Sequence Of Integers Research Articles

On a general divisor problem related to a certain Dedekind zeta-function over a specific sequence of positive integers

We investigate the average behavior of coefficients of the Dirichlet series of positive integral power of the Dedekind zeta-function $\zeta_{\mathbb{K}_3}(s)$ of a non-normal cubic extension $\mathbb{K}_3$ of $\mathbb{Q}$ over a certain sequence of positive integers. More precisely, we prove an asymptotic formula with an error term for the sum\[ \sum_{{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}\leq {x}}\atop{(a_{1},a_{2},a_{3},a_{4},a_{5},a_{6})\in\mathbb{Z}^{6}}}a_{k,\mathbb{K}_3} (a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}),\]where $(\zeta_{\mathbb{K}_3}(s))^{k}:=\sum_{n=1}^{\infty}\frac{a_{k,\mathbb{K}_3}(n)}{n^{s}}$.

On the Lindelöf hypothesis for general sequences

AbstractIn a recent paper, Gonek, Graham, and Lee introduced a notion of the Lindelöf hypothesis (LH) for general sequences that coincides with the usual LH for the Riemann zeta function in the case of the sequence of positive integers. They made two conjectures: that LH should hold for every admissible sequence of positive integers, and that LH should hold for the “generic” admissible sequence of positive real numbers. In this paper, we give counterexamples to the first conjecture, and show that the second conjecture can be either true or false, depending on the meaning of “generic”: we construct probabilistic processes producing sequences satisfying LH with probability 1, and we construct Baire topological spaces of sequences for which the subspace of sequences satisfying LH is meagre. We also extend the main result of Gonek, Graham, and Lee, stating that the Riemann hypothesis is equivalent to LH for the sequence of prime numbers, to the context of Beurling generalized number systems.

A Class of Bounded Iterative Sequences of Integers

In this note, we show that, for any real number τ∈[12,1), any finite set of positive integers K and any integer s1≥2, the sequence of integers s1,s2,s3,… satisfying si+1−si∈K if si is a prime number, and 2≤si+1≤τsi if si is a composite number, is bounded from above. The bound is given in terms of an explicit constant depending on τ,s1 and the maximal element of K only. In particular, if K is a singleton set and for each composite si the integer si+1 in the interval [2,τsi] is chosen by some prescribed rule, e.g., si+1 is the largest prime divisor of si, then the sequence s1,s2,s3,… is periodic. In general, we show that the sequences satisfying the above conditions are all periodic if and only if either K={1} and τ∈[12,34) or K={2} and τ∈[12,59).

A comparison of three evolutionary algorithms for group scheduling in theme parks with multitype facilities.

During the peak tourist season, large theme parks often experience a simultaneous influx of visitors, resulting in prolonged waiting times for popular attractions. This extended waiting significantly reduces tourists' satisfaction and may negatively impact their willingness to revisit the theme park. In Taiwan, schools at all levels often plan graduation trips to theme parks for their students. Students are divided into groups and must enter and exit the theme park at the same time. This article presents a new theme park problem with multitype facilities (TPP-MTF) for student groups. Based on the group's preference for theme park facilities, multitype reserved tickets with popular facilities are designed for groups, so groups do not need to wait for the reserved facilities. Since the waiting time for groups can be reduced, the theme park can also obtain ticket fees in advance and estimate the number of visitors to the theme park, so the theme park and the group can achieve a win-win situation. This article proposes a new decoding approach for a random permutation of integer sequence and embeds it into an immune-based algorithm, genetic algorithm, and particle swarm optimization algorithm to solve the TPP-MTF problem. A theme park in Taiwan was taken as an example and numerical results of the three algorithms were analyzed and compared to verify the effectiveness of the proposed algorithms.

A threshold for the best two-term underapproximation by Egyptian fractions

Let G be the greedy algorithm that, for each θ∈(0,1], produces an infinite sequence of positive integers (an)n=1∞ satisfying ∑n=1∞1/an=θ. For natural numbers p<q, let Υ(p,q) denote the smallest positive integer j such that p divides q+j. Continuing Nathanson’s study of two-term underapproximations, we show that whenever Υ(p,q)⩽3, G gives the (unique) best two-term underapproximation of p/q; i.e., if 1/x1+1/x2<p/q for some x1,x2∈N, then 1/x1+1/x2⩽1/a1+1/a2. However, the same conclusion fails for every Υ(p,q)⩾4. Next, we study stepwise underapproximation by G. Let em=θ−∑n=1m1/an be the mth error term. We compare 1/am to a superior underapproximation of em−1, denoted by N/bm (N∈N⩾2), and characterize when 1/am=N/bm. One characterization is am+1⩾Nam2−am+1. Hence, for rational θ, we only have 1/am=N/bm for finitely many m. However, there are irrational numbers such that 1/am=N/bm for all m. Along the way, various auxiliary results are encountered.

Determinants of Toeplitz–Hessenberg Matrices with Generalized Leonardo Number Entries

Abstract Let un = un (k) denote the generalized Leonardo number defined recursively by un = un− 1 + un− 2 + k for n ≥ 2, where u 0 = u 1 = 1. Terms of the sequence un (1) are referred to simply as Leonardo numbers. In this paper, we find expressions for the determinants of several Toeplitz–Hessenberg matrices having generalized Leonardo number entries. These results are obtained as special cases of more general formulas for the generating function of the corresponding sequence of determinants. Special attention is paid to the cases 1 ≤ k ≤ 7, where several connections are made to entries in the On-Line Encyclopedia of Integer Sequences. By Trudi’s formula, one obtains equivalent multi-sum identities involving sums of products of generalized Leonardo numbers. Finally, in the case k = 1, we also provide combinatorial proofs of the determinant formulas, where we make extensive use of sign-changing involutions on the related structures.

Subdivision algorithms with modular arithmetic

We study the de Casteljau subdivision algorithm for Bezier curves and the Lane-Riesenfeld algorithm for uniform B-spline curves over the integers mod m, where m>2 is an odd integer. We place the integers mod m evenly spaced around a unit circle so that the integer k mod m is located at the position on the unit circle ate2πki/m=cos(2kπ/m)+isin(2kπ/m)↔(cos(2kπ/m),sin(2kπ/m)).Given a sequence of integers (s0,…,sm)mod m, we connect consecutive values sjsj+1 on the unit circle with straight line segments to form a control polygon. We show that if we start these subdivision procedures with the sequence (0,1,…,m)mod m, then the sequences generated by these recursive subdivision algorithms spawn control polygons consisting of the regular m-sided polygon and regular m-pointed stars that repeat with a period equal to the minimal integer k such that 2k=±1modm. Moreover, these control polygons represent the eigenvectors of the associated subdivision matrices corresponding to the eigenvalue 2−1modm. We go on to study the effects of these subdivision procedures on more general initial control polygons, and we show in particular that certain control polygons, including the orbits of regular m-sided polygons and the complete graphs of m-sided polygons, are fixed points of these subdivision procedures.

On sequences of integers with small prime factors

On sequences of integers with small prime factors

Asymptotics of Z-convex polyominoes

The degree of convexity of a convex polyomino P is the smallest integer k such that any two cells of P can be joined by a monotone path inside P with at most k changes of direction. In this paper we show that one can compute in polynomial time the number of polyominoes of area n and degree of convexity at most 2 (the so-called Z-convex polyominoes). The integer sequence that we have computed allows us to conjecture the asymptotic number an of Z-convex polyominoes of area n, ɑn ∼ C·exp(π)√11n/4⁄n3/2.

Modelling the Lengths of Quantitative Motifs in Pseudorandom Integer Sequences

This study explores the behaviour of quantitative motifs in random numerical data, focussing on the property of length. One hundred and twenty sequences of pseudorandom integers were generated and segmented into quantitative motifs using the standard definition. The length–frequency relationship was modelled in each case using the Zipf-Alekseev function. Different combinations of sequence length, integer range, and pseudorandom number generator were trialled. The model-fitting was successful in all cases (𝑅2 1 &gt; 0.9 in 118/120 cases and &gt; 0.8 in 2/120), and the model parameters fell in the same range as those obtained in a previous textual study of motifs. Integer range was the main factor affecting the parameter values, with sequence length affecting the degree of spread. Further comparisons between textual and random data are needed, as well as a theoretical explanation of why motifs in random data demonstrate lawful behaviour. In future textual motif studies, particular attention needs to be paid to possible dependencies on value range and text length.

On the central limit theorem for the elephant random walk with gradually increasing memory and random step size

&lt;abstract&gt;&lt;p&gt;In this paper, we investigate an extended version of the elephant random walk model. Unlike the traditional approach where step sizes remain constant, our model introduces a novel feature: step sizes are generated as a sequence of positive independent and identically distributed random variables, and the step of the walker at time $ n+1 $ depends only on the steps of the walker between times $ 1, ..., m_n $, where $ (m_n)_{n\geqslant 1} $ is a sequence of positive integers growing to infinity as $ n $ goes to infinity. Our main results deal with the validity of the central limit theorem for this new variation of the standard ERW model introduced by Schütz and Trimper in $ 2004 $.&lt;/p&gt;&lt;/abstract&gt;

On a sequence of rational numbers with unusual divisibility by a power of 2

In this note we consider the sequence of rational numbers bn = ∑k=1n2k∕k. We show that the power of 2 in the expansion of bn is unusually large, at least n + 1 − log2(n + 1), and that this bound is best possible. The sequence bn, n = 1,2,3,…, is related to the sequence A0031449 in the On-Line Encyclopedia of Integer Sequences.

Portal-site epinephrine injections improve visualisation in arthroscopic rotator cuff repair

ObjectivesThe aim of this study was to evaluate if portal-site injections of 1:200,000 epinephrine improve intraoperative visualisation in arthroscopic rotator cuff repair. MethodsPatients with partial-thickness supraspinatus tears were selected for the study. They were assigned consecutive numbers and were divided into two groups—control group and intervention group. The surgeries were done by a single surgeon. Every odd-numbered patient was injected. Johnson's visibility classification, surgeon 5-point ordinal Likert scale (LS), and other parameters were recorded. ResultsA total of 221 participants (58.4 ​± ​6.1 years) were selected. Intraoperative visibility was better in the intervention group according to Johnson's classification—satisfactory visibility was achieved in 68 of 110 patients (62%, control group) compared to 89 of 111 patients (80%) (p ​= ​0.003). Surgeon LS was superior in the intervention group, with a notable decrease in worsened visibility cases. The operative time did not alter statistically significantly—82.2 ​± ​14.4 ​min for the control group, compared to 80.9 ​± ​10.8 ​min in the intervention group (p ​= ​0.056). No injection-associated complications were recorded. ConclusionsPortal-site injection of diluted epinephrine solution is safe and improves intraoperative visualisation in arthroscopic rotator cuff repair. This addition does not increase operative time. Level of evidenceLevel 3, case–control study.

MIRROR (LEFT-RECURSIVE) GRAY CODE

Here we consider a version of the Binary Reflected Gray Code (BRGC, or Gray code for short), called the mirror Gray code – in parallel and in comparison to the Gray code. We discuss some sources where the two codes are not distinguished and the reasons why. We present our arguments for treating the two codes as distinct, starting with a definition of the mirror Gray code and showing its main properties. Among the most important of these is the relationship between the two codes – the codewords of the mirror Gray code are “left-right mirror images” of those in the Gray code and vice versa. Other arguments we show are the differences in the algorithms for generating the codewords. We also present another approach that involves representing the codewords by integers (their serial numbers), generating and examining the two codes by using their sequences of serial numbers. For this purpose, we use the bijections that relate the pairs of combinatorial objects: binary vectors in a certain order and the corresponding integer sequences and subsets. In this way, we get different algorithms, sequences, and functions for successor and predecessor, for ranking and unranking at the mirror Gray code. We believe that this article provides at least one more perspective, as well as expands and enriches the knowledge of the Gray code.

Quiddities of polygon dissections and the Conway-Coxeter Frieze equation

We study a $2 \times 2$ matrix equation arising naturally in the theory of Coxeter frieze patterns. It is formulated in terms of the generators of the group $\mathrm{PSL}(2,\mathbb{Z})$ and is closely related to continued fractions. It appears in a number of different areas, for example, toric varieties. We count its positive solutions, obtaining a series of integer sequences, some known and some new. This extends classical work of Conway and Coxeter proving that the first of these sequences is the Catalan numbers.

Proving properties of some greedily-defined integer recurrences via automata theory

Venkatachala on the one hand, and Avdispahić & Zejnulahi on the other, both studied integer sequences with an unusual sum property defined in a greedy way, and proved many results about them. However, their proofs were rather lengthy and required numerous cases. In this paper, I provide a different approach, via finite automata, that can prove the same results (and more) in a simple, unified way. Instead of case analysis, we use a decision procedure implemented in the free software Walnut. Using these ideas, we can prove a conjecture of Quet and find connections between Quet's sequence and the “married” functions of Hofstadter.

Computational ghost imaging encryption using RSA algorithm and discrete wavelet transform

Ghost imaging technology has been widely used in the field of optical image encryption owing to its unique non-local imaging characteristics. However, there are some problems such as long encryption and decrypting time, poor reliability and large number of keys. To solve these problems, this paper proposed a computational ghost imaging encryption scheme based on RSA algorithm and wavelet transform. Firstly, the representation relationship between measurement matrix and random integer sequence is established to reduce the transmission of key quantity. Secondly, RSA encryption algorithm is used to improve the security of the system. Finally, using wavelet decomposition for packet transmission. In addition, the cake cutting matrix is used as the measurement matrix to reduce the sampling times and improve the quality of reconstructed images. The simulation results show that the encryption system has strong information transmission security and good robustness. It provides some reference for image encryption and information security transmission.

Performance of the PERSIANN Family of Products over the Mekong River Basin and Their Application for the Analysis of Trends in Extreme Precipitation Indices

Near-real-time satellite precipitation estimation is indispensable in areas where ground-based measurements are not available. In this study, an evaluation of two near-real-time products from the Center for Hydrometeorology and Remote Sensing at the University of California, Irvine—PERSIANN-CCS (Precipitation Estimation from Remotely Sensed Information using Artificial Neural Networks—Cloud Classification System) and PDIR-Now (PERSIANN-Dynamic Infrared Rain Rate near-real-time)—were compared to each other and evaluated against IMERG Final (Integrated Multi-satellite Retrievals for Global Precipitation Measurement—Final Run) from 2015 to 2020 over the Mekong River Basin and Delta (MRB) using a spatial resolution of 0.1∘ by 0.1∘ and at a daily scale. PERSIANN-CDR (PERSIANN-Climate Data Record) was also included in the evaluation but was not compared against the real-time products. In this evaluation, PDIR-Now exhibited a superior performance to that of PERSIANN-CCS, and the performance of PERSIANN-CDR was deemed satisfactory. The second part of the study entailed performing a Mann–Kendall trend test of extreme precipitation indices using 38 years of PERSIANN-CDR data over the MRB. This annual trend analysis showed that extreme precipitation over the 95th and 99th percentiles has decreased over the Upper Mekong River Basin, and the consecutive number of wet days has increased over the Lower Mekong River Basin.

Match Exposure, Consecutive Match Number, and Recovery Days Affect Match Running During International Women's Soccer Tournaments.

Brown, G, Massard, T, Wignell, T, McCall, A, and Duffield, R. Match exposure, consecutive match number, and recovery days affect match running during international women's soccer tournaments. J Strength Cond Res 38(3): 577-583, 2024-To determine the influence of tournament-specific factors (match exposure, consecutive matches, and days between matches) on match running in women's international soccer players (footballers), 28 players from 1 national team were observed over 4 international tournaments using 10 Hz global positioning system units. Linear mixed models with a forward selection approach were used to analyze the collective effect of tournament-specific factors of match exposure (playing time), consecutive matches (count), days between matches (count) and time on pitch on relative (min -1 ) match running metrics (total distance, high-speed running, very high-speed running, accelerations, and decelerations), and the change in these metrics between halves. The significance level was set at p < 0.05. Days between matches were included in the final models for total distance ( p < 0.0001), high-speed running ( p = 0.002), very high-speed running ( p = 0.025), and change in accelerations ( p < 0.0001) and decelerations ( p = 0.002). Prior match exposure was included in the models for total distance ( p = 0.021), change in total distance ( p < 0.0001), and change in high-speed running ( p = 0.0008). Consecutive match number was included in the model for deceleration ( p = 0.030), change in very high-speed running ( p = 0.016), change in acceleration ( p < 0.0001), and change in deceleration ( p < 0.0001). Time on pitch was included in the final total distance ( p = 0.044), high-speed running ( p < 0.0001), acceleration ( p < 0.0001), and deceleration ( p = 0.0003) models. Overall, reduced match running throughout tournaments was associated with increased time on pitch and ≤3 days between matches. Increasing match exposure and more time between matches minimizes reductions in match running between halves. Practitioners can use these variables to inform player monitoring during international soccer tournaments and on return to clubs.

A study on the sum of ‘n+1’ Consecutive Coral Numbers

A study on the sum of ‘n+1’ Consecutive Coral Numbers