Deterministic Sequences Research Articles

Gravity-guided snapping sequence in 3D modular multistable metamaterials

The snapping sequence of multistable metamaterials is critical for their applications in elastic wave control and energy release. Despite being a fundamental property, the effect of gravity on the snapping sequence has never been studied. This paper investigates the mechanical mechanism how structural gravity affects the snapping sequence of multistable metamaterials to construct deterministic static and dynamic snapping sequences. A biaxial snap-through availability three-dimensional (3D) modular multistable metamaterial (MMM) is developed. The 3D MMM is assembled from unit cells consisting of a dismountable middle bar (M-bar) and a fixed frame containing two bistable curved beams. Except experimental tests and numerical simulations, analytical analyses are also conducted to verify the snapping sequence induced by gravity in the 3D MMM. In addition, given that the M-bar is dismountable, the effects of its length on the mechanical properties and the impact resistance of the 3D MMM are discussed in detail. It is found that gravity can guide both static and dynamic deterministic snapping sequences of the 3D MMM to optimize the process of elastic wave propagation and energy release, and the proposed 3D MMM can enhance structural impact resistance through elastic deformations.

Minimal zero entropy subshifts can be unrestricted along any sparse set

Abstract We present a streamlined proof of a result essentially presented by the author in [Some counterexamples in topological dynamics. Ergod. Th. & Dynam. Sys.28(4) (2008), 1291–1322], namely that for every set $S = \{s_1, s_2, \ldots \} \subset \mathbb {N}$ of zero Banach density and finite set A, there exists a minimal zero-entropy subshift $(X, \sigma )$ so that for every sequence $u \in A^{\mathbb {Z}}$ , there is $x_u \in X$ with $x_u(s_n) = u(n)$ for all $n \in \mathbb {N}$ . Informally, minimal deterministic sequences can achieve completely arbitrary behavior upon restriction to a set of zero Banach density. As a corollary, this provides counterexamples to the polynomial Sarnak conjecture reported by Eisner [A polynomial version of Sarnak’s conjecture. C. R. Math. Acad. Sci. Paris353(7) (2015), 569–572] which are significantly more general than some recently provided by Kanigowski, Lemańczyk and Radziwiłł [Prime number theorem for analytic skew products. Ann. of Math. (2)199 (2024), 591–705] and by Lian and Shi [A counter-example for polynomial version of Sarnak’s conjecture. Adv. Math.384 (2021), Paper no. 107765] and shows that no similar result can hold under only the assumptions of minimality and zero entropy.

Probabilistic prediction and context tree identification in the Goalkeeper game

In this article we address two related issues on the learning of probabilistic sequences of events. First, which features make the sequence of events generated by a stochastic chain more difficult to predict. Second, how to model the procedures employed by different learners to identify the structure of sequences of events. Playing the role of a goalkeeper in a video game, participants were told to predict step by step the successive directions—left, center or right—to which the penalty kicker would send the ball. The sequence of kicks was driven by a stochastic chain with memory of variable length. Results showed that at least three features play a role in the first issue: (1) the shape of the context tree summarizing the dependencies between present and past directions; (2) the entropy of the stochastic chain used to generate the sequences of events; (3) the existence or not of a deterministic periodic sequence underlying the sequences of events. Moreover, evidence suggests that best learners rely less on their own past choices to identify the structure of the sequences of events.

Differential effects of bilateral hippocampal CA3 damage on the implicit learning and recognition of complex event sequences

ABSTRACT Learning regularities in the environment is a fundament of human cognition, which is supported by a network of brain regions that include the hippocampus. In two experiments, we assessed the effects of selective bilateral damage to human hippocampal subregion CA3, which was associated with autobiographical episodic amnesia extending ~50 years prior to the damage, on the ability to recognize complex, deterministic event sequences presented either in a spatial or a non-spatial configuration. In contrast to findings from related paradigms, modalities, and homologue species, hippocampal damage did not preclude recognition memory for an event sequence studied and tested at four spatial locations, whereas recognition memory for an event sequence presented at a single location was at chance. In two additional experiments, recognition memory for novel single-items was intact, whereas the ability to recognize novel single-items in a different location from that presented at study was at chance. The results are at variance with a general role of the hippocampus in the learning and recognition of complex event sequences based on non-adjacent spatial and temporal dependencies. We discuss the impact of the results on established theoretical accounts of the hippocampal contributions to implicit sequence learning and episodic memory.

What is the Simplest Linear Ramp?

We discuss conditions under which a deterministic sequence of real numbers, interpreted as the set of eigenvalues of a Hamiltonian, can exhibit features usually associated to random matrix spectra. A key diagnostic is the spectral form factor (SFF) — a linear ramp in the SFF is often viewed as a signature of random matrix behavior. Based on various explicit examples, we observe conditions for linear and power law ramps to arise in deterministic spectra. We note that a very simple spectrum with a linear ramp is En ~ log n. Despite the presence of ramps, these sequences do not exhibit conventional level repulsion, demonstrating that the lore about their concurrence needs refinement. However, when a small noise correction is added to the spectrum, they lead to clear level repulsion as well as the (linear) ramp. We note some remarkable features of logarithmic spectra, apart from their linear ramps: they are closely related to normal modes of black hole stretched horizons, and their partition function with argument s = β + it is the Riemann zeta function ζ(s). An immediate consequence is that the spectral form factor is simply −ζ|(it)|2. Our observation that log spectra have a linear ramp, is closely related to the Lindelöf hypothesis on the growth of the zeta function. With elementary numerics, we check that the slope of a best fit line through |ζ(it)|2 on a log-log plot is indeed 1, to the fourth decimal. We also note that truncating the Riemann zeta function sum at a finite integer N causes the would-be-eternal ramp to end on a plateau.

Destruction of CPE-normality along deterministic sequences

Destruction of CPE-normality along deterministic sequences

DECENTRALIZED CONDITIONAL GRADIENT METHOD ON TIME-VARIABLE GRAPHS

In this paper, we consider a generalization of the decentralized Frank-Wulff algorithm for network time variables, study the convergence properties of the algorithm, and carry out the corresponding numerical experiments. The changing network is modeled as a deterministic or stochastic sequence of graphs.

Laws of Large Numbers for Weighted Sums of Independent Random Variables: A Game of Mass

We consider weighted sums of independent random variables regulated by an increment sequence and provide operative conditions that ensure a strong law of large numbers for such sums in both the centred and non-centred case. The existing criteria for the strong law are either implicit or based on restrictions on the increment sequence. In our setup we allow for an arbitrary sequence of increments, possibly random, provided the random variables regulated by such increments satisfy some mild concentration conditions. In the non-centred case, convergence can be translated into the behaviour of a deterministic sequence and it becomes a game of mass when the expectation of the random variables is a function of the increment sizes. We identify various classes of increments and illustrate them with a variety of concrete examples.

Optimal pure strategies for a discrete search game

Consider a two-person zero-sum search game between a Hider and a Searcher. The Hider chooses to hide in one of n discrete locations (or “boxes”) and the Searcher chooses a search sequence specifying which order to look in these boxes until finding the Hider. A search at box i takes ti time units and finds the Hider—if hidden there—independently with probability qi, for i=1,…,n. The Searcher wants to minimize the expected total time needed to find the Hider, while the Hider wants to maximize it. It is shown in the literature that the Searcher has an optimal search strategy that mixes up to n distinct search sequences with appropriate probabilities. This paper investigates the existence of optimal pure strategies for the Searcher—a single deterministic search sequence that achieves the optimal expected total search time regardless of where the Hider hides. We identify several cases in which the Searcher has an optimal pure strategy, and several cases in which such optimal pure strategy does not exist. An optimal pure search strategy has significant practical value because the Searcher does not need to randomize their actions and will avoid second guessing themselves if the chosen search sequence from an optimal mixed strategy does not turn out well.

Investment certificates pricing using a Quasi-Monte Carlo framework: Case-studies based on the Italian market

The Monte Carlo method, thanks to its flexibility in designing even extremely complex payoffs, is assuming an increasingly important role in quantitative analysis. Its main limitation is the high computational cost linked to its modest speed of convergence to the fair value of the product. One of the best-known statistical techniques is to replace the random number generator with “low discrepancy” deterministic numerical sequences, producing a Quasi-Monte Carlo. Through its implementation for the analysis of three investment certificates featuring different characteristics and different stochastic processes used for the underlying simulation, the study demonstrates the possibility of achieving interesting results in terms of performance even for pricing these structured products ever more popular in the financial industry.

On arithmetic functions orthogonal to deterministic sequences

We prove the equivalence of Sarnak's conjecture on Möbius orthogonality with a Kolmogorov type property conjectured by Veech for Furstenberg systems of the Möbius function. This yields a combinatorial condition on the Möbius function itself which is equivalent to Sarnak's conjecture. As a matter of fact, our arguments remain valid in a larger context: we characterize all bounded arithmetic functions orthogonal to all topological systems whose all ergodic measures yield systems from a fixed characteristic class (zero entropy class is an example of such a characteristic class) with the characterization persisting in the logarithmic setup. As a corollary, we obtain that the logarithmic Sarnak's conjecture holds if and only if the logarithmic Möbius orthogonality is satisfied for all dynamical systems whose ergodic measures yield nilsystems.

On controllability and observability of a class of fractional-order switched systems with impulse

This paper is devoted to investigating two structural properties for a class of fractional-order switched and impulsive systems. The structural properties, i.e., observability and controllability, are explored mainly based on an algebraic approach. More specifically, firstly, according to the Laplace transform and mathematical induction, the general solution of such hybrid fractional-order systems is obtained over every impulsive interval. Next, applying the solution derived and relevant matrix theory, several necessary and sufficient controllability and observability criteria that take the form of a row of Gramian matrices are analytically established in terms of a deterministic impulse-switching time sequence. Resorting to the property of matrix Mittag-Leffler function, the developed controllability and observability Gramian criteria are further converted to some easy-test Kalman-type rank conditions. Finally, a numerical example illustrating the theoretical controllability and observability conditions is given.

Optimizing crop rotations via Parrondo's paradox for sustainable agriculture.

Crop rotation, a sustainable agricultural technique, has been at humanity's disposal since time immemorial and is practised globally. Switching between cover crops and cash crops helps avoid the adverse effects of intensive farming. Determining the optimum cash-cover rotation schedule for maximizing yield has been tackled on multiple fronts by agricultural scientists, economists, biologists and computer scientists, to name a few. However, considering the uncertainty due to diseases, pests, droughts, floods and impending effects of climate change is essential when designing rotation strategies. Analysing this time-tested technique of crop rotations with a new lens of Parrondo's paradox allows us to optimally use the rotation technique in synchrony with uncertainty. While previous approaches are reactive to the diversity of crop types and environmental uncertainties, we make use of the said uncertainties to enhance crop rotation schedules. We calculate optimum switching probabilities in a randomized cropping sequence and suggest optimum deterministic sequences and judicious use of fertilizers. Our methods demonstrate strategies to enhance crop yield and the eventual profit margins for farmers. Conforming to translational biology, we extend Parrondo's paradox, where two losing situations can be combined eventually into a winning scenario, to agriculture.

Stochasticity in the synchronization of strongly coupled spiking oscillators

Synchronization of electrical oscillators is a crucial step toward practical implementation of oscillator-based and bio-inspired computing. Here, we report the emergence of an unusual stochastic pattern in coupled spiking Mott nanodevices. Although a moderate capacitive coupling results in a deterministic alternating spiking, increasing the coupling strength leads counterintuitively to stochastic disruptions of the alternating spiking sequence. The disruptions of the deterministic spiking sequence are a direct consequence of the small intrinsic stochasticity in electrical triggering of the insulator–metal transition. Although the stochasticity is subtle in individual nanodevices, it becomes dramatically enhanced just in a single pair of coupled oscillators and, thus, dominates the synchronization. This is different from the stochasticity and multimodal coupling, appearing due to collective effects in large oscillator networks. The stochastic spiking pattern in Mott nanodevices results in a discrete inter-spike interval distribution resembling those in biological neurons. Our results advance the understanding of the emergent synchronization properties in spiking oscillators and provide a platform for hardware-level implementation of probabilistic computing and biologically plausible electronic devices.

Programming metastable transition sequences in digital mechanical materials

The realization of unconventional computing methods in soft matter has inspired the creation of mechanological systems capable of processing information. These systems often utilize bistable mechanisms that serve as mechanically abstracted bits to perform digital logic operations. Yet, the input of such operations often requires manual control of complex cyclic loading to reach a desired configuration. This research presents a method to design digital mechanical materials that can enter a programmable sequence of metastable configurations through simple displacement-controlled inputs. Interactions between serially connected bistable units enable the transition and reset behavior of mechanical bits. An analytical model using the principle of minimum total potential energy of a single bit is presented to articulate systematic ways to tune the critical force thresholds and displacements of metastable transition sequences. This work elucidates how the application of a prescribed displacement allows for novel resetting behavior that enables simultaneous transitions of multiple bits. By combining the principles of deterministic transition sequences and mechanical computing, a mechanical analog to digital conversion (ADC) material system is introduced that digitizes a continuous force stimulus into binary configurations that perform computations within the mechanical material based on the applied load. The mechanical ADC establishes a foundation for the integration of environmental sensing, information processing, and response in a soft material system.

Sharp global convergence guarantees for iterative nonconvex optimization with random data

We consider a general class of regression models with normally distributed covariates, and the associated nonconvex problem of fitting these models from data. We develop a general recipe for analyzing the convergence of iterative algorithms for this task from a random initialization. In particular, provided each iteration can be written as the solution to a convex optimization problem satisfying some natural conditions, we leverage Gaussian comparison theorems to derive a deterministic sequence that provides sharp upper and lower bounds on the error of the algorithm with sample splitting. Crucially, this deterministic sequence accurately captures both the convergence rate of the algorithm and the eventual error floor in the finite-sample regime, and is distinct from the commonly used “population” sequence that results from taking the infinite-sample limit. We apply our general framework to derive several concrete consequences for parameter estimation in popular statistical models including phase retrieval and mixtures of regressions. Provided the sample size scales near linearly in the dimension, we show sharp global convergence rates for both higher-order algorithms based on alternating updates and first-order algorithms based on subgradient descent. These corollaries, in turn, reveal multiple nonstandard phenomena that are then corroborated by extensive numerical experiments.

Research on simplification of branches method of accident sequences based on expert knowledge and heuristic optimization algorithm

The simplification of branches method of accident sequences based on expert knowledge and heuristic optimization algorithm is proposed in this paper, to increase the effectiveness of nuclear power plant safety margin analysis. First, the classification criteria for accident sequence and expert knowledge are used to screen out the deterministic failure sequences and deterministic safety sequences that are obviously. Secondly, a heuristic optimization method is used to study the variation in the accident sequences for those accident sequences that are challenging to evaluate with expert knowledge. Moreover, a hybrid particle swarm optimization (HPSO) is proposed to increase the accuracy of key safety parameters. This enables efficient accident sequence type identification, by comparing and evaluating key safety parameters and safety limits. Finally, the accident sequences are simplified using the cut-off frequency. To verify the effectiveness of the method, the small break loss of coolant (SBLOCA) accident in the nuclear power plant is analyzed in this paper. The numerical results show that the proposed HPSO improved the accuracy of solving the key safety parameters of accident sequences compared with the traditional method. By simplifying the accident sequences, the number of accident sequences requiring further analysis was reduced from 64 to 5.

Discriminative Transition Sequences of Origami Metamaterials for Mechanologic

Transitions of multistability in materials are exploited for various functions and applications, such as spectral gap tuning, impact energy trapping, and wave steering. However, a fundamental and comprehensive understanding of the transitions, either quasistatic or dynamic transitions, has not yet been acquired, especially in terms of the sequence predictability and tailoring mechanisms. This research, utilizing the stacked Miura‐ori‐variant (SMOV) structure that has multistable shape reconfigurability as a platform, uncovers the deep knowledge of quasistatic and dynamic transitions and proposes the corresponding versatile formation and tuning of mechanical logic gates. Through theoretical, numerical, and experimental means, discriminative and deterministic quasistatic transition sequences, including reversible and irreversible ones, are uncovered, where they constitute a transition map that is editable upon adjusting the design parameters. Via applying dynamic excitations and tailoring the excitation conditions, reversible transitions between all stable configurations become attainable, generating a fully connected transition map. Benefiting from the nonlinearity of the quasistatic and dynamic transitions, basic and compound mechanical logic gates are achieved. The versatility of the scheme is demonstrated using a single SMOV to realize different complex logic operations without increasing structural complexity, showing its unique computing power and inspiring the avenue for efficient physical intelligence.

Impaired executive functioning mediates the association between aging and deterministic sequence learning

ABSTRACT Sensitivity to the fixed ordering of actions and events, or deterministic sequence learning, is an important skill throughout adulthood. Yet, it remains unclear whether age deficits in sequencing exist, and we lack a firm understanding of which factors might contribute to age-related impairments when they arise. Though debated, executive functioning, governed by the frontal lobe, may underlie age-related sequence learning deficits in older adults. The present study asked if age predicts errors in deterministic sequence learning across the older adult lifespan (ages 55–89), and whether executive functioning accounts for any age-related declines. Healthy older adults completed a comprehensive measure of frontal-based executive abilities as well as a deterministic sequence learning task that required the step-by-step acquisition of associations through trial-and-error feedback. Among those who met a performance-based criterion, increasing age was positively correlated with higher sequencing errors; however, this relationship was no longer significant after controlling for executive functioning. Moreover, frontal-based executive abilities mediated the relationship between age and sequence learning performance. These findings suggest that executive or frontal functioning may underlie age deficits in learning judgment-based, deterministic serial operations.

Generating highly entangled states via discrete-time quantum walks with Parrondo sequences

Quantum entanglement has multiple applications in quantum information processing. Developing methods to generate highly entangled states independent of initial conditions is an essential task. Herein we aim to generate highly entangled states via discrete-time quantum walks. We propose deterministic Parrondo sequences that generate states that are generally much more entangled than states produced by sequences using only one of the two coins. We show that some Parrondo sequences generate highly entangled states, which are independent of the phase of the initial state used and further lead to maximally entangled states in some cases. We study Parrondo sequences for a small number of time steps and the asymptotic limit of a large number of time steps.