Second Root Vectors for Multiparameter Eigenvalue Problems of Fredholm Type

A class of multiparameter eigenvalue problems involving (generally) non self-adjoint and unbounded operators is studied. A basis for the second root subspace, at eigenvalues of Fredholm type, is computed in terms of the underlying multiparameter system. A self-adjoint version of this result is given under a weak definiteness condition, and Sturm-Liouville and finite-dimensional examples are considered.

     

Factorizations along commutative subspace lattices

A theorem of D. R. Larson on the factorization of positive-definite operators along complete, countable nests in a Hilbert space is generalized to the case of commutative subspace lattices: We characterize those selfadjoint, positive-definite operators which can be factored A^* A where A and A^–1 leave invariant the subspaces in a given countable, complete, commutative subspace lattice. Applications to inner-outer factorization theory and factorization of a positive-definite operator with respect an uncountable commutative subspace lattice are given. These results have applications to function theory on the polydisk, nonanticipative representations of Gaussian random fields, and multiparameter systems theory.     

Root Vectors for Geometrically Simple Multiparameter Eigenvalues

A class of multiparameter eigenvalue problems involving (generally) non self-adjoint and unbounded operators is studied. Bases for lower order root subspaces, at geometrically simple eigenvalues of Fredholm type of arbitrary finite index, are computed in terms of the underlying multiparameter system.     

Expansion of solution in terms of generalized eigenfunctions for a hyperbolic system with static boundary condition

This paper studies a linear hyperbolic system with static boundary condition that was first studied in Neves et al. [J. Funct. Anal. 67(1986) 320-344]. It is shown that the spectrum of the system consists of zeros of a sine-type function and the generalized eigenfunctions of the system constitute a Riesz basis with parentheses for the root subspace. The state space thereby decomposes into topological direct sum of root subspace and another invariant subspace in which the associated semigroup is superstable: that is to say, the semigroup is identical to zero after a finite time period.     

On Some Properties of Polynomial Bases of Subspaces over the Field of Rational Functions in Several Variables

Spaces of multiparameter rational vectors, i.e., of vectors whose components are rational functions in several variables, and polynomial bases of their subspaces are considered. The conjecture that any subspace in the space of multiparameter rational vectors possesses a free polynomial basis, i.e., a basis such that the associated basis multiparameter polynomial matrix has no finite regular spectrum, is disproved by an example. Some consequences of this fact are indicated. Simpler proofs of some properties of the singular spectra of basis polynomial matrices corresponding to the null-spaces of a singular polynomial matrix are presented. Bibliography: 5 titles.     

Lusztig isomorphisms for Drinfel'd doubles of bosonizations of Nichols algebras of diagonal type

In the structure theory of quantized enveloping algebras, the algebra isomorphisms determined by Lusztig led to the first general construction of PBW bases of these algebras. Also, they have important applications to the representation theory of these and related algebras. In the present paper the Drinfel'd double for a class of graded Hopf algebras is investigated. Various quantum algebras, including small quantum groups and multiparameter quantizations of semisimple Lie algebras and of Lie superalgebras, are covered by the given definition. For these Drinfel'd doubles Lusztig maps are defined. It is shown that these maps induce isomorphisms between doubles of bosonizations of Nichols algebras of diagonal type. Further, the obtained isomorphisms satisfy Coxeter type relations in a generalized sense. As an application, the Lusztig isomorphisms are used to give a characterization of Nichols algebras of diagonal type with finite root system.     

线性空间的亚子空间

本文给出了线性空间中亚子空间的概念，讨论了亚子空间的性质和维数，并将其应用到有解的非齐次线性方程组中去．     

Multiparameter nonlocal spectral problems for operator-differential equations

We study the spectral properties, the basis property of the system of root vectors, and the smoothness of a solution represented as a series in this system for a class of multiparameter spectral problems for differential-operator equations with nonlocal conditions.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 26–30.     

Root vectors for geometrically simple two-parameter eigenvalues

A class of two-parameter eigenvalue problems involving generally nonselfadjoint and unbounded operators is studied. A basis for the root subspace at a geometrically simple eigenvalue of Fredholm type is computed in terms of the underlying two-parameter system. Comparison with Faierman's work on two-parameter boundary value problems of Sturm-Liouville type is given as an application.

     

Spencer <Emphasis Type="Italic">δ</Emphasis>-cohomology,Restrictions, Characteristics and Involutive Symbolic PDEs

We generalize the notion of involutivity to systems of differential equations of different orders and show that the classical results relating involutivity, restrictions, characteristics and characteristicity, known for first order systems, extend to the general context. This involves, in particular, a new definition of strong characteristicity. The proof exploits a spectral sequence relating Spencer δ-cohomology of a symbolic system and its restriction to a non-characteristic subspace.      

边界剪力反馈下梁振动系统根子空间的完备性

讨论了一个在边界上有剪力反馈控制的Euler-Bernoulli梁方程，证明了其广义本征函数生成的根子空间在能量Hilbert空间中是完备的．     

Differentiability of the C<Subscript>0</Subscript>-semigroup and Failure of Riesz Basis for a Mono-tubular Heat Exchanger Equation with Output Feedback: A Case Study

In this article, we show that the C₀-semigroup associated with the closed-loop system of a mono-tubular heat exchanger equation with static output feedback studied in [System & Control Letters, 50 (2003), 363–369] is differentiable after some finite time period. This result strengthes the main conclusion of the spectrum-determined growth condition in aforementioned paper. Moreover, it is shown that the system is not a Riesz spectral system although the root subspace is complete in the energy Hilbert space. This is in sharp contrast to results for certain second order hyperbolic systems.     

模糊仿射集与模糊向量子空间的再定义

利用模糊直线定义模糊仿射集，研究其有关性质。在此基础上，定义模糊向量子空间，证明一个模糊集为模糊仿射集的充要条件是该模糊集是一个模糊子空间的平移。     

加权Bergman空间中的不变子空间上的根算子

In this paper, for an invariant subspace I of the weighted Bergman space, the weighted root operator is defined. We study the weighted root operator and obtain its fundamental properties when the invariant subspace I has finite index. Furthermore, we give some examples of the root operator and estimate ranks of the operators.     

The tensor t‐function: A definition for functions of third‐order tensors

A definition for functions of multidimensional arrays is presented. The definition is valid for third‐order tensors in the tensor t‐product formalism, which regards third‐order tensors as block circulant matrices. The tensor function definition is shown to have similar properties as standard matrix function definitions in fundamental scenarios. To demonstrate the definition's potential in applications, the notion of network communicability is generalized to third‐order tensors and computed for a small‐scale example via block Krylov subspace methods for matrix functions. A complexity analysis for these methods in the context of tensors is also provided.     

Pointed Hopf Algebras with Triangular Decomposition

In this paper, we present an approach to the definition of multiparameter quantum groups by studying Hopf algebras with triangular decomposition. Classifying all of these Hopf algebras which are of what we call weakly separable type over a group, we obtain a class of pointed Hopf algebras which can be viewed as natural generalizations of multiparameter deformations of universal enveloping algebras of Lie algebras. These Hopf algebras are instances of a new version of braided Drinfeld doubles, which we call asymmetric braided Drinfeld doubles. This is a generalization of an earlier result by Benkart and Witherspoon (Algebr. Represent. Theory 7(3) ? BC) who showed that two-parameter quantum groups are Drinfeld doubles. It is possible to recover a Lie algebra from these doubles in the case where the group is free abelian and the parameters are generic. The Lie algebras arising are generated by Lie subalgebras isomorphic to \(\mathfrak {sl}_{2}\).     

Stäckel transform of Lax equations

We construct a map between Lax equations for pairs of Liouville integrable Hamiltonian systems related by a multiparameter Stäckel transform. Using this map, we construct Lax representation for a wide class of separable systems by applying the multiparameter Stäckel transform to Lax equations of suitably chosen systems from a seed class. For a given separable system, we obtain in this way a set of nonequivalent Lax equations parameterized by an arbitrary function of the spectral parameter, as it is in the case of a related seed system.     

An iterative method for solving the stable subspace of a matrix pencil and its application

This work is to propose an iterative method of choice to compute a stable subspace of a regular matrix pencil. This approach is to define a sequence of matrix pencils via particular left null spaces. We show that this iteration preserves a semigroup property depending only on the initial matrix pencil. Via this recursion relationship, we propose an accelerated iterative method to compute the stable subspace and use it to provide a theoretical result to solve the principal square root of a given matrix, both nonsingular and singular. We show that this method can not only find out the matrix square root, but also construct an iterative approach which converges to the square root with any desired order.     

Lower semicontinuity property of multiparameter optimal stopping value and its application to multiparameter prophet inequalities

This paper is concerned with the optimal stopping problem for discrete time multiparameter stochastic processes with the index set Nd. The optimal stopping value of a discrete time multiparameter integrable stochastic process whose negative part is uniformly integrable, is lower semicontinuous for the topology of convergence in distribution. The multiparameter version of prophet inequality for the one-parameter optimal stopping problem is formulated and the lower semicontinuity property of the optimal stopping value is applied to the multiparameter prophet inequality.     

Interaction of Vortical and Acoustic Waves: From General Equations to Integrable Cases

The equations of the (2+1)-dimensional boundary-layer perturbation split into eigenmodes: a vortex wave and two acoustic waves. We assume that the equations of state (Taylor series approximation) are arbitrary. We realize a mode definition via local-relation equations extracted from the linearization of the general system over the boundary-layer flow. Each such link determines an invariant subspace and the corresponding projector. We examine the nonlinear equation for a vortex wave using a special orthogonal coordinate system based on streamlines. The equations for the orthogonal curves are linked to the Laplace equations via Laplace and Moutard transformations. The nonlinearity determines the proper form of the interaction between vortical and acoustic boundary-layer perturbation fields fixed by projecting to a subspace of the Orr-Sommerfeld equation solutions for the Tollmienn-Schlichting (linear vortical) wave and by the corresponding procedure for the acoustic wave. We suggest a new mechanism for controlling the nonlinear resonance of the Tollmienn-Schlichting wave by sound via a four-wave interaction.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 171–181, July, 2005.