Explore the vast range of titles available.

In matrix-valued datasets the sampled matrices often exhibit correlations among both their rows and their columns. A useful and parsimonious model of such dependence is the matrix normal model, in which the covariances among the elements of a random matrix are parameterized in terms of the Kronecker product of two covariance matrices, one representing row covariances and one representing column covariance. An appealing feature of such a matrix normal model is that the Kronecker covariance structure allows for standard likelihood inference even when only a very small number of data matrices is available. For instance, in some cases a likelihood ratio test of dependence may be performed with a sample size of one. However, more generally the sample size required to ensure boundedness of the matrix normal likelihood or the existence of a unique maximizer depends in a complicated way on the matrix dimensions. This motivates the study of how large a sample size is needed to ensure that maximum likelihood estimators exist, and exist uniquely with probability one. Our main result gives precise sample size thresholds in the paradigm where the number of rows and the number of columns of the data matrices differ by at most a factor of two. Our proof uses invariance properties that allow us to consider data matrices in canonical form, as obtained from the Kronecker canonical form for matrix pencils.

All possible nonlocal versions of the derivative nonlinear Schrödinger equations are derived by the nonlocal reduction from the Chen–Lee–Liu equation, the Kaup–Newell equation and the Gerdjikov–Ivanov equation which are gauge equivalent to each other. Their solutions are obtained by composing constraint conditions on the double Wronskian solution of the Chen–Lee–Liu equation and the nonlocal analogues of the gauge transformations among them. Through the Jordan decomposition theorem, those solutions of the reduced equations from the Chen–Lee–Liu equation can be written as canonical form within real field.

This paper presents a methodology for finding a pseudo-upper or lower triangular state-space realization (SSR) from an incommensurate fractional-order transfer function. This SSR is obtained for a particular case of incommensurate fractional-order systems that can be represented by pseudo-upper or lower triangular multi-order state-space equations, which are derived by drawing the block diagram of the transfer functions. The obtained realization is very similar to the controllability canonical form for ordinary transfer functions. It is demonstrated that the obtained realization is controllable. Thus, the state feedback controllers can be systematically designed for these systems.

A global framework for treating nonlinear differential dynamical systems is presented. It rests on the fact that most systems can be transformed into the quasi-polynomial format. Any system in this format belongs to an infinite equivalence class characterized by two canonical forms, the Lotka-Volterra (LV) and the monomial systems. Both forms allow for finding total or partial integrability conditions, invariants and dimension reductions of the original systems. The LV form also provides Lyapunov functions and systematic tools for stability analysis. An abstract Lie algebra is shown to underlie the whole formalism. This abstract algebra can be expressed in several realizations among which are the bosonic creation-destruction operators. One of these representations allows one to obtain the analytic form of the general coefficient of the Taylor series representing the solution of the original system. This generates a new class of special functions that are solutions of these nonlinear dynamical systems. From the monomial canonical form, one can prove an equivalence relationship between urn processes and dynamical systems. This establishes a new link between nonlinear dynamics and stochastic processes. This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 1)'.

In this paper, interval observers are designed for linear dynamic systems described by continuous-time models with exogenous disturbances, measurement noises, and parametric uncertainties. Jordan canonical form-based relations are presented for an interval observer that estimates the set of admissible values of a given linear function of the system state vector. The theoretical results are illustrated by a practical example.

A new approach to the transformations of the matrices of linear continuous-time systems to their canonical forms with desired eigenvalues is proposed. Conditions for the existence of solutions to the problems were given and illustrated by simple numerical examples.

An effective analytical method is proposed to solve the problem of modal control by output for a wide class of linear time-invariant systems in which the sum of inputs and outputs can be not only greater than or equal to but also less than the dimension of a state vector. The method is based on reducing the modal control by output to modal observation with fewer inputs. At the same time, it is not necessary to additionally ensure the solvability of the equation connecting the matrix of observer matrix and the desired matrix of controller by output. The reduction is performed by constructing a generalized dual canonical form of control using the operations of the block transpose and the rank decomposition of matrices. The method significantly expands the class of systems for which an analytical solution exists compared to the previously proposed approaches, since it is not strictly tied to the control system’s dimension and also does not require mandatory zeroing of the column and obtaining a system with a scalar input. Based on the proposed method, a strict algorithm for the analytical solution of problems from the considered class is formed. A simple and convenient necessary condition of reducibility of modal control by output to modal observation with fewer inputs is also obtained, which allows evaluating the possibility of analytical solution of the original problem basing only on its formulation. Examples of various problems of modal control by output in which the sum of inputs and outputs is less than or equal to the dimension of a state vector are considered in symbolic form. A detailed analytical solution of the considered examples demonstrates the efficiency of the proposed approach practical application.

A unitoid is a square matrix that can be brought to diagonal form by a congruence transformation. Among different diagonal forms of a unitoid  , there is only one, up to the order adopted for the principal diagonal, whose nonzero diagonal entries all have the modulus 1. It is called the congruence canonical form of  , while the arguments of the nonzero diagonal entries are called the canonical angles of . If   is nonsingular then its canonical angles are closely related to the arguments of the eigenvalues of the matrix  , called the cosquare of  . Although the definition of a unitoid reminds the notion of a diagonalizable matrix in the similarity theory, the analogy between these two matrix classes is misleading. We show that the Jordan block  , which is regarded as an antipode of diagonalizability in the similarity theory, is a unitoid. Moreover, its cosquare   has   distinct unimodular eigenvalues. Then we immerse   in the family of the Jordan blocks  , where   is varying in the range  . At some point to the left of 1,   is not a unitoid any longer. We discuss this moment in detail in order to comprehend how it can happen. Similar moments with even smaller   are discussed, and certain remarkable facts about the eigenvalues of cosquares and their condition numbers are pointed out.

Practical algorithms for computing canonical forms of multi-digraphs do not exist in the literature. This paper proposes two practical approaches for finding canonical forms, from the perspective of nD symbolic computation. Initially, the approaches turn the problem of finding canonical forms of multi-digraphs into computing canonical forms of indexed monomials in computer algebra. Then, the first approach utilizes the double coset representative method in computational group theory for canonicalization of indexed monomials and shows that finding the canonical forms of a class of multi-digraphs in practice has polynomial complexity of approximately O((k+p)2) or O(k2.1) by the computer algebra system (CAS) tool Tensor-canonicalizer. The second approach verifies the equivalence of canonicalization of indexed monomials and finding canonical forms of (simple) colored tripartite graphs. It is found that the proposed algorithm takes approximately O((k+2p)4.803) time for a class of multi-digraphs in practical implementation, combined with one of the best known graph isomorphism tools Traces, where k and p are the vertex number and edge number of a multi-digraph, respectively.

This paper aims to study a class of neutral differential equations of higher-order in canonical form. By using the comparison technique, we obtain sufficient conditions to ensure that the studied differential equations are oscillatory. The criteria that we obtained are to improve and extend some of the results in previous literature. In addition, an example is given that shows the applicability of the results we obtained.