Coefficient Constraints Research Articles

Adaptive Multifidelity Constraints Method for Efficient Multidisciplinary Missile Design Framework

The adaptive multifidelity constraints method is developed and proposed to ensure the convergence of significant constraints to high-fidelity results for increasing the reliability and robustness of an optimal configuration at the conceptual design stage without any noticeable turnaround time. The adaptive multifidelity constraints algorithm is demonstrated for two numerical examples, with savings of 58.6 and 64% in high-fidelity evaluations, to obtain the convergence of adaptive constraints to high-fidelity results. The implementation of the adaptive multifidelity constraints algorithm is integrated into the multidisciplinary air-to-ground missile design optimization framework. The lift, drag, and longitudinal control effectiveness coefficient constraints are considered as multifidelity constraints due to their importance in the sizing of air-to-ground missile control surfaces for diving and attacking missions. The optimal air-to-ground missile configuration using adaptive multifidelity constraints yields more reliable and robust results compared with the optimal air-to-ground missile configuration using the low-fidelity analysis only, whereas the adaptive constraints converge into the high-fidelity results.

Bright, dark, and singular solitons in optical fibers with spatio-temporal dispersion and spatially dependent coefficients

Investigated in this work is the nonlinear Schrödinger equation with space-dependent parameters, which models the propagation of optical solitons in spatially inhomogeneous optical fiber with detuning, spatiotemporal dispersion, intermodal dispersion, and fiber gain or loss. Through the ansatz scheme, analytical bell, kink, and singular soliton solutions under certain coefficient constraints are obtained.

Rogue wave solutions for a generalized nonautonomous nonlinear equation in a nonlinear inhomogeneous fiber

In this paper, we investigate a generalized nonautonomous nonlinear equation which describes the ultrashort optical pulse propagating in a nonlinear inhomogeneous fiber. By virtue of the generalized Darboux transformation, the first- and second-order rogue-wave solutions for the generalized nonautonomous nonlinear equation are obtained, under some variable–coefficient constraints. Properties of the first- and second-order rogue waves are graphically presented and analyzed: When the coefficients are all chosen as the constants, we can observe the some functions, the shapes of wave crests and troughs for the first- and second-order rogue waves change. Oscillating behaviors of the first- and second-order rogue waves are observed when the coefficients are the trigonometric functions.

Implementation of continuous [formula omitted]-adaptive finite element spaces without limitations on hanging sides and distribution of approximation orders

Adaptive techniques using hp refinement are known to be one of the most efficient methodologies to accelerate the convergence of finite element algorithms. However, the implementation of computational tools for the development of hp-adaptive algorithms is intricate and depends strongly on the data structure. There exist few computational environments available to the scientific finite element community capable to implement hp-adaptive approximation spaces for the complete family of finite element topologies, and which implement hanging sides. This article describes a methodology for the development of continuous hp-adaptive finite element approximation spaces, without constraints on the refinement strategy concerning the difference of levels and approximation orders between neighboring elements. The shape functions are hierarchical, and the coefficient constraints associated with hanging sides that can occur in non-conformal geometric meshes are defined using L2-projections. The topological and functional aspects of the construction are described in one, two and three dimensions, for a variety of geometric entities (line, triangle, quadrilateral, tetrahedron, pyramid, prism, and hexahedron). The implementation is demonstrated in the object-oriented scientific computational environment NeoPZ (http://github.com/labmec/neopz). NeoPZ is a general finite element approximation software, which incorporates a variety of variational formulations. Validation of the refinement methodology is demonstrated by two and three dimensional numerical experiments.

Performance Analysis of a Reduced Rank Spatial Filter for Interference Cancellation

In the interference mitigation context, it has been shown by simulations in Fety et al. (International symposium on wireless communication systems, pp 241---245, 2012), the outperformance of the coefficient constraints (CC) versus the power constraint on the channel impulse response, in terms of BER about 1---3 dB. However, no theoretical justification has been introduced. In this paper, we have proved theoretically this result. Moreover, we have investigated an interesting issue of the CC constraint concerning the choice of the coefficient position; we have given a full theoretical framework analysis about this feature. Two substantial propositions have been introduced to this end. Theoretical results were validated by simulations.

Multi-Soliton and Rational Solutions for the Extended Fifth-Order KdV Equation in Fluids

Abstract Korteweg–de Vries (KdV)-type equations are used as approximate models governing weakly nonlinear long waves in fluids, where the first-order nonlinear and dispersive terms are retained and in balance. The retained second-order terms can result in the extended fifth-order KdV equation. Through the Darboux transformation (DT), multi-soliton solutions for the extended fifth-order KdV equation with coefficient constraints are constructed. Soliton propagation properties and interactions are studied: except for the velocity, the amplitude and width of the soliton are not influenced by the coefficient of the original equation; the amplitude, velocity, and wave shape of each soltion remain unchanged after the interaction. By virtue of the generalised DT and Taylor expansion of the solutions for the corresponding Lax pair, the first- and second-order rational solutions of the equation are obtained.

Higher-order-effects management of soliton interactions in the Hirota equation.

The study of soliton interactions is of significance for improving pulse qualities in nonlinear optics. In this paper, interaction between two solitons, which is governed by the Hirota equation, is considered. Via use of the Hirota method, an analytic soliton solution is obtained. Then a two-period vibration phenomenon is observed. Moreover, turning points of the coefficients of higher-order terms, which are related with sudden delaying or leading, are found and analyzed. With different coefficient constraints, soliton interactions are discussed by different frequency separation with the split-step Fourier method, and characteristics of soliton interactions are exhibited. Through turning points, we get a pair of solitons which tend to be bound solitons but not exactly. Furthermore, we control a pair of solitons to emit at different emission angles. The stability of the two-period vibration is analyzed. Results in this paper may be helpful for the applications of optical self-routing, waveguiding, and faster switching.

Anti-dark solitons for a variable-coefficient higher-order nonlinear Schrödinger equation in an inhomogeneous optical fiber

Investigated in this paper is a variable-coefficient higher-order nonlinear Schrödinger equation, which can describe the propagation of subpicosecond or femtosecond optical pulse in an inhomogeneous optical fiber. With a set of the Painlevé-integrable coefficient constraints, the equation is transformed into its bilinear forms. Single- and two- anti-dark soliton solutions are constructed via the Hirota method. Based on the solutions, we graphically discuss the features of the anti-dark solitons, as well as their interaction, in the inhomogeneous optical fibers. As shown in our results, the backgrounds of the anti-dark solitons are related to the gain/loss coefficients, while the third-order dispersion coefficients directly influence the propagation trajectories of the anti-dark solitons, which provide a possible way to manage these solitons in the inhomogeneous fiber. On the other hand, overtaking interaction between the two anti-dark solitons is obtained, and seen to be elastic. The frequency shift parameter has almost no effect on the solitons.

On coefficient constraints and evaluation restrictions for linearized polynomials

We provide an elementary approach to compute the monoid structure of q-linearized subfield preserving polynomials having coefficients in a subfield Fqd⊆Fqm. Additionally, we derive generalizations for some classical results by Brawley, Carlitz and Vaughan, of which we provide simpler proofs.

Wronskian and Grammian determinant structure solutions for a variable-coefficient forced Kadomtsev–Petviashvili equation in fluid dynamics

Taking the inhomogeneities of media and nonuniform boundaries into account, the variable-coefficient equations can describe more realistic physical phenomena than their constant-coefficient counterparts. In this paper, a variable-coefficient forced Kadomtsev–Petviashvili equation with inhomogeneous nonlinearity, dispersion, perturbed term and external force is investigated. Using a modified dependent variable transformation, this equation is first bilinearized. Then, the N-soliton solutions in two different kinds of determinant structure, that is the Wronskian and Grammian determinant soliton solutions for the variable-coefficient forced Kadomtsev–Petviashvili equation are presented and verified under certain coefficient constraints. The sample soliton solutions are given by choosing suitable determinant elements, and several kinds of soliton evolution situations are discussed and illustrated.

Types of coefficient constraints of coupled nonlinear Schrödinger equations for elastic and inelastic interactions between spatial solitons with symbolic computation

With symbolic computation and Hirota method, analytic two-soliton solutions for the coupled nonlinear Schrodinger (CNLS) equations, which describe the propagation of spatial solitons in an AlGaAs slab waveguide, are derived. Two types of coefficient constraints of the CNLS equations to distinguish the elastic and inelastic interactions between spatial solitons are obtained for the first time in this paper. Asymptotic analysis is made to investigate the spatial soliton interactions. The inelastic interactions are studied under the obtained coefficient constraints of the CNLS equations. The influences of parameters for the obtained soliton solutions are discussed. All-optical switching and soliton amplification are studied based on the dynamic properties of inelastic interactions between spatial solitons. Numerical simulations are in good agreement with the analytic results. The presented results have applications in the design of birefringence-managed switching architecture.

Subideals of operators

A subideal is an ideal of an ideal of B(H) and a principal subideal is a principal ideal of an ideal of B(H). We determine necessary and sufficient conditions for a principal subideal to be an ideal of B(H). This generalizes to arbitrary ideals the 1983 work of Fong and Radjavi characterizing principal subideals of the ideal of compact operators that are also ideals of B(H). We then characterize all principal subideals. We also investigate the lattice structure of subideals as part of the general study of ideal lattices such as the often studied lattice structure of ideals of B(H). This study of subideals and the study of elementary operators with coefficient constraints are closely related.

Conditions of unicity of the best non-symmetric $L_1$-approximant

The necessary and sufficient conditions of the unicity of the best $(\overline{\alpha},\overline{\beta})$-approximant with coefficient constraints for the continuous vector-valued functions on a metric compact are obtained.

Elastic and inelastic interactions between optical spatial solitons in nonlinear optics

Elastic and inelastic interactions between optical spatial solitons in nonlinear optics, which to our knowledge have not been reported before, are investigated in this paper. Analytic soliton solutions for the coupled nonlinear Schrödinger equation are obtained using the Hirota method. The coefficient constraints, which can be used to determine whether the soliton interactions are elastic or inelastic, are presented. With different coefficient constraints, the characteristics of soliton interactions are exhibited. The results of this paper may have applications in the design of directional couplers and all-optical logical gates.

NRQED Lagrangian at order1/M4

The parity and time-reversal invariant effective Lagrangian for a heavy fermion interacting with an Abelian gauge field, i.e., NRQED, is constructed through order $1/{M}^{4}$. The implementation of Lorentz invariance in the effective theory becomes nontrivial at this order, and a complete solution for Wilson coefficient constraints is obtained. Matching conditions in the one-fermion sector are presented in terms of form factors and two-photon matrix elements of the nucleon. The extension of NRQED to describe interactions of the heavy fermion with a light fermion is introduced. Sample applications are discussed; these include the computation of nuclear structure effects in atomic bound states, the model-independent analysis of radiative corrections to low-energy lepton-nucleon scattering, and the study of static electromagnetic properties of nucleons.

SOLITON INTERACTIONS FOR A HIROTA–MAXWELL–BLOCH SYSTEM IN THE INHOMOGENEOUS ERBIUM-DOPED FIBER

In this paper, we consider a generalized Hirota–Maxwell–Bloch system with the higher-order dispersion and self-steepening effects, which describes the propagation of ultrashort optical pulse in the inhomogeneous erbium-doped fiber. Under certain coefficient constraints, N-soliton solutions are obtained through the Hirota method and symbolic computation. Soliton interactions are graphically presented and analyzed in the different fibers. Compared with the Hirota equation without the Maxwell–Bloch parts, the self-induced transparency effect caused by the doped erbium atoms is found to lead to the change of the soliton velocity and phase. In addition, the amplitudes and widths of solitons are respectively observed to decrease and increase in the dispersion-decreasing and dispersion-increasing fibers. Finally, we give the modulational instability condition through the linear stability analysis.

A nonnegative sparse representation based fuzzy similar neighbor classifier

This paper presents a new classification technique by combining the well-known Sparse Representation based algorithm with the theory of Fuzzy Set. The basic idea of this work is that samples with the same class-labels should be more similar to each other than those with different class-labels. Based on this similarity rule, we first impose the nonnegative coefficient constraints on the sparse representation based algorithm and obtain the desirable similar neighbors. Then by introducing the theory of Fuzzy Set into our work, we construct the fuzzy class membership matrix and then assign the decision membership of the query sample to each class. The class assigned with the dominant decision membership is wanted. The proposed approach is called the Nonnegative Sparse Representation based Fuzzy Similar Neighbor Classification (FSNC). FSNC has the following properties: (a) the neighbor parameter K is not needed to be set in advance, and K is adaptively set by the algorithm itself; (b) similar neighbors are also generated adaptively and contain much more similar properties of the query sample; (c) the degree of similarity of data is clearly recorded in the sparse nonnegative coefficient vector; (d) the fuzzy decision rule is effective and the proposed classifier FSNC is simple. Experimental results conducted on the Wine database from UCI, the AR face database, the CENPARMI handwritten numeral database, and the PolyU palmprint database show that the new proposed classification technique outperforms some other state-of-the-art classifiers.

Multi-Soliton Solutions for an Inhomogeneous Nonlinear Schrödinger– Maxwell–Bloch System in the Erbium-Doped Fiber

Under investigation in this paper is an inhomogeneous nonlinear Schrödinger-Maxwell-Bloch system with variable dispersion and nonlinear effects, which describes the propagation of optical pulses in an inhomogeneous erbium-doped fiber. Under certain coefficient constraints, multi-soliton solutions are obtained by the Hirota method and symbolic computation. Evolution and interaction of the solitons are plotted, and the self-induced transparency effect caused by the doped erbium atoms is found to lead to the change of the soliton velocity and phase. Overall phase shift can be observed when the parameter accounting for the interaction between the silica and doped erbium atoms is taken as a constant.

ANALYTIC DARK SOLITON SOLUTIONS FOR A GENERALIZED VARIABLE-COEFFICIENT HIGHER-ORDER NONLINEAR SCHRÖDINGER EQUATION IN OPTICAL FIBERS USING SYMBOLIC COMPUTATION

In this paper, a generalized variable-coefficient nonlinear Schrödinger equation with higher-order and gain/loss effects which can be used to describe the femtosecond pulse propagation is analytically investigated via symbolic computation. Under sets of coefficient constraints, such an equation is transformed into a completely integrable constant-coefficient higher-order nonlinear Schrödinger equation. Furthermore, through the transformation, the dark one- and two-soliton solutions for the generalized variable-coefficient higher-order nonlinear Schrödinger equation are derived by means of the bilinear method.

George Lorentz and inequalities in approximation

George Lorentz influenced the author’s research on inequalities in approximation in many ways. This is the connecting thread of this survey paper. The themes of the survey are listed at the very beginning of the Introduction. 0. Introduction 1. Bernstein-type inequalities for exponential sums. 2. Remez-type inequalities for exponential sums. 3. Lorentz degree of polynomials. 4. Markovand Bernstein-type inequalities for constrained polynomials. 5. Muntz-type theorems. 6. Remez-type inequalities and Newman’s product problem. 7. Multivariate approximation. 8. Newman’s inequality. 9. Littlewood polynomials. 10. Inequalities for generalized polynomials. 11. Markovand Bernstein-type inequalities for rational functions. 12. Nikolskii-type inequalities for shift-invariant function spaces. 13. Inverse Markovand Bernstein-type inequalities. 14. Ultraflat sequences of unimodular polynomials. 15. Zeros of polynomials with coefficient constraints. 1. Bernstein-type Inequalities for Exponential Sums The results in this section were, in large measure, motivated by the letter of Lorentz below. “Dear Tamas: Feb. 27, 1988 I know you are interested in Bernstein-type inequalities and I am also. In some non-linear cases one has ‖P ‖X ≤ Φ(n)‖P‖Y , where n is the dimension of the set of the P ’s, and the norms are taken in different Banach spaces X and Y . For instance, inequalities of Dolzhenko and Pekarskii for rational 2000 Mathematics Subject Classifications: Primary: 41A17 Typeset by AMS-TEX 1 functions are of this type. I have proved an inequality of this type for exponential functions ∑n 1 aje λjx or the extended exponential sums