Global finite-time stabilization of switched high-order rational power nonlinear systems

This works is concerned with the finite-time optimal stabilization problem for a class of switched non-strict-feedback nonlinear systems whose powers are possibly different positive odd rational numbers in the sense the powers of each subsystem might differ from others. It is well known that high-order nonlinear systems are intrinsically challenging as feedback linearization and backstepping method successfully developed for low-order systems fail to work. To this purpose, the nested saturation homogeneous controller is constructively devised to achieve global finite-time stability. Furthermore, the corresponding design parameters are optimized by minimizing a well-defined cost function, and thus an optimal controller being independent of switching signals is obtained. Simulation results are eventually provided to validate the effectiveness of the proposed control scheme.     

Regenerative structure of Markov chains simulated via common random numbers

A standard strategy in simulation, for comparing two stochastic systems, is to use a common sequence of random numbers to drive both systems. Since regenerative output analysis of the steady-state of a system requires that the process be regenerative, it is of interest to derive conditions under which the method of common random numbers yields a regenerative process. It is shown here that if the stochastic systems are positive recurrent Markov chains with countable state space, then the coupled system is necessarily regenerative; in fact, we allow couplings more general than those induced by common random numbers. An example is given which shows that the regenerative property can fail to hold in general state space, even if the individual systems are regenerative.     

Parity Alternating Permutations and Signed Eulerian Numbers

This paper introduces subgroups of the symmetric group and studies their combinatorial properties. Their elements are called parity alternating, because they are permutations with even and odd entries alternately. The objective of this paper is twofold. The first is to derive several properties of such permutations by subdividing them into even and odd permutations. The second is to discuss their combinatorial properties; among others, relationships between those permutations and signed Eulerian numbers. Divisibility properties by prime powers are also deduced for signed Eulerian numbers and several related numbers.     

A simple method of chaos control for a class of chaotic discrete-time systems

In this paper, a simple method is proposed for chaos control for a class of discrete-time chaotic systems. The proposed method is built upon the state feedback control and the characteristic of ergodicity of chaos. The feedback gain matrix of the controller is designed using a simple criterion, so that control parameters can be selected via the pole placement technique of linear control theory. The new controller has a feature that it only uses the state variable for control and does not require the target equilibrium point in the feedback path. Moreover, the proposed control method cannot only overcome the so-called “odd eigenvalues number limitation” of delayed feedback control, but also control the chaotic systems to the specified equilibrium points. The effectiveness of the proposed method is demonstrated by a two-dimensional discrete-time chaotic system.     

Stability of the Kolmogorov flow and its modifications

Recurrence formulas are obtained for the kth term of the long wavelength asymptotics in the stability problem for general two-dimensional viscous incompressible shear flows. It is shown that the eigenvalues of the linear eigenvalue problem are odd functions of the wave number, while the critical values of viscosity are even functions. If the velocity averaged over the long period is nonzero, then the loss of stability is oscillatory. If the averaged velocity is zero, then the loss of stability can be monotone or oscillatory. If the deviation of the velocity from its period-average value is an odd function of spatial variable about some x ₀, then the expansion coefficients of the velocity perturbations are even functions about x ₀ for even powers of the wave number and odd functions about for x ₀ odd powers of the wave number, while the expansion coefficients of the pressure perturbations have an opposite property. In this case, the eigenvalues can be found precisely. As a result, the monotone loss of stability in the Kolmogorov flow can be substantiated by a method other than those available in the literature.     

一些关于Chebyshef多项式以及它们应用的恒等式(英文)

本文研究了Chebyshef多项式的一类幂和问题.利用初等方法以及Chebyshef多项式的性质,获得了一些有趣的恒等式,推广了Melham关于Lucas数的奇数次幂和的猜想.     

Adaptive tracking control for a class of switched uncertain nonlinear systems under a new state-dependent switching law

This paper deals with the problem of adaptive fuzzy tracking control for a class of switched uncertain nonlinear systems. Fuzzy logic systems are utilized to approximate the unknown nonlinear functions, and the adaptive backstepping and dynamic surface control techniques are adopted. First, a new state-dependent switching method is proposed. By introducing convex combination technique and designing a state-dependent switching law, only the solvability of the adaptive tracking control problem for a convex combination of the subsystems is necessary. Second, a new common Lyapunov function with switched adaptive parameters is constructed to reduce the conservatism. Third, to avoid Zeno behavior, a modified state-dependent switching law with dwell time is proposed. It is shown that under the proposed control and switching laws, all the signals of the closed-loop system are bounded and all the state tracking errors can converge to a priori accuracy, even if some subsystems are uncontrollable. Finally, the effectiveness of the proposed method is illustrated through two simulation examples.     

A quadrature-based approach to improving the collocation method

Summary The collocation method is a popular method for the approximate solution of boundary integral equations, but typically does not achieve the high order of convergence reached by the Galerkin method in appropriate negative norms. In this paper a quadrature-based method for improving upon the collocation method is proposed, and developed in detail for a particular case. That case involves operators with even symbol (such as the logarithmic potential) operating on 1-periodic functions; a smooth-spline trial space of odd degree, with constant mesh spacingh=1/n; and a quadrature rule with 2n points (where ann-point quadrature rule would be equivalent to the collocation method). In this setting it is shown that a special quadrature rule (which depends on the degree of the splines and the order of the operator) can yield a maximum order of convergence two powers ofh higher than the collocation method.     

The number of powers of 2 in a representation of large even integers (I)

Under the Generalized Riemann Hypothesis, it is proved that for any integer k⩾770 there is N_k>0 depending onk only such that every even integer ⩾N_k is a sum of two odd prime numbers andk powers of 2. The research is partially supported by RGC research grant (HKU 518/96P). The first author is supported by Post-Doctoral Fellowship of The University of Hong Kong.     

Completeness of Normal Rational Curves

The completeness of normal rational curves, considered as (q + 1)-arcs in PG(n, q), is investigated. Previous results of Storme and Thas are improved by using a result by Kovács. This solves the problem completely for large prime numbers q and odd nonsquare prime powers q = p ^2h+1 with p prime, , where p ₀(h) is an odd prime number which depends on h.     

Two results on powers of 2 in Waring-Goldbach problem

In this paper, it is proved that every sufficiently large odd integer is a sum of a prime, four cubes of primes and 106 powers of 2. What is more, every sufficiently large even integer is a sum of two squares of primes, four cubes of primes and 211 powers of 2.     

Neither even nor odd: Sixth grade students’ dilemmas regarding the parity of zero

This study investigates two sixth grade students’ dilemmas regarding the parity of zero. Both students originally claimed that zero was neither even nor odd. Interviews revealed a conflict between students’ formal definitions of even numbers and their concept images of even numbers, zero, and division. These images were supported by practically based explanations relying on everyday contexts. By using mathematically based explanations that rely solely on mathematical notions, students were able to correctly conclude that zero is an even number. Extending the natural number system in elementary school to include zero can be used as springboard to encourage the use of mathematically based explanations.     

On the Waring-Goldbach Problem for Fourth and Fifth Powers

It is shown that every sufficiently large integer congruentto 14 modulo 240 may be written as the sum of 14 fourth powersof prime numbers, and that every sufficiently large odd integermay be written as the sum of 21 fifth powers of prime numbers.The respective implicit bounds 14 and 21 improve on the previousbounds 15 (following from work of Davenport) and 23 (due toThanigasalam). These conclusions are established through themedium of the Hardy-Littlewood method, the proofs being somewhatnovel in their use of estimates stemming directly from exponentialsums over prime numbers in combination with the linear sieve,rather than the conventional methods which ‘waste’a variable or two by throwing minor arc estimates down to anauxiliary mean value estimate based on variables not restrictedto be prime numbers. In the work on fifth powers, a switchingprinciple is applied to a cognate problem involving almost primesin order to obtain the desired conclusion involving prime numbersalone. 2000 Mathematics Subject Classification: 11P05, 11N36,11L15, 11P55.     

Wandering sets for a class of borel isomorphisms of [0,1)

A wandering set for a map ϕ is a set containing precisely one element from each orbit of ϕ. We study the existence of Borel wandering sets for piecewise linear isomorphisms. Such sets need not exist even when the parameters involved are rational, but they do exist if in addition all the slopes are powers of 2. For ϕ having at most one discontinuity, the existence of a Borel wandering set is equivalent to rationality of the Poincaré rotation number. We compute the rotation numbers for a special class of such functions. The main result provides a concrete method of connecting certain pairs of wavelet sets.     

On one approach to realization of the branch-and-bound method on a distributed system

The branch-and-bound method is one of most efficient search algorithms. However, parallel decomposition of this method is difficult even on architectures with common memory, not to mention distributed systems. The paper describes an algorithm for solution of the problems that can be solved with the branch-and-bound method on a distributed computer system. The results of testing of the proposed algorithm via solution of the chess-game problem are presented.     

Odd multiperfect numbers of abundancy 4

Euler's structure theorem for any odd perfect number is extended to odd multiperfect numbers of abundancy power of 2. In addition, conditions are found for classes of odd numbers not to be 4-perfect: some types of cube, some numbers divisible by 9 as the maximum power of 3, and numbers where 2 is the maximum even prime power.     

Counteracting the dynamical degradation of digital chaos via hybrid control

Chaotic systems would degrade owing to finite computing precisions, and such degradation often seriously affects the performance of digital chaos-based applications. In this paper, a chaotification method is proposed to solve the dynamical degradation of digital chaotic systems based on a hybrid structure, where a continuous chaotic system is applied to control the digital chaotic system, and a unidirectional coupling controller that combines a linear external state control with a modular function is designed. Moreover, we proof rigorously that a class of digital chaotic systems can be driven to be chaotic in the sense that the system is sensitive to initial conditions. Different from the existing remedies, this method can recover the dynamical properties of system, and even make some properties better than those of the original chaotic system. Thus, this new approach can be applied to the fields of chaotic cryptography and secure communication.     

Periodic structure of transversal maps on sum-free products of spheres

In this article, we study the periodic structure of transversal maps on the product of spheres of different dimensions. In particular, we give sufficient conditions in order that such maps have infinitely many even and odd periods. Moreover, we also provide sufficient conditions for having non-zero Lefschetz numbers of period m for infinitely many m's. We extend these results to transversal maps on rational exterior spaces of rank 1.     

Odd order Brauer–Manin obstruction on diagonal quartic surfaces

We determine the odd order torsion subgroup of the Brauer group of diagonal quartic surfaces over the field of rational numbers. We show that a non-constant Brauer element of odd order always obstructs weak approximation but never the Hasse principle.     

Robust control of chaos in Lorenz systems subject to mismatch uncertainties

This paper is concerned with robust control for a class of Lorenz systems subject to mismatch uncertainties. It is implemented by using variable structure control. The proposed variable structure controller ensures the occurrence of the sliding mode for the error dynamics. It is guaranteed that under the proposed control law, uncertain Lorenz systems can drive the system state exactly to some specific points or in a predictable neighborhood of arbitrary desired points in the state space even with mismatch uncertainties, which is not addressed in the literature.