Nonsymmetric Operators Research Articles

A Perron-type Theorem on the Principal Eigenvalue of Nonsymmetric Elliptic Operators

We provide a proof for a Perron-type theorem on the principal eigenvalue of nonsymmetric elliptic operators based on the strong maximum principle. This proof is modeled after a variational proof of Perron's theorem for matrices with positive entries that does not appeal to Perron-Frobenius theory.

Semi-Dirichlet forms, Feynman–Kac functionals and the Cauchy problem for semilinear parabolic equations

In the first part of the paper we prove various results on regularity of Feynman–Kac functionals of Hunt processes associated with time-dependent semi-Dirichlet forms. In the second part we study the Cauchy problem for semilinear parabolic equations with measure data involving operators associated with time-dependent forms. Model examples are non-symmetric divergence form operators and fractional laplacians with possibly variable exponents. We first introduce a definition of a solution resembling Stampacchia's definition in the sense of duality and then, using the results of the first part, we prove the existence, uniqueness and regularity of solutions of the problem under mild assumptions on the data.

Symmetrization of functions and principal eigenvalues of elliptic operators

In this paper, we consider shape optimization problems for the principal eigen-values of second order uniformly elliptic operators in bounded domains of R n. We first recall the classical Rayleigh-Faber-Krahn problem, that is the minimization of the principal eigenvalue of the Dirichlet Laplacian in a domain with fixed Lebesgue measure. We then consider the case of the Laplacian with a bounded drift, that is the operator −∆ + v · ∇, for which the minimization problem is still well posed. Next, we deal with more general elliptic operators −div(A∇) + v · ∇ + V , for which the coefficients fulfill various pointwise, integral or geometric constraints. In all cases, some operators with radially symmetric coefficients in an equimeasurable ball are shown to have smaller principal eigenvalues. Whereas the Faber-Krahn proof relies on the classical Schwarz symmetrization, another type of symmetrization is defined to handle the case of general (possibly non-symmetric) operators.

Sign-preserving of principal eigenfunctions in P1 finite element approximation of eigenvalue problems of second-order elliptic operators

This paper is concerned with the P1 finite element approximation of the eigenvalue problem of second-order elliptic operators subject to the Dirichlet boundary condition. The focus is on the preservation of the basic properties of the principal eigenvalue and eigenfunctions of the continuous problem. It is shown that when the stiffness matrix is an irreducible M-matrix, the discrete eigenvalue problem maintains almost all of the basic properties such as the smallest eigenvalue being real and simple and the corresponding eigenfunctions being either positive or negative inside the physical domain. Mesh conditions leading to such a stiffness matrix are also studied. A sufficient condition is that the mesh is simplicial, interiorly connected, and acute when measured in the metric specified by the inverse of the diffusion matrix. The acute requirement can be replaced by the Delaunay condition in two dimensions. Numerical results show that when the stiffness matrix is not an M-matrix, a finite element approximation can be structurally different from the continuous eigenvalue problem: the eigenfunctions corresponding to the smallest eigenvalue can change sign over the physical domain and the smallest eigenvalue (in modulus) can even be complex for the case with nonsymmetric operators.

Optimal hardy weight for second-order elliptic operator: An answer to a problem of Agmon

For a general subcritical second-order elliptic operator P in a domain Ω⊂Rn (or noncompact manifold), we construct Hardy-weight W which is optimal in the following sense. The operator P−λW is subcritical in Ω for all λ<1, null-critical in Ω for λ=1, and supercritical near any neighborhood of infinity in Ω for any λ>1. Moreover, if P is symmetric and W>0, then the spectrum and the essential spectrum of W−1P are equal to [1,∞), and the corresponding Agmon metric is complete. Our method is based on the theory of positive solutions and applies to both symmetric and nonsymmetric operators. The constructed Hardy-weight is given by an explicit simple formula involving two distinct positive solutions of the equation Pu=0, the existence of which depends on the subcriticality of P in Ω.

Jordan decomposition and geometric multiplicity for a class of non-symmetric Ornstein-Uhlenbeck operators

Abstract In this paper, we calculate the Jordan decomposition for a class of non-symmetric Ornstein-Uhlenbeck operators with the drift coefficient matrix, being a Jordan block, and the diffusion coefficient matrix, being the identity multiplying a constant. For the 2-dimensional case, we present all the general eigenfunctions by mathematical induction. For the 3-dimensional case, we divide the calculation of the Jordan decomposition into three steps. The key step is to do the canonical projection onto the homogeneous Hermite polynomials, and then use the theory of systems of linear equations. Finally, we get the geometric multiplicity of the eigenvalue of the Ornstein-Uhlenbeck operator.

Adaptive FEM with Optimal Convergence Rates for a Certain Class of Nonsymmetric and Possibly Nonlinear Problems

We analyze adaptive mesh-refining algorithms for conforming finite element discretizations of certain non-linear second-order partial differential equations. We allow continuous polynomials of arbitrary, but fixed polynomial order. The adaptivity is driven by the residual error estimator. We prove convergence even with optimal algebraic convergence rates. In particular, our analysis covers general linear second-order elliptic operators. Unlike prior works for linear non-symmetric operators, our analysis avoids the interior node property for the refinement, and the differential operator has to satisfy a G\r{a}rding inequality only. If the differential operator is uniformly elliptic, no additional assumption on the initial mesh is posed.

Factorization of Non-Symmetric Operators and Exponential H-Theorem

We present an abstract method for deriving decay estimates on the resolvents and semigroups of non-symmetric operators in Banach spaces in terms of estimates in another smaller reference Banach space. This applies to a class of operators writing as a regularizing part, plus a dissipative part. The core of the method is a high-order quantitative factorization argument on the resolvents and semigroups. We then apply this approach to the Fokker-Planck equation, to the kinetic Fokker- Planck equation in the torus, and to the linearized Boltzmann equation in the torus. We finally use this information on the linearized Boltzmann semi- group to study perturbative solutions for the nonlinear Boltzmann equation. We introduce a non-symmetric energy method to prove nonlinear stability in this context in $L^1_v L^\infty _x (1 + |v|^k)$, $k > 2$, with sharp rate of decay in time. As a consequence of these results we obtain the first constructive proof of exponential decay, with sharp rate, towards global equilibrium for the full nonlinear Boltzmann equation for hard spheres, conditionally to some smoothness and (polynomial) moment estimates. This improves the result in [32] where polynomial rates at any order were obtained, and solves the conjecture raised in [91, 29, 86] about the optimal decay rate of the relative entropy in the H-theorem.

Spectral Convex Functions of Operators and Approximate Intertwining Relationships

We develop theoretical tools for the analysis of convex spectral functions of non-symmetric operators on Hilbert spaces. The obtained results are applied to an optimization problem arising from the theory of inverse problems, which involves the notion of intertwining relationship.

Spectral properties and low‐dimensional description of loop and recycle reactors

The spectral properties of the discrete and continuous convection and convection‐diffusion operators with loop or recycle boundary condition are analyzed. It is shown that the spectral properties of these nonsymmetric operators are closely related to the theory of circulant (Toeplitz) matrices and the complex Fourier series, respectively. Although there may be many complex eigenvalues, the smallest eigenvalue is real and approaches zero as the loop circulation or recycle ratio increases. This property is used to simplify nonlinear diffusion‐convection‐reaction models of loop and recycle reactors to obtain two‐mode low‐dimensional averaged models that are accurate in the limit of large recycle ratio. Explicit expressions for the two mixing coefficients that relate the two concentration modes and their dependence on various inlet conditions are also derived. Finally, the application of the low‐dimensional models to determine the impact of macromixing on the conversion, yield, and selectivity for the case of nonlinear kinetics is illustrated. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3365–3377, 2013

The inverse of nonsymmetric two-level Toeplitz operator matrices

The Gohberg–Semencul formula allows one to express the entries of the inverse of a Toeplitz matrix using only a few entries (the first row and the first column) of the inverse matrix, under some nonsingularity condition. In this paper we will provide a two variable generalization of the Gohberg–Semencul formula in the case of a nonsymmetric two-level Toeplitz matrix with a symbol of the form f(z1,z2)=1P(z1,z2)¯Q(z1,z2) where P(z1,z2) and Q(z1,z2) are stable polynomials of two variables. We also consider the case of operator valued two-level Toeplitz matrices. In addition, we propose an equation solver involving two-level Toeplitz matrices. Numerical results are included.

Calculation of degree and global bifurcation for variational inequalities with nonsymmetric operators in Banach spaces

In order to prove global bifurcation for variational inequalities, it is useful to know local mapping degrees for certain maps associated to the inequalities. Some methods are provided to calculate these degrees. In contrast to existing results, no symmetry on the derivatives needs to be assumed to apply these methods. In fact, it is also shown that the whole approach is possible even in the setting of Banach spaces and for noncompact operators with only positively homogeneous “derivatives”.

COHERENT STATE FUNCTIONAL INTEGRAL IN LOOP QUANTUM COSMOLOGY: ALTERNATIVE DYNAMICS

Coherent state functional integral for the minisuperspace model of loop quantum cosmology is studied. By the well-established canonical theory, the transition amplitude in the path integral representation of loop quantum cosmology with alternative dynamics can be formulated through group averaging. The effective action and Hamiltonian with higher-order quantum corrections are thus obtained. It turns out that for a nonsymmetric Hamiltonian constraint operator, the Moyal (star)-product emerges naturally in the effective Hamiltonian. For the corresponding symmetric Hamiltonian operator, the resulted effective theory implies a possible quantum cosmological effect in large scale limit in the alternative dynamical scenario, which coincides with the result in canonical approach. Moreover, the first-order modified Friedmann equation still contains the particular information of alternative dynamics and hence admits the possible phenomenological distinction between the different proposals of quantum dynamics.

Optimal Hardy-type inequalities for elliptic operators

For a general second order elliptic operator P in a domain Ω, we construct a Hardy weight W in the punctured domain Ω⋆:=Ω∖{0} such that P−λW is subcritical in Ω⋆ for λ<1, null-critical in Ω⋆ for λ=1, and supercritical near infinity and near 0 for λ>1. Our method is based on the theory of positive solutions and applies to both symmetric and nonsymmetric operators. The constructed Hardy weight is given by an explicit formula involving the Green function of P and its gradient.

Coherent state functional integrals in quantum cosmology

Coherent state functional integrals for the minisuperspace models of quantum cosmology are studied. By the well-established canonical theories, the transition amplitudes in the path-integral representations of Wheeler-DeWitt quantum cosmology and loop quantum cosmology can be formulated through group averaging. The effective action and Hamiltonian with higher-order quantum corrections are thus obtained in both models. It turns out that for a non-symmetric Hamiltonian constraint operator, the Moyal (star)-product emerges naturally in the effective Hamiltonian. This reveals the intrinsic relation among coherent state functional integral, effective theory and star-product. Moreover, both the resulted effective theories imply a possible quantum cosmological effect in large scale limit.

On the acceleration of explicit finite difference methods for option pricing

Implicit finite difference methods are conventionally preferred over their explicit counterparts for the numerical valuation of options. In large part the reason for this is a severe stability constraint known as the Courant–Friedrichs–Lewy (CFL) condition which limits the latter class's efficiency. Implicit methods, however, are difficult to implement for all but the most simple of pricing models, whereas explicit techniques are easily adapted to complex problems. For the first time in a financial context, we present an acceleration technique, applicable to explicit finite difference schemes describing diffusive processes with symmetric evolution operators, called Super-Time-Stepping. We show that this method can be implemented as part of a more general approach for non-symmetric operators. Formal stability is thereby deduced for the exemplar cases of European and American put options priced under the Black–Scholes equation. Furthermore, we introduce a novel approach to describing the efficiencies of finite difference schemes as semi-empirical power laws relating the minimal real time required to carry out the numerical integration to a solution with a specified accuracy. Tests are described in which the method is shown to significantly ameliorate the severity of the CFL constraint whilst retaining the simplicity of the underlying explicit method. Degrees of acceleration are achieved yielding comparable, or superior, efficiencies to a set of benchmark implicit schemes. We infer that the described method is a powerful tool, the explicit nature of which makes it ideally suited to the treatment of symmetric and non-symmetric diffusion operators describing complex financial instruments including multi-dimensional systems requiring representation on decomposed and/or adaptive meshes.

On the relation between the full Kostant–Toda lattice and multiple orthogonal polynomials

The correspondence between a high-order non-symmetric difference operator with complex coefficients and the evolution of an operator defined by a Lax pair is established. The solution of the discrete dynamical system is studied, giving explicit expressions for the resolvent function and, under some conditions, the representation of the vector of functionals, associated with the solution for our integrable systems. The method of investigation is based on the evolutions of the matrical moments.

Robust stability of nonlinear plants with a non-symmetric Prandtl-Ishlinskii hysteresis model

In this paper, robust stability of nonlinear plants represented by non-symmetric Prandtl-Ishlinskii (PI) hysteresis model is studied. In general, PI hysteresis model is the weighted superposition of play or stop hysteresis operators, and the slopes of the operators are considered to be the same. In order to make a hysteresis model, a modified form of non-symmetric play hysteresis operator with unknown slopes is given. The hysteresis model is described by a generalized Lipschitz operator term and a bounded parasitic term. Since the generalized Lipschitz operator is unknown, a new condition using robust right coprime factorization is proposed to guarantee robust stability of the controlled plant with the hysteresis nonlinearity. As a result, based on the proposed robust condition, a stabilized plant is obtained. A numerical example is presented to validate the effectiveness of the proposed method.

Robust stability and tracking of non-linear systems with modified Prandtl-Ishlinskii hysteresis model

This paper introduces a method to guarantee robust stability of nonlinear systems with modified Prandtl-Ishlinskii (PI) hysteresis model and presents a tracking controller design method to compensate the effect of hysteretic non-linearity. In order to make hysteresis model closer to the real hysteretic behaviour, a modified PI hysteresis model which is described by a non-symmetric play operator with unknown slopes is adopted. Using an operator theoretic approach, the modified PI hysteresis model is represented as a generalised Lipschitz operator and a bounded parasitic term. To guarantee robust stability of the controlled systems with hysteresis, a robust condition using robust right coprime factorisation is used. To ensure the output tracking performance, a non-linear tracking controller is designed. Simulation results are presented to validate the effectiveness of the proposed control design method.

The Effect of the Schwarz Rearrangement on the Periodic Principal Eigenvalue of a Nonsymmetric Operator

This paper is concerned with the periodic principal eigenvalue $k_\lambda(\mu)$ associated with the operator $-\frac{d^2}{dx^2}-2\lambda\frac{d}{dx}-\mu(x)-\lambda^2$, where $\lambda\in\mathbb{R}$ and $\mu$ is continuous and periodic in $x\in\mathbb{R}$. Our main result is that $k_\lambda(\mu^*)\leq k_\lambda(\mu)$, where $\mu^*$ is the Schwarz rearrangement of the function $\mu$. From a population dynamics point of view, using reaction-diffusion modeling, this result means that the fragmentation of the habitat of an invading population slows down the invasion. We prove that this property does not hold in higher dimension if $\mu^*$ is the Steiner symmetrization of $\mu$. For heterogeneous diffusion and advection, we prove that increasing the period of the coefficients decreases $k_\lambda$, and we compute the limit of $k_\lambda$ when the period of the coefficients goes to 0. Last, we prove that in dimension 1, rearranging the diffusion term decreases $k_\lambda$. These results rely on some new formula for the periodic principal eigenvalue of a nonsymmetric operator.