Consecutive Zeros Research Articles

Asymptotic formulae for the zeros of orthogonal polynomials

Let be an algebraic polynomial that is orthonormal with weight on the interval . When is a perturbation (in certain limits) of the Chebyshev weight of the first kind, the zeros of the polynomial and the differences between pairs of (not necessarily consecutive) zeros are shown to satisfy asymptotic formulae as , which hold uniformly with respect to the indices of the zeros. Similar results are also obtained for perturbations of the Chebyshev weight of the second kind. First, some preliminary results on the asymptotic behaviour of the difference between two zeros of an orthogonal trigonometric polynomial, which are needed, are established.Bibliography: 15 titles.

Area bound dynamic time warping based fast and accurate person authentication using a biometric pen

The paper presents a modified dynamic time warping (DTW) technique for person authentication based on time series matching obtained from handwriting. The online data has been acquired by a biometric smart pen device. The proposed method allows fast and accurate classification of human individuals based on handwritten PIN words or signature samples. Although classic DTW provides robust distance measurements essential for accurate classification of sequences, it is computationally expensive. To speed up computations we introduce area bound dynamic time warping (AB_DTW) that divides time series into several areas bounded by segments of consecutive zero crossings including local peaks and valleys. Unlike classic DTW which compares whole signals, the proposed AB_DTW warps areas bounded by the local regions. Two kinds of data abstraction formats of area bound-1 dimensional and 2 dimensional-are evaluated. Experimental results show that because of a higher-level data abstraction, the proposed approach is several times faster than classic DTW. Moreover, AB_DTW does not offer substantial loss of accuracy which is required for authentication performance using handwritten PIN words and signatures sampled by biometric pen device.

Gaps between zeros of second-order half-linear differential equations

In this paper, for second order half-linear differential equations, we will establish some new inequalities of Lyapunov’s type. These inequalities give results related to the spacing between consecutive zeros of a solution and the spacing between a zero of a solution and/or a zero of its derivative. The results also yield conditions for disfocality, disconjugacy and lower bounds for an eigenvalue of a boundary value problem. The main results will be proved by making use of some generalizations of Opial and Wirtinger type inequalities. Some examples are considered to illustrate the main results.

On gaps between zeros of the Riemann zeta-function

Assuming the Riemann Hypothesis, we show that infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at least 2.7327 times the average spacing and infinitely often they differ by at most 0.5154 times the average spacing.

Exact number of local extreme points of curvature function of solutions of second-order linear differential equations

We study local properties of the curvature κy(x) of every nontrivial solution y=y(x) of the second-order linear differential equation (P): (p(x)y′)′+q(x)y=0, x∈(a,b)=I, where p(x) and q(x) are smooth enough functions. It especially includes the Euler, Bessel and other important types of second-order linear differential equations. Some sufficient conditions on the coefficients p(x) and q(x) are given such that the curvature κy(x) of every nontrivial solution y of (P) has exactly one extreme point between each two its consecutive simple zeros. The problem of three local extreme points of κy(x) is also considered but only as an open problem. It seems it is the first paper dealing with this kind of problems. Finally in Appendix, we pay attention to an application of the main results to a study of non-regular points (the cusps) of the e-parallels of graph Γ(y) of y (the offset curves of Γ(y)).

A Reversible Data Hiding for JPEG Images

A novel reversible data hiding technique for JPEG images is proposed in this paper. Consecutive zeros in the tail of DCT coefficient sequence in each block are exploited to embed a number of secret bits by modifying only one coefficient. Thanks to the efficient data embedding in the DCT coefficients, the proposed scheme can provide a good rate-distortion performance. Also, when having an image containing secret data, one can perfectly recover the original image after extracting the embedded data. Experimental results show that the proposed scheme outperforms the previous method.

Small gaps between zeros of twisted L-functions

We use the asymptotic large sieve, developed by the authors, to prove that if the Generalized Riemann Hypothesis is true, then there exist many Dirichlet L-functions that have a pair of consecutive zeros closer together than 0.37 times their average spacing. More generally, we investigate zero spacings within the family of twists by Dirichlet characters of a fixed L-function and give precise bounds for small gaps which depend only on the degree of the L-function.

New Inequalities of Opial′s Type on Time Scales and Some of Their Applications

We will prove some new dynamic inequalities of Opial′s type on time scales. The results not only extend some results in the literature but also improve some of them. Some continuous and discrete inequalities are derived from the main results as special cases. The results will be applied on second‐order half‐linear dynamic equations on time scales to prove several results related to the spacing between consecutive zeros of solutions and the spacing between zeros of a solution and/or its derivative. The results also yield conditions for disfocality of these equations.

Opial's type inequalities on time scales and some applications

We prove some new Opial type inequalities on time scales and employ them to prove several results related to the spacing between consecutive zeros of a solution or between a zero of a solution and a zero of its derivative for second order dynamic equation

A generalization of the Hansen–Mullen conjecture on irreducible polynomials over finite fields

Let q be a prime power and Fq the finite field with q elements. We examine the existence of irreducible polynomials with prescribed coefficients over Fq. We focus on a conjecture by Hansen and Mullen which states that for n⩾3, there exist irreducible polynomials over Fq of degree n, with any one coefficient prescribed to any element of Fq (this being nonzero when the constant coefficient is being prescribed) and was proved by Wan. We introduce a variation of Wanʼs method to give restrictions subject to which this result can be extended to more than one prescribed coefficient; for example we show the asymptotical existence of irreducible polynomials with trace and any other one coefficient prescribed to any value. It also follows from our generalization the existence of irreducible polynomials with sequences of consecutive zero coefficients.

On β-expansions of unity for rational and transcendental numbers β

Abstract We investigate the sequence of integers x 1, x 2, x 3, … lying in {0, 1, …, [β]} in a so-called Rényi β-expansion of unity 1 = $\sum\limits_{j = 1}^\infty {x_j \beta ^{ - j} } $ for rational and transcendental numbers β > 1. In particular, we obtain an upper bound for two strings of consecutive zeros in the β-expansion of unity for rational β. For transcendental numbers β which are badly approximable by algebraic numbers of every large degree and bounded height, we obtain an upper bound for the Diophantine exponent of the sequence X = (xj)j=1∞ in terms of β.

Stieltjes interlacing of zeros of Laguerre polynomials from different sequences

Stieltjes’ Theorem (cf. Szegö (1959) [10]) proves that if { p n } n = 0 ∞ is an orthogonal sequence, then between any two consecutive zeros of p k there is at least one zero of p n for all positive integers k , k < n ; a property called Stieltjes interlacing. We prove that Stieltjes interlacing extends across different sequences of Laguerre polynomials L n α , α > − 1 . In particular, we show that Stieltjes interlacing holds between the zeros of L n − 1 α + t and L n + 1 α , α > − 1 , when t ∈ { 1 , … , 4 } but not in general when t > 4 or t < 0 and provide numerical examples to illustrate the breakdown of interlacing. We conjecture that Stieltjes interlacing holds between the zeros of L n − 1 α + t and those of L n + 1 α for 0 < t < 4 . More generally, we show that Stieltjes interlacing occurs between the zeros of L n + 1 α and the zeros of the k th derivative of L n α , as well as with the zeros of L n − k α + k + t for t ∈ { 1 , 2 } and k ∈ { 1 , 2 , … , n − 1 } . In each case, we identify associated polynomials, analogous to the de Boor–Saff polynomials (cf. de Boor and Saff (1986) [3], Ismail (2005) [6]), that are completely determined by the coefficients in a mixed three-term recurrence relation, whose zeros complete the interlacing process.

Study protocol of physical activity and sedentary behaviour measurement among schoolchildren by accelerometry - Cross-sectional survey as part of the ENERGY-project

BackgroundPhysical activity and sedentary behaviour among children should be measured accurately in order to investigate their relationship with health. Accelerometry provides objective and accurate measurement of body movement, which can be converted to meaningful behavioural outcomes. The aim of this study was to evaluate the best evidence for the decisions on data collection and data processing with accelerometers among children resulting in a standardized protocol for use in the participating countries.Methods/DesignThis cross-sectional accelerometer study was conducted as part of the European ENERGY-project that aimed to produce an obesity prevention intervention among schoolchildren. Five countries, namely Belgium, Greece, Hungary, Switzerland and the Netherlands participated in the accelerometer study. We used three different Actigraph models-Actitrainers (triaxial), GT3Xs and GT1Ms. Children wore the device for six consecutive days including two weekend days. We selected an epoch length of 15 seconds. Accelerometers were placed at children's waist at the right side of the body in an elastic belt.In total, 1082 children participated in the study (mean age = 11.7 ± 0.75 y, 51% girls). Non-wearing time was calculated as periods of more than 20 minutes of consecutive zero counts. The minimum daily wearing time was set to 10 hours for weekdays and 8 hours for weekend days. The inclusion criterion for further analysis was having at least three valid weekdays and one valid weekend day. We selected a cut-point (count per minute (cpm)) of <100 cpm for sedentary behaviour, <3000 cpm for light, <5200 cpm for moderate, and >5200 cpm for vigorous physical activity. We also created time filters for school-time during data cleaning in order to explore school-time physical activity and sedentary behaviour patterns in particular.DiscussionThis paper describes the decisions for data collection and processing. Use of standardized protocols would ease future use of accelerometry and the comparability of results between studies.

Validation of Accelerometer Wear and Nonwear Time Classification Algorithm

the use of movement monitors (accelerometers) for measuring physical activity (PA) in intervention and population-based studies is becoming a standard methodology for the objective measurement of sedentary and active behaviors and for the validation of subjective PA self-reports. A vital step in PA measurement is the classification of daily time into accelerometer wear and nonwear intervals using its recordings (counts) and an accelerometer-specific algorithm. the purpose of this study was to validate and improve a commonly used algorithm for classifying accelerometer wear and nonwear time intervals using objective movement data obtained in the whole-room indirect calorimeter. we conducted a validation study of a wear or nonwear automatic algorithm using data obtained from 49 adults and 76 youth wearing accelerometers during a strictly monitored 24-h stay in a room calorimeter. The accelerometer wear and nonwear time classified by the algorithm was compared with actual wearing time. Potential improvements to the algorithm were examined using the minimum classification error as an optimization target. the recommended elements in the new algorithm are as follows: 1) zero-count threshold during a nonwear time interval, 2) 90-min time window for consecutive zero or nonzero counts, and 3) allowance of 2-min interval of nonzero counts with the upstream or downstream 30-min consecutive zero-count window for detection of artifactual movements. Compared with the true wearing status, improvements to the algorithm decreased nonwear time misclassification during the waking and the 24-h periods (all P values < 0.001). the accelerometer wear or nonwear time algorithm improvements may lead to more accurate estimation of time spent in sedentary and active behaviors.

Embedding And Employing Metadata In Digital Music Using Format Specific Methods

A method of embedding useful data in a digital music file in a way that maintains compatibility with existing digital music formats and players and has no effect on the size of the digital music file or sound quality. A digital music file is analyzed to determine a list of start and end locations of at least n consecutive zeros, defining a list of record locations, where n is an integer greater than two. The file store locations are written in initial record locations of the list of record locations and the data and a start key is written remaining record locations of the list of record locations. A process of retrieving the embedded data is also disclosed.

On the zeros of [formula omitted]-orthogonal polynomials for Freud weights

This paper gives the estimates of the distance between two consecutive zeros of the n th m -orthogonal polynomial P n for a Freud weight W = e − Q as follows. Let { x k n } be the zeros of P n in decreasing order, a n = a n ( Q ) the n th Mhaskar–Rahmanov–Saff number, and ϕ n ( x ) = max { n − 2 / 3 , 1 − | x | / a n } . Assume that Q ∈ C ( R ) is even, Q ( 0 ) = 0 , Q ′ ∈ C [ 0 , ∞ ) , Q ′ ( x ) > 0 , x ∈ ( 0 , ∞ ) , Q ″ ∈ C ( 0 , ∞ ) , and for some A , B > 1 , A ≤ ( x Q ′ ( x ) ) ′ Q ′ ( x ) ≤ B , x ∈ ( 0 , ∞ ) . Then, for 1 ≤ k ≤ n − 1 , x k n − x k + 1 , n ≤ c a n n ϕ n ( x k n ) − 1 / 2 and x k n − x k + 1 , n ≥ { c a n n ϕ n ( x k n ) − 1 / 2 , m = 2 , c a n n ϕ n ( x k n ) ( m − 2 ) / 2 , m ≥ 3 . Moreover, we have − a n < x n n < ⋯ < x 1 n < a n and for even m , − a n [ 1 − c ( W , m ) n − 2 / 3 ] ≤ x n n < ⋯ < x 1 n ≤ a n [ 1 − c ( W , m ) n − 2 / 3 ] . This paper also gives the estimates of Christoffel type functions with odd order and discusses the convergence of Gaussian quadrature formulas for such a weight.

The largest singletons of set partitions

Recently, Deutsch and Elizalde have studied the largest and the smallest fixed points of permutations. Motivated by their work, we consider the analogous problems in set partitions. Let A n , k denote the number of partitions of { 1 , 2 , … , n + 1 } with the largest singleton { k + 1 } for 0 ≤ k ≤ n . In this paper, several explicit formulas for A n , k , involving a Dobinski-type analog, are obtained by algebraic and combinatorial methods. Furthermore, many combinatorial identities involving A n , k and Bell numbers are presented by operator methods, and congruence properties of A n , k are also investigated. It is shown that the sequences ( A n + k , k ) n ≥ 0 and ( A n + k , k ) k ≥ 0 (mod p ) are periodic for any prime p , and contain a string of p − 1 consecutive zeroes. Moreover their minimum periods are conjectured to be N p = p p − 1 p − 1 .

Large gaps between consecutive zeros of the Riemann zeta-function

Combining the amplifiers, we exhibit other choices of coefficients that improve the results on large gaps between the zeros of the Riemann zeta-function. Precisely, assuming the Generalized Riemann Hypothesis (GRH), we show that there exist infinitely many consecutive gaps greater than 3.033 times the average spacing.

A note on the gaps between consecutive zeros of the Riemann zeta-function

Assuming the Riemann Hypothesis, we show that infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at most 0.5155 times the average spacing and that infinitely often they differ by at least 2.6950 times the average spacing.

Applications of Wirtinger Inequalities on the Distribution of Zeros of the Riemann Zeta-Function

On the hypothesis that the th moments of the Hardy -function are correctly predicted by random matrix theory and the moments of the derivative of are correctly predicted by the derivative of the characteristic polynomials of unitary matrices, we establish new large spaces between the zeros of the Riemann zeta-function by employing some Wirtinger-type inequalities. In particular, it is obtained that which means that consecutive nontrivial zeros often differ by at least 6.1392 times the average spacing.