Vassiliev measures of complexity of open and closed curves in 3-space

In this article, we define Vassiliev measures of complexity for open curves in 3-space. These are related to the coefficients of the enhanced Jones polynomial of open curves in 3-space. These Vassiliev measures are continuous functions of the curve coordinates; as the ends of the curve tend to coincide, they converge to the corresponding Vassiliev invariants of the resulting knot. We focus on the second Vassiliev measure from the enhanced Jones polynomial for closed and open curves in 3-space. For closed curves, this second Vassiliev measure can be computed by a Gauss code diagram and it has an integral formulation, the double alternating self-linking integral. The double alternating self-linking integral is a topological invariant of closed curves and a continuous function of the curve coordinates for open curves in 3-space. For polygonal curves, the double alternating self-linking integral obtains a simpler expression in terms of geometric probabilities.

Sequential Simulations of Steel Frame Buildings Under Multi-Phase Hazardous Loads During Earthquake and Tsunami

Based on the experience of the 2011 Great East Japan Earthquake and the following tsunami, this study aims to develop effective analytical tools that can comprehensively be applied to buildings under multi-phase hazardous loads such as seismic motion, fluid force, and debris impact. Simulations by two kinds of analytical tools were conducted. First, a structural collapse analysis of a steel frame building under successive applications of varying loads was performed using the ASI (Adaptively Shifted Integration)-Gauss code, which simulates behaviors of structures by simple modeling. The steel frame building model was first excited under an acceleration record observed in Kesennuma-shi during the earthquake, and fluid forces due to a tsunami wave were applied. Then, the collapse behavior of the building was investigated by implementing a sophisticated contact algorithm in the numerical code to express a collision between debris and a building. It became evident that the damage to the building intensifies if a head-on collision occurs under a tsunami flow with a lower inundation height, and the damage to the building becomes larger if sideway collisions occur under a tsunami flow with a higher inundation height and higher velocity. The second simulation was conducted by using the stabilized finite element method based on the volume of fluid method, to estimate a drag coefficient of an actual tsunami evacuation building with openings. The practicability of an estimated wave force using the drag coefficient was confirmed by comparing with the wave force obtained from the fluid analysis. Finally, a sequential structural analysis, with a debris collision phase at the end, was conducted using the ASI-Gauss code to simulate the washout behavior of the building.

On gauss codes of virtual doodles

We discuss Gauss codes of virtual diagrams and virtual doodles. The notion of a left canonical Gauss code is introduced and it is shown that oriented virtual doodles are uniquely presented by left canonical Gauss codes.

Meander knots and links

We introduced concept of meander knots, 2-component meander links and multi-component meander links and derived different families of meander knots and links from open meanders with n ? 16 crossings. We also defined semi-meander knots (or knots with ordered Gauss code) and their product.

AVERAGE BRIDGE NUMBER OF SHADOW RESOLUTIONS

A Gauss code for a virtual knot diagram is a sequence of crossing labels, each repeated twice and assigned a + or - symbol to identify over and undercrossings. Eliminate these symbols and what remains is a Gauss code for the shadow of the diagram, one type of virtual pseudodiagram. While it is now impossible to determine which particular virtual diagram the shadow resulted from, we can consider the collection of all diagrams, called resolutions of the shadow, that would yield such a code. We compute the average virtual bridge number over all these diagrams and show that for a shadow with n classical precrossings, the average virtual bridge number is [Formula: see text].