Up-down Sequences Research Articles

Mathematical Modeling of a Class of Symmetrical Islamic Design

In this paper, we present a new model for simulating an interesting class of Islamic design. Based on periodic sequences on the one-dimensional manifolds, and from emerging numbers, we construct closed graphs with edges on the unit circle. These graphs build very nice shapes and lead to a symmetrical class of geometric patterns of so-called Islamic design. Moreover, we mathematically characterize and analyze some convergence properties of the used up-down sequences. Finally, four planar type of patterns are simulated.

The effect of intrathecal dexmedetomidine on the dose requirement of hyperbaric bupivacaine in spinal anaesthesia for caesarean section: a prospective, double-blinded, randomized study

BackgroundStudies have shown that dexmedetomidine (Dex) can prolong the duration of analgesia when added to local anaesthetic as an adjuvant in a central or peripheral nerve block. We hypothesized that intrathecal Dex can reduce the ED95 of spinal hyperbaric bupivacaine. Therefore, we conducted this prospective, double-blinded, randomized study to verify our hypothesis.MethodsNinety patients were allocated into the Dexmedetomidine group (received bupivacaine + 5 mcg dexmedetomidine) and the Control group (received bupivacaine + the same volume of saline) using a double-blinded and randomized method. The first patient in each group received 5 mg of IT hyperbaric bupivacaine, and the next dose for the following patient was determined by the probability of successful anaesthesia of the previous neighbouring dose. An improved up-down sequence allocated method combined with probit analysis was used to determine the ED95 of intrathecal hyperbaric bupivacaine for the two groups.ResultsThe ED95 and 95% confidence intervals (95% CI) of IT hyperbaric bupivacaine of the Dex group and Control group were 8.4 mg (95% CI, 6.5~ 13.8 mg) and 12.1 mg (95% CI, 8.3~ 312.8 mg), respectively. The duration of sensory block was longer in the Dex group than in the Control group (110.3 ± 35.3 vs 67.5 ± 26.2). The duration of analgesia was also longer in the Dex group than in the Control group (224.9 ± 45.4 vs 155.1 ± 31.6). The consumption of postoperative rescued sufentanil was significantly higher in the Control group than in the Dex group.ConclusionIntrathecal 5 mcg dexmedetomidine potentiated hyperbaric bupivacaine antinociception by 31% in spinal anaesthesia for patients undergoing caesarean section.Trial registrationWe registered this study in a Chinese Clinical Trial Registry (ChiCTR) centre on Nov 1st 2016 and received the registration number: ChiCTR-IPR-16009699.

Optimum time for intravenous cannulation after induction with sevoflurane, oxygen, and nitrous oxide in children without any premedication.

Intravenous cannulation is usually done in children after inhalational induction with volatile anesthetic agents. The optimum time for safe intravenous cannulation after induction with sevoflurane, oxygen, and nitrous oxide has been studied in premedicated children, but there is no information for the optimum time for cannulation with inhalational induction in children without premedication. The aim of this study was to determine the optimum time for intravenous cannulation after the induction of anesthesia with sevoflurane, oxygen, and nitrous oxide in children without any premedication. This is a prospective, observer-blinded, up-and-down sequential allocation study in unpremedicated ASA grade 1 children aged 2-6years undergoing elective dental surgery. Intravenous cannulation was attempted after inhalational induction with sevoflurane, oxygen, and nitrous oxide. The timing of cannulation was considered adequate if there was no movement, coughing, or laryngospasm. The cannulation attempt for the first child was set at 4minutes after the loss of eyelash reflex and the time for intravenous cannulation was determined by the up-and-down method using 15seconds as step size. Probit test was used to analyze the up-down sequences for the study. The adequate time for effective cannulation after induction with sevoflurane, oxygen, and nitrous oxide in 50% and 95% of patients was 53.02seconds (95% confidence limits, 20.23-67.76seconds) and 87.21seconds (95% confidence limits, 70.77-248.03seconds), respectively. We recommend waiting for 1minute 45seconds (105seconds) after the loss of eyelash reflex before attempting intravenous cannulation in pediatric patients induced with sevoflurane, oxygen, and nitrous oxide without any premedication.

Pair correlation functions of two- and three-dimensional hard-core fluids confined into narrow pores: Exact results from transfer-matrix method

The effect of confinement is studied on the local structure of two- and three-dimensional hard-core fluids. The hard disks are confined between two parallel lines, while the hard spheres are in a cylindrical hard pore. In both cases only nearest neighbour interactions are allowed between the particles. The vertical and longitudinal pair correlation functions are determined by means of the exact transfer-matrix method. The vertical pair correlation function indicates that the wall induced packing constraint gives rise to a zigzag (up-down sequence) shaped close packing structure in both two- and three-dimensional systems. The longitudinal pair correlation function shows that both systems transform continuously from a one-dimensional gas-like behaviour to a zigzag solid-like structure with increasing density.

Minimum local analgesic concentration of ropivacaine for intra-operative caudal analgesia in pre-school and school age children - analysis using the Dixon method

Deng et al. recently quantified the minimal local anaesthetic concentration of ropivacaine for intra-operative caudal analgesia in pre-school and school age children [1]. Although in their methods, the authors stated that the concentration of ropivacaine received by a particular subject was determined according to Dixon’s up and down method, the ED50 presented in the results was calculated using probit and logit regression only. We respectfully suggest that for proper analysis of the data, the ED50 with 95% CI should have been estimated from the up-down sequences using the method of Dixon and Massey. The sequences should then have been subjected to probit regression analysis as a backup or sensitivity test. We calculated the ED50 by averaging the midpoint concentration of all independent pairs of patients involving a crossover according to Dixon and Massey’s method [2, 3] using all 25 patients depicted in both groups. The ED50 (95% CI) values for school age and pre-school children are 0.143% (0.139–0.147) and 0.098% (0.093–0.103), respectively, compared with the authors’ results of 0.143% (95% CI: 0.132–0.157%) and 0.107% (95% CI: 0.089–0.122%). This would not have altered the overall result of the study, but is a more accurate presentation of their results. No external funding and no competing interests declared. Previously posted at the Anaesthesia Correspondence website: http://www.anaesthesiacorrespondence.com.

Dual Time Scales for Categorical Decision Making in Auditory Cortex

Category formation allows us to group perceptual objects into meaningful classes and is fundamental to cognition. Categories can be derived from similarity relationships of object features by using prototypes or multiple exemplars, or from abstract relationships of features and rules . A variety of brain areas have been implicated in categorization processes, but mechanistic insights on the single-cell and local-network level are still rare and limited to the matching of individual objects to categories . For directional categorization of tone steps, as in melody recognition , abstract relationships between sequential events (higher or lower in frequency) have to be formed. To explore the neuronal mechanisms of this categorical identification of step direction, we trained monkeys for more than two years on a contour-discrimination task with multiple tone sequences. In the auditory cortex of these highly trained monkeys, we identified two interrelated types of neuronal firing: Increased phasic responses to tones categorically represented the reward-predicting downward frequency steps and not upward steps; subsequently, slow modulations of tonic firing predicted the behavioral decisions of the monkeys, including errors. Our results on neuronal mechanisms of categorical stimulus identification and of decision making attribute a cognitive role to auditory cortex, in addition to its role in signal processing.

Extrapolation and interpretation

EDITOR: We would like to congratulate Albertin and colleagues on their study into the effects of adding nitrous oxide on the MACBAR of sevoflurane given with two target-controlled concentrations of remifentanil in women [1]. They used a modified up-down sequential allocation technique to determine the MACBAR of sevoflurane, described initially by Jung and Choi [2]. This involved two stages. The first used a standard up-down sequence using predetermined equally spaced test levels until three reversals were observed. For the second stage, the testing interval was reduced, restarting the up-down sequence at the nearest level to the average and continuing for four reversals. The logic is that by reducing the testing interval, the value of the final estimator is more precise and the doses used in the second stage are closer to the true median dose. A number of issues follow. The methods, as described in this paper, do not appear to have been followed consistently in that authors appear to have broken their own rule of achieving three reversals prior to moving on to the second stage of the sequential allocation technique in Figures 2 and 4. Figures 1 and 3, although more in keeping with the stated design, show considerable variability and limited stability in the first stage. They also present estimates of ED95 (should be EC95 for sevoflurane concentration), which are extrapolated well beyond the limits of the data as presented. Such extrapolations are better suited to discussion rather than results. They also report similar estimates of precision for EC50 and EC95, where the latter is usually more imprecise. Also, incidentally Figure 4 appears to have an error in the correct plotting of the MACBAR and error bar. The main issue, however, is how to best present and analyse these data given that the method described by the authors removes up to approximately 60% of the subjects from the analysis. The question is whether we can simply ‘ignore’ the initial data? Intuitively the answer is ‘yes’ for Figures 2 and 4, but perhaps ‘no’ for Figures 1 and 3. Also whilst the simulations by Jung [2] show the expected reductions in mean square error by reducing the testing interval, it is not clear what should be done about the variability of data with larger testing intervals where this is trending or when the reversals are unbalanced as in this study. (The schematic in the Jung paper [2] showed a balanced example). A logical approach is then to consider all the data together. Whilst this will decrease the precision of the estimates, it implies that we are not discarding the variability of otherwise valid observations. The authors present an up-down analysis and correctly use logistic regression as a back-up analysis. In this instance, Table 1 shows the logistic regression results reported in the study and an additional analysis of all the data using a loglogistic maximum likelihood approach. They confirm the highly significant (P = 0.001) effects of nitrous oxide and alfentanil on MACBAR of sevoflurane. However, the fiducial 95% confidence intervals (CI) for the estimates of ED95 are relatively imprecise compared to the original report.Table 1: Original results with 95% CI for EC95.In conclusion, again the authors are to be commended on using a dose-varying methodology. Care must be taken when using regression methods to extrapolate point estimates beyond the data as collected and should not be assigned the same weight as median estimates. M. O. Columb P. Tandon Acute Intensive Care Unit, South Manchester University Hospital, Wythenshawe, Manchester, UK

Minimum Local Analgesic Dose

This double-blind, randomized study was aimed at detecting the effect of three different volumes of intrathecal levobupivacaine on the minimum local analgesic dose in early labor. Ninety-three nulliparous women requesting combined spinal-epidural analgesia, at more than 37 weeks gestation, with spontaneous onset of labor, cervical dilatation from 2 to 5 cm, were enrolled. Parturients received 10 ml (group 10), 5 ml (group 5), or 2.5 ml (group 2.5) of the spinal solution containing plain levobupivacaine diluted with 0.9% wt/vol saline to achieve the desired dose and volume at room temperature. A lumbar epidural catheter was then placed. The initial dose for each group was 2.0 mg, and the following doses were determined by the response of the previous patient using up-down sequential allocation. The authors required the test solution to achieve a visual analog pain score of 10 mm or less to be considered effective. The up-down sequences were analyzed using the Dixon and Massey formula and regression logistic model. The minimum local analgesic dose of spinal levobupivacaine in spontaneously laboring women was 1.35 mg (95% confidence interval, 1.25-1.45 mg) in group 10, 1.63 mg (95% confidence interval, 1.51-1.76 mg) in group 5, and 1.97 mg (95% confidence interval, 1.89-2.05 mg) in group 2.5. A unit change in volume increased the odds of an effective response multiplicatively by a factor of 1.8. Analgesia can be achieved using lower doses and higher volumes even in subarachnoid space. The important role of the volume should be considered not only in epidural but also in spinal analgesia.

Minimum Local Analgesic Dose

Background. This double-blind, randomized study was aimed at detecting the effect of three different volumes of intrathecal levobupivacaine on the minimum local analgesic dose in early labor.Methods. Ninety-three nulliparous women requesting combined spinal-epidural analgesia, at more than 37 weeks gestation, with spontaneous onset of labor, cervical dilatation from 2 to 5 cm, were enrolled. Parturients received 10 ml (group 10), 5 ml (group 5), or 2.5 ml (group 2.5) of the spinal solution containing plain levobupivacaine diluted with 0.9% wt/vol saline to achieve the desired dose and volume at room temperature. A lumbar epidural catheter was then placed. The initial dose for each group was 2.0 mg, and the following doses were determined by the response of the previous patient using up-down sequential allocation. The authors required the test solution to achieve a visual analog pain score of 10 mm or less to be considered effective. The up-down sequences were analyzed using the Dixon and Massey formula and regression logistic model.Results. The minimum local analgesic dose of spinal levobupivacaine in spontaneously laboring women was 1.35 mg (95% confidence interval, 1.25-1.45 mg) in group 10, 1.63 mg (95% confidence interval, 1.51-1.76 mg) in group 5, and 1.97 mg (95% confidence interval, 1.89-2.05 mg) in group 2.5. A unit change in volume increased the odds of an effective response multiplicatively by a factor of 1.8.Conclusions. Analgesia can be achieved using lower doses and higher volumes even in subarachnoid space. The important role of the volume should be considered not only in epidural but also in spinal analgesia.

Periodic de Bruijn triangles: exact and asymptotic results

We study the distribution of the number of permutations with a given periodic up-down sequence w.r.t. the last entry, find exponential generating functions and prove asymptotic formulas for this distribution.

Exploratory Experimentation: Goethe, Land, and Color Theory

The style of investigation exemplified by Goethe’s experiments with color is often undervalued, but has repeatedly proved its worth.

Enumerating pairs of permutations with the same up-down form

For each sequence q = {q i} = ± 1, i = 1, …, n−1 let N q = the number of permutations σ of 1, 2, …, n with up-down sequence sgn( σ i+1 − σ i) = q i, i = 1,…, n−1. Clearly Σ q ( N q/ n!) =1 but what is the probability p n = Σ q ( N q/ n!) 2 that two random permutations have the same up-down sequence? We show that p n = ( K n−1 1,1) where 1 = 1( x, y) ≡ 1 and K n−1 is the iterated integral operator with Kφ( x, y) = ∫ 0 1∫ 0 1 K( x, y; x′, y′) φ( x′, y′) dx′ dy′ on L 2[0, 1] × [0, 1] where K( x, y; x′, y′) is 1 if ( x− x′)( y − y′) > 0 otherwise, and ( f, g) = ∫ 0 1∫ 0 1 fg. The eigenexpression of K yeilds p n ∼ cα n as n → ∞, where c ≈ 1.6, α ≈ 0.55. We also give a recursion formula for a polynomial whose coefficients are the frequencies of all the possible forms.

Maximal chains of subwords and up-down sequences of permutations

The free monoid generated by two letters is a poset for the relation “ u is a subword of v .” Two bijections between permutations and maximal chains starting from the empty word, involving the classical up-down sequence and the so-called odd-even height sequence, are given. For arbitrary alphabet, symmetric polynomials in s variables appear, generalizing the Eulerian polynomials and refining polynomials studied by Carlitz, Gessel, Riordan, and Stanley. The skewhook property of Foulkes is extended.

Équidistribution des permutations ayant une forme donnée selon les avances et coavances

We prove a conjecture of D. Foata and M. P. Schützenberger about a property on equidistribution of permutations with prescribed up-down sequence. For that we give an algorithm of construction and a formula for the number of permutations σ having prescribed up-down sequence, advances (rises of the inverse σ −1), and coadvances (advances of the complementary of the mirror image of σ).

Sedimentary facies in the Archean Timiskaming Group and their tectonic implications, abitibi greenstone belt, Northeastern Ontario, Canada

The Archean Timiskaming Group occupies two east-west trending belts in the southeastern part of the Abitibi greenstone belt in northeastern Ontario, and unconformably overlies a thick sequence of submarine mafic-to-felsic volcanic rocks. Within the Timiskaming Group four sedimentary facies are included in a Non-marine Association, which is dominated by pebble-cobble conglomerate and sandstone of braided river origin. Smaller amounts of floodplain, lacustrine, and eolian deposits are also present. A Resedimented Association consists of four other facies, and includes graded and ungraded conglomerates, sandy turbidites, sandy and silty, laminated turbidites, and oxide iron-formation. These facies represent deposition within submarine fans. Paleocurrent patterns in the Timiskaming Group were established using sedimentary structures, and orientations of gravel-sized clasts and sand grains. These data suggest multiple paleoslopes. Lateral and vertical facies changes between the Non-Marine Association and Resedimented Association are abrupt, and there are no indications of intervening shallow, marine deposition. Various lines of evidence suggest that for the Timiskaming Group fluvial rock bodies represent remnants of volcanic islands, which were separated by deep-water marine basins receiving turbidites and resedimented conglomerates. The vertical sequence defined by submarine volcanic rocks, overlain by Non-marine Association rocks and subaerial volcanic rocks, and capped by components of the Resedimented Association is termed an up-down sequence. Rapid subsidence within the up-down sequence is interpreted to result from foundering of volcanic islands. Depositional environments in the Timiskaming Group and in other Archean greenstone belts in Canada contrast with sedimentary environments from Archean greenstone belts in South Africa, where paralic deposition appears to be widespread. This suggests small, unstable continental fragments in the Canadian Shield during the Archean, and larger, more stable continents in the South African Shield for this time period.

Enumeration of up-down sequences

Generalizing the notion of up-down permutations, the author considers sequences σ = (a 1, a 2, , α N) of length N = s 2 + s 2 ++ s n, where α i ∈ {1, 2,n } and the element j occurs exactly s j times. The repeated elements of a are labeled i, i′, i″, and it is assumed that they occur in a m natural order. Generating functions for the number of up-down sequences are constructed. Making use of the generating functions, explicit formulas for the number of up-down sequences are obtained.

Up-down sequences

Up-down sequences