Consecutive Zeros Research Articles

A comparison of 10 accelerometer non-wear time criteria and logbooks in children

BackgroundThere are many unresolved issues regarding data reduction algorithms for accelerometry. The choice of criterion for removal of non-wear time might have a profound influence on physical activity (PA) and sedentary time (SED) estimates. The aim of the present study was to compare 10 different non-wear criteria and a log of non-wear periods in 11-year-old children.MethodsChildren from the Active Smarter Kids study performed 7-days of hip-worn accelerometer monitoring (Actigraph GT3X+) and logged the number of non-wear periods each day, along with the approximate duration and reason for non-wear. Accelerometers were analyzed using 10 different non-wear criteria: ≥ 10, 20, 30, 45, 60, and 90 min of consecutive zero counts without allowance for interruptions, and ≥60 and 90 min with allowance for 1 and 2 min of interruptions.Results891 children provided 5203 measurement days, and reported 1232 non-wear periods ranging from 0 to 3 periods per day: on most days children reported no non-wear periods (77.1% of days). The maximum number of non-wear periods per day was 2 for the 90-min criterion, 3 to 5 for most criteria, 7 for the 20-min criterion, and 20 for the 10-min criterion. The non-wear criteria influenced overall PA (mean values across all criteria: 591 to 649 cpm; 10% difference) and SED time (461 to 539 min/day; 17% difference) estimates, especially for the most prolonged SED bouts. Estimates were similar for time spent in intensity-specific (light, moderate, vigorous, and moderate-to-vigorous) PA, but varied 6–9% among the non-wear criteria for proportions of time spent in intensity-specific PA (% of total wear time).ConclusionsPopulation level estimates of PA and SED differed between different accelerometer non-wear criteria, meaning that non-wear time algorithms should be standardized across studies to reduce confusion and improve comparability of children’s PA level. Based on the numbers and reasons for non-wear periods, we suggest a 45 or 60-min consecutive zero count-criterion not allowing any interruptions to be applied in future pediatric studies, at least for children older than 10 years.Trial RegistrationThe study is registered in Clinicaltrials.gov with identification number NCT02132494. Registered 7 April 2014.

Linear Sturm-Liouville problems with Riemann-Stieltjes integral boundary conditions

We study second-order linear Sturm-Liouville problems involving general homogeneous linear Riemann-Stieltjes integral boundary conditions. Conditions are obtained for the existence of a sequence of positive eigenvalues with consecutive zero counts of the eigenfunctions. Additionally, we find interlacing relationships between the eigenvalues of such Sturm-Liouville problems and those of Sturm-Liouville problems with certain two-point separated boundary conditions.

Identification of minimal parameters for optimal suppression of chaos in dissipative driven systems

Taming chaos arising from dissipative non-autonomous nonlinear systems by applying additional harmonic excitations is a reliable and widely used procedure nowadays. But the suppressory effectiveness of generic non-harmonic periodic excitations continues to be a significant challenge both to our theoretical understanding and in practical applications. Here we show how the effectiveness of generic suppressory excitations is optimally enhanced when the impulse transmitted by them (time integral over two consecutive zeros) is judiciously controlled in a not obvious way. Specifically, the effective amplitude of the suppressory excitation is minimal when the impulse transmitted is maximum. Also, by lowering the impulse transmitted one obtains larger regularization areas in the initial phase difference-amplitude control plane, the price to be paid being the requirement of larger amplitudes. These two remarkable features, which constitute our definition of optimum control, are demonstrated experimentally by means of an analog version of a paradigmatic model, and confirmed numerically by simulations of such a damped driven system including the presence of noise. Our theoretical analysis shows that the controlling effect of varying the impulse is due to a subsequent variation of the energy transmitted by the suppressory excitation.

THE EXCEPTIONAL SETS ON THE RUN-LENGTH FUNCTION OF β-EXPANSIONS

Let [Formula: see text] and the run-length function [Formula: see text] be the maximal length of consecutive zeros amongst the first [Formula: see text] digits in the [Formula: see text]-expansion of [Formula: see text]. The exceptional set [Formula: see text] is investigated, where [Formula: see text] is a monotonically increasing function with [Formula: see text]. We prove that the set [Formula: see text] is either empty or of full Hausdorff dimension and residual in [Formula: see text] according to the increasing rate of [Formula: see text].

Gaps between zeros of the Riemann zeta-function

We prove that there exist infinitely many consecutive zeros of the Riemann zeta-function on the critical line whose gaps are greater than $3.18$ times the average spacing. Using a modification of our method, we also show that there are even larger gaps between the multiple zeros of the zeta function on the critical line (if such zeros exist).

Lehmer pairs and derivatives of Hardy's Z-function

Occurrences of very close zeros of the Riemann zeta function are strongly connected with Lehmer pairs and with the Riemann Hypothesis. The aim of the present note is to derive a condition for a pair of consecutive simple zeros of the ζ-function to be a Lehmer pair in terms of derivatives of Hardy's Z-function. Furthermore, we connect Newman's conjecture with stationary points of the Z-function, and present some numerical results.

The distribution of zeros of ζ′(s) and gaps between zeros of ζ(s)

Assume the Riemann Hypothesis. We establish a local structure theorem for zeros of the Riemann zeta-function ζ(s) and its derivative ζ′(s). As an application, we prove a stronger form of half of a conjecture of Radziwiłł [18] concerning the global statistics of these zeros. Roughly speaking, we show that on the Riemann Hypothesis, if there occurs a small gap between consecutive zeta zeros, then there is exactly one zero of ζ′(s) lying not only very close to the critical line but also between that pair of zeta zeros. This refines a result of Zhang [22]. Some related results are also shown. For example, we prove a weak form of a conjecture of Soundararajan, and suggest a repulsion phenomena for zeros of ζ′(s).

Satisfiability modulo theory and binary puzzle

The binary puzzle is a sudoku-like puzzle with values in each cell taken from the set {0, 1}. We look at the mathematical theory behind it. A solved binary puzzle is an n × n binary array where n is even that satisfies the following conditions:(1) No three consecutive ones and no three consecutive zeros in each row and each column,(2) Every row and column is balanced, that is the number of ones and zeros must be equal in each row and in each column,(3) Every two rows and every two columns must be distinct.The binary puzzle had been proven to be an NP-complete problem [5].Research concerning the satisfiability of formulas with respect to some background theory is called satisfiability modulo theory (SMT). An SMT solver is an extension of a satisfiability (SAT) solver. The notion of SMT can be used for solving various problem in mathematics and industries such as formula verification and operation research [1, 7].In this paper we apply SMT to solve binary puzzles. In addition, we do an experiment in solving different sizes and different number of blanks. We also made comparison with two other approaches, namely by a SAT solver and exhaustive search.

Solving a Binary Puzzle

A Binary puzzle is a Sudoku-like puzzle with values in each cell taken from the set { 0, 1}. Let n ge 4 be an even integer, a solved binary puzzle is an n times n binary array that satisfies the following conditions: (1) no three consecutive ones and no three consecutive zeros in each row and each column; (2) the number of ones and zeros must be equal in each row and in each column; (3) there can be no repeated row and no repeated column. This paper proposes three approaches to solve the puzzle. The first method is based on a complete backtrack-based search algorithm. The idea is to propagate and fill an unsolved binary puzzle according to the three constraints, followed by a random guess if the puzzle remains unsolved. The second method of solving a binary puzzle is by representing it as an instance of a Boolean satisfiability problem which allows the solution for a binary puzzle to be obtained using SAT solvers. The third approach is based on expressing a binary puzzle as a system of polynomial equations over the binary field mathbb {F}_2. The set of solutions for the equation system implies the solutions for the binary puzzle and it is obtained by computing a Gröbner basis of the ideal generated by the polynomials. We experimentally compare the three approaches with binary puzzles of various sizes and different numbers of empty cells using a computer algebra system.

The search for a Hamiltonian whose energy spectrum coincides with the Riemann zeta zeroes

Inspired by the Hilbert–Polya proposal to prove the Riemann Hypothesis we have studied the Schroedinger QM equation involving a highly nontrivial potential, and whose self-adjoint Hamiltonian operator has for its energy spectrum one which approaches the imaginary parts of the zeta zeroes only in the asymptotic (very large [Formula: see text]) region. The ordinates [Formula: see text] are the positive imaginary parts of the nontrivial zeta zeroes in the critical line :[Formula: see text]. The latter results are consistent with the validity of the Bohr–Sommerfeld semi-classical quantization condition. It is shown how one may modify the parameters which define the potential, and fine tune its values, such that the energy spectrum of the (modified) Hamiltonian matches not only the first two zeroes but the other consecutive zeroes. The highly nontrivial functional form of the potential is found via the Bohr–Sommerfeld quantization formula using the full-fledged Riemann–von Mangoldt counting formula (without any truncations) for the number [Formula: see text] of zeroes in the critical strip with imaginary part greater than [Formula: see text] and less than or equal to [Formula: see text].

Some remarks on the differences between ordinates of consecutive zeta zeros

If $0 < \gamma_1 \le \gamma_2 \le \gamma_3 \le \ldots$ denote ordinates of complex zeros of the Riemann zeta-function $\zeta(s)$, then several results involving the maximal order of $\gamma_{n+1}-\gamma_n$ and the sum $$ \sum_{0<\gamma_n\le T}{(\gamma_{n+1}-\gamma_n)}^k \qquad(k>0) $$ are proved.

Evaluation of oscillatory integrals for analytical groundwater flow and mass transport models

Modeling of transient dynamics of an interface between fluids of identical density and viscosity, but different otherwise, is of great interest in aquifer hydraulic, and advective contaminant transport, and has broad application. Closed-form solutions are often available for problems with simple, practically important geometry, but the integrals that appear in such solutions often have integrands with two or more oscillatory factors. Such integrals pose difficulties for numerical evaluation because the positive and negative contributions of the integrand largely cancel and the integrands decay very slowly in the integration domain. Some problems with integrands with a single oscillatory factor were tackled in the past with an integration/summation/extrapolation (ISE) method: breaking the integrand at consecutive zeros to obtain an alternating series and then using the Shanks algorithm to accelerate convergence of the series. However, this technique is ineffective for problems with multiple oscillatory factors. We present a comprehensive strategy for evaluation of such integrals that includes a better ISE method, an interval truncation method, and long-time asymptotics; this strategy is applicable to a large class of integrals with either single or multiple oscillatory factors that arise in modeling of groundwater flow and transport. The effectiveness of this methodology is illustrated by examples of integrals used in well hydraulics, groundwater recharge design, and particle tracking.

The run-length function of the β-expansion of the unit

For any real number β>1, the run-length function rn(β) is defined as the maximal length of consecutive zero digits amongst the first n digits in the β-expansion of 1. It was known that rn(β) grows to infinity with the speed logβ⁡n for Lebesgue almost all β∈(1,2). In this note, we quantify the size of the set of β for which rn(β) grows to infinity in a general speed. More precisely, for any strictly increasing function ϕ:N→R+ with ϕ(n) tending to +∞ and ϕ(n)/n decreasing to 0 as n→∞, we prove that for any real numbers 0≤a≤b≤+∞, the setEa,b={β∈(1,2): liminfn→∞rn(β)ϕ(n)=a,limsupn→∞rn(β)ϕ(n)=b} is of full Hausdorff dimension.

Thermal stability of phytochemicals, HMF and antioxidant activity in cape gooseberry (Physalis peruviana L.)

Cape gooseberry is a fruit recognised for having relevant contents of health-promoting compounds. Changes in the content of phytochemicals (ascorbic acid, β-carotene, catechin and epicatechin), hydroxymethylfurfural (HMF) and antioxidant activity of this fruit were studied at various temperatures and times. Ascorbic acid degradation was described by a first order reaction. β-carotene was not degraded and followed an isomerization reaction from 80°C onwards. Formation of HMF was described with a consecutive zero together with a first order reaction model. The contents of catechin and epicatechin increased at 40°C. More than three competing reactions did not allow to make kinetic modelling. Antioxidant activity followed fractional first order conversion model. Comparison with kinetics found in other fruits showed that health-promoting compounds of cape gooseberry are relatively more stable to heat treatment. This makes cape gooseberry suitable for the preparation of foods (jam, juices and dehydrated fruit) with relevant health-promoting compounds contents.

Are we missing the sitting? Agreement between accelerometer non-wear time validation methods used with older adults' data.

We used Bland Altman plots to compare agreement between a self-report diary and five different non-wear time algorithms [an algorithm that uses ≥60 min of consecutive zeroes (Troiano) and four variations of an algorithm that uses ≥90 min of consecutive zeroes to define a non-wear period] for estimating community-dwelling older adults’ (n = 106) sedentary behaviour and wear time (min/day) as measured by accelerometry. We found that the Troiano algorithm may overestimate sedentary behaviour and wear time by ≥30 min/day. Algorithms that use ≥90 min of continuous zeroes more closely approximate participants’ sedentary behaviour and wear time. Across the self-report diary vs. ≥90 min algorithm comparisons, mean differences ranged between −4.4 to 8.1 min/day for estimates of sedentary behaviour and between −10.8 to 1.0 min/day for estimates of wear time; all 95% confidence intervals for mean differences crossed zero. We also found that 95% limits of agreement were wide for all comparisons, highlighting the large variation in estimates of sedentary behaviour and wear time. Given the importance of reducing sedentary behaviour and encouraging physical activity for older adult health, we conclude that it is critical to establish accurate approaches for measurement.

The largest gap between zeros of entire L-functions is less than 41.54

Using suitable feasible pairs and convex combinations of Selberg minorant functions, the upper bound under GRH and the Ramanujan hypothesis on the largest gap between consecutive zeros of an entire L-function in Bober, Conrey, Farmer, Fujii, Koutsoliotas, Lemurell, Rubinstein and Yoshida [2] is improved from 45.3236 to 41.54. An application about nonexistence of certain entire L-functions is also provided.

A small improvement in the gaps between consecutive zeros of the Riemann zeta-function

Feng and Wu introduced a new general coefficient sequence into Montgomery and Odlyzko’s method for exhibiting irregularity in the gaps between consecutive zeros of \(\zeta (s)\) assuming the Riemann hypothesis. They used a special case of their sequence to improve upon earlier results on the gaps. In this paper we consider a general sequence related to that of Feng and Wu, and introduce a somewhat less general sequence \(\{a_n\}\) for which we write the Montgomery–Odlyzko expressions explicitly. As an application, we give the following slight improvement of Feng and Wu’s result: infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at most 0.515396 times the average spacing and infinitely often they differ by at least 2.7328 times the average spacing.

Deterministic simulation of highly intermittent hydrologic time series

Application of a deterministic geometric approach for the simulation of highly intermittent hydrologic data is presented. Specifically, adaptations of the fractal-multifractal (FM) method and a Cantorian extension are advanced in order to simulate rainfall records measured at the daily scale and encompassing a water year. It is shown, using as case studies 2 years of rainfall sets gathered in Laikakota, Bolivia and Tinkham, Washington, USA, that the FM approach, relying on only at most 8 parameters, is capable of closely preserving either the whole record’s histogram (therefore including moments), the whole data’s Renyi entropy function and/or the maximum number of consecutive zero values present in the sets, resulting in suitable rainfall simulations, whose overall features and textures are similar to those of the observed sets. The study hence establishes the possibility of simulating highly intermittent sets in time in a deterministic and holistic way as a novel parsimonious methodology to supplement available stochastic frameworks.

Sturmian comparison theory for half‐linear and nonlinear differential equations via Picone identity

In this paper, Sturmian comparison theory is developed for the pair of second‐order differential equations; first of which is the nonlinear differential equations of the form urn:x-wiley:mma:media:mma4224:mma4224-math-0001 and the second is the half‐linear differential equations urn:x-wiley:mma:media:mma4224:mma4224-math-0002 where Φα(s) = |s|α − 1s and α1 &gt; ⋯ &gt; αm &gt; β &gt; αm + 1 &gt; ⋯ &gt; αn &gt; 0. Under the assumption that the solution of has two consecutive zeros, we obtain Sturm–Picone type and Leighton type comparison theorems for by employing the new nonlinear version of Picone formula that we derive. Wirtinger type inequalities and several oscillation criteria are also attained for . Examples are given to illustrate the relevance of the results. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Lehmer Pairs Revisited

ABSTRACTWe seek to understand how the technical definition of a Lehmer pair can be related to more analytic properties of the Riemann zeta function, particularly the location of the zeros of ζ′(s). Because we are interested in the connection [Csordas et al. 94] between Lehmer pairs and the de Bruijn–Newman constant Λ, we assume the Riemann hypothesis throughout. We define strong Lehmer pairs via an inequality on the derivative of the pre-Schwarzian of Riemann’s function Ξ(t), evaluated at consecutive zeros:Theorem 1 shows that strong Lehmer pairs are Lehmer pairs. Theorem 2 describes PΞ′(γ) in terms of ζ′(ρ) where ρ = 1/2 + iγ. Theorem 3 expresses PΞ′(γ+) + PΞ′(γ−) in terms of nearby zeros ρ′ of ζ′(s). We examine 114, 661 pairs of zeros of ζ(s) around height t = 106, finding 855 strong Lehmer pairs. These are compared to the corresponding zeros of ζ′(s) in the same range.