Multiple Knots Research Articles

Knots of the Umbilical Cord: Incidence, Diagnosis, and Management

(Int J Gynecol Obstet. 2024;00:1–8. doi: 10.1002/ijgo.15373) The umbilical cord is surrounded by Wharton jelly and is frequently referred to as the “lifeline” because of its essential function in transporting nutrients and expelling fetal waste. Factors such as the length and coiling of the cord impact its health and functionality. An essential aspect of umbilical cord function is an arterial anastomosis, known as “Hyrtl anastomosis,” which helps regulate and distribute blood flow to the placenta. This structure, typically located 3 cm before the cord enters the placenta, is vital for fetal well-being, and abnormalities can increase fetal stress or, in severe cases, lead to fetal demise. Umbilical cord knots (UCK) are among the observed abnormalities, often detected accidentally during routine sonography. True knots range from 0.3% to 2.1%, and while usually solitary, multiple knots can occur. Such knots have also been observed in monkey fetuses. The clinical significance of a knot depends on whether it is tight or loose; a tight knot, especially in a hyper-coiled cord, is less flexible and more prone to complications during birth.

Multiple Multicolored 3D Polarization Knots Arranged along Light Propagation.

Polarization and color play essential roles in understanding optical phenomena and practical applications. Customized three-dimensional (3D) light fields, characterized by specific polarization and color distributions, have garnered growing interest owing to their unique optical attributes and expanded capacity for information encoding. To align with the ongoing trend of compactness and integration, it is desirable to develop lightweight optical elements that can simultaneously control polarization and color in 3D space. Although engineering longitudinally variable 3D optical structures with predesigned color and polarization information can add more degrees of freedom and additional capacity for information encoding, it has not been reported. We propose a metasurface approach to generating multiple 3D polarization knots along the light propagation direction. Each knot features two colors and an engineered 3D polarization profile. Different multicolored 3D polarization knots are obtained by controlling the observation region along the light propagation. Our approach simultaneously combines polarization, color, and longitudinal control in 3D environment, offering extra degrees of freedom for engineering complex vector beams. The unique properties of the developed metadevices, together with the design flexibility and compactness of metasurface, pave the way for polarization systems with small volumes applicable to some areas such as complex structured beams and encryption.

Percutaneous femoral access: Stuck guide wire, decannulation difficulty due to unravelling and knotting.

Percutaneous techniques for femoral arterial access are increasingly being performed due to advances in endovascular cerebral procedures, as they provide a less morbid and minimally invasive approach than open procedures. Common complications associated with this peripheral puncture include hematoma, bleeding, pseudoaneurysm, arteriovenous fistula, retroperitoneal bleeding, inadvertent venous puncture, dissection, etc. The retrograde femoral access is currently the most frequently used arterial access as it is technically straightforward, allows for the use of larger size sheaths and catheters, allows repeated attempts, etc. Although being technically less challenging, grave complications can occur due to hardware failure. Here, we present a case of unruptured posterior inferior cerebellar artery (PICA) aneurysm, who underwent uneventful diagnostic cerebral digital substraction angiography (DSA) via right femoral artery route on first attempt, but on second attempt for therapeutic intervention, landed up with stuck guide wire and faced decannulation difficulty due to unravelling of guide wire and multiple knot formation, which was finally removed after multiple attempts at pulling and improvised manoeuvres. Such cannulation and decannulation difficulties have been reported multiple times for central venous access, but extremely rarely for femoral routes, making this case a rarity and worth reporting.

Fish Mouth Closure of Horizontal Cleavage Tear of the Meniscus using Self-locking Modified Racking Hitch Knot (HitPat knot). - A Technical Note.

Horizontal cleavage tears (HCT) of the meniscus are treated with fish-mouth closure with sutures using different methods. Using a no. 0 fiber wire and taking bites through both the flaps using a suture passing device and tying multiple knots is a cost-effective technique. A racking hitch knot is an option for all inside repairs of HCT. It needs multiple half hitches for locking the knot. We used a modification of the racking hitch knot (HP knot), making it a self-locking knot, so it does not need half hitches. Loop of no. 0 fiber wire is passed across both the flaps of the meniscus and a modified racking hitch knot tied in one goes closing the fish mouth. This modification (HP knot) saves operative time, it is cost-effective and has the distinct advantage of a reduction in the size of the final knot construct.

Hermite–Birkhoff spline Quasi-Interpolation with application as dense output for Gauss–Legendre and Gauss–Lobatto Runge–Kutta schemes

Spline Quasi-Interpolation (QI) of even degree 2R on general partitions is introduced, where derivative information up to order R≥1 at the spline breakpoints is required and maximal convergence order can be proved. Relying on the B-spline basis with possible multiple inner knots, a family of quasi-interpolating splines with smoothness of order R is associated with each R≥1, since there is the possibility of using different local sequences of breakpoints to define each QI spline coefficient. By using suitable finite differences approximations of the necessary discrete derivative information, each QI spline in this family can be associated also with a twin approximant belonging to the same spline space, but requiring just function information at the breakpoints.Among possible different applications of the introduced QI scheme, a smooth continuous extension of the numerical solution of Gauss-Lobatto and Gauss-Legendre Runge-Kutta methods is here considered. When R>1, such extension is based on the use of the variant of the QI scheme with derivative approximation which preserves the approximation power of the original Runge-Kutta scheme.

Validation of a hybrid multi-phase field model for fracture of wood

Fracture mechanics simulations are crucial for understanding the failure of wood, where the microstructure strongly influences mechanical properties. In particular, complex crack topologies present a challenge for which not many modeling approaches are suited. Therefore, a hybrid multi-phase field model, supporting cohesive failure in orthotropic materials with favorable fracture planes, is used. We show that the model agrees well with the investigated experiments, with tensile strengths and fracture energy release rates in a reasonable range for wood. Those findings allow the application of this model to more complex situations like wooden boards with multiple knots and complex fiber courses and show that the model’s input parameters are not purely numerical values but rather material parameters.

Multilevel quadratic spline integration

In this paper we present new quadratures based on both quasi-interpolation and multilevel methods by using bivariate quadratic B-spline functions, defined on simple and multiple knot type-2 triangulations, improving classical quadratures, based on quasi-interpolating splines. We also prove some symmetry properties that simplify their expression, study their approximation performances, propose some numerical results and a comparison with other known multilevel spline quadratures.

A practical method for computing with piecewise Chebyshevian splines

A piecewise Chebyshevian spline space is good for design when it possesses a B-spline basis and this property is preserved under knot insertion. The interest in such kind of spaces is justified by the fact that, similarly as for polynomial splines, the related parametric curves exhibit the desired properties of convex hull inclusion, variation diminution and intuitive relation between the curve shape and the location of the control points. For a good-for-design space, in this paper we construct a set of functions, called transition functions, which allow for efficient computation of the B-spline basis, even in the case of nonuniform and multiple knots. Moreover, we show how the spline coefficients of the representations associated with a refined knot partition and with a raised order can conveniently be expressed by means of transition functions. This result allows us to provide effective procedures that generalize the classical knot insertion and degree raising algorithms for polynomial splines. We further discuss how the approach can straightforwardly be generalized to deal with geometrically continuous piecewise Chebyshevian splines as well as with splines having section spaces of different dimensions. From a numerical point of view, we show that the proposed evaluation method is easier to implement and has higher accuracy than other existing algorithms.

A Reverse Non-Stationary Generalized B-Splines Subdivision Scheme

In this paper, two new families of non-stationary subdivision schemes are introduced. The schemes are constructed from uniform generalized B-splines with multiple knots of orders 3 and 4, respectively. Then, we construct a third-order reverse subdivision framework. For that, we derive a generalized multi-resolution mask based on their third-order subdivision filters. For the reverse of the fourth-order scheme, two methods are used; the first one is based on least-squares formulation and the second one is based on solving a linear optimization problem. Numerical examples are given to show the performance of the new schemes in reproducing different shapes of initial control polygons.

High-Resolution CT Findings of Leukemia Pulmonary Infiltration and Chemotherapy Outcome and In Vitro Study of Human Lung Tissue

In order to explore the high-resolution CT findings of leukemia pulmonary infiltration and chemotherapy outcomes and the in vitro study of human lung tissue, this paper selected a total of 120 clinically or surgically confirmed leukemia patients at the designated hospital of the study from December 2014 to December 2018, and divided them into three groups according to the random number table method: pulmonary infiltration group, chemotherapy outcome group and in vitro study group, with 40 cases in each group. The CT imaging features of the three groups of patients were observed and summarized respectively; the anomalous evaluation indexes of pulmonary parenchyma tissue abnormalities included CT halo sign, air crescent sign, lung segment consolidation, bronchial vascular bundle and nodules; the CT abnormalities such as thickening of the interlobular septum, bronchial interstitial thickening, nodular shadow, ground glassy change, and air cavity consolidation were selected as observation indicators. The results show that all cases have multiple solid nodules or multiple plaques, varying in number, size and distribution, in which 13 cases have multiple patchy shadows, 9 cases have multiple knots and 11 cases have multiple plaques and nodules; lesions are mainly distributed along the bronchial vessels in 21 cases, and 9 cases are along the center of the small leaves and 5 cases are randomly distributed; there are 13 cases that have frosted glass, in which 4 cases with pleural effusion, 9 cases with mold infection, show multiple patchy shadows with halo signs and layered mold balls. In summary, leukemia pulmonary infiltration has polymorphic high-resolution CT findings and chemotherapy outcomes; high-resolution CT imaging and in vitro studies of human lung tissue have important clinical and pathological research value for leukemia infiltration and chemotherapy outcome. The results of this study provide a reference for the further researches on high-resolution CT findings of pulmonary infiltration and chemotherapy outcomes and in vitro studies of human lung tissue.

A Knot Invariant Defined Based on the Skein Relation with Two Equations

Knot theory is a branch of the geometric topology, the core question of knot theory is to explore the equivalence classification of knots; In other words, for a knot, how to determine whether the knot is an unknot; giving two knots, how to determine whether the two knots are equivalent. To prove that two knots are equivalent, it is necessary to turn one knot into another through the same mark transformation, but to show that two knots are unequal, the problem is not as simple as people think. We cannot say that they are unequal because we can't see the deformation between them. For the equivalence classification problem of knots, we mainly find equivalent invariants between knots. Currently, scholars have also defined multiple knot invariants, but they also have certain limitations, and even more difficult to understand. In this paper, based on existing theoretical results, we define a knot invariant through the skein relation with two equations. To prove this knot invariant, we define a function <i>f</i>(<I>L</I>), and to prove <i>f</i>(<I>L</I>) to be a homology invariant of a non-directed link, we need to show that it remains constant under the Reideminster moves. This article first defines the <i>f_k</i>(<I>L</I>), the property of <i>f</i>(<I>L</I>) is obtained by using the properties of <i>f_k</i>(<I>L</I>). In the process of proof, the induction method has been used many times. The proof process is somewhat complicated, but it is easier to understand. And the common knot invariant is defined by one equation, which defining the knot invariant with two equations in this paper.

Comparison of Hook Plate Fixation Versus Arthroscopic Coracoclavicular Fixation Using Multiple Soft Anchor Knots for the Treatment of Acute High-Grade Acromioclavicular Joint Dislocations

To compare the clinical and radiologic outcomes of arthroscopically assisted coracoclavicular (CC) fixation using multiple soft anchor knots versus hook plate fixation in patients with acute high-grade Rockwood type III and V acromioclavicular (AC) joint dislocations. This retrospective study included 22 patients with acute Rockwood type III and V AC joint dislocations who underwent arthroscopic fixation or hook plate fixation surgery between February 2016 and March 2018. Patients were categorized into 2 groups: arthroscopically assisted CC fixation using multiple soft anchor knots group (AR, n= 12) and hook plate fixation group (HO, n= 10). We measured the CC distances (CCDs) and CCD ratio at 6 months, 1 year, and last follow-up postoperatively to compare the radiologic results between the groups. Clinical outcomes were assessed at 1 year postoperatively and at the last follow-up using the Visual Analog Scale, American Shoulder and Elbow Surgeons (ASES) scores, and Shoulder Pain and Disability Index (SPADI) scores, and the Disabilities of the Arm, Shoulder, and Hand (DASH) questionnaire. Magnetic resonance imaging after hook plate removal was used to evaluate the healing ligaments and tendon-bone interface. The patients in the AR group had better ASES, SPADI, and Quick DASH scores than the patients in the HO group at 1 year postoperatively and at last follow-up. The CCD and CCD ratio were significantly better in the AR group than in the HO group at the last follow-up period (P= .007/0.029). Magnetic resonance imaging findings showed grade I in 60% of patients in the AR group and grade III in 60% of patients in the HO group. AC joint arthritic change was observed in 40% of the HO group. The CC fixation method using multiple soft anchor knots showed satisfactory results and had superior CC ligament healing ability and maintenance of CCD than hook fixation. Level III, retrospective therapeutic comparative investigation.

Complex Twisted Knots of Umbilical Cord in a Monochorionic-Diamniotic Twin Gestation: A Case Report

Background: True knots and tight loops of umbilical cord can cause serious fetal complications in monochorionic-monoamniotic twins but are usually unexpected in Monochorionic-diamniotic twins because of the presence of the intertwin membrane. This report presents a case of monochorionic-diamniotic twin gestation with a complex cord knots. Case report: A 31-year-old G2Ab1 with monochorionic-diamniotic twin pregnancy in the gestational age of 30 weeks presented with ruptured membrane since 3weeks before delivery. At the delivery time, multiple umbilical cord knots was found. Conclusion: Premature ruptured membrane can cause septostomy of the intertwin membrane, multiple umbilical cord knots and its complications. Therefore, these cases should be considered for evaluation of the presence of intertwin membrane and umbilical cord knots in each sonography examination. [GMJ.2020;9:e1878]

Task Space Trajectory Planning for Robot Manipulators to Follow 3-D Curved Contours

The demand for robots has increased in the industrial field, where robots are utilized in tasks that require them to move through complex paths. In the motion planning of a manipulator, path planning is carried out to determine a series of the positions of robot end effectors without collision. Therefore, it is necessary to carry out trajectory planning to determine position, velocity, and acceleration over time and to control an actual industrial manipulator. Although several methods have already been introduced for point-to-point trajectory planning, a trajectory plan which moves through multiple knots is required to allow robots to adapt to more complicated tasks. In this study, a trajectory planning based on the Catmull–Rom spline is proposed to allow a robot to move via several points in a task space. A method is presented to assign intermediate velocities and time to satisfy the velocity conditions of initial and final knots. To optimize the motion of the robot, a time-scaling method is presented to minimize the margin between the physical maximum values of velocity and acceleration in real robots and the planned trajectory, respectively. A simulation is then performed to verify that the proposed method can plan the trajectory for moving multiple knots without stopping, and also to check the effects of control parameters. The results obtained show that the proposed methods are applicable to trajectory planning and require less computation compared with the cubic spline method. Furthermore, the robot follows the planned trajectory, and its motion does not exceed the maximum values of velocity and acceleration. An experiment is also executed to prove that the proposed method can be applied to real robotic tasks to dispense glue onto the sole in the shoe manufacturing process. The results from this experiment show that the robot can follow the 3-D curved contour in uniform speed using the proposed method.

Tying different knots in a molecular strand.

The properties of knots are exploited in a range of applications, from shoelaces to the knots used for climbing, fishing and sailing1. Although knots are found in DNA and proteins2, and form randomly in other long polymer chains3,4, methods for tying5 different sorts of knots in a synthetic nanoscale strand are lacking. Molecular knots of high symmetry have previously been synthesized by using non-covalent interactions to assemble and entangle molecular chains6-15, but in such instances the template and/or strand structure intrinsically determines topology, which means that only one type of knot is usually possible. Here we show that interspersing coordination sites for different metal ions within an artificial molecular strand enables it to be tied into multiple knots. Three topoisomers-an unknot (01) macrocycle, a trefoil (31) knot6-15, and a three-twist (52) knot-were each selectively prepared from the same molecular strand by using transition-metal and lanthanide ions to guide chain folding in a manner reminiscent of the action of protein chaperones16. We find that the metal-ion-induced folding can proceed with stereoinduction: in the case of one knot, a lanthanide(III)-coordinated crossing pattern formed only with a copper(I)-coordinated crossing of particular handedness. In an unanticipated finding, metal-ion coordination was also found to translocate an entanglement from one region of a knotted molecular structure to another, resulting in an increase in writhe (topological strain) in the new knotted conformation. The knot topology affects the chemical properties of the strand: whereas the tighter 52 knot can bind two different metal ions simultaneously, the looser 31 isomer can bind only either one copper(I) ion or one lutetium(III) ion. The ability to tie nanoscale chains into different knots offers opportunities to explore the modification of the structure and properties of synthetic oligomers, polymers and supramolecules.

Piecewise latent growth models: beyond modeling linear-linear processes.

Piecewise latent growth models (LGMs) for linear-linear processes have been well-documented and studied in recent years. However, in the latent growth modeling literature, advancements to other functional forms as well as to multiple changepoints or knots have been nearly non-existent. This manuscript deals with three extensions. The first is to a piecewise latent growth model incorporating higher-order polynomials. The second is to extend the basic framework to three phases. The last extension is to inherently nonlinear functions. In these extensions, the changepoint(s) is a parameter to be estimated and may be fixed or allowed to vary across subjects as an application warrants. The approaches are developed and two illustrative empirical examples from psychology are used to highlight the methodological nuances. Annotated statistical software is provided to make these elaborations accessible to practitioners and methodologists.

Multilevel quadratic spline quasi-interpolation

In this paper we present new approximation schemes based on both quasi-interpolation and multilevel methods by using bivariate quadratic B-spline functions, defined on simple and multiple knot type-2 triangulations, improving the classical quasi-interpolating spline results. We also prove polynomial reproduction, optimal approximation order and propose some numerical results and applications.

Probing an X-Ray Flare Pattern in Mrk 421 Induced by Multiple Stationary Shocks: A Solution to the Bulk Lorentz Factor Crisis

Abstract The common observations of multiple radio VLBI stationary knots in high-frequency-peaked BL Lacs (HBLs) can be interpreted as multiple recollimation shocks accelerating particles along jets. This approach can resolve the so-called “bulk Lorentz factor crisis” of sources with a high Lorentz factor deduced from maximum γ–γ opacity and fast variability and apparently inconsistent slow/stationary radio knots. It also suggests that a unique pattern of the nonthermal emission variability should appear after each strong flare. Taking advantage of the 13 yr of observation of the HBL Mrk 421 by the X-ray Telescope on the Neil Gehrels Swift Observatory (Swift-XRT), we probe for such an intrinsic variability pattern. Its significance is then statistically estimated via comparisons with numerous similar simulated light curves. A suggested variability pattern is identified, consistent with a main flare emission zone located in the most upstream 15.3 GHz radio knot at 0.38 mas from the core. Subsequent flux excesses in the light curve are consistent with a perturbation crossing all of the downstream radio knots with a constant apparent speed of 45c. The significance of the observed variability pattern not arising from stochastic processes is found above three standard deviations, opening a promising path for further investigations in other blazars and with other energy bands. In addition to highlighting the role of stationary radio knots as high-energy particle accelerators in jets, the developed method allows estimates of the apparent speed and size of a jet perturbation without the need to directly observe any motion in jets.

Optical interleaver based on nested multiple knot microfiber resonators.

A novel design of nested optical fiber based multiple knot resonators is presented. The design consists of three knot resonators, two of which share a significant fraction of their optical path. The relationship between the knots' diameter ratio and the transmission spectrum is investigated. The output spectrum is theoretically analyzed using transfer matrix analysis and the calculated results exhibit good agreement with experimental results. The free spectral range (FSR) is varied by simply fine-tuning the diameter of the small knot. The periodic spectrum of this optical microfiber based photonic device has a number of applications in the sensing and communications field, e.g.,optical interleavers, frequency combs, filters, and fiber lasers. This Letter demonstrates that the variation of the output spectrum can be implemented simply by changing the knot sizes and coupling coefficients.

Extrapolation of Cubic Splines in General with Multiple Knots

Extrapolation of Cubic Splines in General with Multiple Knots