetry on proton–(anti)proton colliders are presented.

Integration of Pressure Transient Data into Reservoir Models Using the Fast Marching Method

SummaryCalibration of reservoir model properties by integration of well-test data remains an important research topic. Well-test data have been recognized as an effective tool to describe transient flow behavior in petroleum reservoirs. It is also closely related to the drainage volume of the well and the pressure-front propagation in the subsurface. Traditional analytic means of estimating reservoir permeability relies on an interpretation of the diagnostic plot of the well pressure and production data, which usually leads to a bulk average estimation of the reservoir permeability. When more detailed characterization is needed, a forward model that is sensitive to the reservoir heterogeneity needs to be established, and a numerical inversion technique is required.We use the concept of the diffusive time of flight (DTOF) to formulate an asymptotic solution of the diffusivity equation that describes transient flow behavior in heterogeneous petroleum reservoirs. The DTOF is obtained from the solution of the Eikonal equation using the fast marching method (FMM). It can be used as a spatial coordinate that reduces the 3D diffusivity equation to an equivalent 1D formulation. We investigate the drainage-volume evolution as a function of time in terms of the DTOF. The drainage volume might be directly related to the well-test derivative, which can be used in an inversion calculation to calibrate reservoir model parameters.The analytic sensitivity coefficients of the well-test derivative with respect to reservoir permeability are derived and incorporated into an objective function to perform model calibration. The key to formulating the sensitivity coefficients is to use the functional derivative of the Eikonal equation to derive the analytic sensitivity of the DTOF to reservoir permeability. Its solution is implemented by tracking the characteristic trajectory of the local Eikonal solver within the FMM. The major advantage of calculating the sensitivity coefficients using the FMM is its significant computational efficiency during the iterative inversion process.This inverse-modeling approach is tested on a 2D synthetic heterogeneous reservoir model and then applied to the 3D Brugge Field, where a single well with constant flow rate is simulated. The well-test derivative is shown to be inversely proportional to the drainage volume and is treated as the objective function for inversion. With an additional constraint to honor the prior model, our inverse-modeling approach will adjust the reservoir model to obtain permeability as a function of distance from the well within the drainage volume. It provides a modification of reservoir permeability both within and beyond the depth of investigation (DOI).

Solving Second-Order Linear Differential Equations with Random Analytic Coefficients about Regular-Singular Points

In this contribution, we construct approximations for the density associated with the solution of second-order linear differential equations whose coefficients are analytic stochastic processes about regular-singular points. Our analysis is based on the combination of a random Fröbenius technique together with the random variable transformation technique assuming mild probabilistic conditions on the initial conditions and coefficients. The new results complete the ones recently established by the authors for the same class of stochastic differential equations, but about regular points. In this way, this new contribution allows us to study, for example, the important randomized Bessel differential equation.

Analytic wavelet packets for robust ultrasonic flow measurement

AbstractA key element in robust transit-time ultrasonic flow measurement is the accurate estimation of the transit-time difference. Conventional methods, such as cross-correlation or the estimation in the phase domain, are limited in their robustness against signal distortions, interfering signals or noise. In this work, we present a novel method to estimate the transit-time difference through the fusion of selected analytic wavelet packet coefficients. The combination of the complex coefficients, which represent a projection of the signal on analytic wavelets, with a configurable time-frequency resolution allows a sub-sample estimation at the frequency of interest. After giving an introduction into the fundamentals of analytic wavelet packets based on multi-scale filtering, we introduce two features that correlate strongly with the transit-time difference. The selection and fusion of these features is done by using correlation coefficients with a calibration set and principal component analysis. Finally, using a clamp-on flow measurement system, the robustness against temperature variation and measurement noise is shown and compared with conventional methods.

Multiparameter perturbation theory of matrices and linear operators

We show that a normal matrix A A with coefficients in C [ [ X ] ] \mathbb {C}[[X]] , X = ( X 1 , … , X n ) X=(X_1, \ldots , X_n) , can be diagonalized, provided the discriminant Δ A \Delta _A of its characteristic polynomial is a monomial times a unit. The proof is an adaptation of our proof of the Abhyankar-Jung Theorem. As a corollary we obtain the singular value decomposition for an arbitrary matrix A A with coefficient in C [ [ X ] ] \mathbb {C}[[X]] under a similar assumption on Δ A A ∗ \Delta _{AA^*} and Δ A ∗ A \Delta _{A^*A} . We also show real versions of these results, i.e., for coefficients in R [ [ X ] ] \mathbb {R}[[X]] , and deduce several results on multiparameter perturbation theory for normal matrices with real analytic, quasi-analytic, or Nash coefficients.

Painlevé analysis for new (3 + 1)-dimensional Boiti–Leon–Manna–Pempinelli equations with constant and time-dependent coefficients

PurposeThe purpose of this paper is to introduce two new (3 + 1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equations, the first with constant coefficients and the other with time-dependent coefficients. The author obtains multiple soliton solutions and multiple complex soliton solutions for the two developed models.Design/methodology/approachThe newly developed models with constant coefficients and with time-dependent coefficients have been handled by using the simplified Hirota’s method. The author also uses the complex Hirota’s criteria for deriving multiple complex soliton solutions.FindingsThe two developed BLMP models exhibit complete integrability for any constant coefficient and any analytic time-dependent coefficients by investigating the compatibility conditions for each developed model.Research limitations/implicationsThe paper presents an efficient algorithm for handling integrable equations with constant and analytic time-dependent coefficients.Practical implicationsThe paper presents two new integrable equations with a variety of coefficients. The author showed that integrable equations with constant and time-dependent coefficients give real and complex soliton solutions.Social implicationsThe paper presents useful algorithms for finding and studying integrable equations with constant and time-dependent coefficients.Originality/valueThe paper presents an original work with a variety of useful findings.

On uniqueness of continuation for polynomials of solutions to second-order elliptic PDE

Let be an elliptic operator with smooth enough coefficients, u a solution to the equation Lu = 0 in , an open set. As is well known, if then u = 0 everywhere in Ω. Let P be a polynomial of variables , functions the solutions to Lu = 0; let . If the coefficients of the operator are (real) analytic then are analytic and implies in Ω. Is the same true for smooth but not analytic coefficients? The question also concerns the polynomials of harmonic quaternion fields. In general, the answer turns out to be negative: the paper provides the relevant counterexamples.

Local estimates for Hörmander’s operators of first kind with analytic Gevrey coefficients and application to the regularity of their Gevrey vectors

Local estimates for Hörmander’s operators of first kind with analytic Gevrey coefficients and application to the regularity of their Gevrey vectors

Linear directional differential equations in the unit ball: solutions of bounded L-index

AbstractWe study sufficient conditions of boundedness ofL-index in a directionb∈ ℂn∖ {0} for analytic solutions in the unit ball of a linear higher order non-homogeneous differential equation with directional derivatives. These conditions are restrictions by the analytic coefficients in the unit ball of the equation. Also we investigate asymptotic behavior of analytic functions of boundedL-index in the direction and estimate its growth. The results are generalizations of known propositions for entire functions of several variables.

Painlevé analysis for three integrable shallow water waves equations with time-dependent coefficients

Purpose The purpose of this paper is concerned with investigating three integrable shallow water waves equations with time-dependent coefficients. The author obtains multiple soliton solutions and multiple complex soliton solutions for these three models. Design/methodology/approach The newly developed equations with time-dependent coefficients have been handled by using Hirota’s direct method. The author also uses the complex Hirota’s criteria for deriving multiple complex soliton solutions. Findings The developed integrable models exhibit complete integrability for any analytic time-dependent coefficients defined though compatibility conditions. Research limitations/implications The paper presents an efficient algorithm for handling time-dependent integrable equations with analytic time-dependent coefficients. Practical implications This study introduces three new integrable shallow water waves equations with time-dependent coefficients. These models represent more specific data than the related equations with constant coefficients. The author shows that integrable equations with time-dependent coefficients give real and complex soliton solutions. Social implications The paper presents useful algorithms for finding integrable equations with time-dependent coefficients. Originality/value The paper presents an original work with a variety of useful findings.

Biological variation of resting measures of ventilation and gas exchange in a large healthy cohort.

Resting measures of ventilation and gas exchange are impacted by a variety of physiological stressors, such as those resulting from a research intervention or an extreme environment. However, the biological variation of these parameters, an important statistical consideration for identifying a meaningful physiological change, has not been quantified. We performed a retrospective analysis of 21 studies completed by the U.S. Army Research Institute of Environmental Medicine (USARIEM) from 1985 to present, totaling 411 healthy volunteers. First, we determined the intraindividual, interindividual, and analytic coefficients of variation (CVI, CVG, and CVA, respectively) and subsequently the index of individuality and heterogeneity (II and IH, respectively). Second, when deemed appropriate via these outcomes, we defined the accompanying static and dynamic thresholds, beyond which a significant deviation from normal is indicated. End-tidal partial pressure of oxygen (PETO2) and the respiratory exchange ratio (RER) approached the II threshold required to be considered useful in the static assessment of physiological deviations from normal. PETO2 and peripheral oxygen saturation (SpO2) approached the IH threshold required to be considered useful in the dynamic assessment of physiological deviations from normal. This analysis identifies RER and PETO2 as parameters that might be most useful when aiming to identify a meaningful ventilatory change following a research intervention or stressor. Alternatively, other parameters of ventilation and gas exchange, such as PETCO2 and VE, may be less useful for observing an anticipated physiological change.

Value Distribution and Arbitrary-Order Derivatives of Meromorphic Solutions of Complex Linear Differential Equations in the Unit Disc

In this paper, we investigate the value distribution of meromorphic solutions and their arbitrary-order derivatives of the complex linear differential equation f ′ ′ + A ( z ) f ′ + B ( z ) f = F ( z ) in Δ with analytic or meromorphic coefficients of finite iterated p-order, and obtain some results on the estimates of the iterated exponent of convergence of meromorphic solutions and their arbitrary-order derivatives taking small function values.

Gevrey Properties and Summability of Formal Power Series Solutions of Some Inhomogeneous Linear Cauchy-Goursat Problems

In this article, we investigate the Gevrey and summability properties of the formal power series solutions of some inhomogeneous linear Cauchy-Goursat problems with analytic coefficients in a neighborhood of $(0,0)\in \mathbb {C}^{2}$. In particular, we give necessary and sufficient conditions under which these solutions are convergent or are k-summable, for a convenient positive rational number k, in a given direction.

Construction of Exact Solutions of Irregularly Degenerate Elliptic Equations with Analytic Coefficients

We solve a boundary value problem (problem E in the sense of M.V. Keldysh) for an irregularly degenerate elliptic operator in a rectangle. The exact solution of the problem is constructed as a series in the eigenfunctions of the limit operator. The method of spectral isolation of singularities, which generalizes the method of regularization of singular perturbations to the case of degenerate elliptic equations, is developed.

Polynomial integrals of magnetic geodesic flows on the 2-torus on several energy levels

In this paper the geodesic flow on the 2-torus in a non-zero magnetic field is considered. Suppose that this flow admits an additional first integral $ F $ on $ N+2 $ different energy levels which is polynomial in momenta of an arbitrary degree $ N $ with analytic periodic coefficients. It is proved that in this case the magnetic field and the metric are functions of one variable and there exists a linear in momenta first integral on all energy levels.

Spectral Portraits in the Semi-Classical Approximation of the Sturm-Liouville Problem with a Complex Potential

We study the problem on the limit behavior of the discrete spectrum of the Sturm-Liouville problem − ε y ″ ( x ) + P ( x λ ) y ( x ) = 0 y ( a ) = y ( b ) = 0 , provided that the physical parameter ϵ tends to zero. It is assumed that λ is the spectral parameter, the function P is polynomial on x with analytic coefficients on λ varying on a domain G in the complex λ-plane ℂ. The case P(x, λ) = p(x) - λ with complex valued function p corresponds to the usual linear spectral problem. Boundary conditions are formed by arbitrary complex numbers a, b or ±∞. We shall show that in this case the eigenvalues are concentrated along the so-called limit spectral graph as ϵ → 0. We define three type of curves, forming this graph and find the asymptotic formulae for the eigenvalue distribution along the curves of various types. For the case P(x, λ) = p(x) - λ with real polynomial p(x) these formulae coincide with well known Bohr-Sommerfeld quantization formulae. Non-self-adjoint Sturm-Liouville problems are often found in mathematical physics. We demonstrate this considering the well-known in hydromechanics Orr-Sommerfeld spectral problem.

Quantitative unique continuation for operators with partially analytic coefficients. Application to approximate control for waves

In this article, we first prove quantitative estimates associated to the unique continuation theorems for operators with partially analytic coefficients of Tataru [Tat95, Tat99b], Robbiano–Zuily [RZ98] and Hörmander [Hör97]. We provide local stability estimates that can be propagated, leading to global ones. Then, we specify the previous results to the wave operator on a Riemannian manifold \mathcal M with boundary. For this operator, we also prove Carleman estimates and local quantitative unique continuation from and up to the boundary \partial \mathcal M . This allows us to obtain a global stability estimate from any open set \Gamma of \mathcal M or \partial \mathcal M , with the optimal time and dependence on the observation. As a first application, we compute a sharp lower estimate of the intensity of waves in the shadow of an obstacle. We also provide the cost of approximate controllability on the compact manifold \mathcal M : for any T > 2\: \mathrm {sup}_{x \in \mathcal M} \mathrm {dist}(x,\Gamma) , we can drive any data of H^1_0 \times L^2 in time T to an \varepsilon -neighborhood of zero in L^2 \times H^{-1} , with a control located in \Gamma , at cost e^{C/\varepsilon} . We finally obtain related results for the Schrödinger equation.