Simulative Analysis of a Family of DNA Tetrahedrons Produced by Changing the Twisting Number of Each Double Helix

In this paper, three tetrahedral nanocages, composed of six DNA double helix edges with all having the twist number 1, 2 or 3, have been characterized using classical molecular dynamics simulation to measure the specific structural and conformational features produced by only changing the twisting number of each double helix. The simulation result indicates that three tetrahedral cages are relatively stable and are maintained along the entire trajectory. Each double helix is more inclined to behave as a whole in the 2TD and 3TD cages than in the 1TD cage according to the cross-correlation maps for three nanocages, and also their local motions are more easily induced by the conformational variability of the thymidine linkers due to the increased flexibility of each helix. Hence, the double helices become the important factors on the structural stability of total cages with the DNA twisting number, and also give the signification contributions to the sizes of these cages conferring the larger spaces of the 2TD and 3TD cages than the 1TD cage. Our result provides an insight into which roles the double helix edges play in assembling DNA polyhedron, and also contribute to improving the loading capacity of DNA tetrahedron in drug delivery.

Crosscap number of knots and volume bounds

In this paper, we obtain the crosscap number of any alternating knots by using our recently-introduced diagrammatic knot invariant (Theorem 1). The proof is given by properties of chord diagrams (Kindred proved Theorem 1 independently via other techniques). For non-alternating knots, we give Theorem 2 that generalizes Theorem 1. We also improve known formulas to obtain upper bounds of the crosscap number of knots (alternating or non-alternating) (Theorem 3). As a corollary, this paper connects crosscap numbers and our invariant with other knot invariants such as the Jones polynomial, twist number, crossing number, and hyperbolic volume (Corollaries 1–7). In Appendix A, using Theorem 1, we complete giving the crosscap numbers of the alternating knots with up to 11 crossings including those of the previously unknown values for [Formula: see text] knots (Tables A.1).

Twist number and the alternating volume of knots

It was previously shown by the first author that every knot in [Formula: see text] is ambient isotopic to one component of a two-component, alternating, hyperbolic link. In this paper, we define the alternating volume of a knot [Formula: see text] to be the minimum volume of any link [Formula: see text] in a natural class of alternating, hyperbolic links such that [Formula: see text] is ambient isotopic to a component of [Formula: see text]. Our main result shows that the alternating volume of a knot is coarsely equivalent to the twist number of a knot.

Effect of Various Chemical Treatments on Handling of Wool Fabric

It is well known that fabric handle is controllable by controlling and adjusting mechanical properties of the fabric. The handling of fabrics, on one hand, is mostly decided by the structure of the fabric, i.e. quality of yarn, yarn count, twist number, density of warp and weft, fabric weight and weave design. Such a fabric structure is represented in terms of Cover Factor. On the other hand, the handling of fabrics can also be changed through the mechanical properties by dyeing and finishing processes after weaving. The present work investigated how much influence is exerted on wool fabric by 11 sorts of chemicals generally used in dyeing and finishing processes, and how much change is exerted on the handle of processed fabrics. The experimental results were compared with the original fabric with no treatment in terms of mechanical properties relevant to KES. It was confirmed that the fabric handle greatly depended on fabric structure. Besides, the differences in the degree of damage and the hydrophilicity of wool fiber arisen from the treatments using chemicals were examined. The effect of chemicals used in dyeing and finishing processes was also investigated on the environment. Keywords: Wool, Chemical treatment, Fabric handle, KES, Environment.

Combinatorics of link diagrams and volume

We show that the volumes of certain hyperbolic A-adequate links can be bounded (above and) below in terms of two diagrammatic quantities: the twist number and the number of certain alternating tangles in an A-adequate diagram. We then restrict our attention to plat closures of certain braids, a rich family of links whose volumes can be bounded in terms of the twist number alone. Furthermore, in the absence of special tangles, our volume bounds can be expressed in terms of a single stable coefficient of the colored Jones polynomial. Consequently, we are able to provide a new collection of links that satisfy a Coarse Volume Conjecture.

Estimate of Number of Periodic Solutions of Second-Order Asymptotically Linear Difference System

We investigate the number of periodic solutions of second-order asymptotically linear difference system. The main tools are Morse theory and twist number, and the discussion in this paper is divided into three cases. As the system is resonant at infinity, we use perturbation method to study the compactness condition of functional. We obtain some new results concerning the lower bounds of the nonconstant periodic solutions for discrete system.

ON THE Δ-UNKNOTTING NUMBER OF WHITEHEAD DOUBLES OF KNOTS

In this paper, we give a bound for the Δ-unknotting number of a Whitehead double in terms of the unknotting number and a certain integral invariant of its companion knot. As applications, we show that the Δ-unknotting number of m-twisted Whitehead doubles of certain knots does not remember its companion knot, and is equal to the twist number m. We also give possible Δ-unknotting number of m-twisted Whitehead doubles whose companions are knots with unknotting number 1, certain twist knots, amphicheiral knots, and positive knots.