Pointwise Stabilizer Research Articles

Stability of A Class of Relay Feedback Systems Arising in Controlled Mechanical Systems with Ideal Coulomb Friction

Coulomb friction is inevitable in every mechanical system with contact motion. When the mechanical system with Coulomb friction is under feedback control, it can destabilize the system by generating limit cycles. Controlled mechanical systems with ideal Coulomb friction can be viewed as a particular class of relay feedback systems characterized by the zero DC gain property and the positivity of the first Markov parameter. This paper elaborates recent results on sufficient conditions to guarantee the global pointwise stability of such systems. The scope of analysis has been kept broad so that the results apply to systems with multiple inertia elements and multiple Coulomb friction sources. To employ the recent advances in the absolute stability theory, the limiting arguments are adopted to approximate the relay elements to continuous functions. As a result, a new sufficient condition on the global pointwise stability of the systems with multiple Coulomb friction sources is derived by extending the existing result with a single Coulomb friction source when the stiction level is larger than the Coulomb friction level. Also, it has been shown that the describing function criterion is indeed an exact condition when the order of the closed-loop system is 3. Simulation results are presented with a flexible joint mechanism to illustrate the main points.

On the Structure of Framed Vertex Operator Algebras and Their Pointwise Frame Stabilizers

In this paper, we study the structure of a general framed vertex operator algebra. We show that the structure codes (C,D) of a framed VOA V satisfy certain duality conditions. As a consequence, we prove that every framed VOA is a simple current extension of the associated binary code VOA V_C. This result would give a prospect on the classification of framed vertex operator algebras. In addition, the pointwise frame stabilizer of V is studied. We completely determine all automorphisms in this pointwise stabilizer, which are of order 1, 2 or 4. The 4A-twisted sector and the 4A-twisted orbifold theory of the famous Moonshine VOA are also constructed explicitly. We verify that the top module of this twisted sector is of dimension 1 and of weight 3/4 and the VOA obtained by 4A-twisted orbifold construction of the moonshine VOA is isomorphic to the moonshine VOA itself.

On base sizes for actions of finite classical groups

AbstractLet G be a finite almost simple classical group and let Ω be a faithful primitive non-standard G-set. A subset of Ω is a base for G if its pointwise stabilizer in G is trivial. Let b(G) be the minimal size of a base for G. A well-known conjecture of Cameron and Kantor asserts that there exists an absolute constant c such that b(G) ≤ slant c for all such groups G, and the existence of such an undetermined constant has been established by Liebeck and Shalev. In this paper we prove that either b(G) ≤ 4, or G = U 6 (2) · 2, G ω = U 4 (3) · 2 2 and b(G) = 5. The proof is probabilistic, using bounds on fixed point ratios.

Pointwise Stabilization of a Hybrid System and Optimal Location of Actuator

We consider a pointwise stabilization problem for a model arising in the control of noise. We prove that we have exponential stability for the low frequencies but not for the high frequencies. Thus, we give an explicit polynomial decay estimation at high frequencies that is valid for regular initial data while clarifying that the behavior of the constant which intervenes in this estimation there, functions as the frequency of cut. We propose a numerical approximation of the model and study numerically the best location of the actuator at low frequencies.

Closed subgroups of the infinite symmetric group

Let S = Sym(Ω) be the group of all permutations of a countably infinite set Ω, and for subgroups G1, G2 ≤ S let us write G1 ≈ G2 if there exists a finite set $$ U \subseteq S $$ such that $$ \left\langle {G_1 \cup U } \right\rangle = \left\langle {G_2 \cup U} \right\rangle $$ . It is shown that the subgroups closed in the function topology on S lie in precisely four equivalence classes under this relation. Which of these classes a closed subgroup G belongs to depends on which of the following statements about pointwise stabilizer subgroups $$ G_{(\Gamma )} $$ of finite subsets $$ \Gamma \subseteq \Omega $$ holds:

PARTITION ACTIONS OF SYMMETRIC GROUPS AND REGULAR BIPARTITE GRAPHS

A base of an action of a group G on a set $\Omega$ is a subset $B \subseteq \Omega$ such that the pointwise stabiliser of $B$ in $G$ is the identity. We prove that if $\Omega$ is the set of partitions of $[ 1, kl ]$ into $l$ subsets of size $k$ , then the action of $S_{kl}$ on $\Omega$ has a base of size two if and only if $k \geq 3$ and $l \geq \max \{ k + 3, 8 \}$ . This result completes a classification of the primitive base 2 actions of the symmetric groups. During the proof we show that there exists a $k$ -regular bipartite graph $\mathcal{G}$ on $2l$ vertices with no non-trivial automorphisms fixing the bipartite blocks if and only if $k \geq 3$ and $l \geq \max \{ k + 3, 8 \}$ .

Jordan groups and limits of betweenness relations

Jordan groups are transitive permutation groups for which, in a non-trivial way, the pointwise stabilizer of a subset is transitive on its complement. In the first instance I intend to further clarify the programme of classification of infinite primitive Jordan groups as worked on, in various publications, by Sampson Adeleke, Dugald Macpherson, Peter Neumann, Meenaxi Bhattacharjee and Keith Johnson. In particular, I construct limits of betweenness relations using generalised Frass amalgamation techniques.

A measuring argument for finite permutation groups

LetG be a finite transitive permutation group on a finite setS. LetA be a nonempty subset ofS and denote the pointwise stabilizer ofA inG byC G (A). Our main result is the following inequality: [G :C G (A)]≥|G||A|/|S|.

Stability of undercompressive shock profiles

Using a simplified pointwise iteration scheme, we establish nonlinear phase-asymptotic orbital stability of large-amplitude Lax, undercompressive, overcompressive, and mixed under–overcompressive type shock profiles of strictly parabolic systems of conservation laws with respect to initial perturbations | u 0 ( x ) | ⩽ E 0 ( 1 + | x | ) − 3 / 2 in C 0 + α , E 0 sufficiently small, under the necessary conditions of spectral and hyperbolic stability together with transversality of the connecting profile. This completes the program initiated by Zumbrun and Howard in [K. Zumbrun, P. Howard, Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J. 47 (4) (1998) 741–871], extending to the general undercompressive case results obtained for Lax and overcompressive shock profiles in [A. Szepessy, Z. Xin, Nonlinear stability of viscous shock waves, Arch. Ration. Mech. Anal. 122 (1993) 53–103; T.-P. Liu, Pointwise convergence to shock waves for viscous conservation laws, Comm. Pure Appl. Math. 50 (11) (1997) 1113–1182; K. Zumbrun, P. Howard, Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J. 47 (4) (1998) 741–871; K. Zumbrun, Refined wave-tracking and nonlinear stability of viscous Lax shocks, Methods Appl. Anal. 7 (2000) 747–768; M.-R. Raoofi, L 1 -asymptotic behavior of perturbed viscous shock profiles, thesis, Indiana Univ., 2004; C. Mascia, K. Zumbrun, Pointwise Green's function bounds and stability of relaxation shocks, Indiana Univ. Math. J. 51 (4) (2002) 773–904; C. Mascia, K. Zumbrun, Stability of small-amplitude shock profiles of symmetric hyperbolic–parabolic systems, Comm. Pure Appl. Math. 57 (7) (2004) 841–876; C. Mascia, K. Zumbrun, Pointwise Green's function bounds for shock profiles with degenerate viscosity, Arch. Ration. Mech. Anal. 169 (3) (2003) 177–263; C. Mascia, K. Zumbrun, Stability of large-amplitude shock profiles of hyperbolic–parabolic systems, Arch. Ration. Mech. Anal. 172 (1) (2004) 93–131; C. Mascia, K. Zumbrun, Stability of large-amplitude shock profiles of general relaxation systems, SIAM J. Math. Anal., in press], and for special “weakly coupled” (respectively scalar diffusive–dispersive) undercompressive profiles in [T.P. Liu, K. Zumbrun, Nonlinear stability of an undercompressive shock for complex Burgers equation, Comm. Math. Phys. 168 (1) (1995) 163–186; T.P. Liu, K. Zumbrun, On nonlinear stability of general undercompressive viscous shock waves, Comm. Math. Phys. 174 (2) (1995) 319–345] (respectively [P. Howard, K. Zumbrun, Pointwise estimates for dispersive–diffusive shock waves, Arch. Ration. Mech. Anal. 155 (2000) 85–169]). In particular, together with spectral results of [K. Zumbrun, Dynamical stability of phase transitions in the p-system with viscosity–capillarity, SIAM J. Appl. Math. 60 (2000) 1913–1924], our results yield nonlinear stability of large-amplitude undercompressive phase-transitional profiles near equilibrium of Slemrod's model [M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Arch. Ration. Mech. Anal. 81 (4) (1983) 301–315] for van der Waal gas dynamics or elasticity with viscosity–capillarity.

Pointwise stabilization of strings with non‐linear feedback

AbstractWe consider an initial and boundary value problem for a homogenous string subject to an internal pointwise control. The solution resulting from a non‐linear feedback is studied. We give various explicit decay estimates depending on the control position and the feedback non‐linearity. Copyright © 2004 John Wiley & Sons, Ltd.

Nonlinear stability of a convective motion in a porous layer driven by a horizontally periodic temperature gradient

The nonlinear global exponential pointwise stability of a vertical steady flow driven by a horizontal periodic temperature gradient in a porous layer is performed. It is shown that the stability threshold depends on the supremum of a quadratic functional, having non constant coefficients, and new in the literature on the convection problem. In solving the variational problem, a suitable functional transformation is used.

An algebraic theory of portfolio allocation

Using group and majorization theory, we explore what can be established about allocation of funds among assets when asymmetries in the returns vector are carefully controlled. The key insight is that preferences over allocations can be partially ordered via majorized convex hulls that have been generated by a permutation group. Group transitivity suffices to ensure complete portfolio diversification. Point-wise stabilizer subgroups admit sectoral separability in fund allocations. We also bound the admissible allocation vector by a set of linear constraints the coefficients of which are determined by group operations on location and scale asymmetries in the rate of returns vector. For a distribution that is symmetric under a reflection group, the linear constraints may be further strengthened whenever there exists an hyperplane that separates convex sets.

ℵ0-categorical structures with arbitrarily fast growth of algebraic closure

Various results have been proved about growth rates of certain sequences of integers associated with infinite permutation groups. Most of these concern the number of orbits of the automorphism group of an ℵ0-categorical structure on the set of unordered n-subsets or on the set of n-tuples of elements of . (Recall that by the Ryll-Nardzewski Theorem, if is countable and ℵ0-categorical, the number of the orbits of its automorphism group Aut() on the set of n-tuples from is finite and equals the number of complete n-types consistent with the theory of .) The book [Ca90] is a convenient reference for these results. One of the oldest (in the realms of ‘folklore’) is that for any sequence (Kn)n∈ℕ of natural numbers there is a countable ℵ0-categorical structure such that the number of orbits of Aut() on the set of n-tuples from is greater than kn for all n.These investigations suggested the study of the growth rate of another sequence. Let be an ℵ0-categorical structure and X be a finite subset of . Let acl(X) be the algebraic closure of X, that is, the union of the finite X-definable subsets of . Equivalently, this is the union of the finite orbits on of Aut()(X), the pointwise stabiliser of X in Aut(). Define

Bases of primitive linear groups

Let V be a finite vector space and G⩽GL(V) a linear group. A base of G is a set of vectors whose pointwise stabiliser in G is trivial. We prove that if G is irreducible and primitive on V, then G has a base of size at most 18log|G|/log|V|+c, where c is an absolute constant. This verifies part of a conjecture of Pyber on base sizes of primitive permutation groups.

Loci in quotients by finite groups, pointwise stabilizers and the Buchsbaum property

Article Loci in quotients by finite groups, pointwise stabilizers and the Buchsbaum property was published on April 29, 2002 in the journal Journal für die reine und angewandte Mathematik (volume 2002, issue 547).

Pointwise estimates and stability for degenerate viscous shock waves

Pointwise estimates and stability for degenerate viscous shock waves

Errata to: 'Pointwise semigroup methods and stability of viscous shock waves'

Errata to: 'Pointwise semigroup methods and stability of viscous shock waves'

Automorphism groups of differentially closed fields

We examine the connections between several automorphism groups associated with a saturated differentially closed field U of characteristic zero. These groups are: Γ, the automorphism group of U; the automorphism group of Γ; Aut( G(U)) , the automorphism group of the differential combinatorial geometry of U and Aut G (U −) , the group of field automorphisms of U that respect differential closure. Our main results are: • If U is of cardinality λ +=2 λ for some infinite regular cardinal λ, then the set of subgroups of Γ consisting of all pointwise stabilizers of Dcl's of finite subsets of U is invariant under Aut( Γ) (Theorem 2.10). • If U is of arbitrary infinite cardinality, then each automorphism of the differential combinatorial geometry is induced by a field automorphism of U that respects differential closure (Theorem 3.1). • If U is of cardinality λ +=2 λ for some infinite regular cardinal λ, then each automorphism of Γ is induced by an element of Aut G (U −) acting on Γ by conjugation (Theorem 4.10). • If U is of cardinality λ +=2 λ for some infinite regular cardinal λ, then the outer automorphism group of Γ is isomorphic to the multiplicative group of the rationals (Theorem 4.18).

Generalizations of the sampling theorem: Seven decades after Nyquist

The sampling theorem is one of the most basic and fascinating topics in engineering sciences. The most well-known form is Shannon's uniform-sampling theorem for bandlimited signals. Extensions of this to bandpass signals and multiband signals, and to nonuniform sampling are also well-known. The connection between such extensions and the theory of filter banks in DSP has been well established. This paper presents some of the less known aspects of sampling, with special emphasis on non bandlimited signals, pointwise stability of reconstruction, and reconstruction from nonuniform samples. Applications in multiresolution computation and in digital spline interpolation are also reviewed.

Optimal location of the actuator for the pointwise stabilization of a string

Abstract We study the large time behavior of the solutions of a homogeneous string equation with a homogeneous Dirichlet boundary condition at the left end and a homogeneous Neumann boundary condition at the right end. A pointwise interior actuator gives a linear viscous damping term. We give a complete characterization of the positions of the actuator for which the system becomes exponentially stable in the energy space. Moreover, we show that the fastest decay rate is obtained if the actuator is located at the middle of the string.