I-limit points and I-cluster points of multiset sequences

A sufficient condition for the kinematic stability of a robot hybrid position/force control scheme is defined. The test for the sufficient condition is a technique that uses only the kinematic information to determine system stability or to identify situations where possible stability problems may occur. The sufficient condition test is applied to an example, using the original hybrid position/force control scheme of J.J. Craig and M.H. Raibert (1979) and the corrected hybrid position/force control scheme proposed by W.D. Fisher and M.S. Mujtaba (1992). A root locus analysis showed the original hybrid control approach to be unstable. It is shown that the sufficient condition is also violated for this situation. In the corrected hybrid control approach, the sufficient condition is never violated for any manipulator and the same example is shown to be always stable. A root locus plot for the hybrid control approaches is included to confirm the results of the sufficient stability condition. >

In this paper, we characterize local circularity for the Laspeyres, Paasche, and Fisher price indexes. In the first two cases, we begin by deriving a sufficient condition for achieving circularity that establishes that at least one of two proposed equalities must hold. We end up showing that the sufficient condition is also necessary. We continue with the Fisher price index that is the geometric mean of the two, and we find a sufficient circularity condition that is a direct consequence of the corresponding sufficient conditions for its two component indexes. However, we also show that, unlike its Laspeyres and Paasche components, this sufficient circularity condition for the Fisher price index is not necessary. We reach different conclusions when we extend our investigation to the circularity properties of the geometric Laspeyres, geometric Paasche, and Tornqvist price indexes, for which none of the proposed sufficient conditions is necessary. Throughout, we distinguish local circularity, which all six price indexes satisfy, from global circularity, which none of the price indexes satisfies.

This thesis is concerned with the dissipative control and filtering problems of singular systems. Four classes of singular systems are considered: delay-free singular systems, singular systems with constant time-delay, uncertain singular systems with time-varying delay and sensor failures, and singular Markovian jump systems with actuator failures.
\nFor delay-free singular systems, the system augmentation approach is employed to study the dissipative control and filtering problems. First, the approach is used to solve the dissipative control problem by static output-feedback for standard state-space systems which are the special cases of singular systems. For a continuous-time standard state-space system, the closed-loop system is represented in an augmented system form. Based on the augmented system, a necessary and sufficient dissipativity condition is proposed, which decouples the Lyapunov matrix and controller matrix. To further separate the Lyapunov matrix and the system matrices, an equivalent condition is obtained by introducing some slack matrices. Then, a necessary and sufficient condition for the existence of a static output-feedback controller is proposed, and an iterative algorithm is given to solve the condition. For discrete-time singular systems, by giving an equivalent representation of the solution set, a necessary and sufficient dissipativity condition is proposed in terms of strict linear matrix inequality (LMI) which can be easily solved by standard commercial software. Then a state-feedback controller design method is given based on the augmentation system approach. The method is extended to the static output-feedback control problem and the reduced-order dissipative filtering problem.
\nFor continuous-time singular time-delay systems, the problem of state-feedback dissipative control is considered. An improved delay-dependent dissipativity condition in terms of LMIs is established by employing the delay-partitioning technique, which guarantees a singular system to be admissible and dissipative. Based on this, a delay-dependent sufficient condition for the existence of a state-feedback controller is proposed to guarantee the admissibility and dissipativity of the closed-loop system. In addition to delay-dependence, the obtained results are also dependent on the level of dissipativity. Moreover, the results obtained unify existing results on H∞ performance analysis and passivity analysis for singular systems.
\nFor discrete-time singular systems with polytopic uncertainties, time-varying delay and sensor failures, the problem of robust reliable dissipative filtering is considered. The filter is designed by the reciprocally convex approach such that the filtering error singular system is admissible and strictly (Q, S, R)-dissipative. For singular systems with time-varying delay and sensor failures, a sufficient condition of reliable dissipative analysis is obtained in terms of LMIs. Then the result is extended to the uncertain case by introducing some variables to decouple the Lyapunov matrices and the filtering error system matrices. Moreover, a desired filter for uncertain singular systems with time-varying delay and sensor failures is obtained by solving a set of LMIs.
\nFor continuous-time singular Markovian jump systems with actuator failures, the problem of reliable dissipative control is addressed. Attention is focused on the state-feedback controller design method such that the closed-loop system is admissible and strictly (Q, S, R)-dissipative. A sufficient condition is obtained in terms of strict LMIs. Moreover, the results obtained unify existing results on H∞control and passive control on singular Markovian jump systems.

Necessary and sufficient stability conditions for spatially homogeneous stochastic regimes are considered in the chains of unidirectionally coupled one-dimensional maps. According to the form of a linearised system, two types of coupling between the elements of the chain are distinguished: dissipative and quasiinertial coupling. For the dissipative coupling, necessary and sufficient stability conditions are found for two different types of stability: homogeneous (when initial excitations are damped in all elements of the chain simultaneously) and weak stability (when the time of the perturbation damping depends significantly on the departure of the elements from the end of the chain). For the quasi-inertial coupling, necessary and sufficient conditions are found for time-periodic spatially homogeneous solutions. Sufficient stability conditions for stochastic regimes in the general case were not obtained. Numerical experiment revealed stable stochastic regimes in the parameter regions predicted by the theory. In chains with quasi-inertial coupling stable spatially homogeneous regimes were not observed.

Necessary and sufficient conditions for a second-order Wiener-Itô integral process to be mixing

Distinguishing necessary and sufficient conditions can be challenging to undergraduate logic and critical thinking students. Explaining J. L. Mackie’s notion of INUS conditions—insufficient but necessary parts of unnecessary but sufficient conditions—is an even more difficult concept to understand. It is helpful to have memorable examples that not only clarify the concept, but make it easy to remember. Law student turned stand-up comedian Demetri Martin uses necessary, sufficient, and INUS conditions to construct absurdist jokes. These jokes provide effective tools for making Mackie’s notion understandable and memorable.

In this paper we propose the following approach to the dimensioning of the radio part of the downlink in OFDMA networks. First, we use information theory to characterize the bit-rate in the channel from a base station to its mobile. It depends on the power and bandwidth allocated to this mobile. Then, we describe the resource (power and bandwidth) allocation problem and characterise feasible configurations of bit-rates of all users. As the key element, we propose some particular sufficient condition (in a multi-Erlang form) for a given configuration of bit-rates to be feasible. Finally, we consider an Erlang's loss model, in which streaming arrivals whose admission would lead to the violation of this sufficient condition are blocked and lost. In this model, the blocking probabilities can be calculated using Kaufman-Roberts algorithm. We propose it to evaluate the minimal density of base stations assuring acceptable blocking probabilities for a streaming traffic of a given load per surface unit. We validate this approach by comparison of the blocking probabilities to these simulated in the similar model in which the admission control is based on the original feasibility property (instead of its sufficient condition). Our sufficient bit-rate feasibility condition can also be used to dimension the network with respect to the elastic traffic.

From the energy principle a sufficient stability condition can be derived that involves a disposable function. The existence of some choice of the disposable function such as to satisfy the sufficient condition is a necessary condition for stability. The disposable function may also be chosen so as to yield sufficient conditions that are not necessary, but that may be useful from the points of view of simplicity and generality. This technique is illustrated for ``maximal-force configurations,'' where the current density is everywhere orthogonal to the magnetic field.

Wang et al. have proposed collision attacks for various hash functions. Their approach is to first construct a differential path, and then determine the conditions (sufficient conditions) that maintain the differential path. If a message that satisfies all sufficient conditions is found, a collision can be generated. Therefore, in order to apply the attack of Wang et al., we need techniques for constructing differential paths and for determining sufficient conditions. In this paper, we propose the “SC algorithm”, an algorithm that can automatically determine the sufficient conditions. The input of the SC algorithm is a differential path, that is, all message differentials and differentials of the chaining variables. The SC algorithm then outputs the sufficient conditions. The computation time of the SC algorithm is within few seconds. In applying the method of Wang et al. to MD5, there are 3 types of sufficient conditions: conditions for controlling the carry length when differentials appear in the chaining variables, conditions for controlling the output differentials of the Boolean function when the input variables of the function have differentials and conditions for controlling the relationship between the carry effect and left rotation operation. Sufficient conditions for SHA-1, SHA-0 and MD4 consist of only Type 1 and Type 2. Type 3 is unique to MD5. The SC algorithm can construct Type 1 and Type 2 conditions; we use the method of Liang et al. to construct Type 3 conditions. The complexity of the collision attack depends on the number of sufficient conditions needed. The SC algorithm constructs the fewest possible sufficient conditions. To check the feasibility of the SC algorithm, we apply it to the differential path of MD5 given by Wang et al. It is shown to yield 12 fewer conditions than the latest work on MD5. The SC algorithm is applicable to the MD-family and the SHA-family. This paper focuses on the sufficient conditions of MD5, but only as an example.

The use of effective distance functions has been explored for many data mining problems including clustering, nearest neighbor search, and indexing. Recent research results show that if the Pearson variation of the distance distribution converges to zero with increasing dimensionality, the distance function will become unstable (or meaningless) in high dimensional space even with the commonly used Lp metric on the Euclidean space. This result has spawned many subsequent studies. We first comment that although the prior work provided the sufficient condition for the unstability of a distance function, the corresponding proof has some defects. Also, the necessary condition for unstability (i.e., the negation of the sufficient condition for the stability) of a distance function, which is required for function design, remains unknown. Consequently, we first provide in this paper a general proof for the sufficient condition of unstability. More importantly, we go further to prove that the rapid degradation of Pearson variation for a distance distribution is in fact a necessary condition of the resulting unstability. With the result, we will then have the necessary and the sufficient conditions for unstability, which in turn imply the sufficient and necessary conditions for stability. This theoretical result derived leads to a powerful means to design a meaningful distance function. Explicitly, in light of our results, we design in this paper a meaningful distance function, called Shrinkage-Divergence Proximity (abbreviated as SDP), based on a given distance function. It is empirically shown that the SDP significantly outperforms prior measures for its being stable in high dimensional data space and robust to noise, and is thus deemed more suitable for distance-based clustering applications than the priorly

Studies stability properties of linear time-invariant delay systems. The specific notion under consideration is asymptotic stability independent of delay. As an attempt to achieve a compromise between the complexity and tightness of various stability tests, the authors present a number of sufficient stability conditions which improve several previously available sufficient conditions and which are also much easier to verify than the known necessary and sufficient conditions.

The dynamic behavior of a physical system can concisely be described by the least action principle. In the centrum of its mathematical presentations is a specific function of the coordinates and velocities, i.e., the Lagrangian. If the integral of a Lagrangian is stationary, then the system is moving along an extremal path through the phase space. All Lie symmetries of a Lagrangian correspond to a conserved quantity, and the conservation principle is explained by variation symmetry. Briefly, that is the meaning of Nother’s theorem. After showing that Goodwin’s cyclical growth model has a Lagrangian, we introduce the generalized Nother’s theorem and apply it to Goodwin’s 2D model in order to get its Hamiltonian. We prove that the cyclical motion in his model derives from its dynamic Lie symmetries. These cyclical trajectories are extremal trajectories in the phase space, and along these trajectories the first integral of the model’s Lagrangian is stationary, which by the principle of least action also means that they satisfy the first-order necessary conditions. The optimality still needs to satisfy the sufficient condition. Since the Legendre’s second-order sufficient condition is not applicable here, we show it satisfies another systems of sufficient conditions, the local convex surface of Lagrangian with the minimum non-trivial fixed-point and contour lines. Our conclusion is that all systems’ solutions described by the other first-order nonlinear ordinary differential equations are optimal if they have a Lagrangian that satisfies the sufficient and necessary conditions.

The limit circle and limit point criteria for second-order linear difference equations

SUMMARY The LQ+ problem, i.e. the finite-horizon linear quadratic optimal control problem with nonnegative state constraints, is studied for positive linear systems in continuous time and in discrete time. Necessary and sufficient optimality conditions are obtained by using the maximum principle. These conditions lead to a computational method for the solution of the LQ+ problem by means of a corresponding Hamiltonian system. In addition, the necessary and sufficient conditions are proved for the LQ+-optimal control to be given by the standard LQ-optimal state feedback law. Then sufficient conditions are established for the positivity of the LQ-optimal closed-loop system. In particular, such conditions are obtained for the problem of minimal energy control with penalization of the final state. Moreover, a positivity criterion for the LQ-optimal closed-loop system is derived for positive discrete-time systems with a positively invertible (dynamics) generator. The main results are illustrated by numerical examples. Copyright q 2010 John Wiley & Sons, Ltd.

A set $S\subseteq\mathbb{R}^n$ is called semidefinite programming (SDP) representable or semidefinite representable if S equals the projection of a set in higher dimensional space which is describable by some linear matrix inequality (LMI). Clearly, if S is SDP representable, then S must be convex and semialgebraic (it is describable by conjunctions and disjunctions of polynomial equalities or inequalities). This paper proves sufficient conditions and necessary conditions for SDP representability of convex sets and convex hulls by proposing a new approach to constructing SDP representations. The contributions of this paper are: (i) For bounded SDP representable sets $W_1,\dots,W_m$, we give an explicit construction of an SDP representation for $\mathrm{conv}(\cup_{k=1}^mW_k)$. This provides a technique for building global SDP representations from the local ones. (ii) For the SDP representability of a compact convex semialgebraic set S, we prove sufficient: the boundary $\partial S$ is nonsingular and positively curved, while necessary is: $\partial S$ has nonnegative curvature at each nonsingular point. In terms of defining polynomials for S, nonsingular boundary amounts to them having nonvanishing gradient at each point on $\partial S$ and the curvature condition can be expressed as their strict versus nonstrict quasi-concavity at those points on $\partial S$ where they vanish. The gaps between them are $\partial S$ having or not having singular points either of the gradient or of the curvature's positivity. A sufficient condition bypassing the gaps is when some defining polynomials of S satisfy an algebraic condition called sos-concavity. (iii) For the SDP representability of the convex hull of a compact nonconvex semialgebraic set T, we find that the critical object is $\partial_c T$, the maximum subset of $\partial T$ contained in $\partial\mathrm{conv}(T)$. We prove sufficient for SDP representability: $\partial_c T$ is nonsingular and positively curved, and necessary is: $\partial_c T$ has nonnegative curvature at nonsingular points. The gaps between our sufficient and necessary conditions are similar to case (ii). The positive definite Lagrange Hessian (PDLH) condition, which meshes well with constructions, is also discussed.