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We consider a nonlinear discrete-time quantum walk whose coin operator temporally changes according to periodic, Fibonacci quasiperiodic, and random sequences, respectively. Three distinguished nonlinearity regimes are numerically found, where transport spreading behaviors are similar to that of linear quantum walks, traveling soliton-like behavior, and first self-trapped near original site then multiple-traveling-soliton-like behavior, respectively. We contrast the differences among the three kinds of quantum walks and give some reasonable explanations. [ABSTRACT FROM AUTHOR]

Purpose: Nowadays, in the millimeter frequency range, the dielectric waveguides of various modifications have certain advantages over the standard metal waveguides, primarily due to the possibility of creating functional units based on them. This is due to the relative simplicity and low cost of manufacturing the dielectric waveguides and functional units using them, the high degree of their integration with active elements, the use in their manufacture of different dielectrics and polymers with a wide range of material constants and a variety of mechanical properties (in particular, some materials have a significant flexibility). After making a series of physical experiments we have found the possibility of implementing the frequency selection and radiation into free space of electromagnetic waves by a hybrid metal-dielectric structure. Design/methodology/approach: The studied electrodynamic structure belongs to the class of hybrid metal-dielectric structures. It includes a modified inverted dielectric waveguide with a periodic sequence on the dielectric plate of fifteen dielectric rods with metallized coating on one of the faces placed outwards. The structure efficiency was estimated by the voltage standing-wave ratio (VSWR) values and power attenuation in the duct. The measurements were made with the reflectometer method. To estimate the degree of electromagnetic field concentration near the rod inhomogeneities in the near zone, the mobile probe method was used. The field structures were visualized with the method of isolines. Findings: The results of a series of experimental investigations showed the possibility of matching the structure with the external waveguides in the frequency range of 26.5-32.5 GHz with the voltage standing-wave ratio (VSWR) less than 1.8. The frequency dependence of attenuation is oscillatory with clearly expressed frequency ranges with small and large attenuation values. Moreover, the dependence is almost periodic, which is typical of periodic structures. The frequency response slope in the transition zones can be quite high and reach values of 41.26 dB/GHz. The degree of concentration of the electric field near the waveguide dielectric rod and the degree of excitation of the dielectric inhomogeneities was found by directly measured electric field strength in the near zone. Measurements of energy characteristics made under the short-circuit conditions for the main guide and in the mode of matched load of the main guide showed both the ability to control the polarization characteristics and the ability to change the appearance of the pattern and its orientation in space. Conclusions: It has been experimentally proven that a hybrid metal-dielectric structure, being a modified inverted dielectric waveguide with a periodic sequence on the dielectric plate of fifteen dielectric rods with metalized coating on one of the faces placed outwards, can be effectively integrated into a standard transmission line. It is found that this structure can be matched with the external circuits in a fairly wide frequency range. It is also found that in different frequency ranges this hybrid metal-dielectric structure shows the possibility of both efficient frequency selection and radiation in free space. Antenna measurements have shown the beam pattern shape controllability.

The method of estimating the period of an almost periodic sequence is proposed. The period of an almost periodic sequence is the length of the smallest periodic pattern that composes the corresponding completely periodic sequence by repetition. The method can be applied to corrupted periodic sequences that originated from a periodic sequence that consists of no less than eight complete periodic patterns. In the corresponding periodic sequence with noise of insertion, deletion or change, some periodic patterns might be corrupted by noise intrusion. The level of noise is supposed to be less than 10%, this assumption allows us to use a shift operator with the window of width 16 so that we can observe at least more than twice every uncorrupted pattern of length 16 that the sequence under consideration contains. The method is based on the counting of the number of symbols in the word w between the two first symbols of the closest equal patterns of length 16. Only patterns that were encountered in word more than twice are used for counting the differences between the left positions of the neighbor equal patterns in the sequence under consideration. All the differences counted are arranged in the ascending order and the 25% quantile and the median of the sequence of the differences are calculated. The computational experiment shows that in most cases the 25% quantile is an appropriate estimation of the length of the periodic pattern when the noise level is no more than 5%. Sometimes the method provides a sufficient result in case of noise level between 5 and 10%. The dependence of the percentage of sufficient estimations of periodic pattern length on the noise level was studied in cases of only one type of noise and the noise of all three types in equal proportions. The computational experiment showed that in all situations 25% quantile provides more sufficient estima-tions than the median. The method is supposed to be improved, in order to restore the whole corresponding periodic sequence using only the sequence with noise.

Extreme events are a major concern in statistical modeling. Random missing data can constitute a problem when modeling such rare events. Imputation is crucial in these situations and therefore models that describe different imputation functions enhance possible applications and enlarge the few known families of models that cover these situations. In this paper we consider a family of models { Y n } , n ≥ 1 , that can be associated to automatic systems which have a periodic control, in the sense that at instants multiple of T, T ≥ 2 , no value is lost. Random missing values are here replaced by the biggest of the previous observations up to the one surely registered. We prove that when the underlying sequence is stationary, { Y n } is T-periodic and, if it also verifies some local dependence conditions, then { Y n } verifies one of the well known D T (s) (u n) , s ≥ 1 , dependence conditions for T-periodic sequences. We also obtain the extremal index of { Y n } and relate it to the extremal index of the underlying sequence. A consistent estimator for the parameter that "controls" the missing values is here proposed and its finite sample properties are analysed. The obtained results are illustrated with Markovian sequences of recognized interest in applications. [ABSTRACT FROM AUTHOR]

The construction of almost periodic sequences is needed for the analysis of the cycles’ detection methods, identification of symbolic sequences’ features and sensitivity analysis. Two probabilistic models of noise are proposed for constructing almost periodic symbolic sequences. Models provide various types of noise in a periodic sequence, such as changing, adding and deleting characters. Thus, on the basis of a periodic symbolic sequence, an almost periodic sequence is constructed. It is necessary to ensure a given level of noise in the constructed almost periodic sequence. The required level of simulated noise is guaranteed by a two-level model, in which the positions for simulated noise are determined at the first level, based on the discrete random variable, chosen by the researcher. At the second level, the necessary changes are made at the corresponding positions using a random variable that simulates the noise itself. The second model is based on simulated noise with probability (depending on the noise level) in each element of the sequence. The observed noise level is calculated based on the Levenshtein distance between the original periodic sequence and the almost periodic sequence that we constructed. The observed noise level is always somewhat less than the level of the noise simulated, since when calculating the Levenshtein distance, a shorter way of obtaining a noisy sequence from a periodic one can be found than the one that is used in constructing an almost periodic sequence. A comparison of the proposed models for the proximity of the noise level they provide to a given noise level is made. The computational experiment showed that the observed noise level is closer to that provided by the two-level model. The software implementation of these models will be used in future to study the algorithms for finding cycles in noisy periodic symbolic sequences.

The introduction of Parikh matrices by Mateescu et al. in 2001 has sparked numerous new investigations in the theory of formal languages by various researchers, among whom is Şerbănuţă. Recently, a decade-old conjecture by Şerbǎnuţǎ on the M-ambiguity of words has disproved, leading to new possibilities in the study of such words. In this paper, we investigate how selective repeated duplications of letters in a word affect the M-ambiguity of the resulting words. The corresponding M-ambiguity of each of those words is then presented in sequences, which we term as M-ambiguity sequences. We show that nearly all patterns of M-ambiguity sequences are attainable. Finally, by employing certain algebraic approach and some underlying theory in integer programming, we show that repeated periodic duplications of letters of the same type in a word result in an M-ambiguity sequence that is ultimately periodic. [ABSTRACT FROM AUTHOR]

A subtraction game is an impartial combinatorial game involving a finite set S of positive integers. The nim-sequence G S associated with this game is ultimately periodic. In this paper, we study the nim-sequence G S ∪ { c } where S is fixed and c varies. We conjecture that there is a multiple q of the period of G S , such that for sufficiently large c , the pre-period and period of G S ∪ { c } are linear in c if c modulo q is fixed. We prove it in several cases. We also give new examples with period 2 inspired by this conjecture. [ABSTRACT FROM AUTHOR]

Abnormal patterns detection in data (abnormal detection) is a very important research direction in the field of data analysis, and especially the anomaly detection of time series is one of the difficulties. At present, there are many researches on anomaly detection of time series data. Different techniques such as sliding window, wavelet analysis, Probabilistic Graphical Model (PGM), and Recurrent Neural Network (RNN) are used to detect the abnormal patterns. However, these techniques still have some deficiencies in processing this problem, which cannot guarantee real-time efficiency and accuracy in real engineering. In this paper, an anomaly detection algorithm based on Attention-GRU and iForest is proposed for periodic time series. The model is constructed based on Attention-GRU to predict long data sequence and ensure the detection efficiency in engineering. iForest is used to establish a normal data fluctuation range, avoiding the error caused by hypothesis testing. Practice verification shows that this model can improve the anomaly detection efficiency of periodic time series data in actual engineering, and has better recall rate and accuracy. [ABSTRACT FROM AUTHOR]

Finite automata transform periodic sequences into periodic ones. The period of the output sequence is bounded from above by a linear function of input period. It is known that pushdown automata also preserve the set of periodic sequences. We prove that the output period for one-counter pushdown automata is bounded from above by a quadratic function of input period. We also give an example of an automaton with a quadratic lower bound on output period. [ABSTRACT FROM AUTHOR]

In this paper, periodic sequences with period N and nonlinear complexity N − 2 are investigated. A necessary and sufficient condition for characterizing such sequences is established, and a recursive method is proposed to generate all possible binary sequences with period N and nonlinear complexity N − 2. The exact number of such sequences is also determined. [ABSTRACT FROM AUTHOR]

Quaternary sequences of both even and odd period having low autocorrelation are studied. We construct new families of balanced quaternary sequences of odd period and low autocorrelation using cyclotomic classes of order eight, as well as investigate the linear complexity of some known quaternary sequences of odd period. We discuss a construction given by Chung et al. in "New Quaternary Sequences with Even Period and Three-Valued Autocorrelation" (IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E93-A(1), 309-315 2010) first by pointing out a slight modification and then by showing that, in certain cases, this slight modification generalizes the construction given by Shen et al. in "New Families of Balanced Quaternary Sequences of Even Period with Three-level Optimal Autocorrelation" (IEEE Commun. Lett. 2017(10), 2146-2149 2017). We investigate the linear complexity of these sequences as well. [ABSTRACT FROM AUTHOR]

Š. Schwarz in his paper [SCHWARZ, Š.: Zur Theorie der Halbgruppen, Sborník prác Prírodovedeckej fakulty Slovenskej univerzity v Bratislave, Vol. VI, Bratislava, 1943, 64 pp.] proved the existence of maximal subgroups in periodic semigroups and a decade later he brought into play the maximal subsemigroups and thus he embodied the idempotents in the structural description of semigroups [SCHWARZ, Š.: Contribution to the theory of torsion semigroups, Czechoslovak Math. J. 3(1) (1953), 7–21]. Later in his papers he showed that a proper description of these structural elements can be used to (re)prove many useful and important results in algebra and number theory. The present paper gives a survey of selected results scattered throughout the literature where an semigroup approach based on tools like idempotent, maximal subgroup or maximal subsemigroup either led to a new insight into the substance of the known results or helped to discover new approach to solve problems. Special attention will be given to some disregarded historical connections between semigroup and ring theory. [ABSTRACT FROM AUTHOR]

We determine the cycle structure of linear feedback shift register with arbitrary monic characteristic polynomial over any finite field. For each cycle, a method to find a state and a new way to represent the state are proposed. [ABSTRACT FROM AUTHOR]

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