H(2)-unknotting Number of a Knot

An H（2）-move is a local move of a knot which is performed by adding a half-twisted band. It is known an H（2）-move is an unknotting operation. We define the H（2）-unknotting number of a knot K to be the minimum number of H（2）-moves needed to transform K into a trivial knot. We give several methods to estimate the H（2）-unknotting number of a knot. Then we give tables of H（2）-unknotting numbers of knots with up to 9 crossings.

Intrinsic Knotting of Almost Complete Partite Graphs

Let G be a complete p-partite graph with 2 edges removed, p ≥ 7, which is intrinsically knotted. Let J represent any graph obtained from G by a finite sequence of △-Y exchanges and/or vertex expansions. In the present paper, we show that the removal of any vertex of J and all edges incident to that vertex produces an intrinsically linked graph. This result offers more intrinsically knotted graphs which hold for the conjecture presented in Adams＇ book （Adams C. The Knot Book. New York： W. H. Freeman and Company, 1994）, that is, the removal of any vertex from an intrinsically knotted graph yields an intrinsically linked graph.

Unique method for removal of knotted lumbar epidural catheter: A case report

BACKGROUND Combined spinal-epidural(CSE)anesthesia is the preferred anesthesia method for cesarean delivery.The use of an epidural catheter is essential for administering additional drugs intraoperatively and managing postoperative pain.However,the insertion of epidural catheters is associated with various complications,such as total spinal anesthesia,symptoms indicative of spinal nerve root irritation,and challenges in epidural catheter removal.CASE SUMMARY We present a case report of a challenging epidural catheter removal due to knotting.The lumbar computed tomography scan results revealed that the catheter formed a tight knot in the epidural space.We used a novel extubation method and successfully removed the catheter.CONCLUSION The operator can use opposite forces to"spiral"apart the spinal joints by positioning the patient's body in a specific position.The findings indicate that,when combined with imaging examination results,this method is effective for the removal of epidural catheters.

Three-dimensional knotting of W_(17)O_(47)@PEDOT:PSS nanowires enables high-performance flexible cathode for dual-functional electrochromic and electrochemical device

An efficient integration of electrochromic and electrochemical devices into one flexible entity enables both energy storage and energy-saving dual-functionalities.For this purpose,achieving both high electrochromic and electrochemical performance is the key aspect.Herein,a new 3D architecture is successfully made by knotting W_(17)O_(47)@PEDOT(poly(3,4-ethylenedioxythiophene)):PSS(poly(styrenesulfonate))nanowires with NaWO_(3)nanoknots,and interestingly,the 3D W_(17)O_(47)/(NaWO_(3)-knots)@PEDOT:PSS cathode thus-made simultaneously exhibits a large optical modulation(79.7%at 633 nm),an ultra-long cycling life(76%of original optical modulation retained after 12400 cycles),and a high areal capacitance(55.1 mF cm^(-2)at 0.1 mA cm^(-2)).Our density functional theory(DFT)calculations demonstrate that the much improved dual-functional performance is correlated to the raised electronic conductivity and ion adsorption at the W_(17)O_(47)/(NaWO_(3)nanoknots)interface,together with the ion adsorption of PEDOT:PSS in the 3D-knotted architecture.As a proof-of-concept application,different-sized flexible dual-functional electrochromic/electrochemical devices(FDEDs)were assembled and investigated for various application scenarios,including a smart window(15 cm×10 cm),a wearable wristband(20 cm×2.5 cm),and a smart eyeglass.The smart window made of the FDED enables a large temperature difference of 27.6℃ confirm-tested in model houses,where the energy source also powers three light-emitting diodes(LEDs).The understandings of the key governing principles in the electrodes and dual-functionalities provide a timely foundation for the new generation flexible multifunctional devices.

Influences of flexible defect on the interplay of supercoiling and knotting of circular DNA

Knots are discovered in biophysical systems,such as DNA and proteins.Knotted portions in knotted DNA are significantly bent and their corresponding bending angles are comparable with or larger than the sharp bending angle resulting in flexible defects.The role of flexible defects in the interplay of supercoiling and knotting of circular DNA were predicted by a Monte Carlo simulation.In knotted DNA with a particular knot type,a flexible defect noticeably enhances the supercoiling of the knotted DNA and the decreasing excitation energy makes the knotted portion more compact.A reduction in twist rigidity and unwinding of flexible defects are incorporated into the numerical simulations,so that interplay of supercoiling and knotting of circular DNA is studied under torsional conditions.Increasing unwinding not only results in a wider linking number distribution,but also leads to a drift of the distribution to lower values.A flexible defect has obvious effects on knotting probability.The summation of equilibrium distribution probability for nontrivial knotted DNA with different contour length does not change with excitation energy monotonically and has a maximum at an intermediate value of excitation energy around 5 kBT.In the phase space of knot length and gyration radius of knotted DNA,knot length does not anticorrelate with its gyration radius,which is attributed to the flexible defect in the knotted portion,which leads to the release of bending energy and inhibited the competition between entropy and bending energy.

New Perspective to Isogeometric Analysis:Solving Isogeometric Analysis Problem by Fitting Load Function

Isogeometric analysis(IGA)is introduced to establish the direct link between computer-aided design and analysis.It is commonly implemented by Galerkin formulations(isogeometric Galerkin,IGA-G)through the use of nonuniform rational B-splines(NURBS)basis functions for geometric design and analysis.Another promising approach,isogeometric collocation(IGA-C),working directly with the strong form of the partial differential equation(PDE)over the physical domain defined by NURBS geometry,calculates the derivatives of the numerical solution at the chosen collocation points.In a typical IGA,the knot vector of the NURBS numerical solution is only determined by the physical domain.A new perspective on the IGAmethod is proposed in this study to improve the accuracy and convergence of the solution.Solving the PDE with IGA can be regarded as fitting the load function defined on the NURBS geometry(right-hand side)with derivatives of the NURBS numerical solution(left-hand side).Moreover,the design of the knot vector has a close relationship to theNURBS functions to be fitted in the area of data fitting in geometric design.Therefore,the detected feature points of the load function are integrated into the initial knot vector of the physical domainto construct thenewknot vector of thenumerical solution.Then,they are connected seamlessly with the IGA-C framework for its great potential combining the accuracy and smoothness merits with the computational efficiency,which we call isogeometric collocation by fitting load function(IGACL).In numerical experiments,we implement our method to solve 1D,2D,and 3D PDEs and demonstrate the improvement in accuracy by comparing it with the standard IGA-C method.We also verify the superiority in the accuracy of our knot selection scheme when employed in the IGA-G method,which we call isogeometric Galerkin by fitting load function(IGA-GL).

Three-dimensional inversion of knot defects recognition in timber cutting

The comprehensive utilization of wood is the main goal of log cutting,but knot defects increase the diffi-culty of rationally optimizing cutting.Due to the lack of real shape data of knot defects in logs,it is diffi cult for detection methods to establish a correlation between signal and defect morphology.An image-processing method is proposed for knot inversion based on distance regularized level set segmentation(DRLSE)and spatial vertex clustering,and with the inversion of the defects existing relative board position in the log,an inversion model of the knot defect is established.First,the defect edges of the top and bottom images of the boards are extracted by DRLSE and ellipse fi tting,and the major axes of the ellipses made coplanar by angle correction;second,the coordinate points of the top and bottom ellipse edges are extracted to form a spatial straight line;third,to solve the intersection dispersion of spatial straight lines and the major axis plane,K-medoids clustering is used to locate the vertex.Finally,with the vertex and the large ellipse,a 3D cone model is constructed which can be used to invert the shape of knots in the board.The experiment was conducted on ten defective larch boards,and the experimental results showed that this method can accurately invert the shapes of defects in solid wood boards with the advantages of low cost and easy operation.

A Novel Localized Meshless Method for Solving Transient Heat Conduction Problems in Complicated Domains

This paper first attempts to solve the transient heat conduction problem by combining the recently proposed local knot method(LKM)with the dual reciprocity method(DRM).Firstly,the temporal derivative is discretized by a finite difference scheme,and thus the governing equation of transient heat transfer is transformed into a non-homogeneous modified Helmholtz equation.Secondly,the solution of the non-homogeneous modified Helmholtz equation is decomposed into a particular solution and a homogeneous solution.And then,the DRM and LKM are used to solve the particular solution of the non-homogeneous equation and the homogeneous solution of the modified Helmholtz equation,respectively.The LKM is a recently proposed local radial basis function collocationmethod with themerits of being simple,accurate,and free ofmesh and integration.Compared with the traditional domain-type and boundary-type schemes,the present coupling algorithm could be treated as a really good alternative for the analysis of transient heat conduction on high-dimensional and complicated domains.Numerical experiments,including two-and three-dimensional heat transfer models,demonstrated the effectiveness and accuracy of the new methodology.

Rhino中NURBS曲线的探讨

NURBS曲线在CAD软件开发中被广泛应用。Rhino软件中的NURBS曲线,涉及到NURBS曲线的构成元素、属性、关键定义项目、缺陷和用NURBS曲线构建曲面,对于进行工业产品造型设计有着较强的现实意义。

A Comparison between the Mechanical Behaviour of Steel Wires and Ultra High Molecular Weight Poly Ethylene Cables for Sternum Closure

Sternum closure after open heart surgery is typically done with steel wires. Final approximation of sternal parts and connection is achieved by twisting the ends of the wire and bending the twisted assembly towards the sternum in order to minimize outward protrusion. Though this routine procedure is highly effective, some failures do occur, e.g. due to wire fracture. Fatigue fracture of the wires, e.g. due to coughing implies a failure risk. An alternative development is to make cables from gel spun Ultra High Molecular Weight Poly Ethylene (UHMWPE) fibres, such fibres are extremely strong, yet flexible, and if made as a very pure grade, they are highly bio compatible. The optimal connection technique will be different from that of steel. Connection will rather be with knotting than twisting. A new sternum closure and fixation technique has been developed for the sternum. Additionally, a testing technique was developed, for a connection of simulated sternum parts, using different materials according to their respective optimal connection method and subsequently testing the mechanical properties of the connection. Substantial differences were observed. The mechanical behaviour of twisted steel wire connection showed more scatter than the knotted UHMWPE cables and some initial slack was sometimes present in the twisted cables. The maximum attainable force in the steel wires was determined by “untwisting” due to the external load. The maximum force in the UHMWPE cables was determined by the knot strength, either slipping for small knots, or breaking of the cables at the knots for slip-improved knots. The maximum force on the knotted UHMWPE cables was substantially larger than the maximum force on the twisted steel wires. Fatigue tests were performed on both the steel solution and the UHMWPE cables solution. The performance was about similar, although the simulated sternum opening was smaller for the UHMWPE cables at higher load levels. Summarizing, the UHMWPE cables show two advantages namely higher maximum load and more reproducible mechanical behaviour due to less scatter in the mechanical behaviour. On the other hand, the connection by knotting UHMWPE cables is somewhat more elaborate than the simple twisting connection of steel wires.

KNOT认证加密算法的零和区分器分析

KNOT认证加密算法是国际轻量级密码算法征集竞赛活动第2轮入围的候选算法之一。该算法具有软硬件实现快、资源消耗低等特点,其安全性受到了全球广泛的关注。基于标志位技术,提出了密码S盒的新可分性模型构建方法。同时,利用KNOT-256的算法结构,构建了KNOT-256密码算法新的可分性的混合整数线性规划模型,并由此设计了该算法零和区分器的自动化新搜索方法。研究结果表明:KNOT-256置换存在30轮的零和区分器,尽管该区分器对KNOT认证加密算法(分组长度为256的版本的初始化轮数是52轮)的安全性没有造成实际的威胁,但得到的结果证实了新的零和区分器构造方法是有效的。

Hydrogen Bond Networks' QSAR and Topologica Analysis of Konjac Glucomannan Chains

Based on the knot theory and researching of network structures of glucomannan molecules, the polysaccharides were analyzed. The link prediction analysis is to further reveal the interactions between polysaccharides, to elaborate QSAR of polysaccharides, and to analyze the network conformation relationships among polysaccharides. We made a classification for glucomannan molecules based on the related domestic and international theories, and investigated their network structures and application prospects. The knot theory and the link predictions not only simplify the glucomannan microscopic descriptions but also play a guiding role in predicting and regulating the structures.

FREE VIBRATION ANALYSIS OF ARBITRARY SHAPED PLATES BY BOUNDARY KNOT METHOD

The boundary knot method （BKM） is a truly meshless boundary-type radial basis function （RBF） collocation scheme, where the general solution is employed instead of the fundamental solution to avoid the fictitious outside boundary of the physical domain of interest. In this study, the BKM is first used to calculate the free vibration of free and simply-upported thin plates. Compared with the analytical solution and ANSYS （a commercial FEM code） results, the present BKM is highly accurate and fast convergent.

Nice knot缝线技术治疗骨性锤状指

2017年4月~2019年2月,我科采用Nice knot缝线技术治疗15例骨性锤状指患者,疗效满意,报道如下。1材料与方法1.1病例资料本组15例,男9例,女6例,年龄19~45岁。典型表现为锤状指畸形,X线检查显示远节指骨基底部背侧撕脱骨折。均为新鲜闭合损伤。伤后至手术时间3~7 d。1.2治疗方法臂丛麻醉下手术。

Zeros of the Jones Polynomial for Torus Knots and 2-bridge Knots

We study zeros of the Jones polynomial and their distributions for torus knots and 2-bridge knots. We prove that e(2m+1)πi/2and e(2m+1)πi/4(m is a positive integer)can not be the zeros of Jones polynomial for torus knots T p,q by the knowledge of the trigonometric function. We elicit the normal form of Jones polynomials of the 2-bridge knot C(-2, 2, ···,(-1)r2) by the recursive form and discuss the distribution of their zeros.

A recent review on phytochemical constituents and medicinal properties of kesum(Polygonum minus Huds.)

Medicinal plants and herbal preparations are gaining renowned interest in scientific communities nowadays due to their reliable pharmacological actions and affordability to common people which makes them effective in control of various diseases.Polygonum minus(Polygonaceae)locally known as kesuni is an aromatic plant commonly used in Malay delicacies.The plant is having potential applications due to its high volatile oil constituents in perfumes and powerful antioxidant activity.It has been used traditionally to treat various ailments including dandruff.The research has been carried out by various researchers using different in vitro and in vivo models for biological evaluations to support these claims.This review paper may help upcoming research activities on Polygonum minus by giving up to date information on the phytochemical constituents and medicinal properties of kesum to a possible extent with relevant data.

Exogenous Peroxidase Mitigates Cadmium Toxicity, Enhances Rhizobial Population and Lowers Root Knot Formation in Rice Seedlings

Soil cadmium (Cd) causes toxicity and oxidative stress, alters biochemical processes and rootknot formation in rice. Irrigation of exogenous peroxidase (POX) together with its co-substrate H2O2(POXRice + H2O2), is likely to have protective effect upon the biochemical and nodular changes in ricegrown in Cd-rich soil. Exposure to Cd concentration of 1.00 mg/L increased oxidative stress, loss of cellviability, electrolyte leakage and root knot formation, whereas it significantly lowered the chlorophyll leveland rhizobium growth in rice. Irrigation of exogenous POXRice + H2O2 to Cd-stressed rice seedlingsreversed the Cd-induced alterations in rice to levels similar in control (non-stressed) seedlings. Resultsprovided strong evidence of exogenous POXRice + H2O2-mediated reversal and restoration of physiologicaland biochemical processes as well as increased resistance of rice seedlings to root knot formation.Irrigation with POXRice + H2O2 appeared to contribute towards bringing normoxic conditions in the otherwisehypoxic soil environment by enhancing the O2 in pot-experiments due to reduced Cd uptake, enhancedmineral homeostasis of essential elements viz. P, Fe, Mo, Mg and Mn for maintenance of root architecturedamaged by lipid peroxidation and reduction in oxidative stress by reducing Cd-induced reactive oxygenspecies generation. Therefore, the mitigation of Cd-toxicity in rice through this novel approach appeared tobe a promising mode to limit Cd-uptake, modulate protective and tolerance mechanisms for sustainablerice yield in Cd-contaminated rice-croplands and prevent nematode attack in rice, however, more detailedstudies are needed prior to large scale applications.

Predicting Financial Contagion and Crisis by Using Jones, Alexander Polynomial and Knot Theory

Topological methods are rapidly developing and are becoming more used in physics, biology and chemistry. One area of topology has showed its immense potential in explaining potential financial contagion and financial crisis in financial markets. The aforementioned method is knot theory. The movement of stock price has been marked and braids and knots have been noted. By analysing the knots and braids using Jones polynomial, it is tried to find if there exists an untrivial knot equal to unknot? After thorough analysis, possible financial contagion and financial crisis prediction are analysed by using instruments of knot theory pertaining in that sense to Jones, Laurent and Alexander polynomial. It is proved that it is possible to predict financial disruptions by observing possible knots in the graphs and finding appropriate polynomials. In order to analyse knot formation, the following approach is used: “Knot formation in three-dimensional space is considered and the equations about knot forming and its disentangling are considered”. After having defined the equations in three-dimensional space, the definition of Brownian bridge concerning formation of knots in three-dimensional space is defined. Using analogy method, the notion of Brownian bridge is translated into 2-dimensional space and the foundations for the application of knot theory in 2-dimensional space have been set up. At the same time, the aforementioned approach is innovative and it could be used in accordance with stochastic analysis and quantum finance.

Nice Knot缝合结合张力带技术治疗髌骨下极骨折

目的:探讨Nice Knot缝合结合张力带技术治疗髌骨下极骨折的临床疗效。方法:2016年1月至2017年12月收治12例闭合性髌骨下极骨折患者。男5例,女7例。年龄30~60岁,中位数45岁。均为单侧损伤,左侧5例、右侧7例。均不合并膝骨关节炎、类风湿关节炎、先天性髌骨发育不良、膝关节畸形,均无同侧膝关节骨折及手术史。受伤至手术时间0~7 d,中位数4 d。均采用Nice Knot缝合结合张力带技术治疗。术后定期随访,观察患者的髌骨骨折愈合情况,评定膝关节活动度和B9stman髌骨骨折疗效评分。结果:1例患者失访;11例患者获得随访,随访时间3~6个月,中位数4个月。11例患者的髌骨骨折均愈合良好,骨折临床愈合时间8~16周,中位数12周。术后1个月,患侧膝关节活动度小于健侧(111.81°±6.03°,131.81°±3.37°,t=-14.832,P=0.000);末次随访时,双侧膝关节活动度比较,差异无统计学意义(129.54°±6.50°,131.81°±3.37°,t=-1.838,P=0.096)。术后1个月和末次随访时,11例患者的B9stman髌骨骨折疗效评分分别为(23.81±0.40)分、(28.72±1.19)分;按照末次随访时的B9stman髌骨骨折疗效评分评定,优9例、良2例。结论:采用Nice Knot缝合结合张力带技术治疗髌骨下极骨折,临床疗效好,值得临床推广应用。

KNOT STRENGTH OF INDUSTRIAL TEXTILE FIBRES

In this paper, the knot strengths of the seven strands which were made of polyethleneterephthalate(PET), nylon 6, polyvinyl formal fibre, polypropylene and polyethylene with differentspecification used for industrial purposes were tested and discussed. The results of experimentshow: the knot strength loss does not only depend on the breaking elongation and the diameter ofsample, but also on the shape of the load-extension curve and twist factor of sample and other fac-tors; the range of the knot strength loss and the breaking energy loss vary in a wide range, in thepresent case. the former is from 2.7% to 58.14% and the latter from 16.36% to 78.76%. Thestrength loss of the polypropylene filament is the least among the samples investigated.