A Study on Generalized Balancing Numbers

In this paper, we investigate properties of the generalized balancing sequence and we deal with, in detail, namely, balancing, modified Lucas-balancing and Lucas-balancing sequences. We present Binet’s formulas, generating functions and Simson formulas for these sequences. We also present sum formulas of these sequences. We provide the proofs to indicate how the sum formulas, in general, were discovered. Of course, all the listed sum formulas may be proved by induction, but that method of proof gives no clue about their discovery. Moreover, we consider generalized balancing sequence at negative indices and construct the relationship between the sequence and itself at positive indices. This illustrates the recurrence property of the sequence at the negative index. Meanwhile, this connection holds for all integers. Furthermore, we give some identities and matrices related with these sequences.

On B- Covariant Derivative of First Order for Some Tensors in different Spaces

In this paper, we study the relationship between Cartan's second curvature tensor $P_{jkh}^{i}$ and $(h) hv-$torsion tensor $C_{jk}^{i}$ in sense of Berwald. Moreover, we discuss the necessary and sufficient condition for some tensors which satisfy a recurrence property in $BC$-$RF_{n}$, $P2$-Like-$BC$-$RF_{n}$, $P^{\ast }$-$BC$-$RF_{n}$ and $P$-reducible-$BC-RF_{n}$.

Jℱ-Class Weighted Backward Shifts

In this article [Formula: see text]-class operators are introduced and some basic properties of [Formula: see text]-vectors are given. The [Formula: see text]-class operators include the [Formula: see text]-class operators and [Formula: see text]-class operators introduced by Costakis and Manoussos in 2008. This class also includes the [Formula: see text]-class and [Formula: see text]-class operators defined by Zhang [2012]. Furthermore, for the unilateral weighted backward shifts on a Fréchet sequence space, we establish a criterion under which the shift operators belong to the [Formula: see text]-class. From the criterion it is easy to obtain the existing criteria of hypercyclic backward shifts and of the topological mixing backward shifts. The obtained criterion also reveals the characteristic of [Formula: see text]-class shift operators by the recurrence property. Meanwhile, we obtain infinite topological entropy when the shifts have stronger recurrence property, which generalizes the related results by Brian et al. in 2017.

THEORETICAL AND METHODOLOGICAL BACKGROUND TO THE DEVELOPMENT OF COUNTERCYCLICAL POLICY

The theoretical model revealing the nature of cyclic processes in economy is constructed on the basis of the specified treatment of the interphase recurrence property concept. According to it, the directions of anti-cyclic management taking into account the contents and features of course of each phase of a business cycle are revealed. The author's concept of a state policy of regulation of economic development taking into account its phase is offered.

Research on Characteristics of Human Gait by Electrostatic Detection

Electrostatic detection is remarkably developed and has been employed to detect human activity for years. In this paper, an induction electrostatic detector is designed, and used to measure human walking signals. The gait signals of totally six segments of the same object are measured in the experiment. An algorithm is proposed to obtain accurate gait cycle. The original signals are transformed to correlation coefficient series. Peaks of correlation coefficient series is picked out as the same phase point instead of peaks of walking signal. The time between very second peaks is defined as gait cycle of object. Based on the gait cycle, we discussed the recurrence property as the characteristics of human beings. It is expected that the slope of the recurrence line can be used as an index, and will indicate the personal particularity of objects in detection and their physical conditions.

Global properties of a family of piecewise isometries

We investigate a basic system of a piecewise rotations acting on two half-planes. We prove that for invertible systems, an arbitrary neighbourhood of infinity contains infinitely many periodic points surrounded by periodic cells. In the case where the underlying rotation is rational, we show that all orbits remain bounded, whereas in the case where the underlying rotation is irrational, we show that the map is conservative (satisfies the Poincaré recurrence property). A key part of the proof is the construction of periodic orbits that shadow orbits for certain rational rotations of the plane.