scholarly journals On the convexity of the reachable set with respect to a part of coordinates at small time intervals

2021 ◽  
Vol 31 (2) ◽  
pp. 210-225
Author(s):  
I.O. Osipov
Keyword(s):  
Numerical Modeling ◽  
Sufficient Conditions ◽  
Small Time ◽  
The Other ◽  
Reachable Sets ◽  
Third Order ◽  
Time Intervals ◽  
Linearized System ◽  
Asymptotics Of The Eigenvalues ◽  
Convexity Conditions

We investigate the convexity of the reachable sets for some of the coordinates of nonlinear systems with integral constraints on the control at small time intervals. We have proved sufficient convexity conditions in the form of constraints on the asymptotics of the eigenvalues of the Gramian of the controllability of a linearized system for some of the coordinates. There are two nonlinear third-order systems under study as examples. The system linearized along a trajectory generated by zero control is uncontrollable, and the system in the other example is completely controllable. We investigate the sufficient conditions for convexity of projection of reachable sets. Numerical modeling has been carried out, demonstrating the non-convexity of some projections even for small time intervals.

scholarly journals THE LIMITS OF APPLICABILITY OF THE LINEARIZATION METHOD IN CALCULATING SMALL–TIME REACHABLE SETS

2020 ◽  
Vol 6 (1) ◽  
pp. 71
Author(s):  
Mikhail I. Gusev
Keyword(s):  
Asymptotic Behavior ◽  
Nonlinear Control ◽  
Sufficient Conditions ◽  
Linearization Method ◽  
Small Time ◽  
Reachable Set ◽  
Reachable Sets ◽  
Time Interval ◽  
Initial State ◽  
Linearized System

The reachable sets of nonlinear systems are usually quite complicated. They, as a rule, are non-convex and arranged to have rather complex behavior. In this paper, the asymptotic behavior of reachable sets of nonlinear control-affine systems on small time intervals is studied. We assume that the initial state of the system is fixed, and the control is bounded in the \(\mathbb{L}_2\)-norm. The subject of the study is the applicability of the linearization method for a sufficiently small length of the time interval. We provide sufficient conditions under which the reachable set of a nonlinear system is convex and asymptotically equal to the reachable set of a linearized system. The concept of asymptotic equality is defined in terms of the Banach-Mazur metric in the space of sets.  The conditions depend on the behavior of the controllability Gramian of the linearized system – the smallest eigenvalue of the Gramian should not tend to zero too quickly when the length of the time interval tends to zero.  The indicated asymptotic behavior occurs for a reasonably wide class of second-order nonlinear control systems but can be violated for systems of higher dimension.  The results of numerical simulation illustrate the theoretical conclusions of the paper.

scholarly journals Ascaris lumbricoides: reinfection in children bearing an established worm burden

1991 ◽  
Vol 24 (4) ◽  
pp. 217-221 ◽  
Author(s):  
Fausto E. Lima Pereira ◽  
Andrea P. Sampaio ◽  
Carlos Musso ◽  
Jane S. Castelo
Keyword(s):  
Standard Deviation ◽  
Ascaris Lumbricoides ◽  
Worm Burden ◽  
Bimodal Distribution ◽  
Small Time ◽  
The Other ◽  
Time Intervals ◽  
Distribution Of Values ◽  
Established Infection ◽  
Mature Population

To clarify the existance of reinfection in children bearing an established Ascaris lumbricoides infection, the authors evaluated the weight and the length of worms collected from ten cases of ascaridiasis. The worm burden was greater than 27 worms in nine cases. In seven cases the weight and the length of worms showed little variation, with unimodal distribution of values, suggesting that all the worms in each case belong to the same population, originated from a single brood infection or from successive infections over small time intervals. In three cases there was great variation in worm size indicated by the different values for the means and medians and by the high values for the standard deviation and coefficient of variation. In these three cases there was a bimodal distribution of worm's size suggesting the coexistance of two distinct populations: one, less numerous, composed of mature worms and the other, more numerous, composed ofimmature worms, in two cases, and two distinct populations of immature worms in one case. The existance of worms in different stages of maturation indicates that the less mature population was acquired when the mature worms were established in the gut. These results indicate that the reinfection with Ascaris in children bearing an established infection is not rare and resistance induced by a preexisting infection is not the rule.

Caputo fractional differential equations with non-instantaneous impulses and strict stability by Lyapunov functions

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5217-5239 ◽  
Author(s):  
Ravi Agarwal ◽  
Snehana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Time Intervals ◽  
Dini Derivative ◽  
Comparison Results ◽  
Strict Stability ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential

In this paper the statement of initial value problems for fractional differential equations with noninstantaneous impulses is given. These equations are adequate models for phenomena that are characterized by impulsive actions starting at arbitrary fixed points and remaining active on finite time intervals. Strict stability properties of fractional differential equations with non-instantaneous impulses by the Lyapunov approach is studied. An appropriate definition (based on the Caputo fractional Dini derivative of a function) for the derivative of Lyapunov functions among the Caputo fractional differential equations with non-instantaneous impulses is presented. Comparison results using this definition and scalar fractional differential equations with non-instantaneous impulses are presented and sufficient conditions for strict stability and uniform strict stability are given. Examples are given to illustrate the theory.

scholarly journals Jordan products of quantum channels and their compatibility

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Mark Girard ◽  
Martin Plávala ◽  
Jamie Sikora
Keyword(s):  
Quantum State ◽  
Sufficient Conditions ◽  
The Other ◽  
Independent Interest ◽  
Quantum Channels ◽  
Semidefinite Programs ◽  
Jordan Product ◽  
Marginal Problem ◽  
Jordan Products ◽  
Product Of Matrices

AbstractGiven two quantum channels, we examine the task of determining whether they are compatible—meaning that one can perform both channels simultaneously but, in the future, choose exactly one channel whose output is desired (while forfeiting the output of the other channel). Here, we present several results concerning this task. First, we show it is equivalent to the quantum state marginal problem, i.e., every quantum state marginal problem can be recast as the compatibility of two channels, and vice versa. Second, we show that compatible measure-and-prepare channels (i.e., entanglement-breaking channels) do not necessarily have a measure-and-prepare compatibilizing channel. Third, we extend the notion of the Jordan product of matrices to quantum channels and present sufficient conditions for channel compatibility. These Jordan products and their generalizations might be of independent interest. Last, we formulate the different notions of compatibility as semidefinite programs and numerically test when families of partially dephasing-depolarizing channels are compatible.

scholarly journals On a New Geometric Constant Related to the Euler-Lagrange Type Identity in Banach Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 116
Author(s):  
Qi Liu ◽  
Yongjin Li
Keyword(s):  
Banach Spaces ◽  
Product Space ◽  
Sufficient Conditions ◽  
Normal Structure ◽  
The Other ◽  
Inner Product ◽  
Equivalent Characterization ◽  
Geometric Constant ◽  
Geometric Constants

In this paper, we will introduce a new geometric constant LYJ(λ,μ,X) based on an equivalent characterization of inner product space, which was proposed by Moslehian and Rassias. We first discuss some equivalent forms of the proposed constant. Next, a characterization of uniformly non-square is given. Moreover, some sufficient conditions which imply weak normal structure are presented. Finally, we obtain some relationship between the other well-known geometric constants and LYJ(λ,μ,X). Also, this new coefficient is computed for X being concrete space.

Near-Infrared Combination and Overtone Bands of CH in CHX3, CHX2—CHX2, and CHX2—CX2—CHX2

2005 ◽  
Vol 59 (11) ◽  
pp. 1393-1398 ◽  
Author(s):  
Reikichi Iwamoto ◽  
Akishi Nara ◽  
Toshihiko Matsuda
Keyword(s):  
Excited State ◽  
Near Infrared ◽  
Present Report ◽  
Spectral Characteristics ◽  
The Other ◽  
Coupled Modes ◽  
Third Order ◽  
The Third ◽  
Combination Bands ◽  
Deformation Modes

In the present report we studied spectral characteristics of the near-infrared combination and overtone bands of CH vibrations of a CH sequence. The near-infrared bands of the CH in CHX3 (X, halogen), which were interpreted in terms of the CH stretching and CH deformation fundamentals without any ambiguity, typically showed how the frequency and intensity of a combination or an overtone depend on the vibrational excited state. In the CH–C–CH of CHX2CX2CHX2, the vibrations of one CH are isolated from those of the other CH, and the combination and overtone bands were similarly interpreted as those of the CH, although each of the combination bands was split into two because of non-degeneracy of the CH deformation. In the CH–CH of CHX2CHX2, the CH deformations only have coupled modes. The first combination showed four narrowly separate bands, which were reasonably interpreted on the basis of the CH stretching and the coupled CH deformation modes. We demonstrated that the first combination of coupled modes as well as the combination of up to, at least, the third order of isolated modes have the nature of the characteristic bands.

scholarly journals Experiment and numerical modeling of asymmetrically initiated hexagonal prism warhead

2017 ◽  
Vol 9 (1) ◽  
pp. 168781401668796 ◽  
Author(s):  
Yuan Li ◽  
Yuquan Wen
Keyword(s):  
Numerical Modeling ◽  
The Other ◽  
Energy Utilization ◽  
Hexagonal Prism ◽  
Cylindrical Structure ◽  
Damage Probability ◽  
Other Hand ◽  
Density Enhancement ◽  
Fragment Distribution ◽  
Velocity Enhancement

Traditional fragmentation warheads are usually initiated on the axis, producing a uniform fragment distribution around the warhead, but only little portion of them can be imposed on the targets. The aimable warhead, on the other hand, using the off-axis initiations or the structure deforming, can improve the warhead energy utilization highly. Seeking methods to both enhance the fragment velocity and density has significant value for improving the target damage probability. In this article, a warhead shaped as hexagonal prism was studied using arena experiment and numerical modeling and compared with the traditional cylindrical structure. The fragment velocities and target hit patterns of the two types of warheads under axial initiation and asymmetrical initiation are obtained. It is revealed that for the hexagonal prism warhead, the asymmetrical initiation can enhance the fragment velocities by 27.71% and enhance fragment density by 34.09% compared to the axial initiation. The fragment velocity enhancement is close to that of the cylindrical warhead, but the fragment density enhancement is far above the cylindrical warhead. This indicates that the asymmetrically initiated hexagonal prism warhead is a very effective aimable warhead.

Point process models of rainfall: developments for fine-scale structure

2007 ◽  
Vol 463 (2086) ◽  
pp. 2569-2587 ◽  
Author(s):  
Paul Cowpertwait ◽  
Valerie Isham ◽  
Christian Onof
Keyword(s):  
Poisson Process ◽  
Time Scales ◽  
Rain Gauge ◽  
Historical Record ◽  
Rainfall Data ◽  
Process Models ◽  
Third Order ◽  
Time Intervals ◽  
Historical Series ◽  
Derived Properties

A conceptual stochastic model of rainfall is proposed in which storm origins occur in a Poisson process, where each storm has a random lifetime during which rain cell origins occur in a secondary Poisson process. In addition, each cell has a random lifetime during which instantaneous random depths (or ‘pulses’) of rain occur in a further Poisson process. A key motivation behind the model formulation is to account for the variability in rainfall data over small (e.g. 5 min) and larger time intervals. Time-series properties are derived to enable the model to be fitted to aggregated rain gauge data. These properties include moments up to third order, the probability that an interval is dry, and the autocovariance function. To allow for distinct storm types (e.g. convective and stratiform), several processes may be superposed. Using the derived properties, a model consisting of two storm types is fitted to 60 years of 5 min rainfall data taken from a site near Wellington, New Zealand, using sample estimates taken at 5 min, 1 hour, 6 hours and daily levels of aggregation. The model is found to fit moments of the depth distribution up to third order very well at these time scales. Using the fitted model, 5 min series are simulated, and annual maxima are extracted and compared with equivalent values taken from the historical record. A good fit in the extremes is found at both 1 and 24 hour levels of aggregation, although at the 5 min level there is some underestimation of the historical values. Proportions of time intervals with depths below various low thresholds are extracted from the simulated and historical series and compared. A tendency for underestimation of the historical values is evident at some time scales, with a close fit being obtained as the threshold is increased.

scholarly journals Existence and Lyapunov Stability of Positive Periodic Solutions for a Third-Order Neutral Differential Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-28 ◽  
Author(s):  
Jingli Ren ◽  
Zhibo Cheng ◽  
Yueli Chen
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Periodic Solutions ◽  
Lyapunov Stability ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Neutral Differential Equation ◽  
Positive Periodic Solutions ◽  
Third Order

By applying Green's function of third-order differential equation and a fixed point theorem in cones, we obtain some sufficient conditions for existence, nonexistence, multiplicity, and Lyapunov stability of positive periodic solutions for a third-order neutral differential equation.

scholarly journals New Oscillatory Behavior of Third-Order Nonlinear Delay Dynamic Equations on Time Scales

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Li Gao ◽  
Quanxin Zhang ◽  
Shouhua Liu
Keyword(s):  
Time Scales ◽  
Sufficient Conditions ◽  
Oscillatory Behavior ◽  
Dynamic Equations ◽  
Riccati Transformation ◽  
Third Order ◽  
Generalized Riccati Transformation ◽  
Delay Dynamic Equations ◽  
Inequality Technique ◽  
Nonlinear Delay

A class of third-order nonlinear delay dynamic equations on time scales is studied. By using the generalized Riccati transformation and the inequality technique, four new sufficient conditions which ensure that every solution is oscillatory or converges to zero are established. The results obtained essentially improve earlier ones. Some examples are considered to illustrate the main results.

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