Molecular Knots. D. A. Leigh et al. show in their Communication on page 10484 ff., how the tying of a stopper knot with lanthanide ions can prevent a crown ether from slipping off a molecular thread. ...Image credit: Johannes Richers.
Coloring in graphs of twist knots Şahin, Abdulgani
Numerical methods for partial differential equations,
July 2022, Letnik:
38, Številka:
4
Journal Article
Recenzirano
Let Tn be a twist knot with n half‐twists and Gn be the graph of Tn. The closed neighborhood Nv of a vertex v in Gn, which included at least one colored vertex for each color in a proper n‐coloring ...of Gn, is called a rainbow neighborhood. There are different types of graph coloring in the literature. We consider some of these types in here. In this paper, we determine the chromatic number of graphs of twist knots and study rainbow neighborhood of graphs of twist knots. We determine the rainbow neighborhood number and the fading number of them. Furthermore, we determine coupon coloring and the coupon coloring number of graphs of twist knots.
A spider web collects water by its capture silk for recovering the daytime‐distorted shape during night through water‐sensitive shape memory effect. This unique smart function and geometrical ...structure of spider‐capture‐silk inspires the development of artificial fibers with periodic knots for directional water collection with vast potential applications in water scarce regions. Existing such fibers are mainly based on nylon filaments coated with petroleum‐originated synthetic polymer solutions. Distinct from using synthetic materials, an all silk‐protein fiber (ASPF) with periodic knots endows extremely high volume‐to‐mass water collection capability. This fiber has a main body consisting of B. mori degummed silk coated with recombinant engineered major ampullate spidroin 2 of spider dragline silk. It is 252 times lighter than synthetic polymer coated nylon fibers that once was reported to have the highest water collection performance. The ASPF collects a maximum water volume of 6.6 µL and has a 100 times higher water collection efficiency compared to existing best water collection artificial fibers in terms of volume‐to‐mass index at the shortest length (0.8 mm) of three‐phase contact line. Since silkworm silks are available abundantly, effective use of recombinant spidroins tandemly shows great potential for scalability.
All silk‐protein fiber is presented here for its excellent high‐volume directional water collection by using degummed silk and recombinant spidroin engineered major ampullate spidroin 2 as the coating material to fabricate periodic knots. In comparison to previously reported synthetic fibers, remarkable water collection efficiency is observed in terms of volume‐to‐mass index.
We introduce shadow structures for singular knot theory. Precisely, we define two invariants of singular knots and links. First, we introduce a notion of action of a singquandle on a set to define a ...shadow counting invariant of singular links which generalize the classical shadow colorings of knots by quandles. We then define a shadow polynomial invariant for shadow structures. Lastly, we enhance the shadow counting invariant by combining both the shadow counting invariant and the shadow polynomial invariant. Explicit examples of computations are given.
Knots have a twisted history in quantum physics. They were abandoned as failed models of atoms. Only much later was the connection between knot invariants and Wilson loops in topological quantum ...field theory discovered. Here we show that knots tied by the eigenenergy strings provide a complete topological classification of one-dimensional non-Hermitian (NH) Hamiltonians with separable bands. A Z_{2} knot invariant, the global biorthogonal Berry phase Q as the sum of the Wilson loop eigenphases, is proved to be equal to the permutation parity of the NH bands. We show the transition between two phases characterized by distinct knots occur through exceptional points and come in two types. We further develop an algorithm to construct the corresponding tight-binding NH Hamiltonian for any desired knot, and propose a scheme to probe the knot structure via quantum quench. The theory and algorithm are demonstrated by model Hamiltonians that feature, for example, the Hopf link, the trefoil knot, the figure-8 knot, and the Whitehead link.
Gini ratio is an indicator to measure income inequality. Gini ratio of Indonesia in 2017 is 0.391, still far from Gini ratio target by Bappenas in 2019, that is 0.36. The Gini ratio modeling in this ...study uses a nonparametric regression approach because the form of the regression curve between the Gini ratio and its predictive variables is unknown. One of the estimators in nonparametric regression is spline truncated. Spline truncated has a knot that adjusts to the local characteristics of a function or data more effectively. The number of knots and their location affect the form of regression curve estimation, so it's important to obtain optimal knot. There are methods for selecting optimal knots, such as Generalized Cross Validation (GCV) and Unbiased Risk (UBR). This study compares GCV and UBR in selecting optimal knots on Gini Ratio data in Indonesia 2017. The criteria of the best model are based on Mean Squared Error (MSE) and R2 values. From the result, the optimal knot from GCV was a combination of 3-2-2-3 knot with MSE of 0.00085 and R2 of 79.18%. Meanwhile, by using UBR, the optimal knot is three knots with MSE of 0.00095 and R2 of 66.42%. In conclusion, GCV generated better model than UBR.
We discuss a semi-implicit numerical scheme that allows for minimizing the bending energy of curves within certain isotopy classes. To this end we consider a weighted sum of the bending energy and ...the tangent-point functional.
Based on estimates for the second derivative of the latter and a uniform bi-Lipschitz radius, we prove a stability result implying energy decay during the evolution as well as maintenance of arclength parametrization.
Finally we present some numerical experiments exploring the energy landscape, targeted to the question how to obtain global minimizers of the bending energy in knot classes, so-called elastic knots.
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