First-order Operators Research Articles

Transfer Function of Macro-Micro Manipulation on a PDMS Microfluidic Chip

To achieve fast and accurate cell manipulation in a microfluidic channel, it is essential to know the true nature of its input-output relationship. This paper aims to reveal the transfer function of such a micro manipulation controlled by a macro actuator. Both a theoretical model and experimental results for the manipulation are presented. A second-order transfer function is derived based on the proposed model, where the polydimethylsiloxane (PDMS) deformation plays an important role in the manipulation. Experiments are conducted with input frequencies up to 300 Hz. An interesting observation from the experimental results is that the frequency responses of the transfer function behave just like a first-order integration operator in the system. The role of PDMS deformation for the transfer function is discussed based on the experimentally-determined parameters and the proposed model.

Viscoelastic reverse time migration with attenuation compensation

We have developed a theory of viscoelastic reverse time migration (RTM). The main feature of viscoelastic RTM is a compensation for P- and S-wave attenuation effects in seismic images during migration. The forward modeling engine is based on a viscoelastic wave equation involving fractional Laplacians. Because of the decoupled attenuation property, wave propagation can be simulated in three scenarios, i.e., only the amplitude loss effect, only the phase dispersion effect, or both effects simultaneously. This separation brings practical flexibility to studying attenuation effects on wave propagation and imaging. The backward modeling operator is constructed by reversing the sign of first-order time derivative amplitude loss operators. Synthetic examples determine the ability of viscoelastic RTM to illuminate degraded areas and shadow zones caused by attenuation. Numerical experiments also reveal that [Formula: see text]-compensated imaging is noticeably more accurate in kinematics and dynamics than elastic imaging in the presence of high attenuation. Results from a synthetic 3D model determine the superiority of viscoelastic RTM over elastic RTM in imaging salt flanks and delineation of salt boundaries, which are dimmed in elastic images.

Higher-order modification of Steffensen’s method for solving system of nonlinear equations

In this work, we propose a higher-order derivative free family for solving non-linear systems. The local order of convergence of the constructed family is first determined using first-order divided difference operators for functions of several variables and direct computation by Taylor’s series expansion. Computational efficiencies of the developed scheme is considered and compared with their closest competitors. Moreover, numerical experiments are performed on several large and complex non-linear systems. The results are found to be effective and comparable to other existing methods which also support the theoretical results.

On the Lie Symmetry Algebras of the Stationary Schrödinger and Pauli Equations

A general method for constructing first-order symmetry operators for the stationary Schrodinger and Pauli equations is proposed. It is proven that the Lie algebra of these symmetry operators is a one-dimensional extension of some subalgebra of an e(3) algebra. We also assemble a classification of stationary electromagnetic fields for which the Schrodinger (or Pauli) equation admits a Lie algebra of first-order symmetry operators.

Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation

We extend the De Giorgi–Nash–Moser theory to a class of kinetic Fokker-Planck equations and deduce new results on the Landau-Coulomb equation. More precisely, we first study the Holder regularity and establish a Harnack inequality for solutions to a general linear equation of Fokker-Planck type whose coefficients are merely measurable and essentially bounded, i.e. assuming no regularity on the coefficients in order to later derive results for non-linear problems. This general equation has the formal structure of the hypoelliptic equations of type II , sometimes also called ultraparabolic equations of Kolmogorov type, but with rough coefficients: it combines a first-order skew-symmetric operator with a second-order elliptic operator involving derivatives along only part of the coordinates and with rough coefficients. These general results are then applied to the non-negative essentially bounded weak solutions of the Landau equation with inverse-power law γ ∈ [−d, 1] whose mass, energy and entropy density are bounded and mass is bounded away from 0, and we deduce the Holder regularity of these solutions.

Odd Laplacians: geometrical meaning of potential and modular class

A second-order self-adjoint operator \(\Delta =S\partial ^2+U\) is uniquely defined by its principal symbol S and potential U if it acts on half-densities. We analyse the potential U as a compensating field (gauge field) in the sense that it compensates the action of coordinate transformations on the second derivatives in the same way as an affine connection compensates the action of coordinate transformations on first derivatives in the first-order operator, a covariant derivative, \(\nabla =\partial +\Gamma \). Usually a potential U is derived from other geometrical constructions such as a volume form, an affine connection, or a Riemannian structure, etc. The story is different if \(\Delta \) is an odd operator on a supermanifold. In this case, the second-order potential becomes a primary object. For example, in the case of an odd symplectic supermanifold, the compensating field of the canonical odd Laplacian depends only on this symplectic structure and can be expressed by the formula obtained by K. Bering. We also study modular classes of odd Poisson manifolds via \(\Delta \)-operators, and consider an example of a non-trivial modular class which is related with the Nijenhuis bracket.

Design of an Optimal Preview Controller for a Class of Linear Discrete-Time Descriptor Systems

The preview control problem of a class of linear discrete-time descriptor systems is studied. Firstly, the descriptor system is decomposed into a normal system and an algebraic equation by the method of the constrained equivalent transformation. Secondly, by applying the first-order forward difference operator to the state equation, combined with the error equation, the error system is obtained. The tracking problem is transformed into the optimal preview control problem of the error system. Finally, the optimal controller of the error system is obtained by using the related results and the optimal preview controller of the original system is gained. In this paper, we propose a numerical simulation method for descriptor systems. The method does not depend on the restricted equivalent transformation.

The study of structure in 224–234 thorium nuclei within the framework IBM

An investigation has been made of the behaviour of nuclear structure as a function of an increase in neutron number from 224 Th to 234 Th. Thorium of mass number 234 is a typical rotor nucleus that can be explained by the SU(3) limit of the interacting boson model(IBM) in the algebraic nuclear model. Furthermore, 224−232 Th lie on the path of the symmetry-breaking phase transition. Moreover, the nuclear structure of 224 Th can be explained using X(5) symmetry. However, as 226−230 Th nuclei are not fully symmetrical nuclei, they can be represented by adding a perturbed term to express symmetry breaking. Through the following three calculation steps, we identified the tendency of change in nuclear structure. Firstly, the structure of 232 Th is described using the matrix elements of the Hamiltonian and the electric quadrupole operator between basis states of the SU(3) limit in IBM. Secondly, the low-lying energy levels and E2 transition ratios corresponding to the observable physical values are calculated by adding a perturbed term with the first-order Casimir operator of the U(5) limit to the SU(3) Hamiltonian in IBM. We compared the results with experimental data of 224−234 Th. Lastly, the potential of the Bohr Hamiltonian is represented by a harmonic oscillator, as a result of which the structure of 224−234 Th could be expressed in closed form by an approximate separation of variables. The results of these theoretical predictions clarify nuclear structure changes in Thorium nuclei over mass numbers of practical significance.

Differential Sequences Generated by First-order Recursion Operators

Differential Sequences Generated by First-order Recursion Operators

About peculiarities of application of the method of fast expansions in the solution of the Navier-Stokes equations

The brief presentation of the method of fast expansions is given to solve nonlinear differential equations. Application rules of the operator of fast expansions are specified for solving differential equations. According to the method of fast expansions, an unknown function can be represented as the sum of the boundary function and Fourier series sines and cosines for one variable. The special construction of the boundary functions leads to reasonably fast convergence of the Fourier series, so that for engineering calculations, it is sufficient to consider only the first three members. The method is applicable both to linear and nonlinear integro-differential systems. By means of applying the method of fast expansions to nonlinear Navier-Stokes equations the problem is reduced to a closed system of ordinary differential equations, which solution doesn't represent special difficulties. We can reapply the method of fast expansions to the resulting system of differential equations and reduce the original problem to a system of algebraic equations. If the problem is n -dimensional, then after n -fold application of the method of fast expansions the problem will be reduced to a closed algebraic system. Finally, we obtain an analytic-form solution of complicated boundary value problem in partial derivatives. The flow of an incompressible viscous fluid of Navier–Stokes is considered in a curvilinear pipe. The problem is reduced to solving a closed system of ordinary differential equations with boundary conditions by the method of fast expansions. The article considers peculiarities of finding the coefficients of boundary functions and Fourier coefficients for the zero-order and first-order operators of fast expansions. Obtaining the analytic-form solution is of great interest, because it allows to analyze and to investigate the influence of various factors on the properties of the viscous fluid in specific cases.

Noncommutative Integration and Symmetry Algebra of the Dirac Equation on the Lie Groups

The algebra of first-order symmetry operators of the Dirac equation on four-dimensional Lie groups with right-invariant metric is investigated. It is shown that the algebra of symmetry operators is in general not a Lie algebra. Noncommutative reduction mediated by spin symmetry operators is investigated. For the Dirac equation on the Lie group SO(2,1) a parametric family of particular solutions obtained by the method of noncommutative integration over a subalgebra containing a spin symmetry operator is constructed.

Lie derivatives and structure Jacobi operator on real hypersurfaces in complex projective spaces

On a real hypersurface M in a complex projective space we can consider the Levi-Civita connection and for any nonnull constant k the k-th g-Tanaka–Webster connection. Associated to g-Tanaka–Webster connection we can define a differential operator of first order. We classify real hypersurfaces such that both the Lie derivative and this differential operator, either in the direction of the structure vector field ξ or in any direction of the maximal holomorphic distribution coincide when we apply them to the structure Jacobi operator of M.

Inverse problems for first-order integro-differential operators

Inverse spectral problems for first-order integro-differential operators on a finite interval are studied, the properties of spectral characteristics are established, and uniqueness theorems for solutions of this class of inverse problems are proved.

Universal Positive Mass Theorems

In this paper, we develop a general study of contributions at infinity of Bochner–Weitzenbock-type formulas on asymptotically flat manifolds, inspired by Witten’s proof of the positive mass theorem. As an application, we show that similar proofs can be obtained in a much more general setting as any choice of an irreducible natural bundle and a very large choice of first-order operators may lead to a positive mass theorem along the same lines if the necessary curvature conditions are satisfied.

Symmetry operators of Killing spinors and superalgebras in AdS5

We construct the first-order symmetry operators of Killing spinor equation in terms of odd Killing-Yano forms. By modifying the Schouten-Nijenhuis bracket of Killing-Yano forms, we show that the symmetry operators of Killing spinors close into an algebra in AdS5 spacetime. Since the symmetry operator algebra of Killing spinors corresponds to a Jacobi identity in extended Killing superalgebras, we investigate the possible extensions of Killing superalgebras to include higher-degree Killing-Yano forms. We found that there is a superalgebra extension but no Lie superalgebra extension of the Killing superalgebra constructed out of Killing spinors and odd Killing-Yano forms in AdS5 background.

Two Efficient Derivative-Free Iterative Methods for Solving Nonlinear Systems

In this work, two multi-step derivative-free iterative methods are presented for solving system of nonlinear equations. The new methods have high computational efficiency and low computational cost. The order of convergence of the new methods is proved by a development of an inverse first-order divided difference operator. The computational efficiency is compared with the existing methods. Numerical experiments support the theoretical results. Experimental results show that the new methods remarkably reduce the computing time in the process of high-precision computing.

Boundary value problems for first order elliptic wedge operators

We develop an elliptic theory based in $L^2$ of boundary value problems for general wedge differential operators of first order under only mild assumptions on the boundary spectrum. In particular, we do not require the indicial roots to be constant along the base of the boundary fibration. Our theory includes as a special case the classical theory of elliptic boundary value problems for first order operators with and without the Shapiro-Lopatinskii condition, and can be thought of as a natural extension of that theory to the geometrically and analytically relevant class of wedge operators. Wedge operators arise in the global analysis on manifolds with incomplete edge singularities. Our theory settles, in the first order case, the long-standing open problem to develop a robust elliptic theory of boundary value problems for such operators.

ОБ ОДНОЙ ЗАДАЧЕ ТИПА ДИРИХЛЕ ДЛЯ УРАВНЕНИЯ СОСТАВНОГО ТИПА

The aim of this paper is to prove the existence and uniqueness of smooth solutions to the Dirichlet type problem for one class of third-order equations that do not belong to any of the classic types. One of the main classes of non-classical equations is third-order composite type equations, the operator of which is a composition of first-order hyperbolic operator and an elliptic operator in the main part. A number of boundary value problems for the model composite type equations with the Laplace operator were investigated by T.D. Dzhuraev. Many studies have proved the existence of solutions to boundary value problems upon fulfillment of conditions of the convexity of area boundary. The method of proof used in this paper is similar to the method used in the research paper of the author mentioned above. For the research of composite type linear equations a combination of the method of potentials (Green's function) and integral identities is applied. The research method is based on reducing the studied problem with the help of the Green's function to an integral equation, the proof of its solubility and thus - the proof of the solvability of original problem. Upon fulfillment of certain conditions on given functions, a third-order equation reduces to a second-order equation of elliptic type with an unknown right-hand side and the boundary function. With the help of Green's functions for elliptic equations, the studied problem is reduced to a second-order equivalent integral equation, the solvability follows from Fredholm alternative and the theorem of uniqueness of the solution of the original problem

Spherical Dirac equation on the lattice and the problem of the spurious states

With the development of radioactive ion beam facilities, the study of exotic nuclei with unusual N/Z ratio has attracted much attention. Compared with the stable nuclei, the exotic nuclei have many novel features, such as the halo phenomenon. In order to describe the halo phenomenon with the diffused density distribution, the correct asymptotic behaviors of wave functions should be treated properly. The relativistic continuum Hartree-Bogoliubov (RCHB) theory which provides a unified and self-consistent description of mean field, pair correlation and continuum has achieved great success in describing the spherical exotic nuclei. In order to study the halo phenomenon in deformed nuclei, it is necessary to extend RCHB theory to the deformed case. However, solving the relativistic Hartree-Bogoliubov equation in space is extremely difficult and time consuming. Imaginary time step method is an efficient method to solve differential equations in coordinate space. It has been used extensively in the nonrelativistic case. For Dirac equation, it is very challenging to use the imaginary time step method due to the Dirac sea. This problem can be solved by the inverse Hamiltonian method. However, the problem of spurious states comes out. In this paper, we solve the radial Dirac equation by the imaginary time step method in coordinate space and study the problem of spurious states. It can be proved that for any potential, when using the three-point differential formula to discretize the first-order derivative operator, the energies of the single-particle states respectively with quantum numbers and - are identical. One of them is a physical state and the other is a spurious state. Although they have the same energies, their wave functions have different behaviors. The wave function of physical state is smooth in space while that of spurious state fluctuates dramatically. Following the method in lattice quantum chromodynamics calculation, the spurious state in radial Dirac equation can be removed by introducing the Wilson term. Taking Woods-Saxon potential for example, the imaginary time step method with the Wilson term is implanted successfully and provides the same results as those from the shooting method, which demonstrates its future application to solving the Dirac equation in coordinate space.

The Multiplicity of Eigenvalues of the Hodge Laplacian on 5-Dimensional Compact Manifolds

We study the multiplicity of the eigenvalues of the Hodge Laplacian on smooth, compact Riemannian manifolds of dimension five for generic families of metrics. We prove that generically the Hodge Laplacian, restricted to the subspace of co-exact two-forms, has nonzero eigenvalues of multiplicity two. The proof is based on the fact that the Hodge Laplacian restricted to the subspace of co-exact two-forms is minus the square of the Beltrami operator, a first-order operator. We prove that for generic metrics the spectrum of the Beltrami operator is simple. Because the Beltrami operator in this setting is a skew-adjoint operator, this implies the main result for the Hodge Laplacian.