Group Bn Research Articles

Necessary and sufficient conditions for the irreducibility of a linear representation of the braid group Bn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_n$$\\end{document}

Valerij G. Bardakov and P. Bellingeri introduced a new linear representation ρ¯F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bar{\\rho }_F$$\\end{document} of degree n+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n+1$$\\end{document} of the braid group Bn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_n$$\\end{document}. We study the irreducibility of this representation. We prove that ρ¯F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bar{\\rho }_F$$\\end{document} is reducible to the degree n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n-1$$\\end{document}. Moreover, we give necessary and sufficient conditions for the irreducibility of the complex specialization of its n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n-1$$\\end{document} degree composition factor ϕ¯F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bar{\\phi }_F$$\\end{document}.

On the irreducibility of local representations of the Braid group Bn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_n$$\\end{document}

We prove that any homogeneous local representation φ:Bn→GLn(C)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi :B_n \\rightarrow GL_n(\\mathbb {C})$$\\end{document} of type 1 or 2 of dimension n≥6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 6$$\\end{document} is reducible. Then, we prove that any representation φ:Bn→GLn(C)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi :B_n \\rightarrow GL_n(\\mathbb {C})$$\\end{document} of type 3 is equivalent to a complex specialization of the standard representation τn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au _n$$\\end{document}. Also, we study the irreducibility of all local linear representations of the braid group B3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_3$$\\end{document} of degree 3. We prove that any local representation of type 1 of B3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_3$$\\end{document} is reducible to a Burau type representation and that any local representation of type 2 of B3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_3$$\\end{document} is equivalent to a complex specialization of the standard representation. Moreover, we construct a representation of B3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_3$$\\end{document} of degree 6 using the tensor product of local representations of type 2. Let ui\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_i$$\\end{document}, i=1,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$i=1,2$$\\end{document}, be non-zero complex numbers on the unit circle. We determine a necessary and sufficient condition that guarantees the irreducibility of the obtained representation.

Representations of Flat Virtual Braids by Automorphisms of Free Group

Representations of braid group Bn on n≥2 strands by automorphisms of a free group of rank n go back to Artin. In 1991, Kauffman introduced a theory of virtual braids, virtual knots, and links. The virtual braid group VBn on n≥2 strands is an extension of the classical braid group Bn by the symmetric group Sn. In this paper, we consider flat virtual braid groups FVBn on n≥2 strands and construct a family of representations of FVBn by automorphisms of free groups of rank 2n. It has been established that these representations do not preserve the forbidden relations between classical and virtual generators. We investigated some algebraic properties of the constructed representations. In particular, we established conditions of faithfulness in case n=2 and proved that the kernel contains a free group of rank two for n≥3.

Addendum to “On some torus knot groups and submonoids of the braid groups” [J. Algebra 607 (Part B) (2022) 260–289

In [3], we gave a conjectural presentation of the submonoid of the (n+1)-strand braid group Bn+1 (n≥3) generated by σ1,σ1σ2,…,σ1σ2⋯σn, generalizing a conjecture given by Dehornoy for n=3. We show that this conjecture is false for all n≥3, by showing that the presentation which we exhibited defines a monoid which is not left-cancellative.

Top to random shuffles on colored permutations

A deck of n cards are shuffled by repeatedly taking off the top card, flipping it with probability 1/2, and inserting it back into the deck at a random position. This process can be considered as a Markov chain on the group Bn of signed permutations. We show that the eigenvalues of the transition probability matrix are 0,1/n,2/n,…,(n−1)/n,1 and the multiplicity of the eigenvalue i/n is equal to the number of the signed permutation having exactly i fixed points. We show the similar results hold also for the colored permutations. Further, we show that the mixing time of this Markov chain is nlogn and exhibits cut off, same as the ordinary ’top to random’ shuffles without flipping the cards. The cut off is also analyzed by the strong stationary time as well as the asymptotic formula of the Stirling numbers of the second kind.

Dubrovin–Frobenius manifolds associated with Bn and the constrained KP hierarchy

In this paper, we will show that the Dubrovin–Frobenius prepotentials on the orbit space of the Coxeter group Bn constructed by Arsie et al. [Sel. Math. New Ser. 29, 1 (2023)] coincide with the solutions of Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations associated with the constrained KP hierarchy.

Requirements in structure for chiral recognition of chitosan derivatives

In order to investigate the influence of a minor variation in structure of N-acyl chitosan derivatives on enantioseparation, chiral selectors (CSs) of chitosan 3,6-bis(phenylcarbamate)-2-(phenylacetamide)s and chitosan 3,6-bis(phenylcarbamate)-2-(cyclohexylacetamide)s were synthesized. The corresponding chiral stationary phases (N-PhAc CSPs and N-cHeAc CSPs) were also prepared, respectively, with the two series of CSs. Enantioseparation results revealed that the N-PhAc CSPs were better than the N-cHeAc ones in enantioseparation. Thus, benzyl group (Bn) at C2 should be more preferable to enantioseparation than cyclohexylmethyl (cyclohexyl–CH2–) at the same position. Because N-PhAc CSPs exhibited higher enantioseparation capability than chitosan 3,6-bis(phenylcarbamate)-2-(benzamide) based CSPs (N-Bz CSPs), the Bn should also be more beneficial to enantioseparation than phenyl group (Ph) at C2 in N-Bz CSPs. In addition, it was found that, the cyclohexyl group at C2 in chitosan 3,6-bis(phenylcarbamate)-2-(cyclohexylformamide) CSPs was better than cyclohexyl–CH2– in N-cHeAc CSPs to enantioseparation. In a word, a minor variation at C2 of chitosan derivatives significantly affected enantioseparation. After the prepared CSPs were stood for six months, their enantioseparation capabilities were changed obviously, and the changes were probably related to nature, position and number of a substituent on Ph connected to carbamates at C3 and C6 of the CSs.

Actions of the hyperoctahedral group to compute minimal contractors

The hyperoctahedral group Bn is the group of symmetries of the hypercube [−1,1]n of Rn. For instance permutations, or symmetries along each of the n canonical planes of Rn all belong to Bn. Now, many sets of equations contain symmetries in Bn. This is the case of the addition constraint: x1+x2=x3 or the multiplication x1⋅x2=x3. In robotics, many specific geometrical constraints such as for instance constraints involving distances or angles used for localization also have these symmetries. This paper shows the fundamental role of the hyperoctahedral group for interval-based methods. These methods use operators, called contractors, which contract axis-aligned boxes, without removing any point of the solution set defined by a conjunction of constraints (typically equations, or inequalities). More precisely, the paper presents an algorithm which allows us to build minimal contractors associated to constraints with symmetries in Bn. As an application, we will consider the geometrical constraint associated to the angle between vectors. The corresponding contractor will then be used in a constraint propagation framework in order to localize a robot using several radars.

A comparative study between the efficacy of bupivacaine plus nalbuphine versus bupivacaine alone in pectoral nerve block for management of postoperative pain in patients undergoing breast fibroadenoma excision

BackgroundBreast surgeries, even the relatively minor ones, can be associated with significant postoperative pain affecting patients’ physical and psychological well-being. As a result, regional anaesthetic approaches have been used for breast surgery anaesthesia and analgesia. PECS I and II blocks have been proven to be relatively easy and safe in such regards. Moreover, using adjuvants, such as nalbuphine, have been proposed by many studies to intensify the block and decrease postoperative analgesic intake.This current study aims to assess the effects of using nalbuphine (20 mg) as a bupivacaine 0.25% (10 ml) adjuvant in PECS I block compared to injecting bupivacaine 0.25% (10 ml) alone in fibroadenoma excision patients.This trial enrolled 60 patients, 30 of whom received bupivacaine plus nalbuphine (group BN) and 30 in the bupivacaine only (group B). Postoperative VAS, SpO2 and vital signs at 0, 1, 2, 4, 6, 12 and 24 h as well as the amount of morphine consumed, the time it took for the first analgesic to be administered and adverse effects were both recorded and compared.ResultsRegarding postoperative outcomes, a statistically significant difference between the two groups was not detected regarding VAS, SpO2, vital signs and adverse effects. Similarly, there was no statistically significant difference in total morphine intake in the 24 h following surgery between the two groups (p-value = 0.65). Only 9 out of 60 patients needed analgesia, 5 in the BN group and 4 in the B group, and time to the first analgesic was not of statistically significant difference between BN and B groups (211.8 ± 71.29 and 183.5 ± 29.872 min respectively) (p-value = 0.73). In terms of postoperative complications, there was no statistically significant difference between the two groups.ConclusionsAdding nalbuphine (20 mg) to bupivacaine in PECS I block in fibroadenoma patients undergoing excision is not accompanied by a difference with statistical significance in postoperative VAS, morphine consumption and duration to the first analgesia when compared to bupivacaine alone.

Study to compare the effect of nalbuphine and butorphanol as an adjuvant to intrathecal 0.5% hyperbaric bupivacaine for lower limb surgery

Introduction: Spinal Anaesthesia is routinely used for surgeries performed on lower abdomen, pelvis and lower limbs. Adjuvants allow multiple benefits, most significant being provision of post-operative analgesia. Aims: This observational study was conducted to compare intrathecal nalbuphine versus butorphanol with hyperbaric bupivacaine respectively to assess onset and duration of sensory and motor block, hemodynamic parameters, adverse drug reactions and duration of post-operative analgesia. Materials and Methods: Fifty participants between 20-60 years belonging to ASA grade I/II posted for elective surgery were assigned to : group BN (n=25) : Bupivacaine (0.5%) hyperbaric (3ml) + nalbuphine 1mg (0.1ml) + normal saline 0.1ml and group BB (n=25) : Bupivacaine (0.5%) hyperbaric (3ml) + butorphanol 200mcg (0.2ml). Mean period of sensory/motor blockade, hemodynamic parameters, adverse drug reactions and analgesic requirement post surgery were compared. Results: Onset of sensory blockade was comparable while onset of motor blockade was significantly earlier in nalbuphine than butorphanol. Duration of block and post-operative analgesia was significantly higher for nalbuphine. Hemodynamic parameters and adverse effects were comparable. Conclusion: Nalbuphine is more efficacious than butorphanol due to provision of early onset of motor blockade and prolonged period of sensory/motor blockade with delayed requirement of rescue analgesia.

A Comparative Study of Nalbuphine and Fentanyl as Adjuvants to Bupivacaine in Spinal Anesthesia in Lower Limb Orthopedic Surgeries

Background: Subarachnoid block is the most common technique employed for lower limb orthopedic surgeries. Adjuvants such as fentanyl and nalbuphine are added to increase the duration of post-operative analgesia. This study aimed to compare the efficacy of nalbuphine and fentanyl as adjuvants to bupivacaine in providing intra-operative anesthesia and post-operative analgesia in lower limb orthopedic surgeries. Objectives: The aim of this article is to compare the characteristics of sensory and motor block, hemodynamic parameter, time of first rescue analgesia, and adverse effects between the two groups. Materials and Methods: In this clinical trial, 80 patients undergoing elective lower limb orthopedic surgeries under spinal anesthesia were randomly allocated into two groups. In group BN, the patients received 0.5% 3 mL (H) bupivacaine + 800 mcg nalbuphine. In group BF, the patients received 0.5% 3 mL (H) bupivacaine + 25 mcg fentanyl. In both the groups, characteristics of sensory and motor block and time for first rescue analgesia were compared. Results: The onset of motor block, the maximum level of the block, and time to reach the peak level of the block were significantly faster in the BF group. Duration of motor block and time for first rescue analgesia were significantly prolonged in the BN group. However, there was no significant difference in time for two-segment regression [P = 0.157 (NS)] and hemodynamic changes. Conclusion: We conclude that combination of fentanyl as adjuvant to bupivacaine provides higher segmental level sensory blockage and faster sensory and motor blockage than nalbuphine. But nalbuphine gives longer time of post-operative analgesia than fentanyl did.

Comparing intrathecal dexmedetomidine and fentanyl as adjuvants to hyperbaric bupivacaine in elective casarian section, a study.

Objective: The design of present study was to explore and compare different aspects of fentanyl and DEX when used as adjuvants to hyperbaric bupivacaine in neurexial anesthesia in cases of cesarean sections. Study Design: Randomized Single Blinded study. Setting: Department of Anesthesia, QAMC, Bahawalpur. Period: January, 2019 to December, 2019. Material & Methods: They were divided in three groups, each group consists of 35 patients having the name of group BN, group BF and group BD. Patients in group BN was given the injection bupivacaine alone, group BF, administered injection bupivacaine along with fentanyl 25 mg and group BD given DEX 10 mg with bupivacaine intrathecally between L4 and L5 intervertebral disc. Scrutiny of onset of sensory block to T5, along with time required to attain Bromage O scale (motor block). Regression of sensory block (recovery of sensory function) and time required to reach Bromage 3 scale (recovery of motor function) were also recorded. Hemodynamic parameters such as heart rate, systolic and mean arterial pressures along with Ramsay sedation score were also taken into account. Results: Similar demographic profile has been observed in all groups. All three groups differ in terms of onset of sensory and motor block with p value 0.00 when BN was compared with Group BD and BF. Statistically significant results also observed between group BF and group BD with group BD showing shortest time required for initiation of sensory block (p value .04) and time to reach Bromage O scale (p value .02) . The duration for regression of four sensory segments shows the statistical significance (P 0.000) when all three groups were compared, however, no difference found between BF (175+12.85 min) and BD (171.88 + 12.33 min) which showed a P-value of 0.240. The time required to reach Bromage 3 score was statistically significant between all three groups (p value 0.00) and was also statistically significant between group BF and group BD (p value 0.00) with longest time taken by BD group to reach BROMAGE 3 scale. Both two research groups showed same degree of sedation and comfort for patient. Conclusion: Hyperbaric bupivacaine, in conjunction with dexmedetomidine and fentanyl produced satisfactory results in terms of faster sensory and motor block onset and prolonged time to take in regression of sensory and motor block in comparison to bupivacaine alone. Dexmedetomidine was better among the two adjuncts. Both adjuvants produce same degree of sedation and comfort for patient and relieved apprehension.

Bn−1-bundles on the flag variety, I

In this paper, we consider the orbits of a Borel subgroup Bn−1 of Gn−1=GL(n−1) (respectively SO(n−1)) acting on the flag variety Bn of G=GL(n) (resp. SO(n)). The group Bn−1 acts on Bn with finitely many orbits, and we use the known description of Gn−1-orbits on B to study these orbits. In particular, we show that a Bn−1-orbit is a fibre bundle over a Bn−1-orbit in a generalized flag variety of Gn−1 with fibre a Bm−1-orbit on Bm for some m<n. We further use this fibre bundle structure to study the orbits inductively and describe the monoid action on the fibre bundle. In a sequel to this paper, we use these results to give a complete combinatorial model for these orbits and show how to understand the closure relations on these orbits in terms of the monoid action.

Block number, descents and Schur positivity of fully commutative elements in [formula omitted

The distribution of Coxeter descents and block number over the set of fully commutative elements in the hyperoctahedral group Bn, FC(Bn), is studied in this paper. We prove that the associated Chow quasi-symmetric generating function is equal to a non-negative sum of products of two Schur functions. The proof involves a decomposition of FC(Bn) into a disjoint union of two-sided Barbash–Vogan combinatorial cells, a type B extension of Rubey’s descent preserving involution on 321-avoiding permutations and a detailed study of the intersection of FC(Bn) with Sn-cosets which yields a new decomposition of FC(Bn) into disjoint subsets called fibers. We also compare two different type B Schur-positivity notions, arising from works of Chow and Poirier.

Nano-photoluminescence of natural anyon molecules and topological quantum computation

The proposal of fault-tolerant quantum computations, which promise to dramatically improve the operation of quantum computers and to accelerate the development of the compact hardware for them, is based on topological quantum field theories, which rely on the existence in Nature of physical systems described by a Lagrangian containing a non-Abelian (NA) topological term. These are solid-state systems having two-dimensional electrons, which are coupled to magnetic-flux-quanta vortexes, forming complex particles, known as anyons. Topological quantum computing (TQC) operations thus represent a physical realization of the mathematical operations involving NA representations of a braid group Bn, generated by a set of n localized anyons, which can be braided and fused using a “tweezer” and controlled by a detector. For most of the potential TQC material systems known so far, which are 2D-electron–gas semiconductor structure at high magnetic field and a variety of hybrid superconductor/topological-material heterostructures, the realization of anyon localization versus tweezing and detecting meets serious obstacles, chief among which are the necessity of using current control, i.e., mobile particles, of the TQC operations and high density electron puddles (containing thousands of electrons) to generate a single vortex. Here we demonstrate a novel system, in which these obstacles can be overcome, and in which vortexes are generated by a single electron. This is a ~ 150 nm size many electron InP/GaInP2 self-organized quantum dot, in which molecules, consisting of a few localized anyons, are naturally formed and exist at zero external magnetic field. We used high-spatial-resolution scanning magneto-photoluminescence spectroscopy measurements of a set of the dots having five and six electrons, together with many-body quantum mechanical calculations to demonstrate spontaneous formation of the anyon magneto-electron particles (eν) having fractional charge ν = n/k, where n = 1–4 and k = 3–15 are the number of electrons and vortexes, respectively, arranged in molecular structures having a built-in (internal) magnetic field of 6–12 T. Using direct imaging of the molecular configurations we observed fusion and braiding of eν-anyons under photo-excitation and revealed the possibility of using charge sensing for their control. Our investigations show that InP/GaInP2 anyon-molecule QDs, which have intrinsic transformations of localized eν-anyons compatible with TQC operations and capable of being probed by charge sensing, are very promising for the realization of TQC.

Effect of intrathecal Dexmedetomidine on preventing shivering after spinal anesthesia in caesarian section

Abstract Background Shivering is among the common troublesome complications of spinal anesthesia (SA), and causes discomfort and discontentment in parturients undergoing cesarean section (CS). Objective to investigate the effects of intrathecal dexmedetomidine (DEX) as an adjuvant to heavy bupivacaine in the prevention of shivering in those who underwent CS under SA. Methods A prospective double-blind conducted at Ain Shams University hospitals on 100 parturients who are undergoing CS under Spinal anesthesia over a period of six months starting from December 2018 to May 2019 at Ain shams university hospitals. Patients were randomly subdivided into 2 groups of 50 patients each, Group BD: 5 micrograms DEX was added to 12.5 mg 0.5% heavy bupivacaine. Group BN: 0.5 mL 0.9% normal saline was added to the 12.5mg 0.5% of heavy bupivacaine. Results The incidence of shivering was significantly higher in the BN group (56%) than the BD group (24%). Likewise, the intensity of shivering was significantly higher in the BN group than the BD group. Conclusion Intrathecal DEX in conjugation with heavy bupivacaine reduced the incidence and intensity of shivering in patients undergoing cesarean section after spinal anesthesia and did not increase the incidence of hypotension or bradycardia.

A COMPARISION OF INTRATHECAL HYPERBARIC BUPUVACAINE WITH BUPRENORPHINE AND HYPERBARIC BUPIVACAINE ALONE IN GYNAECOLOGICAL SURGERIES FOR POSTOPERATIVE ANALGESIA

This study was performed at a tertiary care centre after the approval of the Institutional Ethical Committee and obtaining written informed consent from all patients. Sixty ASA I and II, aged 18- 65yrs, bodyweight 45-70kgs scheduled for gynaecological surgeries under spinal anaesthesia were chosen for the study and were divided into two groups named Group B and Group BN each comprising 30 patients. Group B received 3ml of 0.5% hyperbaric bupivacaine with 0.5 ml normal saline and Group BN received 3ml of 0.5% hyperbaric bupivacaine with 0.5ml (150mcg) of buprenorphine. Vital parameters like pulse rate, blood pressure, respiratory rate, SpO2 were recorded at 0 (basal) 15, 30, 45, 90 and 180 minutes. Postoperatively heart rate, blood pressure, respiratory rate and SP02 were monitored at 3 ,6 ,12 and 24 hrs. The mean age, height, weight, duration of surgery were comparable. Time of onset of sensory blockade and motor blockade were noted. The time for rescue medication was 909.0±216.9 min in group BN with a range from 690 min to 1500 min and in group B it was 412.0±89.28 min with a range from 130 min to 195 min. Comparing both groups duration of effective analgesia was significantly higher in group BN with P&lt;0.0001.thus, it can be concluded that addition of buprenorphine as an adjuvant in spinal anaesthesia excellently prolongs duration of analgesia in postoperative period with minimal side effects.

Evaluation of the Effects of Ultra-Low Dose Naloxone on Brachial Plexus Block with Bupivacaine

Introduction: Adjuvants are used in conjunction with local anaesthetics for brachial plexus block to quicken onset, prolong duration and augment the quality of anaesthesia.&#x0D; Objectives: The study aims to evaluate the effect of adding ultra-low dose naloxone to bupivacaine on the duration of postoperative analgesia.&#x0D; Methodology: A prospective, randomized, double-blind study was conducted including forty American Society of Anesthesiologists Physical Status (ASA PS) I and II patients of either sex between 15 and 65 years of age scheduled for upper limb surgery. They were randomized into two groups. Group BN received a mixture of 0.33% plain bupivacaine with ultra-low dose naloxone (100 ng) and group BS received 0.33% plain bupivacaine with saline. The procedure was standardized in both groups. The onset of sensory and motor block were assessed with pinprick sensation and muscle power respectively. Visual Analogue Score (VAS) was used to assess the pain at 1, 2, 4, 8 and 12 hours postoperatively.&#x0D; Result: The patients were comparable with respect to age, sex, ASA PS, baseline hemodynamic and respiratory parameters and duration of surgery. The mean duration of analgesia in group BN was significantly longer (610.7±125.4 mins) as compared to group BS (354.3±59.6 mins) (p&lt;0.001). No complications were noted in both the groups.&#x0D; Conclusion: This study demonstrated that 100 ng naloxone, when used with 0.33% plain bupivacaine for supraclavicular brachial plexus block, prolongs postoperative analgesia significantly after upper limb surgeries, without any significant side effects.

English

BACKGROUND Clonidine and dexmedetomidine are two α-2 adrenergic receptor agonists which have been used as an adjuvant to intrathecal local anaesthetics. Both drugs act by different mechanisms and potentiate the effect of each other. The purpose of the study was to assess the efficacy and safety of adding clonidine and dexmedetomidine to intrathecal hyperbaric bupivacaine on spinal block characteristics in gynaecological procedures. METHODS This study was conducted as a prospective comparative, randomized, double blind, placebo-controlled trial. Based on exclusion and inclusion criteria, 90 patients were included in this study by using a computer generated randomization. Patients were randomly divided into three groups each as follows: Gr BD: 0.5 % hyperbaric bupivacaine (3.5 ml) + 3 μg preservative free dexmedetomidine (total volume of 4 ml), group BC: 0.5 % hyperbaric bupivacaine (3.5 ml) + 0.5 ml clonidine (30 μg) (total volume of 4 ml), group BN (control): 3.5 ml of 0.5 % hyperbaric bupivacaine + same volume normal saline. The spinal block characteristics, hemodynamic stability, and side effects were compared. RESULTS Time to reach sensory block to T10 was 2.52 ± 0.33 minutes in group BN, 1.95 ± 0.38 ms in gr BC and 1.53 ± 0.24 minutes in group BD. The span of sensory block was longer in group BD in comparison to group BC (453.76 ± 52.78 v 389.36 ± 37.4) and group BN (453.76 ± 52.78 vs. 173.33 ± 14.56). Time for rescue analgesia was 169.12 ± 21.14 minutes in group BN, 245.21 ± 32.42 minutes in group BC and 289 ± 24 minutes in group BD. CONCLUSIONS When α-2 adrenergic receptor agonist is used intrathecally with bupivacaine, early motor and sensory block can be achieved along with long span of analgesia. KEYWORDS Bupivacaine, Dexmedetomidine, Clonidine, Spinal Anaesthesia

On the Burau representation of B4

In 1936 W. Burau discovered an interesting family of n×n matrices that give a linear representation of Artin’s classical braid group Bn, n=1,2,…. A natural question followed very quickly: is the so-called Burau representation faithful? Over the years it was proved to be faithful for n≤3, nonfaithful for n≥5, but the case of n=4 remains open to this day, in spite of many papers on the topic. This paper introduces braid groups, describes the problem in ways that make it accessible to readers with a minimal background, reviews the literature, and makes a contribution that reinforces conjectures that the Burau representation of B4 is faithful.