Consecutive Zeros Research Articles

A proof of concept for wear/non-wear classification using accelerometer data in daily activity recording: Synthetic algorithm leveraging probability and continuity of zero counts.

Wearable devices are increasingly utilized to monitor physical activity and sedentary behaviors. Accurately determining wear/non-wear time is complicated by zero counts, where the acceleration-based indexes do not estimate activity intensity, often leading to misclassifications. We propose a novel synthetic classification algorithm that leverages both the probability and continuity of zero counts, aiming to enhance the accuracy of activity estimation. The physical activity data were obtained from 12 office workers wearing wearable devices with 3-axis accelerometers. The wear/non-wear times are classified by the commonly used current method (zero counts lasting longer than 60 minutes are classified as non-wear) and the proposed method. In the proposed method, only times that satisfy the following two criteria are classified as the wear time. (1) The appearance probability preceding and following 60 minutes must be less than the threshold value. (2) The number of consecutive zeros must be less than 10 minutes. The effectiveness of both the current and proposed classification methods was evaluated against the actual behavioral records. This evaluation utilized simulation-based augmented data, which was implemented to address the limited variability inherent in the original dataset. The range of recall, specificity, precisions, and accuracy classified by the current method were 0.93-1.00, 0.93-0.96, 0.85-0.88, and 0.94-0.97, respectively. Indeed, the proposed method shows 0.95-1.00, 0.99-1.00, 0.97-1.00, and 0.98-1.00, respectively. The reduction of misclassification of non-wear time as wear time was achieved by the synthetic classification algorithm. The performance of the proposed approach showed accurate classification of the wear/non-wear time of wearable sensors in office workers.

Perennial and Non‐Perennial Streamflow Regime Shifts Across California, USA

AbstractDespite rises in drought frequency and human water demands, streamflow regime shifts from perennial to non‐perennial have not been evaluated in many arid/semi‐arid regions. To document shifts, we created a methodology that classifies streams as naturally perennial or non‐perennial. Our classification used historical, minimally disturbed‐quality USGS streamflow gages (1950–2015) across California. The number of consecutive zero flow days (≥5 days) was used to classify 61% (96/158) and 39% (62/158) of gages as perennial and non‐perennial, respectively. We developed a random forest model to predict flow regime class based on climate and watershed characteristics. To identify regime shifts, we compared the observed class of contemporary (1980–2023) minimally disturbed and disturbed gages with their modeled, natural class. For most minimally disturbed gages, the observed and natural predicted classes were the same, but 13% (7/52) of gages had a modeled perennial regime with an observed non‐perennial class, indicating a drying trend. Among disturbed gages, 22% (64/290) shifted from perennial to non‐perennial and 7% (21/290) from non‐perennial to perennial. Trends in the minimum 7‐day moving average and number of zero‐flow days provided further evidence of drying at minimally disturbed streams, but no pattern at disturbed gages. Our results indicate that few minimally disturbed perennial streams have become non‐perennial to date, but many streams have experienced drying from climate. Streams impacted by human activities had greater drying rates, but regulation has caused some non‐perennial streams to become perennial. By quantifying expected natural streamflow regimes, this work can help monitor, manage, and conserve stream ecosystems.

Concentration and non-concentration of eigenfunctions of second-order elliptic operators in layered media

This work is concerned with operators of the type A=-\tilde{c}\Delta acting in domains \Omega\coloneq \Omega'\times (0,H)\subseteq\mathbb{R}^{d}\times\mathbb{R}_{+}. The diffusion coefficient \tilde{c}>0 depends on one coordinate y\in (0,H) and is bounded but may be discontinuous. This corresponds to the physical model of “layered media,” appearing in acoustics, elasticity, optical fibers… . Dirichlet boundary conditions are assumed. In general, for each \varepsilon>0, the set of eigenfunctions is divided into a disjoint union of three subsets: \mathfrak{F}_{NG} (non-guided), \mathfrak{F}_{G} (guided) and \mathfrak{F}_{\textup{res}} (residual). The residual set shrinks as \varepsilon \to 0. The customary physical terminology of guided/non-guided is often replaced in the mathematical literature by concentrating/non-concentrating solutions, respectively. For guided waves, the assumption of “layered media” enables us to obtain rigorous estimates of their exponential decay away from concentration zones. The case of non-guided waves has attracted less attention in the literature. While it is not so closely connected to physical models, it leads to some very interesting questions concerning oscillatory solutions and their asymptotic properties. Classical asymptotic methods are available for c(y)\in C^{2} but a lesser degree of regularity excludes such methods. The associated eigenfunctions (in \mathfrak{F}_{NG} ) are oscillatory. However, this fact by itself does not exclude the possibility of “flattening out” of the solution between two consecutive zeros, leading to concentration in the complementary segment. Here we show it cannot happen when c(y) is of bounded variation, by proving a “minimal amplitude hypothesis.” However the validity of such results when c(y) is not of bounded variation (even if it is continuous) remains an open problem.

New results for the upper bounds of the distance between adjacent zeros of first-order differential equations with several variable delays

The distance between consecutive zeros of a first-order differential equation with several variable delays is studied. Here, we show that the distribution of zeros of differential equations with variable delays is not an easy extension of the case of constant delays. We obtain new upper bounds for the distance between zeros of all solutions of a differential equation with several delays, which extend and improve some existing results. Two illustrative examples are given to show the advantages of the proposed results over the known ones.

On the Montgomery–Odlyzko method regarding gaps between zeros of the zeta-function

Assuming the Riemann Hypothesis, it is known that there are infinitely many consecutive pairs of zeros of the Riemann zeta-function within 0.515396 times the average spacing. This is obtained using the method of Montgomery and Odlyzko. We prove that this method can never find infinitely many pairs of consecutive zeros within 0.5042 times the average spacing.

A note on the zeros of the derivatives of Hardy's function Z(t)$Z(t)$

AbstractUsing the twisted fourth moment of the Riemann zeta‐function, we study large gaps between consecutive zeros of the derivatives of Hardy's function , improving upon previous results of Conrey and Ghosh (J. Lond. Math. Soc. 32 (1985) 193–202), and of the second named author (Acta Arith. 111 (2004) 125–140). We also exhibit small distances between the zeros of and the zeros of for every , in support of our numerical observation that the zeros of and , when k and ℓ have the same parity, seem to come in pairs that are very close to each other. The latter result is obtained using the mollified discrete second moment of the Riemann zeta‐function.

APPROXIMATION ORDERS OF A REAL NUMBER IN A FAMILY OF BETA-DYNAMICAL SYSTEMS

In this paper, we study the approximation orders of a real number [Formula: see text] by the partial sums of its [Formula: see text]-expansions as [Formula: see text] varies in the parameter space [Formula: see text]. More precisely, letting [Formula: see text] be the partial sum of the first [Formula: see text] items of the [Formula: see text]-expansion of [Formula: see text], we prove that for any real number [Formula: see text], the approximation order of [Formula: see text] by [Formula: see text] is [Formula: see text] for Lebesgue almost all [Formula: see text]. Moreover, we obtain the size of the set of [Formula: see text] for which [Formula: see text] can be approximated with a more general order [Formula: see text], where [Formula: see text] is a positive function. We also determine the Hausdorff dimension of the set [Formula: see text] where [Formula: see text] is the number of the longest consecutive zeros just after the [Formula: see text]th digit in the [Formula: see text]-expansion of [Formula: see text].

On the distribution of adjacent zeros of solutions to first-order neutral differential equations

The purpose of this paper is to study the distribution of zeros of solutions to a first-order neutral differential equation of the form \begin{equation*} \left[x(t) + p(t) x(t-\tau)\right]' + q(t) x(t-\sigma) = 0, \quad t \geq t_0, \end{equation*} where $p\in C([t_0,\infty),[0,\infty))$, $q \in C([t_0,\infty),(0,\infty))$, $\tau,\sigma>0$, and $\sigma>\tau$. We obtain new upper bound estimates for the distance between consecutive zeros of solutions, which improve upon many of the previously known ones. The results are formulated so that they can be generalized without much effort to equations for which the distribution of zeros problem is related to the study of this property for a first-order delay differential inequality. The strength of our results is demonstrated viatwo illustrative examples.

Multiplicity of Solutions for an Elliptic Kirchhoff Equation

In this paper we study the existence of positive solution to the Kirchhoff elliptic problem -1+γG′‖∇u‖L2(Ω)2Δu=λf(u)inΩ,u=0on∂Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} \\displaystyle -\\left( 1+\\gamma G'\\left( \\Vert \ abla u\\Vert ^2_{L^2(\\Omega )}\\right) \\right) \\Delta u = \\lambda f(u) &{} \ ext{ in } \\; \\Omega ,\\\\ u = 0 &{} \ ext{ on } \\; \\partial \\Omega ,\\\\ \\end{array}\\right. } \\end{aligned}$$\\end{document}where Omega is an open, bounded subset of mathbb {R}^N (Nge 3), f is a locally Lipschitz continuous real function, f(0)ge 0, G'in C(mathbb {R}^+) and G'ge 0. We prove the existence of at least two solutions with L^infty (Omega ) norm between two consecutive zeroes of f for large lambda .

The Use of Accelerometers in Young Children: A Methodological Scoping Review

The purpose of this scoping review was to examine publications using accelerometers in children aged 6 months to <6 years and report on current methodologies used for data collection and analyses. We examined device make and model, device placement, sampling frequency, data collection protocol, definition of nonwear time, inclusion criteria, epoch duration, and cut points. Five online databases and three gray literature databases were searched. Studies were included if they were published in English between January 2009 and March 2021. A total of 627 articles were included for descriptive analyses. Of the reviewed articles, 75% used ActiGraph devices. The most common device placement was hip or waist. More than 80% of articles did not report a sampling frequency, and 7-day protocols during only waking hours were the most frequently reported. Fifteen-second epoch durations and the cut points developed by Pate et al. in 2006 were the most common. A total of 203 articles did not report which definition of nonwear time was used; when reported, “20 minutes of consecutive zeros” was the most frequently used. Finally, the most common inclusion criteria were “greater or equal to 10 hr/day for at least 3 days” for studies conducted in free-living environments and “greater than 50% of the school day” for studies conducted in preschool or childcare environments. Results demonstrated a major lack of reporting of methods used to analyze accelerometer data from young children. A list of recommended reporting practices was developed to encourage increased reporting of key methodological details for research in this area.

Sedentary And Physical Activity Patterns In Children With Intellectual Disabilities: Differences Between Levels Of Intellectual Disability

PURPOSE: To analyze the potential severity difference of children with intellectual disabilities in sedentary patterns and physical activity (PA) levels. METHODS: A cross-sectional descriptive study was completed by 153 children with mild to severe ID (mild ID = 48, moderate ID =60, severe ID = 45) aged 8.0 to 14.0 years. Children wore an ActiGraph GT3+ accelerometer over a consecutive 7 days to assess minutes per day of sedentary time (ST), light physical activity (LPA) and moderate to vigorous physical activity (MVPA). A valid day was defined as recording at 8 hours of measured wear time between 7:00 am and 20:00 pm and non-wear time was defined as >60 min of consecutive zeros from Boddy’ s finding. The intensity cutoffs of the measured PA levels are as follows: LPA (<2,800 count per min (CPM)), MVPA (>4000 CPM) recommended for this population. Multivariate analysis of covariance (MANCOVA) was used to compare the ST and PA levels between different level of ID. RESULTS: Controlling for participants gender, age and BMI. ID level was significantly associated with ST, LPA and MVPA among children with ID, Pillai’s λ = 0.53, F(6,290) = 18.379, η2 = 0.28, p < 0.01. Children with severe ID (711.06 ± 28.84 min/day) had significantly longer ST than those with moderate ID (699.86 ± 123.94 min/day) and those with mild ID (642.64 ± 26.77 min/day). Children with severe ID (60.18 ± 22.18 min/day) had significantly lower LPA than those with moderate ID (76.59 ± 25.82 min/day) and those with mild ID (108.05 ± 26.54 min/day). Children with severe ID (20.75 ± 6.98 min/day) had significantly lower MVPA than those with moderate ID (38.55 ± 17.14 min/day) and those with mild ID (43.95 ± 14.70 min/day). CONCLUSIONS: ST, LPA and MVPA differ significantly among children with different levels of ID, with severer ID level had longer ST and lower PA levels. These findings suggest that population specific interventions are critical to promote physical activity levels and improve health in children with different levels of ID.

On explicit estimates for S(t), S1(t), and [formula omitted] under the Riemann Hypothesis

Assuming the Riemann Hypothesis, we provide explicit upper bounds for moduli of S(t), S1(t), and ζ(1/2+it) while comparing them with recently proven unconditional ones. As a corollary we obtain a conditional explicit bound on gaps between consecutive zeros of the Riemann zeta-function.

A Scalable Bidimensional Randomization Scheme for TLC 3D NAND Flash Memories.

Data randomization has been a widely adopted Flash Signal Processing technique for reducing or suppressing errors since the inception of mass storage platforms based on planar NAND Flash technology. However, the paradigm change represented by the 3D memory integration concept has complicated the randomization task due to the increased dimensions of the memory array, especially along the bitlines. In this work, we propose an easy to implement, cost effective, and fully scalable with memory dimensions, randomization scheme that guarantees optimal randomization along the wordline and the bitline dimensions. At the same time, we guarantee an upper bound on the maximum length of consecutive ones and zeros along the bitline to improve the memory reliability. Our method has been validated on commercial off-the-shelf TLC 3D NAND Flash memory with respect to the Raw Bit Error Rate metric extracted in different memory working conditions.

Localization of Fučík curves for the second order discrete Dirichlet operator

In this paper, we deal with the second order difference equation with asymmetric nonlinearities on the integer lattice and we investigate the distribution of zeros for continuous extensions of positive semi-waves. The distance between two consecutive zeros of two different positive semi-waves depends not only on the parameters of the problem but also on the position of one of these zeros with respect to the integer lattice. We provide an explicit formula for this distance, which allows us to obtain a new simple implicit description of all non-trivial Fučík curves for the discrete Dirichlet operator. Moreover, for fixed parameters of the problem, we show that this distance is bounded and attains its global extrema that are explicitly described in terms of Chebyshev polynomials of the second kind. Finally, for each non-trivial Fučík curve, we provide suitable bounds by two curves with a simple description similar to the description of the first non-trivial Fučík curve.

Timing-Optimized Hardware Implementation to Accelerate Polynomial Multiplication in the NTRU Algorithm

Post-quantum cryptographic algorithms have emerged to secure communication channels between electronic devices faced with the advent of quantum computers. The performance of post-quantum cryptographic algorithms on embedded systems has to be evaluated to achieve a good trade-off between required resources (area) and timing. This work presents two optimized implementations to speed up the NTRUEncrypt algorithm on a system-on-chip. The strategy is based on accelerating the most time-consuming operation that is the truncated polynomial multiplication. Hardware dedicated modules for multiplication are designed by exploiting the presence of consecutive zeros in the coefficients of the blinding polynomial. The results are validated on a PYNQ-Z2 platform that includes a Zynq-7000 SoC from Xilinx and supports a Python-based programming environment. The optimized version that exploits the presence of double, triple, and quadruple consecutive zeros offers the best performance in timing, in addition to considerably reducing the possibility of an information leakage against an eventual attack on the device, making it practically negligible.

Hausdorff dimension of frequency sets of univoque sequences

We study the set consisting of univoque sequences, the set consisting of sequences in which the lengths of consecutive zeros and consecutive ones are bounded, and their frequency subsets , , and , , consisting of sequences respectively in and with frequency, lower frequency and upper frequency of zeros equal to some . The Hausdorff dimension of all these sets are obtained by studying the dynamical system where is the shift map and for integer , studying the Bernoulli-type measures on and finding out the unique equivalent -invariant ergodic probability measure.

Явный вид распределения избранных двумерных и трехмерных статистик (0,1)-последовательности

Рассмотрены совместимые распределения заданного числа 2-цепочек и заданного числа 3-цепочек фиксированного вида случайной битовой последовательности, позволяющие осуществлять статистический анализ локальных участков этой последовательности. В качестве 2-цепочек выступают все конфигурации, состоящие из двух подряд или нулей, или единиц битовой последовательности заданной длины. В свою очередь, 3-цепочками являются все конфигурации, состоящие из трех подряд или единиц (при условии, что 2-цепочки являются нулевыми), или нулей (при условии, что 2-цепочки единичны), а также в качестве 3 -цепочек рассматриваются все конфигурации, состоящие либо из трех подряд цифр: один, ноль и один (при условии, что 2-цепочки нулевые), либо из трех подряд цифр: ноль, один и ноль (при условии, что 2-цепочки единичные). Установлены явные выражения двумерных и трехмерных совместимых распределений событий, отражающих количество некоторых комбинаций указанных цепочек в конечной случайной битовой последовательности. Одно из основных предположений состоит в том, что нули и единицы в битовой последовательности — независимые одинаково распределенные случайные величины. Доказательства формул для распределений указанных событий построены на подсчете числа ответных благоприятных событий при условии, что битовая последовательность содержит фиксированное количество нулей и единиц. В качестве примеров использования явных выражений совместимых распределений приведены таблицы, в которых размещены значения вероятностей вышеперечисленных событий для случайной битовой последовательности длины 40 (табл. 1–3) и длины 24 (табл. 4) для некоторых фиксированных значений числа 2-цепочек и числа 3-цепочек в предположении, что нули и единицы появляются независимо и равновероятно. табл. 1–3 проиллюстрированы пузырьковыми диаграммами. Найденные формулы могут представлять интерес для задач тестирования локальных участков, формируемых на выходе генераторов псевдослучайных чисел, для некоторых задач защиты информации от несанкционированного доступа, а также в других сферах, где возникает необходимость в анализе битовых последовательностей.

The Two-Layer Hierarchical Distribution Model of Zeros of Riemann’s Zeta Function along the Critical Line

This article numerically analyzes the distribution of the zeros of Riemann’s zeta function along the critical line (CL). The zeros are distributed according to a hierarchical two-layered model, one deterministic, the other stochastic. Following a complex plane anamorphosis involving the Lambert function, the distribution of zeros along the transformed CL follows the realization of a stochastic process of regularly spaced independent Gaussian random variables, each linked to a zero. The value of the standard deviation allows the possible overlapping of adjacent realizations of the random variables, over a narrow confidence interval. The hierarchical model splits the ζ function into sequential equivalence classes, with the range of probability densities of realizations coinciding with the spectrum of behavioral styles of the classes. The model aims to express, on the CL, the coordinates of the alternating cancellations of the real and imaginary parts of the ζ function, to dissect the formula for the number of zeros below a threshold, to estimate the statistical laws of two consecutive zeros, of function maxima and moments. This also helps explain the absence of multiple roots.

Probabilistic Forecasting Based Joint Detection and Imputation of Clustered Bad Data in Residential Electricity Loads

Residential electricity load data can include numerous types of bad data, even clustered bad data, as they that are typically captured by simple measurement instruments. For example, in the case of a time-series of Not-a-Number (NaN) errors, the values before or next to a NaN may appear as the sum of actual values during the times of the NaN series. To utilize load data that includes such erroneous data for prediction or data mining analysis, customized detection and imputation should be conducted. This study proposes a new joint detection and imputation method for handling clustered bad data in residential electricity loads. Examples of these data are known invalid data points, such as consecutive NaN or zero values followed by or being ahead of an outlier. The proposed joint detection and imputation scheme first investigates the neighbors of the invalid data points, using probabilistic forecasting techniques. These techniques are implemented by the next valid neighbors to determine whether there is an anomaly or not. Then, adaptive imputations are applied on the basis of the detection, the candidate point should be imputed simultaneously or not. To assess the potential of the newly proposed scheme to characterize the clustered bad data, we analyzed the electricity loads of 354 households. Moreover, joint detection and imputations are conducted to test with the randomly injected synthesized clustered bad data (containing NaNs of various lengths) that is followed by the summation of the actual NaN values. The proposed scheme succeeded in detecting clustered bad data with an accuracy of 95.5% and a false alarm rate of 3.6% for all households in the dataset. Outlier detection-assisted imputation schemes are evaluated for NaNs with optional outliers. Results demonstrate that these schemes improve the overall accuracy significantly compared to schemes without outlier detection.

Identifying ActiGraph non-wear time in pregnant women with overweight or obesity

ObjectivesNon-wear time algorithms have not been validated in pregnant women with overweight/obesity (PW-OW/OB), potentially leading to misclassification of sedentary/activity data, and inaccurate estimates of how physical activity is associated with pregnancy outcomes. We examined: (1) validity/reliability of non-wear time algorithms in PW-OW/OB by comparing wear time from five algorithms to a self-report criterion and (2) whether these algorithms over- or underestimated sedentary behaviors. DesignPW-OW/OB (N = 19) from the Healthy Mom Zone randomized controlled trial wore an ActiGraph GT3x + for 7 consecutive days between 8–12 weeks gestation. MethodsNon-wear algorithms (i.e., consecutive strings of zero acceleration in 60-second epochs) were tested at 60, 90, 120, 150, and 180-min. The monitor registered sedentary minutes as activity counts 0−99. Women completed daily self-report logs to report wear time. ResultsIntraclass correlation coefficients for each algorithm were 0.96−0.97; Bland–Altman plots revealed no bias; mean absolute percent errors were <10%. Compared to self-report (M = 829.5, SD = 62.1), equivalency testing revealed algorithm wear times (min/day) were equivalent: 60- (M = 816.4, SD = 58.4), 90- (M = 827.5, SD = 61.4), 120- (M = 830.8, SD = 65.2), 150- (M = 833.8, SD = 64.6) and 180-min (M = 837.4, SD = 65.4). Repeated measures ANOVA showed 60- and 90-min algorithms may underestimate sedentary minutes compared to 150- and 180-min algorithms. ConclusionsThe 60, 90, 120, 150, and 180-min algorithms are valid and reliable for estimating wear time in PW-OW/OB. However, implementing algorithms with a higher threshold for consecutive zero counts (i.e., ≥150-min) can avoid the risk of misclassifying sedentary data.