Sequence Of Integers Research Articles

Some inner metric parameters of a digraph: iterated line digraphs and integer sequences

AbstractIn this paper, we first give a new result characterizing the strongly connected digraphs with a diameter equal to that of their line digraphs. Then we introduce the concepts of the inner diameter and inner radius of a digraph and study their behaviors in its iterated line digraphs. Furthermore, we provide a method to characterize sequences of integers (corresponding to the inner diameter or the number of vertices of a digraph and its iterated line digraphs) that satisfy some conditions. Among other examples, we apply the method to the cyclic Kautz digraphs, square-free digraphs, and the subdigraphs of De Bruijn digraphs. Finally, we present some tables with new sequences that do not belong to The On-Line Encyclopedia of Integer Sequences.

On a union of homogeneous sequences

A sequence of positive integers is said to be quasi-primitive if there do not exist three distinct integers such that the greatest common divisor of two elements is not the third. It is clear that the sequence of prime numbers is quasi-primitive. The Erdős quasi-primitive conjecture states that the sum of the reciprocal of the product of an element by its logarithm over any quasi-primitive sequence is maximized by the sum over the sequence of all powers of prime numbers. Note that Erdős, before introduce his conjecture, showed that for any quasi-primitive sequence different from one, this sum is convergent and bounded by a real constant. In this article, by studying the analogous Erdős conjecture, we prove that when the real is sufficiently large, there exists a quasi-primitive sequence such that the sum of the reciprocal of the product of an element by the sum of its logarithm and , over this quasi-primitive sequence is very large than the sum over the sequence of all powers of prime numbers. Obviously, if the case where equals zero is also true, then the conjecture is false.

On the existence of certain Lehmer numbers modulo a prime

A Lehmer number modulo an odd prime numberp is a residue class a∈Fp× whose multiplicative inverse ā has opposite parity. Lehmer numbers that are also primitive roots are called Lehmer primitive roots. Analogously, in this article we say that a residue class a∈Fp× is a Lehmer non-primitive root modulop if a is Lehmer number modulo p which is not a primitive root. We provide explicit estimates for the difference between the number of Lehmer non-primitive roots modulo a prime p and their “expected number”, which is p−1−ϕ(p−1)2. Similar explicit bounds are also provided for the number of k-consecutive Lehmer numbers modulo a prime, and k-consecutive Lehmer primitive roots We also prove that for any prime number p>3.05×1014, there exists a Lehmer non-primitive root modulo p. Moreover, we show that for any positive integer k≥2 (respectively, k≥5) and for all primes p≥exp(122k3) (respectively, p≥exp(6.87k)), there exist k consecutive Lehmer numbers modulo p (respectively, k consecutive Lehmer primitive roots modulo p). For large primes p, these theorems generalize two results which were proven in a paper by Cohen and Trudgian appeared in the Journal of Number Theory in 2019.

Transcendence and Normality of Complex Numbers via Hurwitz Continued Fractions

Abstract We study the topological, dynamical, and descriptive set-theoretic properties of Hurwitz continued fractions. Hurwitz continued fractions associate an infinite sequence of Gaussian integers to every complex number that is not a Gaussian rational. The resulting space of sequences of Gaussian integers $\Omega $ is not closed. Using an iterative procedure, we show that $\Omega $ contains a natural subset whose closure $\overline{\textsf{R}}$ encodes continued fraction expansions of complex numbers that are not Gaussian rationals. We prove that $(\overline{\textsf{R}}, \sigma )$ is a subshift with a feeble specification property. As an application, we determine the rank in the Borel hierarchy of the set of Hurwitz normal numbers with respect to the complex Gauss measure. We also construct a family of complex transcendental numbers with bounded partial quotients.

A month of diagnostic imaging studies in an acute supportive/palliative care unit.

The aim was to assess the characteristics of patients who required imaging studies during admission to an acute supportive palliative care unit (ASPCU). A consecutive number of patients who performed imaging studies during ASPCU admission in a month period were assessed. Epidemiological data, ongoing anticancer treatment, cancer diagnosis, reasons for admission, referral, and type of imaging study were recorded. Indications, findings, consequent actions for treatment, prognosis, and discharge were also collected. Twenty-one of 56 patients admitted to ASPCU in the period taken into consideration were assessed. Pain and deterioration of the general condition were the most frequent indications for admission. Computed tomography (CT) was the most frequent imaging study performed. Indications for performing imaging studies depended on individual clinical needs. Findings suggested different clinical decisions, after a comprehensive oncological and palliative care assessment and family conference, the most frequent of which was to withdraw oncological treatments. The majority of patients underwent transition to palliative care and were discharged home or to hospice. Imaging studies were of paramount importance for clinical treatment and decision-making process in an intensive ASPCU. There is the need to explore the need and the possible outcomes of imaging studies, as well cost-effectiveness in any ASPCU.

Random balancing-like sequences

AbstractA balancing-like sequence is a binary recurrence sequence which generalizes the balancing sequence and the sequence of nonnegative integers. This sequence, under certain assumptions, may be used to describe the growth of fortune of a person engaged in some business or profession. Since, in any business or profession, the growth is influenced by many uncertainties, it is more natural to induce some sort of randomness in the balancing-like sequences. If, in a balancing-like sequence, the growth rate is assumed to be a random variable, the resulting sequence will be a stochastic process and the sequence of expectations, in many cases, cannot be described as a binary recurrence sequence. In some cases, the growth rate of expectations increases without limit while, in some cases, it remains finite.

An almost sure limit theorem for the maximum interpoint distance of random vectors in spaces of growing dimension

Let { p n ; n ≥ 1 } be a sequence of positive integers. Let X 1 , … , X p n be i.i.d. random vectors in ℝ n with i.i.d. components that are centered and have variance 1. Consider the maximum interpoint distance M n = max 1 ≤ i < j ≤ p n ‖ X i − X j ‖ 2 . This article presents the almost sure limit theorem for M n under the polynomial and the exponential growth regimes for the dimension. The proofs rely on the moderate deviation principle of the partial sum of i.i.d. random variables, the Chen–Stein Poisson approximation method, and Gaussian approximation techniques.

Lattice points on polyominoes of inversion sequences

Inversion sequences of length n are positive integer sequences e 1 e 2 …e n such that 1 ≤ ei ≤ i for all 1 ≤ i ≤ n. These sequences are in bijection with the permutations of [n]. This paper focuses on the polyomino or barpgraph representation of the inversion sequences. More precisely, we study the distribution of lattice points on these polyominoes. We find the generating functions respect to the length, the number of interior vertices, corners, and vertices of a given degree. We also give simple explicit formulas for the total values of these parameters over all polyominoes of inversion sequences of length n. Throughout this work, we use symbolic computer algebra to facilitate the computations.

Drugs with glutamate-based mechanisms of action in psychiatry.

Psychopharmacotherapy of major psychiatric disorders is mostly based on drugs that modulate serotonergic, dopaminergic, or noradrenergic neurotransmission, either by inhibiting their reuptake or by acting as agonists or antagonists on specific monoamine receptors. The effectiveness of this approach is limited by a significant delay in the therapeutic mechanism and self-perpetuating growth of treatment resistance with a consecutive number of ineffective trials. A growing number of studies suggest that drugs targeting glutamate receptors offer an opportunity for rapid therapeutic effect that may overcome the limitations of monoaminergic drugs. In this article, we present a review of glutamate-modulating drugs, their mechanism of action, as well as preclinical and clinical studies of their efficacy in treating mental disorders. Observations of the rapid, robust, and long-lasting effects of ketamine and ketamine encourages further research on drugs targeting glutamatergic transmission. A growing number of studies support the use of memantine and minocycline in major depressive disorder and schizophrenia. Amantadine, zinc, and Crocus sativus extracts yield the potential to ameliorate depressive symptoms in patients with affective disorders. Drugs with mechanisms of action based on glutamate constitute a promising pharmacological group in the treatment of mental disorders that do not respond to standard methods of therapy. However, further research is needed on their efficacy, safety, dosage, interactions, and side effects, to determine their optimal clinical use.

Conjectural Criteria for the Most Singular Points of the Hilbert Schemes of Points

We provide conjectural necessary and (separately) sufficient conditions for the Hilbert scheme of points of a given length to have the maximum dimension tangent space at a point. The sufficient condition is claimed for 3D and reduces the original problem to a problem in convex geometry. Proving either of the two conjectural statements will in particular resolve a long-standing conjecture by Briançon and Iarrobino back in the ’70s for the case of the powers of the maximal ideal. Furthermore, for specific classes of lengths, we conjecturally classify points satisfying the conjectural sufficient conditions. This in particular (conjecturally) provides many new explicit families of examples of maximum dimension tangent space at a point of the Hilbert schemes of points of lengths strictly between two consecutive tetrahedral numbers ( 3 + k 3 ) .

On trilinear and quadrilinear equations associated with the lattice Gel’fand–Dikii hierarchy

Introduced in Zhang et al. (2012), the trilinear Boussinesq equation is the natural form of the equation for the τ-function of the lattice Boussinesq system. In this paper we study various aspects of this equation: its highly nontrivial derivation from the bilinear lattice AKP equation under dimensional reduction, a quadrilinear dual lattice equation, conservation laws, and periodic reductions leading to higher-dimensional integrable maps and their Laurent property. Furthermore, we consider a higher Gel’fand–Dikii lattice system, its periodic reductions and Laurent property. As a special application, from both a trilinear Boussinesq recurrence as well as a higher Gel’fand–Dikii system of three bilinear recurrences, we establish Somos-like integer sequences.

Triangular Number Products

Summary We show visually that twice the product of a pair of consecutive triangular numbers is triangular, and discuss some consequences.

On the correspondence between integer sequences and vacillating tableaux

On the correspondence between integer sequences and vacillating tableaux

Domination and Independence in Fuzzy Semigraphs

Objectives: To identify dominating and independent sets in fuzzy semigraphs by developing suitable methods. To examine dominating and independent sets in fuzzy semigraphs for their characteristics and behaviors. The goal is to examine the use of the concepts of domination and independence in the fields of analysis of social networks, security of networks, and allocation of resources. Methods: This study introduces the concept of strong arc fuzzy semigraph and explores various parameters such as dominating and independent sets. The Minimum Adjacent Dominating Number (ad-number) \gamma_a(\operatorname{𝓖}) and Maximum Adjacent Dominating Number (ad-number) \Gamma_a(\operatorname{𝓖}) , Minimum End Node Adjacent Dominating Number (ead-number) \gamma_{ea}(\operatorname{𝓖}) and Maximum End Node Adjacent Dominating Number (ead-number) \Gamma_{ae}(\operatorname{𝓖}) as well as Minimum Consecutive Adjacent Dominating Number (cad-number) \gamma_{ca}(\operatorname{𝓖}) and Maximum Consecutive Adjacent Dominating Number (cad-number) \Gamma_{ca}(\operatorname{𝓖}) . These parameters represent the respective minimum and maximum cardinalities taken over all minimal dominating sets of nodes of \operatorname{𝓖} . Additionally, it explores the e-independence dominating number i_e and e-independence number \beta_e(\operatorname{𝓖}) , strongly independence dominating number i_s and strongly independence number \beta_s(\operatorname{𝓖}) and consecutive adjacent independence dominating number i_{ca} and consecutive adjacent independence number \beta_{ca}(\operatorname{𝓖}) . These parameters represent the respective minimum and maximum cardinalities derived from all maximal independent sets of nodes in \operatorname{𝓖} . Findings: The study develops a thorough correlation between fuzzy dominant sets and independent sets in the fuzzy semigraph \operatorname{𝓖} . The statement illustrates the way in which the introduced factors interact and impact the structure and behaviour of the fuzzy semigraph. Novelty: The introduction of strong arc fuzzy semigraphs is a novel notion in the examination of fuzzy graphs. The study involves the creation and examination of novel parameters associated with dominant and independent sets in fuzzy semigraphs. This study aims to establish novel theoretical linkages between fuzzy domination and independent sets, therefore expanding our knowledge of the structural features of fuzzy semigraphs. Keywords: Domination in fuzzy semigraphs, Independent in fuzzy semigraphs, Adjacent-domination, ea-domination, ca-domination, e-independent, s-independent, ca-independent

Associations of night shift work with weight gain among female nurses in The Netherlands: results of a prospective cohort study.

This study aimed to prospectively investigate associations of working night shifts with weight gain in the Nightingale Study, a large cohort of female nurses. This study included 36 273 registered nurses, who completed questionnaires in 2011 and 2017. Cumulative number of nights, mean number of nights/month and consecutive number of nights/month in 2007-2011 were assessed. We used Poisson regression to estimate multivariable-adjusted incidence rate ratios (IRR) of >5% weight gain from 2011 to 2017 among all participants and assess risk of development of overweight/obesity (BMI≥25 kg/m2) among women with healthy baseline body mass index. The reference group consisted of women who never worked nights. Overall, working night shifts in 2007-2011 was associated with >5% weight gain [IRR 1.07, 95% confidence interval (CI) 1.01-1.13]. Associations differed by menopausal status in 2011, with an increased risk of gaining >5% weight limited to postmenopausal women who worked nights (IRR 1.23, 95% CI 1.10-1.38). Postmenopausal women had an increased risk of >5% weight gain when they worked on average ≥4 nights/month (4-5: IRR 1.29, 95% CI 1.09-1.52, ≥6: IRR 1.27, 95% CI 1.11-1.47) or ≥4 consecutive nights/month (IRR 1.37, 95% CI 1.19-1.58), compared to postmenopausal women who never worked nights. For postmenopausal women with healthy weight at baseline, night shift work was associated with an increased risk of overweight/obesity at follow-up (IRR 1.24, 95% CI 1.03-1.50). Working night shifts was associated with a slightly increased risk of weight gain and overweight/obesity development among women who were postmenopausal at study inclusion. Our findings emphasize the importance of health promotion to maintain a healthy weight among (postmenopausal) night workers.

Counting U(N)⊗r⊗O(N)⊗q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U(N)^{\\otimes r}\\otimes O(N)^{\\otimes q}$$\\end{document} invariants and tensor model observables

U(N)⊗r⊗O(N)⊗q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U(N)^{\\otimes r} \\otimes O(N)^{\\otimes q}$$\\end{document} invariants are constructed by contractions of complex tensors of order r+q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r+q$$\\end{document}, also denoted (r, q). These tensors transform under r fundamental representations of the unitary group U(N) and q fundamental representations of the orthogonal group O(N). Therefore, U(N)⊗r⊗O(N)⊗q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U(N)^{\\otimes r} \\otimes O(N)^{\\otimes q}$$\\end{document} invariants are tensor model observables endowed with a tensor field of order (r, q). We enumerate these observables using group theoretic formulae, for tensor fields of arbitrary order (r, q). Inspecting lower-order cases reveals that, at order (1, 1), the number of invariants corresponds to a number of 2- or 4-ary necklaces that exhibit pattern avoidance, offering insights into enumerative combinatorics. For a general order (r, q), the counting can be interpreted as the partition function of a topological quantum field theory (TQFT) with the symmetric group serving as gauge group. We identify the 2-complex pertaining to the enumeration of the invariants, which in turn defines the TQFT, and establish a correspondence with countings associated with covers of diverse topologies. For r>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r>1$$\\end{document}, the number of invariants matches the number of (q-dependent) weighted equivalence classes of branched covers of the 2-sphere with r branched points. At r=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r=1$$\\end{document}, the counting maps to the enumeration of branched covers of the 2-sphere with q+3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q+3$$\\end{document} branched points. The formalism unveils a wide array of novel integer sequences that have not been previously documented. We also provide various codes for running computational experiments.

A diluted version of the problem of the existence of the Hofstadter sequence

We investigate the conditions on an integer sequence f ( n ) , n ∈ N , with f ( 1 ) = 0 , such that the sequence q ( n ) , computed recursively via q ( n ) = q ( n − q ( n − 1 ) ) + f ( n ) , with q ( 1 ) = 1 , exists. We prove that f ( n ) is ‘slow’, that is, f ( n + 1 ) − f ( n ) ∈ { 0 , 1 } , n ≥ 1 , is a sufficient but not necessary condition for the existence of sequence q. Sequences q defined in this way typically display non-trivial dynamics: in particular, they are generally aperiodic with no obvious patterns. We discuss and illustrate this behavior with some examples.

How Long Can k-Göbel Sequences Remain Integers?

Inspired by Episode 3 of the Japanese manga “Seisu-tan” by Doom Kobayashi and Shin-ichiro Seki, we investigate the k-Göbel sequence ( g k , n ) n named after Fritz Göbel. Although the sequence is generally defined as rational, quite a few initial terms behave like an integer sequence. This article addresses a question raised in Seisu-tan and shows that g k , n is always an integer for any k ≥ 2 and 0 ≤ n ≤ 18 .

Characterized Subgroups Related to some Non-arithmetic Sequence of Integers

Characterized Subgroups Related to some Non-arithmetic Sequence of Integers

Characteristics of Patients With Amyotrophic Lateral Sclerosis Followed by a Home Palliative Care Team.

Information about patients with amyothrophic lateral sclerosis (ALS) followed at home is limited. To assess patients's characteristics at admission to a home palliative care program based on a multidisciplinary team, and the temporal course along the trajectory of ALS disease. Retrospective. Setting/subjects: Charts of a consecutive number of ALS patients who were referred to a specialistic home palliative care were reviewed. General data, referral, start of home palliative care, use of ventilator support and nutritional support, were recorded. The existence of advance directives and shared care planning was also collected. 82 patients were examined; 31 patients died before the term of the study and 51 patients were still living. No patient anticipately expressed their will regarding their treatments. However, a certain number of patients shared a care planning with ALS team, generally after starting home care. Most patients did not have ventilatory support at the beginning of home care assistance, but progressively received ventilatory support by NIV or MV, particularly those who were still living. NIV at start of home care was negatively correlated to frontotemporal dementia. (P = 0.015), and directly correlated to referral from hospital and GP (P = 0.031) and awareness of disease (P = 0.034). Gastrostomy at start of home care was positively correlated to referral from hospital (P = 0.046). Gastrostomy during home care was correlated to bulbar SLA (P = 0.017). The use of NIV during home care was positively correlated to shared care planning (P = 0.001). The continuous presence of a multi-specialist team may provide timely intervention, guarantee and trust on the part of the patient and family members.