Nonlinear Algebra Research Articles

Nonlinear thermal coherent states

Coherent states and their generalizations are normally appropriate candidates for describing radiation fields. Despite this, the effect of thermal noise on generalized coherent states has not been extensively studied. In fact, thermal effects are unavoidable at finite temperatures, and they should be taken into account to have a better agreement with experimental results. In this regard, we use the concept of thermal coherent states, which are indeed the standard coherent states including thermal effects. So, in this paper, by using a nonlinear coherent states approach, we generalize the thermal coherent states to their nonlinear counterparts. In other words, we find a natural link between the thermal coherent states and the nonlinear coherent states associated with nonlinear oscillator algebra. Afterwards, the nonclassicality features of the obtained states are numerically investigated to explore the roles of both nonlinearity and thermal noise in physical properties. The results show that the thermal effects lead to the transition from nonclassical states to classical ones. Moreover, it is seen that the operator-valued intensity-dependent function plays a leading role in controlling the depth as well as the domain of nonclassicality aspects.

Application of Robotic Software on Effective Diabetes Control (GH-Method: Math-Physical Medicine)

This paper focuses on the author’s invented robotic software technology, the artificial intelligence glucometer (AIG) product, to provide a diagnosis for diabetes disease and glucose control. From 2010–2013, he self-studied internal medicine and food nutrition. In 2014, he further utilized topology concept, partial differential equation, non-linear algebra, and finite element engineering concept to develop a human metabolism’s mathematical model. It consists of 10 categories and ~500 elements with ~1.5 million collected data of his own body health, disease conditions, and lifestyle details. Starting from 2015, he focused on the root cause of diabetes, which is “glucose”. By applying wave theory, signal processing, energy theory, optical physics, structural & fluid dynamics from physics and engineering modeling; pattern and segmentation analysis, time/space/frequency domain analyses, big data analytics, machine learning and self-correction, prediction equations from mathematics and computer science, he decided to utilize his robotic software as the foundation to further build up his needed medical research and clinical tools. By using the artificial intelligence (AI) robotic software, the author’s average glucose decreased from 280 mg/dL to 118 mg/dL and his hemoglobin A1C (HbA1C or A1C) reduced from 10%+ to below 6.5%, without diabetes medications. All his diabetes complications are either under control or have subsided. This innovative technology of his robotic software for glucose prediction and diabetes control has also been proven by many other patients, who have achieved equally remarkable medical results.

Response spectral density determination for nonlinear systems endowed with fractional derivatives and subject to colored noise

A computationally efficient method for determining the response of non-linear stochastic dynamic systems endowed with fractional derivative elements subject to stochastic excitation is presented. The method relies on a spectral representation both for the system excitation and its response. Specifically, first the ordinary non-linear differential equation of motion is transferred into a set of non-linear algebra equations by employing the harmonic balance method. Next, the response Fourier coefficients are determined by solving these non-linear equations. Finally, repeated use of the proposed procedure yields the response power spectral density. Pertinent numerical examples, including a fractional Duffing and a bilinear oscillator, demonstrate the accuracy of the proposed method.

Bayesian WIV Estimators for 3-D Bearings-Only TMA With Speed Constraints

This paper develops Bayesian weighted instrumental variable (WIV) estimators under equality and inequality speed constraints for the three-dimensional (3-D) bearings-only target motion analysis (TMA) problem, which show improved estimation accuracies compared to their unconstrained counterpart. Incorporating the speed constraint information into our previously proposed Bayesian WIV estimator imposes a quadratic constraint on this problem. Finding an optimal solution for the resulting non-convex problem is by no means straightforward. The main problem with iterative optimization methods such as the maximum a posteriori is that they require initialization and may converge to a local minimum if poorly initialized. Using advanced linear and nonlinear algebra techniques, non-iterative accurate estimators under equality and inequality constraints are developed for this problem. Furthermore, it is proven that the proposed equality constrained estimator is approximately asymptotically unbiased and that the inequality constrained estimator is the optimal solution. An approximate covariance matrix has also been developed for the constrained WIV under equality constraint. Simulation results indicate that the equality and inequality constrained estimators outperform their unconstrained counterpart in the simulated scenarios.

An Efficient Finite Volume Method for Nonlinear Distributed-Order Space-Fractional Diffusion Equations in Three Space Dimensions

A Crank–Nicolson finite volume approximation for the nonlinear distributed-order space-fractional diffusion equations in three space dimensions is developed, and a resulting nonlinear algebra system with Kronecker-product type coefficient matrix is formulated. The finite volume scheme is proved to be unconditionally stable and convergent with second-order accuracy in terms of the step sizes in time and distributed orders, and $$\min \{3-{\hat{\alpha }}, 3-{\hat{\beta }}, 3-{\hat{\gamma \}}}$$ -order accuracy in space with respect to a discrete norm. At each time step, the Newton’s iterative method is employed as a nonlinear solver to handle the resulting nonlinear algebra system. Moreover, during each Newton’s iteration, an efficient biconjugate gradient stabilized method (BiCGSTAB) is developed, in which both the matrix storage and matrix-vector multiplications are efficiently conducted by using the Toeplitz sub-structure of the coefficient matrices. It is proved that the BiCGSTAB method only requires the storage of $$\mathcal {O}(N)$$ and the computational cost of $$\mathcal {O}(N \log N)$$ per iteration, while no accuracy is lost compared with the Gaussian elimination method. Thus, an efficient finite volume method is developed, and it is well suitable for large-scale modeling and simulations, especially for multi-dimensional problems. Numerical experiments are presented to verify the theoretical results and show strong potential of the efficient method.

CPN -Rosochatius system, superintegrability, and supersymmetry

We propose new superintegrable mechanical system on the complex projective space $\mathbb{CP}^N$ involving a potential term together with coupling to a constant magnetic fields. This system can be viewed as a $\mathbb{CP}^N$-analog of both the flat singular oscillator and its spherical analog known as "Rosochatius system". We find its constants of motion and calculate their (highly nonlinear) algebra. We also present its classical and quantum solutions. The system belongs to the class of "K\"ahler oscillators" admitting $SU(2|1)$ supersymmetric extension. We show that, in the absence of magnetic field and with the special choice of the characteristic parameters, one can construct $\mathcal{N}=4, d=1$ Poinacar\'e supersymmetric extension of the system considered.

Hidden symmetries of rationally deformed superconformal mechanics

We study the spectrum generating closed nonlinear superconformal algebra that describes $\mathcal{N}=2$ super-extensions of rationally deformed quantum harmonic oscillator and conformal mechanics models with coupling constant $g=m(m+1)$, $m\in {\mathbb N}$. It has a nature of a nonlinear finite $W$ superalgebra being generated by higher derivative integrals, and generally contains several different copies of either deformed superconformal $\mathfrak{osp}(2|2)$ algebra in the case of super-extended rationally deformed conformal mechanics models, or deformed super-Schrodinger algebra in the case of super-extension of rationally deformed harmonic oscillator systems.

On light-like deformations of the Poincar\xe9 algebra

We investigate the observational consequences of the light-like deformations of the Poincaré algebra induced by the jordanian and the extended jordanian classes of Drinfel’d twists. Twist-deformed generators belonging to a Universal Enveloping Algebra close nonlinear algebras. In some cases the nonlinear algebra is responsible for the existence of bounded domains of the deformed generators. The Hopf algebra coproduct implies associative nonlinear additivity of the multi-particle states. A subalgebra of twist-deformed observables is recovered whenever the twist-deformed generators are either hermitian or pseudo-hermitian with respect to a common invertible hermitian operator.

Higher spin currents with manifest SO(4) symmetry in the large 𝒩 = 4 holography

The large [Formula: see text] nonlinear superconformal algebra is generated by six spin-[Formula: see text] currents, four spin-[Formula: see text] currents and one spin-[Formula: see text] current. The simplest extension of these [Formula: see text] currents is described by the [Formula: see text] higher spin currents of spins [Formula: see text]. In this paper, by using the defining operator product expansions (OPEs) between the [Formula: see text] currents and [Formula: see text] higher spin currents, we determine the [Formula: see text] higher spin currents (the higher spin-[Formula: see text] currents were found previously) in terms of affine Kac–Moody spin-[Formula: see text], one currents in the Wolf space coset model completely. An antisymmetric second rank tensor, three antisymmetric almost complex structures or the structure constant are contracted with the multiple product of spin-[Formula: see text] currents. The eigenvalues are computed for coset representations containing at most four boxes, at finite [Formula: see text] and [Formula: see text]. After calculating the eigenvalues of the zeromode of the higher spin-[Formula: see text] current acting on the higher representations up to three (or four) boxes of Young tableaux in [Formula: see text] in the Wolf space coset, we obtain the corresponding three-point functions with two scalar operators at finite [Formula: see text]. Furthermore, under the large [Formula: see text] ’t Hooft-like limit, the eigenvalues associated with any boxes of Young tableaux are obtained and the corresponding three-point functions are written in terms of the ’t Hooft coupling constant in simple form in addition to the two-point functions of scalars and the number of boxes.

A two-grid discontinuous Galerkin method for a kind of nonlinear parabolic problems

A discontinuous Galerkin approximation for the space variables with the backward Euler time discretisation for a kind of parabolic problems with the nonlinear diffusion, convection and source terms is investigated. To solve the strongly nonlinear algebra system, a two-grid method is proposed. With this algorithm, solving a nonlinear system on the fine discontinuous finite element space is reduced into solving a nonlinear problem on a coarse gird of size and solving a linear problem on a fine grid of size. Convergence estimates in H1-norm are obtained. The numerical experiments are provided to confirm our theoretical analysis.

Microformal Geometry and Homotopy Algebras

We extend the category of (super)manifolds and their smooth mappings by introducing a notion of microformal or "thick" morphisms. They are formal canonical relations of a special form, constructed with the help of formal power expansions in cotangent directions. The result is a formal category so that its composition law is also specified by a formal power series. A microformal morphism acts on functions by an operation of pullback, which is in general a nonlinear transformation. More precisely, it is a formal mapping of formal manifolds of even functions (bosonic fields), which has the property that its derivative for every function is a ring homomorphism. This suggests an abstract notion of a "nonlinear algebra homomorphism" and the corresponding extension of the classical "algebraic-functional" duality. There is a parallel fermionic version. The obtained formalism provides a general construction of $L_{\infty}$-morphisms for functions on homotopy Poisson ($P_{\infty}$-) or homotopy Schouten ($S_{\infty}$-) manifolds as pullbacks by Poisson microformal morphisms. We also show that the notion of the adjoint can be generalized to nonlinear operators as a microformal morphism. By applying this to $L_{\infty}$-algebroids, we show that an $L_{\infty}$-morphism of $L_{\infty}$-algebroids induces an $L_{\infty}$-morphism of the "homotopy Lie--Poisson" brackets for functions on the dual vector bundles. We apply this construction to higher Koszul brackets on differential forms and to triangular $L_{\infty}$-bialgebroids. We also develop a quantum version (for the bosonic case), whose relation with the classical version is like that of the Schr\"odinger equation with the Hamilton--Jacobi equation. We show that the nonlinear pullbacks by microformal morphisms are the limits at $\hbar\to 0$ of certain "quantum pullbacks", which are defined as special form Fourier integral operators.

A bulk localized state and new holographic renormalization group flow in 3D spin-3 gravity

We construct a localized state of a scalar field in 3D spin-3 gravity. 3D spin-3 gravity is thought to be holographically dual to [Formula: see text]-extended CFT on a boundary at infinity. It is known that while [Formula: see text] algebra is a nonlinear algebra, in the limit of large central charge [Formula: see text] a linear finite-dimensional subalgebra generated by [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text] is singled out. The localized state is constructed in terms of these generators. To write down an equation of motion for a scalar field which is satisfied by this localized state, it is necessary to introduce new variables for an internal space [Formula: see text], [Formula: see text], [Formula: see text], in addition to ordinary coordinates [Formula: see text] and [Formula: see text]. The higher-dimensional space, which combines the bulk space–time with the “internal space,” which is an analog of superspace in supersymmetric theory, is introduced. The “physical bulk space–time” is a 3D hypersurface with constant [Formula: see text], [Formula: see text] and [Formula: see text] embedded in this space. We will work in Poincaré coordinates of AdS space and consider [Formula: see text]-quasi-primary operators [Formula: see text] with a conformal weight [Formula: see text] in the boundary and study two and three point functions of [Formula: see text]-quasi-primary operators transformed as [Formula: see text]. Here, [Formula: see text] and [Formula: see text] are [Formula: see text] generators in the hyperbolic basis for Poincaré coordinates. It is shown that in the [Formula: see text] limit, the conformal weight changes to a new value [Formula: see text]. This may be regarded as a Renormalization Group (RG) flow. It is argued that this RG flow will be triggered by terms [Formula: see text] added to the action.

Statistical Aspects of Coherent States of the Higgs Algebra

We construct and study various aspects of coherent states of a polynomial angular momentum algebra. The coherent states are constructed using a new unitary representation of the nonlinear algebra. The new representation involves a parameter γ that shifts the eigenvalues of the diagonal operator J0.

Momentum Maps and Stochastic Clebsch Action Principles

We derive stochastic differential equations whose solutions follow the flow of a stochastic nonlinear Lie algebra operation on a configuration manifold. For this purpose, we develop a stochastic Clebsch action principle, in which the noise couples to the phase space variables through a momentum map. This special coupling simplifies the structure of the resulting stochastic Hamilton equations for the momentum map. In particular, these stochastic Hamilton equations collectivize for Hamiltonians that depend only on the momentum map variable. The Stratonovich equations are derived from the Clebsch variational principle and then converted into It\^o form. In comparing the Stratonovich and It\^o forms of the stochastic dynamical equations governing the components of the momentum map, we find that the It\^o contraction term turns out to be a double Poisson bracket. Finally, we present the stochastic Hamiltonian formulation of the collectivized momentum map dynamics and derive the corresponding Kolmogorov forward and backward equations.

Bi-orthogonal approach to non-Hermitian Hamiltonians with the oscillator spectrum: Generalized coherent states for nonlinear algebras

A set of Hamiltonians that are not self-adjoint but have the spectrum of the harmonic oscillator is studied. The eigenvectors of these operators and those of their Hermitian conjugates form a bi-orthogonal system that provides a mathematical procedure to satisfy the superposition principle. In this form the non-Hermitian oscillators can be studied in much the same way as in the Hermitian approaches. Two different nonlinear algebras generated by properly constructed ladder operators are found and the corresponding generalized coherent states are obtained. The non-Hermitian oscillators can be steered to the conventional one by the appropriate selection of parameters. In such limit, the generators of the nonlinear algebras converge to generalized ladder operators that would represent either intensity-dependent interactions or multi-photon processes if the oscillator is associated with single mode photon fields in nonlinear media.

Nonlinear internal wave at the interface of two-layer liquid due to a moving hydrofoil

This paper is concerned with the internal wave at the interface of two layers of liquids due to a hydrofoil in the lower layer liquid. The two-layer fluid is assumed moving parallel to the interface at different velocities. The stratified flow is modeled based on the incompressible potential flow theory, with the nonlinear boundary conditions at the interface. Boundary integral equations are formulated for the fully nonlinear interfacial wave generated by the hydrofoil. The numerical model results in a set of nonlinear algebra equations, which are solved using the quasi-Newton method. We show that the quasi-Newton method is more efficient than Newton’s method, which is often used for solving these types of equations in the literature. The wave profiles were analyzed in terms of the location and thickness of the hydrofoil, the Froude number, and the ratio of the densities of the two fluids. The computations show that the interfacial wave amplitude showed a trend first of increase and then of decrease with the distance between the hydrofoil and the still interface.

Higher spin currents in the enhanced N = 3 $$ \mathcal{N}=3 $$ Kazama-Suzuki model

The N=3 Kazama-Suzuki model at the `critical' level has been found by Creutzig, Hikida and Ronne. We construct the lowest higher spin currents of spins (3/2, 2,2,2,5/2, 5/2, 5/2, 3) in terms of various fermions. In order to obtain the operator product expansions (OPEs) between these higher spin currents, we describe three N=2 OPEs between the two N=2 higher spin currents denoted by (3/2, 2, 2, 5/2) and (2, 5/2, 5/2, 3) (corresponding 36 OPEs in the component approach). Using the various Jacobi identities, the coefficient functions appearing on the right hand side of these N=2 OPEs are determined in terms of central charge completely. Then we describe them as one single N=3 OPE in the N=3 superspace. The right hand side of this N=3 OPE contains the SO(3)-singlet N=3 higher spin multiplet of spins (2, 5/2, 5/2, 5/2, 3,3,3, 7/2), the SO(3)-singlet N=3 higher spin multiplet of spins (5/2, 3,3,3, 7/2, 7/2, 7/2, 4), and the SO(3)-triplet N=3 higher spin multiplets where each multiplet has the spins (3, 7/2, 7/2, 7/2, 4,4,4, 9/2), in addition to N=3 superconformal family of the identity operator. Finally, by factoring out the spin-1/2 current of N=3 linear superconformal algebra generated by eight currents of spins (1/2, 1,1,1, 3/2, 3/2, 3/2, 2), we obtain the extension of so-called SO(3) nonlinear Knizhnik Bershadsky algebra.

W $$ \mathcal{W} $$ -symmetry, topological vertex and affine Yangian

We discuss the representation theory of non-linear chiral algebra $\mathcal{W}_{1+\infty}$ of Gaberdiel and Gopakumar and its connection to Yangian of $\hat{\mathfrak{u}(1)}$ whose presentation was given by Tsymbaliuk. The characters of completely degenerate representations of $\mathcal{W}_{1+\infty}$ are for generic values of parameters given by the topological vertex. The Yangian picture provides an infinite number of commuting charges which can be explicitly diagonalized in $\mathcal{W}_{1+\infty}$ highest weight representations. Many properties that are difficult to study in $\mathcal{W}_{1+\infty}$ picture turn out to have a simple combinatorial interpretation.

An efficient numerical method for reactive flow with general equation of states

SummaryThis paper presents an efficient method to simulate the reactive flow for general equation of states with the compressible fluid model coupled with reactive rate equation. The important aspect is to deal with mixture of different phases in one cell, which will inevitably happen in the Eulerian method for reactive flow. Physical variables such as the pressure,velocity, and speed of sound in each cell need to be reconstructed for the Harten‐Lax‐Leer‐Contact (HLLC) Riemann solver, which will result in nonlinear algebra equations, and these reconstructed variables are used to obtain the flux. Numerical examples of stable and unstable detonations with different equation of states demonstrate the accuracy and efficiency of this method. Copyright © 2016 John Wiley & Sons, Ltd.

Extended supersymmetry and hidden symmetries in one-dimensional matrix quantum mechanics

We study properties of nonlinear supersymmetry algebras realized in the one-dimensional quantum mechanics of matrix systems. Supercharges of these algebras are differential operators of a finite order in derivatives. In special cases, there exist independent supercharges realizing an (extended) supersymmetry of the same super-Hamiltonian. The extended supersymmetry generates hidden symmetries of the super-Hamiltonian. Such symmetries have been found in models with (2×2)-matrix potentials.