Non-periodic Sequences Research Articles

Application of Genetic Algorithms for Periodicity Recognition and Finite Sequences Sorting

Unpredictable strings are sequences of data with complex and erratic behavior, which makes them an object of interest in various scientific fields. Unpredictable strings related to chaos theory was investigated using a genetic algorithm. This paper presents a new genetic algorithm for converting large binary sequences into their periodic form. The MakePeriod method is also presented, which is aimed at optimizing the search for such periodic sequences, which significantly reduces the number of generations to achieve the result of the problem under consideration. The analysis of the deviation of a nonperiodic sequence from its considered periodic transformation was carried out, and methods of crossover and mutation were investigated. The proposed algorithm and its associated conclusions can be applied to processing large sequences and different values of the period, and also emphasize the importance of choosing the right methods of crossover and mutation when applying genetic algorithms to this task.

Non‐covalent Intramolecular Interactions Induced High‐Performance Terpolymer Donors

AbstractTernary polymerization are demonstrated as a feasible and successful method of optimizing absorption, energy level, and crystallinity by maintaining the excellent properties of a polymer matrix. However, the third unit inevitably interrupts the periodic sequence distribution of the ordering packing of conjugated backbones, and thus balancing the trade‐off between the advantages of the third component and the disruption caused by the nonperiodic sequence distribution remains a great challenge. Herein, two terpolymer donors L1 and L2 are developed by introducing diethyl‐5,5′“‐dibromo‐4,4′”‐bis(2‐ethylhexyl)‐[2,2′:5′,2′'‐terthiophene]‐3′,4′‐dicarboxylate (DDT) unit as the third component. Owing to the electron‐deficient ability and S···O non‐covalent intramolecular interactions of DDT, terpolymers L1 and L2 displays deeper energy levels, enhanced molecular rigidity, and planarity than PM6. After blending with L8‐BO, the L1‐based polymer solar cells exhibit well miscibility with L8‐BO, leading to a more well‐defined nanofibrous morphology, face‐on orientation, and a slight energy loss in the blend films. As a result, the optimized L1:L8‐BO‐based device achieves a record power conversion efficiency (PCE) as high as 18.75%. This work provides a feasible strategy to develop high‐performance polymer donors via utilizing S···O non‐covalent intramolecular interactions to improve the planarity of polymer donors.

Derived from the traditional principles of Islamic geometry, a methodology for generating non-periodic long-range sequences in one-dimension for 8-fold, 10-fold, and 12-fold rotational symmetries

This paper explores the use of traditional principles of Islamic geometry to generate the long-range order of different non-period sequences in one-dimension. The discovery of quasicrystals with forbidden long-range order has uncovered a range of puzzling non-periodic symmetries, triggering a variety of investigations into understanding their complicated structural logic. The discovery of historical patterns in Islamic architecture with similar quasi-periodic formations is providing new insights into understanding the mathematical properties of these complicated formations. This paper argues that the hierarchical self-similar logic underlying the construction process of the historical patterns can serve as the key ingredient for resolving the long-range order of the one-dimensional non-periodic sequences. A case study approach is employed to test the hypotheses. Three historical patterns with non-period symmetries were tested: 10-fold rotational symmetry, 8-fold rotational symmetry and 12-fold rotational symmetry. The results show that the methodology is successful in generating three different non-periodic sequences in one-dimension.

Three‐dimensional spatially resolved phase graph framework

An open-source spatially resolved phase graph framework is proposed for simulating arbitrary pulse sequences in the presence of piece-wise constant gradients with arbitrary orientations in three dimensions. It generalizes the extended phase graph algorithm for analysis of nonperiodic sequences while preserving its efficiency, and is able to estimate the signal modulation in the 3D spatial domain. The framework extends the recursive magnetization-evolution algorithm to account for anisotropic diffusion and exploits a novel 3D k-space grid-merging method to balance the computational effort and memory requirements against acceptable simulation errors. A new postsimulation module is proposed to track and visualize the signal evolution both in the k-space and in the image domain, which can be used for simulating image artifacts or finding frequency-response profiles. To illustrate the developed technique, three examples are presented: (1) fast off-resonance calculation for dictionary building in MR fingerprinting, (2) validation of a steady-state sequence with quasi-isotropic diffusion weighting, and (3) investigation of the magnetization evolution in PRESS-based spectroscopic imaging. The grid-merging algorithm of the proposed framework demonstrates high calculation efficiency exemplified by frequency-response simulation of pseudo steady-state or diffusion-weighted steady-state sequences. It further helps to visualize the signal evolution in PRESS-based sequences. The proposed simulation framework has been validated based on several different example applications for analyzing signal evolution in the frequency and spatial domain.

ON THE -ERROR LINEAR COMPLEXITY OF SEQUENCES FROM FUNCTION FIELDS

The linear complexity and the error linear complexity are two important security measures for stream ciphers. We construct periodic sequences from function fields and show that the error linear complexity of these periodic sequences is large. We also give a lower bound for the error linear complexity of a class of nonperiodic sequences.

An algorithm for periodicity and almost periodicity of uniform substitution sequences and its implications on aperiodic spin models

Substitution sequences are sequences generated by successive applications of deterministic substitution rules on n letters that are, in general, neither regular nor disordered. We propose a simple algorithm for classifying uniform substitutions sequences concerning periodicity and/or almost periodicity, based only on the substitution rules, whose implementation was recently registered. That classification is based on two theorems – that are also proved – about almost periodicity (extended for non-uniform substitutions) and periodicity, respectively. It recognizes automatically the character of the sequences concerning the periodicity and almost periodicity: purely periodic, ultimately periodic, weakly non periodic, and strongly non-periodic. That recognization selects the non-periodic sequences; among them, it also selects automatically the ones that promote relevance of geometrical fluctuations on the critical behavior of spin models in relation to the homogeneous system.

β- and δ-Al-Fe-Si intermetallic phase, their intergrowth and polytype formation

Crystal structure characteristics of coexisting β and δ phase were investigated in a purely intermetallic Al-Fe-Si alloy and in plate-shaped intermetallic particles formed in a model secondary Al7.1Si1.2Fe0.3Mn alloy. The β- and the δ-Al-Fe-Si intermetallic phase are thermodynamic and crystallographic distinct phases despite some structural similarities. Nevertheless, both phases can appear severely intergrown within single plate-shaped particles after solidification of Fe-containing secondary Al-Si alloys under non-equilibrium conditions which is attributed to the good fit along the (001) planes of both phases. Thereby, few layers of the δ-like stacking sequences within the β phase can be considered as a typical planar defect of the β phase. The β phase exhibits polytype structures for which structural models have been formulated based on ordering of the constituting double layers being shifted by a/2 or b/2 displacements leading to 4 different double layer positions A, B, C, and D. An AB polytype with idealized Acam symmetry actually shows a monoclinic distortion towards A12/a1 symmetry. An ABCD polytype with idealized I41/acd symmetry has been derived, too. Next to the ordered polytypes, stackings with non-periodic sequences of A, B, C and D double layers exist.

Rendimiento de Enfoques de Control Auto-Disparado


 
 
 El control auto-disparado produce secuencias de muestreo no periódico que varían dependiendo de factores de diseño relacionados con la estabilidad y el rendimiento del sistema a ser controlado. Dentro de este marco, recientemente han sido desarrollados dos enfoques dirigidos a minimizar un costo cuadrático, considerando un rendimiento óptimo y persiguiendo el mismo objetivo de control; cada una de ellos sigue una regla de muestreo diferente. Un enfoque se basa en mantener el valor de control actual tanto tiempo como sea posible, mientras que un umbral de rendimiento óptimo no se traspase. El otro enfoque se basa en la generación de una señal de control a trozos que se aproxima a una señal de control óptima continua, sujeta a determinadas limitaciones. Este artículo presenta un estudio comparativo entre los dos enfoques, proporcionando una percepción útil para la realización de futuras investigaciones. Como métricas de interés, el rendimiento de control y la utilización de recursos fueron considerados, y para evaluarlos, se hizo uso del intervalo de muestreo promedio y del costo normalizado. Se demostró que el diferente espacio de búsqueda de cada enfoque plantea un desafío para diseñar un marco de comparación equitativo, y que ambos enfoques superan al muestreo periódico. 
 
 

О ПРЕОБРАЗОВАНИЯХ ПЕРИОДИЧЕСКИХ ПОСЛЕДОВАТЕЛЬНОСТЕЙ

One of essential problems in generating pseudo-random numbers is the problem of periodicity of the resulting numbers. Some generators output periodic sequences. To avoid it several ways are used. Here we present the following approach: supposed we have some order in the considered set. Let's invent some algorithm which produces disorder in the set. E.g. if we have a periodic sequence of integers, let's construct an irrational number implying the given set. Then the figures of the resulting number form a non-periodic sequence. Here we can use continued fractions and Lagrange's theorem asserts that the resulting number is irrational. Another approach is to use series of the form \(\sum_{n=0}^\infty \frac{a_n}{n!}\) with a periodic sequence of integers \(\{a_n\}, a_{n+T}=a_n\) which is irrational. Here we consider polyadic series \(\sum_{n=0}^\infty a_n n!\) with a periodic sequence of positive integers \(\{a_n\},a_{n+T} = a_n\) and describe some of their properties.

Dynamics of red blood cells in oscillating shear flow

We present a three-dimensional computational study of fully deformable red blood cells of the biconcave resting shape subject to sinusoidally oscillating shear flow. A comprehensive analysis of the cell dynamics and deformation response is considered over a wide range of flow frequency, shear rate amplitude and viscosity ratio. We observe that the cell exhibits either a periodic motion or a chaotic motion. In the periodic motion, the cell reverses its orientation either by passing through the flow direction (horizontal axis) or by passing through the flow gradient (vertical axis). The chaotic dynamics is characterized by a non-periodic sequence of horizontal and vertical reversals. The study provides the first conclusive evidence of the chaotic dynamics of fully deformable cells in oscillating flow using a deterministic numerical model without the introduction of any stochastic noise. In certain regimes of the periodic motion, the initial conditions are completely forgotten and the cells become entrained in the same sequence of horizontal reversals. We show that chaos is only possible in certain frequency bands when the cell membrane can rotate by a certain amount, allowing the cells to swing near the maximum shear rate. As such, the bifurcation between the horizontal and vertical attractors in phase space always occurs via a swinging inflection. While the reversal sequence evolves in an unpredictable way in the chaotic regime, we find a novel result that there exists a critical inclination angle at the instant of flow reversal which determines whether a vertical or horizontal reversal takes place, and is independent of the flow frequency. The chaotic dynamics, however, occurs at a viscosity ratio less than the physiological values. We further show that the cell shape in oscillatory shear at large amplitude exhibits a remarkable departure from the biconcave shape, and that the deformation is significantly greater than that in steady shear flow. A large compression of the cells occurs during the reversals which leads to over/undershoots in the deformation parameter. We show that due to the large deformation experienced by the cells, the regions of chaos in parameter space diminish and eventually disappear at high shear rate, in contradiction to the prediction of reduced-order models. While the findings bolster support for reduced-order models at low shear rate, they also underscore the important role that the cell deformation plays in large-amplitude oscillatory flows.

Expansion complexity and linear complexity of sequences over finite fields

The linear complexity is a measure for the unpredictability of a sequence over a finite field and thus for its suitability in cryptography. In 2012, Diem introduced a new figure of merit for cryptographic sequences called expansion complexity. We study the relationship between linear complexity and expansion complexity. In particular, we show that for purely periodic sequences both figures of merit provide essentially the same quality test for a sufficiently long part of the sequence. However, if we study shorter parts of the period or nonperiodic sequences, then we can show, roughly speaking, that the expansion complexity provides a stronger test. We demonstrate this by analyzing a sequence of binomial coefficients modulo $p$. Finally, we establish a probabilistic result on the behavior of the expansion complexity of random sequences over a finite field.

A Corticostriatal Neural System Enhances Auditory Perception through Temporal Context Processing

The temporal context of an acoustic signal can greatly influence its perception. The present study investigated the neural correlates underlying perceptual facilitation by regular temporal contexts in humans. Participants listened to temporally regular (periodic) or temporally irregular (nonperiodic) sequences of tones while performing an intensity discrimination task. Participants performed significantly better on intensity discrimination during periodic than nonperiodic tone sequences. There was greater activation in the putamen for periodic than nonperiodic sequences. Conversely, there was greater activation in bilateral primary and secondary auditory cortices (planum polare and planum temporale) for nonperiodic than periodic sequences. Across individuals, greater putamen activation correlated with lesser auditory cortical activation in both right and left hemispheres. These findings suggest that temporal regularity is detected in the putamen, and that such detection facilitates temporal-lobe cortical processing associated with superior auditory perception. Thus, this study reveals a corticostriatal system associated with contextual facilitation for auditory perception through temporal regularity processing.

Return time complexity of Sturmian sequences

Let R n ( s ) be the recurrence time of the initial n -word of an infinite sequence s . We have that s is periodic if and only if there exists M such that R n ( s ) ≤ n for all n > M . For non-periodic sequences, therefore, R n ( s ) ≤ n + 1 is the possible lowest upper bound for R n . We show that if R n ( s ) ≤ n + 1 for all n , then s is Sturmian. The opposite does not hold in general and we completely classify such Sturmian sequences.

Electromagnetic Modes in Hybrid Periodic-non-periodic Dielectric Porous Silicon Multilayers

In this work, we study the main features of the spectra of electromagnetic modes propagating in porous Si layered systems. We consider the case of the combination of pe- riodic (Bragg-like) and quasiregular layered components, taking into a quasiregular-periodic- quasiregular geometry. The non-periodic sequences used to simulate the quasiregular regions are of the Fibonacci type. Normal incidence transfer matrix formalism is used to calculate the elec- tromagnetic modes as well as the electric fleld amplitudes corresponding to each of them. The existence of a selectively localized behavior of some optical modes in the structures is particularly discussed.

Balanced words and dynamical systems

This article is devoted to the description of all nonperiodic balanced words with n different letters. A superword W is called balanced if the numbers of equal letters in any two of its factors (subwords) u1 and u2 of equal length differ by at most 1. Balanced words are one of the possible generalizations of Sturmian words. We give a geometric interpretation of nonperiodic balanced sequences over an n-letter alphabet.

Integrality and the Laurent phenomenon for Somos 4 and Somos 5 sequences

AbstractSomos 4 sequences are a family of sequences defined by a fourth-order quadratic recurrence relation with constant coefficients. For particular choices of the coefficients and the four initial data, such recurrences can yield sequences of integers. Fomin and Zelevinsky have used the theory of cluster algebras to prove that these recurrences also provide one of the simplest examples of the Laurent phenomenon: all the terms of a Somos 4 sequence are Laurent polynomials in the initial data. The integrality of certain Somos 4 sequences has previously been understood in terms of the Laurent phenomenon. However, each of the authors of this paper has independently established the precise correspondence between Somos 4 sequences and sequences of points on elliptic curves. This connection is Here we show that these sequences satisfy a stronger condition than the Laurent property, and hence establish a broad set of sufficient conditions for integrality. As a by-product, non-periodic sequences provide infinitely many solutions of an associated quartic Diophantine equation in four variables. The analogous results for Somos 5 sequences are also presented, as well as various examples, including parameter families of Somos 4 integer sequences.

Playing off-line games with bounded rationality

We study a two-person zero-sum game where players simultaneously choose sequences of actions, and the overall payoff is the average of a one-shot payoff over the joint sequence. We consider the maxmin value of the game played in pure strategies by boundedly rational players and model bounded rationality by introducing complexity limitations. First we define the complexity of a sequence by its smallest period (a nonperiodic sequence being of infinite complexity) and study the maxmin of the game where player 1 is restricted to strategies with complexity at most n and player 2 is restricted to strategies with complexity at most m . We study the asymptotics of this value and a complete characterization in the matching pennies case. We extend the analysis of matching pennies to strategies with bounded recall.

The asymptotic behavior of N-adic complexity

We study the asymptotic behavior of stream cipher security mea-sures associated with classes of sequence generators such as linear feedback shift registers and feedback with carry shift registers. For nonperiodic sequences we consider normalized measures and study the set of accumulation points for a fixed sequence. We see that the set of accumulation points is always a closed subinterval of $[0, 1]$. For binary or ternary FCSRs we see that this interval is of the form $[B, 1-B]$, a result that is an analog of an earlier result by Dai, Jiang, Imamura, and Gong for LFSRs.

On a sequence related to that of Thue–Morse and its applications

It is known that the sequence 1 , 2 , 1 , 1 , 2 , 2 , 2 , 1 , 1 , 2 , 1 , 1 , 2 , 1 , 1 , 2 , 2 , … of lengths of blocks of identical symbols in the Thue–Morse sequence has several extremal properties among all non-periodic sequences of the symbols 1 and 2 . Its generating function W ( x ) is equal to ∏ k = 1 ∞ ( 1 + x ( 2 k + ( - 1 ) k - 1 ) / 3 ) . In terms of combinatorics on words, for any given x ∈ ( 0 , 1 ) and ε > 0 , we prove that every non-periodic word of an alphabet { 1 , 2 } has a suffix s whose generating function S ( x ) satisfies the inequality xS ( - x ) > 1 - W ( - x ) - ε . Using this, we prove several bounds for the largest and the smallest limit points of the sequence of fractional parts { ξ b n } , n = 0 , 1 , 2 , … , where b < - 1 is a negative rational number and ξ is a real number. Our results show, for example, that, for any real number ξ ≠ 0 , the sequence of fractional parts { ξ ( - 3 / 2 ) n } , n = 0 , 1 , 2 , … , has a limit point greater than 0.466452 . Furthermore, for each integer b ⩽ - 2 and each real number ξ ∉ Q , we prove that lim inf n → ∞ { ξ b n } ⩽ ∏ k = 1 ∞ ( 1 - | b | - ( 2 k + ( - 1 ) k - 1 ) / 3 ) and show that this inequality is sharp.

Non-Periodic Binary Sequences for Spread Spectrum Applications

For a multiple access communication system it is desirable to have a set of sequences such that (a) each sequence has as peaky an auto-correlation as possible and (b) each pair of sequence has as negligible a cross-correlation as possible. Peakiness of the auto-correlation of a sequence is measured in terms of its discrimination, which is to be maximized. The negligibility of a cross-correlation is judged on the basis of the energy in the cross-correlation which is to be minimized. The satisfaction of the criterion (a) is achievable by maximizing the discrimination of the sequence having its minimum value in the set. The performance along the criterion (b) is improved by seeking to minimize the maximum value of the energy of the cross correlation among all pairs of sequences. These are two established optimization problems, first of a maxmin type and the second of a minimax type. In this work they are combined into a single optimization problem by Simonization, a procedure introduced earlier for simpler situations. Sets of four binary sequences of lengths up to 1000 have been obtained by combinatorial search based on non-local optimization techniques such as Hamming scan, Backtracking, and Sidetracking, introduced earlier in simpler contexts.