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We derive the general shape equations in terms of Euler angles for an elastic model of uniform ribbon with noncircular cross section and vanishing spontaneous curvatures. We show that it has in general not a planar solution for a closed ribbon free of external force and torque. We study the conditions to form a helix with the axis along the direction of the applied force for a ribbon under external force and twisting. We find that if the bending rigidity is greater than the twisting rigidity, then no such helical rod can exist. Our stability analysis shows that a helical ribbon is in general stable or at least metastable under arbitrary force and torque. We find that the extension of the ribbon may undergo a discontinuous transition from a twisted straight rod to a helical ribbon. The intrinsic asymmetric elasticity of a helical ribbon under external torque is also studied.

The transport properties of graphene T-type structure (GTS), both armchair-edged and zigzag-edged, are studied systematically within Landauer formalism in terms of tight binding Green's function method. For armchair-edged GTS (AGTS), a phase diagram of conductance in parameter space of Fermi energy and the stub width is presented. When a gate voltage is applied in the stub, a perfect linear dependence of the gap position shift of conductance on the gate voltage is found. The origin of the conductance gap as well as such linear behavior is explained. For the zigzag-edged GTS (ZGTS), it is found that the dependence of the transport properties on the stub height can be classified into two groups that tightly depend on whether the stub width equals 3n-1 or not.

Nd₂Fe${}_{13.6}$Zr${}_{0.4}$B hard magnetic material were prepared using arc-melting technique on a water-cooled copper hearth kept under argon gas atmosphere. The prepared samples, Nd₂Fe${}_{13.6}$Zr${}_{0.4}$B ingot and ribbon are characterized using X-ray diffraction (XRD), scanning electron microscopy (SEM) for crystal structure determination and morphological studies, respectively. The magnetic properties of the samples have been explored using vibrating sample magnetometer (VSM).

The lattice constants slightly increased due to the difference in the ionic radii of Fe and that of Zr. The bulk density decreased due to smaller molar weight and low density of Zr as compared to that of Fe. Ingot sample shows almost single crystalline phase with larger crystallite sizes whereas ribbon sample shows a mixture of amorphous and crystalline phases with smaller crystallite sizes. The crystallinity of the material was highly affected with high thermal treatments. Magnetic measurements show noticeable variation in magnetic behavior with the change in crystallite size. The sample prepared in ingot type shows soft while ribbon shows hard magnetic behavior.

We first introduce the null-homotopically peripheral quadratic function of a surface-link to obtain a lot of pseudo-ribbon, non-ribbon surface-links, generalizing a known property of the turned spun torus-knot of a non-trivial knot. Next, we study the torsion linking of a surface-link to show that the torsion linking of every pseudo-ribbon surface-link is the zero form, generalizing a known property of a ribbon surface-link. Further, we introduce and algebraically estimate the triple point cancelling number of a surface-link.

We shall study several circles in the 3-sphere called a link which has "high" splitness properties. We offer several kind of those links and study relations among them. Alexander polynomial, Conway polynomial and Milnor μ and invariants did not work for those links as vanishing cause of high splitness. We use higher order elementary ideals to distinguish those links.

We introduce the notions of "k-connected-slice" and "π₁-slice", interpolating between "homotopy ribbon" and "slice". Every high-dimensional knot group π is the group of an (n - 1)-connected-slice n-knot, for all n ≥ 3. However, if π is the group of an n-connected-slice n-knot, the augmentation ideal I(π) has deficiency 1 as a module, while (n + 1)-connected-slice n-knots are trivial. If π is the group of a π₁-slice 2-knot and π' is finitely generated, then π' is free, and so def(π) = 1.

We give a simple argument to show that every polynomial f(t) ∈ ℤ[t] such that f(1) = 1 is the Alexander polynomial of some ribbon 2-knot whose group is a 1-relator group, and we extend this result to links.

It is known that there are 21 ribbon knots with 10 crossings or fewer. We show that for every ribbon knot, there exists a tangle that satisfies two properties associated with the knot. First, under a specific closure, the closed tangle is equivalent to its corresponding knot. Second, under a different closure, the closed tangle is equivalent to the unlink. For each of these 21 ribbon knots, we present a 4-strand tangle that satisfies these properties. We provide diagrams of these tangles and also express them in planar diagram notation.

In this paper, we generalize the well-known constructions of invariants of framed links and 3-manifolds based on Hopf algebras to multiplier Hopf algebras. Results include a structural characterization of ribbon multiplier Hopf algebras from which invariants of framed links arise from the usual functorial construction to the ribbon category of modules, and the extension of Hennings construction of invariants of framed links and 3-manifolds obtained from Hopf algebras to algebraic quantum groups equipped with an appropriate trace.

Carbon Nanotube (CNT) ribbon is a thin layer of aligned, partially overlapping CNTs drawn from a forest of CNTs grown on a substrate. The electrical properties of the ribbon must be understood to put this material into multifunctional applications. Measurements show that CNT ribbon exhibits interesting characteristics including frequency-dependent electrical impedance. The impedance is mainly a combination of resistive and capacitive impedance. The magnitude of the impedance of ribbon increases moderately with increasing frequency then decreases significantly at higher frequency, MHz and above. An electrical model was developed to approximate the electrical impedance of the CNT ribbon. Based on this model, some important properties of the CNT ribbon can be understood. The ribbon capacitance, CNT–CNT contact resistance and resistivity can be approximated using the model. This information is useful in determining the suitability of ribbon for different applications. Methods to improve the electrical conduction of CNT ribbon are also discussed.