On the reflection invariance of residuated chains

It is shown that, under certain conditions, a subset of the graph of a commutative residuated chain is invariant under a geometric reflection. This result implies that a certain part of the graph of the monoidal operation of a commutative residuated chain determines another part of the graph via the reflection on one hand, and tells us about the structure of continuity points of the monoidal operation on the other. Finally, these results are applied for the subdomains of uniqueness problem, yielding new results.     

Uniform invariance principle of discontinuous systems with parameter variation

An extension of the invariance principle for a class of discontinuous righthand sides systems with parameter variation in the Filippov sense is proposed. This extension allows the derivative of an auxiliary function V, also called a Lyapunov-like function, along the solutions of the discontinuous system to be positive on some sets. The uniform estimates of attractors and basin of attractions with respect to parameters are also obtained. To this end, we use locally Lipschitz continuous and regular Lyapunov functions, as well as Filippov theory. The obtained results settled in the general context of differential inclusions, and through a uniform version of the LaSalle invariance principle. An illustrative example shows the potential of the theoretical results in providing information on the asymptotic behavior of discontinuous systems.     

Covariational reasoning and invariance among coordinate systems

Researchers continue to emphasize the importance of covariational reasoning in the context of students’ function concept, particularly when graphing in the Cartesian coordinate system (CCS). In this article, we extend the body of literature on function by characterizing two pre-service teachers’ thinking during a teaching experiment focused on graphing in the polar coordinate system (PCS). We illustrate how the participants engaged in covariational reasoning to make sense of graphing in the PCS and make connections with graphing in the CCS. By foregrounding covariational relationships, the students came to understand graphs in different coordinate systems as representative of the same relationship despite differences in the perceptual shapes of these graphs. In synthesizing the students’ activity, we provide remarks on instructional approaches to graphing and how the PCS forms a potential context for promoting covariational reasoning.     

Hyperbolic systems of conservation laws with some special invariance properties

The Euler equations of fluid dynamics are an example of a very special class of nonlinear, hyperbolic systems of conservation laws, in particular those satisfying conditions of reflection and Galilean invariance. These invariance properties are directly responsible for several of the attractive structural features of this system.     

Energy balance invariance for interacting particle systems

This paper studies the principle of invariance of balance of energy and its consequences for a system of interacting particles under groups of transformations. Balance of energy and its invariance is first examined in Euclidean space. Unlike the case of continuous media, it is shown that conservation and balance laws do not follow from the assumption of invariance of balance of energy under time-dependent isometries of the ambient space. However, the postulate of invariance of balance of energy under arbitrary diffeomorphisms of the ambient (Euclidean) space, does yield the correct conservation and balance laws. These ideas are then extended to the case when the ambient space is a Riemannian manifold. Pairwise interactions in the case of geodesically complete Riemannian ambient manifolds are defined by assuming that the interaction potential explicitly depends on the pairwise distances of particles. Postulating balance of energy and its invariance under arbitrary time-dependent spatial diffeomorphisms yields balance of linear momentum. It is seen that pairwise forces are directed along tangents to geodesics at their end points. One also obtains a discrete version of the Doyle–Ericksen formula, which relates the magnitude of internal forces to the rate of change of the interatomic energy with respect to a discrete metric that is related to the background metric. Dedicated to the memory of Shahram Kavianpour (1975-2007). Jerrold E. Marsden: Research partially supported by the California Institute of Technology and NSF-ITR Grant ACI-0204932. Arash Yavari: Research supported by the Georgia Institute of Technology.     

Energy balance invariance for interacting particle systems

This paper studies the principle of invariance of balance of energy and its consequences for a system of interacting particles under groups of transformations. Balance of energy and its invariance is first examined in Euclidean space. Unlike the case of continuous media, it is shown that conservation and balance laws do not follow from the assumption of invariance of balance of energy under time-dependent isometries of the ambient space. However, the postulate of invariance of balance of energy under arbitrary diffeomorphisms of the ambient (Euclidean) space, does yield the correct conservation and balance laws. These ideas are then extended to the case when the ambient space is a Riemannian manifold. Pairwise interactions in the case of geodesically complete Riemannian ambient manifolds are defined by assuming that the interaction potential explicitly depends on the pairwise distances of particles. Postulating balance of energy and its invariance under arbitrary time-dependent spatial diffeomorphisms yields balance of linear momentum. It is seen that pairwise forces are directed along tangents to geodesics at their end points. One also obtains a discrete version of the Doyle–Ericksen formula, which relates the magnitude of internal forces to the rate of change of the interatomic energy with respect to a discrete metric that is related to the background metric.     

An invariance principle for discrete-time nonlinear systems

In this paper, we derive an invariance principle generalizing LaSalle's invariance principle for discrete-time nonlinear systems. Next, using the invariance principle, we develop a series of results relating stability, observability, and converse Lyapunov theorems for discrete-time nonlinear systems.     

Non-autonomous systems: asymptotic behaviour and weak invariance principles

Results pertaining to asymptotic behaviour of solutions of non-autonomous ordinary differential equations with locally integrably bounded right-hand sides are presented. Ramifications for weakly asymptotically autonomous systems and adaptively controlled systems are highlighted.     

Monotonicity and positive invariance of linear systems over dioids

In this paper, we make connections between two apparently different concepts. The first concept is the (linear) monotonicity of a given matrix which is usually used in order to compare Markov chains. This concept is involved in the simplification of complex stochastic systems in order to control the approximation error made. The second concept is the positive invariance of sets by a (linear) map. The properties of positively invariant sets are involved in many different problems in classical control theory, such as constrained control, robustness analysis, optimisation, and also in aggregation of Markov chains (namely strong lumpability and coherency).

In the context of linear dynamical systems over semirings which play an important role in the study of discrete event systems, we establish links between monotone (or isotone) linear maps and linear maps which admit some special families of positively invariant sets.     

Equi-invariability,bounded invariance complexity and L-stability for control systems

In this paper, we introduce the notions of bounded invariance complexity, bounded invariance complexity in the mean and mean Lyapunov-stability for control systems. Then we characterize these notions by introducing six types of equi-invariability. As a by-product, two new dichotomy theorems for the control system on the control sets are established.     

Supersymmetric invariance and universal central extensions of lie supertriple systems

In this article, we discuss some properties of a supersymmetric invariant bilinear form on Lie supertriple systems. In particular, a supersymmetric invariant bilinear form on Lie supertriple systems can be extended to its standard imbedding Lie superalgebras. Furthermore, we generalize Garland's theory of universal central extensions for Lie supertriple systems following the classical one for Lie superalgebras. We solve the problems of lifting automorphisms and lifting derivations.     

Limit differential inclusions and the invariance principle for nonautonomous systems

Considering nonautonomous differential inclusions we introduce the concept of limit differential inclusions, study their properties and invariance-type properties of the ω-limit sets of solutions, and establish an analog of La Salle’s invariance principle using Lyapunov functions with the derivatives of constant sign. The method is equally applicable to differential equations and, under appropriate assumptions, yields some previously-available results.     

Translation invariance of two-dimensional Gibbsian systems of particles with internal degrees of freedom

One of the main objectives of equilibrium state statistical physics is to analyze which symmetries of an interacting particle system in equilibrium are broken or conserved. Here we present a general result on the conservation of translational symmetry for two-dimensional Gibbsian particle systems. The result applies to particles with internal degrees of freedom and fairly arbitrary interaction, including the interesting cases of discontinuous, singular, and hard core interaction. In particular we thus show the conservation of translational symmetry for the continuum Widom–Rowlinson model and a class of continuum Potts type models.     

Gaussian dominance and phase transitions in systems with continuous symmetry

V. A. Steklov Mathematics Institute, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 79, No. 3, pp. 460–472, June, 1989.     

Tracking and invariance problems for some classes of discrete delay systems

Consider the system with perturbation g _k ∈ ℝ ⁿ and output z _k = Cx _k . Here, A _k ,A _k ^(s) ∈ ℝ ^{n × n} , B _k ⁽¹⁾ ∈ ℝ ^{n × p} , B _k ⁽²⁾ ∈ ℝ ^{n × m} , C ∈ ℝ ^{p × n} . We construct a special Lyapunov-Krasovskii functional in order to synthesize controls u _k ⁽¹⁾ and u _k ⁽²⁾ for which the following properties are satisfied:

Rigidity and gamma convergence for solid‐solid phase transitions with SO(2) invariance

The singularly perturbed two‐well problem in the theory of solid‐solid phase transitions takes the form where u : Ω ? ?ⁿ → ?ⁿ is the deformation, and W vanishes for all matrices in K = SO(n)A ∪ SO(n)B. We focus on the case n = 2 and derive, by means of Gamma convergence, a sharp‐interface limit for I_ε. The proof is based on a rigidity estimate for low‐energy functions. Our rigidity argument also gives an optimal two‐well Liouville estimate: if ?u has a small BV norm (compared to the diameter of the domain), then, in the L¹ sense, either the distance of ?u from SO(2)A or the one from SO(2)B is controlled by the distance of ?u from K. This implies that the oscillation of ?u in weak L¹ is controlled by the L¹ norm of the distance of ?u to K. © 2006 Wiley Periodicals, Inc.