Algebraic Sufficient Conditions Research Articles

Correction of improper linear programming problems in canonical form by applying the minimax criterion

Given an inconsistent system of linear algebraic equations, necessary and sufficient conditions are established for the solvability of the problem of its matrix correction by applying the minimax criterion with the assumption that the solution is nonnegative. The form of the solution to the corrected system is presented. Two formulations of the problem are considered, specifically, the correction of both sides of the original system and correction with the right-hand-side vector being fixed. The minimax-criterion correction of an improper linear programming problem is reduced to a linear programming problem, which is solved numerically in MATLAB.

New results on sampled-data feedback stabilization for autonomous nonlinear systems

Sufficient conditions are established for sampled-data feedback global asymptotic stabilization for nonlinear autonomous systems. One of our main results is an extension of the well known Artstein–Sontag theorem on feedback stabilization concerning affine in the control systems. A second aim of the present work is to provide sufficient conditions for sampled-data feedback asymptotic stabilization for two interconnected nonlinear systems. Lie algebraic sufficient conditions are derived for the case of affine in the control interconnected systems without drift terms.

GENERICITY OF FILLING ELEMENTS

An element of a finitely generated non-Abelian free group F(X) is said to be filling if that element has positive translation length in every very small minimal isometric action of F(X) on an ℝ-tree. We give a proof that the set of filling elements of F(X) is exponentially F(X)-generic in the sense of Arzhantseva and Ol'shanskiı. We also provide an algebraic sufficient condition for an element to be filling and show that there exists an exponentially F(X)-generic subset consisting only of filling elements and whose membership problem has linear time complexity.

Sharp algebraic periodicity conditions for linear higher order difference equations

In this paper we give easily verifiable, but sharp (in most cases necessary and sufficient) algebraic conditions for the solutions of systems of higher order linear difference equations to be periodic. The main tool in our investigation is a transformation, recently introduced by the authors, which formulates a given higher order recursion as a first order difference equation in the phase space. The periodicity conditions are formulated in terms of the so-called companion matrices and the coefficients of the given higher order equation, as well.

Positive Invariance of Constrained Affine Dynamics and Its Applications to Hybrid Systems and Safety Verification

Motivated by long-time dynamic analysis of hybrid systems and safety verification problems, this paper addresses fundamental positive invariance issues of an affine dynamical system on a general polyhedron and their applications. Necessary and sufficient algebraic conditions are established for the existence of a positively invariant set of an affine system on a polyhedron using the tools of lexicographic relation and long-time oscillatory dynamic analysis. A linear program based algorithm is proposed to verify these conditions, and its computational complexity is analyzed. The positive invariance results are applied to obtain an explicit characterization of global switching behaviors of piecewise affine systems. Further, the positive invariance techniques developed in this paper are exploited to show the decidability of safety verification of a class of affine dynamics on semialgebraic sets.

Real radical initial ideals

We explore the consequences of an ideal I ⊂ R [ x 1 , … , x n ] having a real radical initial ideal, both for the geometry of the real variety of I and as an application to sums of squares representations of polynomials. We show that if in w ( I ) is real radical for a vector w in the tropical variety, then w is in the logarithmic set of the real variety. We also give algebraic sufficient conditions for w to be in the logarithmic limit set of a more general semialgebraic set. If in addition w ∈ ( R > 0 ) n , then the corresponding quadratic module is stable. In particular, if in w ( I ) is real radical for some w ∈ ( R > 0 ) n then ∑ R [ x 1 , … , x n ] 2 + I is stable. This provides a method for checking the conditions for stability given by Powers and Scheiderer (2001) [PS].

Structure of planar integral self‐affine tilings

AbstractFor a self‐affine tile in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {R}^2$\end{document} generated by an expanding matrix \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$A\in M_2(\mathbf {Z})$\end{document} and an integral consecutive collinear digit set \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal D}$\end{document}, Leung and Lau [Trans. Amer. Math. Soc. 359, 3337–3355 (2007).] provided a necessary and sufficient algebraic condition for it to be disklike. They also characterized the neighborhood structure of all disklike tiles in terms of the algebraic data A and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal D}$\end{document}. In this paper, we completely characterize the neighborhood structure of those non‐disklike tiles. While disklike tiles can only have either six or eight edge or vertex neighbors, non‐disklike tiles have much richer neighborhood structure. In particular, other than a finite set, a Cantor set, or a set containing a nontrivial continuum, neighbors can intersect in a union of a Cantor set and a countable set.

REGULAR HOMOTOPIC DEFORMATION OF COMPACT SURFACE WITH BOUNDARY AND MAPPING CLASS GROUP

A necessary and sufficient algebraic condition for a diffeomorphism over a surface embedded in S3 to be induced by a regular homotopic deformation is discussed, and a formula for the number of signed pass moves needed for this regular homotopy is given.

Small-Gain Type Sufficient Conditions for Sampled-Data Feedback Stabilization for Autonomous Composite Systems

Small-gain type sufficient conditions are established guaranteeing weak global asymptotic stabilization for composite autonomous systems by means of a sampled-data feedback. Our main result generalizes the well known Small-Gain Theorem for global asymptotic stability for composite systems with no controls, under the presence of the input-to-state-stability. The sufficient conditions we propose enable us to derive a set of algebraic sufficient conditions for the existence of a sampled-data feedback stabilizer with zero-order hold. The work is a continuation of author's work published in same journal.

Sufficient algebraic conditions of parametric invariance of the output of a linear stationary system in the first approximation

A problem of generating sufficient algebraic conditions of parametric invariance of the output of a linear stationary system with uncertain parameters in the first approximation is solved based on analytical capabilities of the trajectory sensitivity technique. An example demonstrating the application of the obtained conditions is given.

THE DECOHERENCE-FREE SUBALGEBRA OF A QUANTUM MARKOV SEMIGROUP WITH UNBOUNDED GENERATOR

Let [Formula: see text] be a quantum Markov semigroup on [Formula: see text] with a faithful normal invariant state ρ. The decoherence-free subalgebra [Formula: see text] of [Formula: see text] is the biggest subalgebra of [Formula: see text] where the completely positive maps [Formula: see text] act as homomorphisms. When [Formula: see text] is the minimal semigroup whose generator is represented in a generalised GKSL form [Formula: see text], with possibly unbounded H, Lℓ, we show that [Formula: see text] coincides with the generalised commutator of [Formula: see text] under some natural regularity conditions. As a corollary we derive simple sufficient algebraic conditions for convergence towards a steady state based on multiple commutators of H and Lℓ. We give examples of quantum Markov semigroups [Formula: see text], with h infinite-dimensional, having a non-trivial decoherence-free subalgebra.

Rules of Designing Routing Metrics for Greedy, Face, and Combined Greedy-Face Routing

Different geographic routing protocols have different requirements on routing metric designs to ensure proper operation. Combining a wrong type of routing metrics with a geographic routing protocol may produce unexpected results, such as geographic routing loops and unreachable nodes. In this paper, we propose a novel routing algebra system to investigate the compatibilities between routing metrics and three geographic routing protocols including greedy, face, and combined greedy-face routing. Five important algebraic properties, respectively, named odd symmetry, transitivity, strict order, source independence, and local minimum freeness, are defined in this algebra system. Based on these algebraic properties, the necessary and sufficient conditions for loop-free, delivery-guaranteed, and consistent routing are derived when greedy, face, and combined greedy-face routing serve as packet forwarding schemes or as path discovery algorithms, respectively. Our work provides essential criteria for evaluating and designing geographic routing protocols.

Stability criteria for a class of fractional order systems

This paper deals with the stability problem in LTI fractional order systems having fractional orders between 1 and 1.5. Some sufficient algebraic conditions to guarantee the BIBO stability in such systems are obtained. The obtained conditions directly depend on the coefficients of the system equations, and consequently using them is easier than the use of conditions constructed based on solving the characteristic equation of the system. Some illustrations are presented to show the applicability of the obtained conditions. For example, it is shown that these conditions may be useful in stabilization of unstable fractional order systems or in taming fractional order chaotic systems.

Algebraic criteria for consensus problems of continuous-time networked systems

This article addresses algebraic criteria for consensus problems of continuous-time networked systems with fixed and switching topology. A special eigenvector ω of the Laplacian matrix is first constructed and correlated with the connectivity of digraph. And then, some necessary and/or sufficient algebraic conditions are presented by employing the vector ω, which can directly determine whether the consensus problem can be solved. More importantly, for the switching topology case, the obtained algebraic conditions for the average consensus problem are necessary and sufficient. Furthermore, the presented results clearly show that only the agents corresponding to the positive elements of ω contribute to the group decision value and decide the collective behaviour of the system. Particularly for the fixed topology case, the role of each agent is exactly measured by the value of the corresponding element of ω. Based on these results, some extended protocols are further proposed to solve the average consensus problem, in which the interaction digraphs are not needed to be balanced. Numerical examples are included to illustrate the obtained results.

On Singerman symmetries of a class of Belyi Riemann surfaces

By virtue of the Belyi Theorem an algebraic curve can be defined over the algebraic numbers if and only if the corresponding Riemann surface can be uniformized by a subgroup of a Fuchsian triangle group. Such surfaces are known as Belyi surfaces and an important class of them consists of Riemann surfaces having the so-called large group of automorphisms. Necessary and sufficient algebraic conditions for these surfaces to be symmetric were found by Singerman in the middle of the seventies and, by a recent result of Köck and Singerman, the algebraic numbers above can be chosen to be real if and only if the respective surface is symmetric. The aim of this paper is to give, in similar terms, the formulas for the number of ovals of the corresponding symmetries, which we refer to as the Singerman symmetries.

Algebraic Criteria for Consensus Problems of Discrete-Time Networked Systems

This paper is mainly devoted to the algebraic criteria for consensus problems of discrete-time networked systems with the fixed and switching topology. A special eigenvector ω of the Laplacian matrix is first correlated with the connectivity of a digraph, and then the relations between a class of Laplacian-type matrix and the stochastic matrix are established. Based on these tools, some necessary and/or sufficient algebraic conditions are proposed, which can directly determine whether the consensus problem can be solved or not. Furthermore, it is proved that only the agents corresponding to the positive elements of ω contribute to the group decision value and decide the collective behavior of the system. Particularly for the fixed topology case, it is shown that not only the role of each agent is exactly proportional to the value of the corresponding element of ω but also the group decision value can be calculated by such a vector and the initial states of all agents.

Collinear Systems and Normal Contractive Projections on JBW*-Triples

Given a family {x k }k∈K of elements x k in the predual A * of a JBW*-triple A, such that the support tripotents e k of x k form a collinear system in the sense of [31], necessary and sufficient criteria for the existence of a contractive projection from A *. onto the subspace $$\overline{\rm lin\{x_{k} : k \in K\}}^{n}$$ are provided. Preparatory to these results, and interesting in itself, is a set of necessary and sufficient algebraic conditions upon a contractive projection P on A for its range PA to be a subtriple. The results also provide criteria for the range of a normal contractive projection on A to be a Hilbert space.

Robust absolute stability of Lurie interval control systems

AbstractThis paper considers robust absolute stability of Lurie control systems. Particular attention is given to the systems with parameters having uncertain, but bounded values. Such so‐called Lurie interval control systems have wide applications in practice. In this paper, a number of sufficient and necessary conditions are derived by using the theories of Hurwitz matrix, M matrix and partial variable absolute stability. Moreover, several algebraic sufficient and necessary conditions are provided for the robust absolute stability of Lurie interval control systems. These algebraic conditions are easy to be verified and convenient to be used in applications. Three mathematical examples and a practical engineering problem are presented to show the applicability of theoretical results. Numerical simulation results are also given to verify the analytical predictions. Copyright © 2007 John Wiley & Sons, Ltd.

A necessary and sufficient condition of a mixed NE in bimatrix games

In the paper a mixed Nash equilibrium (NE) in bimatrix games is considered. Developing an algebraic approach, a necessary and sufficient condition for existence of the NE, the equilibrium profile, the value of its payoff are expressed with saddle point matrices of a special form, i.e. the block matrices with two vector blocks of ones. Mathematics Subject Classification: 91A05, 91A10, 15A24

Criteria for the Lp-dissipativity of systems of second order differential equations

We give complete algebraic characterizations of the L p -dissipativity of the Dirichlet problem for some systems of partial differential operators of the form $\partial_{h}({\mathop{\cal A}\nolimits}^{hk}(x)\partial_{k})$ , where ${\mathop{\cal A}\nolimits}^{hk}(x)$ are m× m matrices. First, we determine the sharp angle of dissipativity for a general scalar operator with complex coefficients. Next we prove that the two-dimensional elasticity operator is L p -dissipative if and only if $$ \left({1\over 2}-{1\over p}\right)^{2} \leq {2(\nu-1)(2\nu-1)\over (3-4\nu)^{2}}, $$ ν being the Poisson ratio. Finally we find a necessary and sufficient algebraic condition for the L p -dissipativity of the operator $\partial_{h} ({\mathop{\cal A}\nolimits}^{h}(x)\partial_{h})$ , where ${\mathop{\cal A}\nolimits}^{h}(x)$ are m× m matrices with complex L1loc entries, and we describe the maximum angle of L p -dissipativity for this operator. Keywords: L p -dissipativity, Algebraic conditions, Elasticity system Mathematics Subject Classification (2000): 47D03, 47D06, 47B44, 74B05