Gauss Code Research Articles

Numerical investigation on mechanical behavior of door systems during seismic excitation

The opening and closing of doors in buildings likely become challenging when earthquakes occur owing to the deformation of the door frames. Although this may delay evacuation, there are no anti-seismic standards for doors defined in the Japanese Building Standards Act, which was amended in 1981. Discussions on doors or door frames from a structural point of view are still rare, even though the behaviors of doors, door frames, and hinges under seismic excitation are critical for avoiding secondary disasters after earthquakes. In this study, a seismic response analysis of a building model with door frames was performed using the adaptively shifted integration (ASI)-Gauss code to find an approach to reduce the deformation of door frames. Numerical results confirmed that the severe damage to the non-structural wall caused by the interlayer deformation of the building allowed both door frames and doors to be severely deformed. The in-plane deformation angles exceeded the angles causing difficulty in opening the doors. A door acted as an element resisting the door frame from deforming on the upper floors, and the door function would be lost on the lower floors by deforming in the out-of-plane direction. Furthermore, the overall deformations of the door frames decreased significantly even on the lower floors, where the interlayer drift angle of the building was larger, by placing the door system in a setback position. This effect was greater when the setback length was longer.

Explosive demolition planning of building structures using key element index

A systematic and appropriate selection of columns to be removed should be conducted to ensure the safe and reliable demolition of old or uninhabitable buildings. However, the explosive demolition techniques used at present mostly rely on experiences and know-how built up by predecessors and are not fully open to general users. Therefore, a clear selection criterion for columns to be removed should be developed based on a quantitative value that indicates the contribution of each column in a building structure. This study applied a key element index to evaluate the contribution of columns and their variance to multistage explosive demolition planning. An adaptively shifted integration (ASI)-Gauss code was used to simulate and investigate the explosive demolition sequences of a 10-story steel-framed building. Various timings between the multistage blasts were evaluated by comparing the efficiencies and levels of safety during demolition. By removing the groups of columns from those that had a greater influence on the variance of the key element index values, a large difference was obtained in the distribution of the index values, especially in the lower structure. The results showed that if a building is not too weakened in the initial blast, the 2nd blast of the main blasts should be conducted at the moment of impact between the upper and lower structures or the following short time to make the most of the impact. The results also showed that too much weakening of a building at the preceding stage might not provide full advantage of the impact that occurs between the main blasts.

Numerical Investigation on the Collapse of the ChamplainTowers South in Florida

In this study, the building collapse of the Champlain Towers South in Florida allegedly caused by deck collapse was examined through the finite element analysis based on the existing hypotheses and investigation. For the numerical analysis, the Adaptively Shifted Integration (ASI) – Gauss code was applied. In addition, the member strength failure algorithm of RC components was modified and the flat-plate structure was introduced into the numerical model of the building. According to the numerical results, the building did not collapse when the external deck of the building connecting the columns, so called as key elements, were removed by the collapse of the external deck.

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.

A Suggestive Interface for Untangling Mathematical Knots.

In this paper we present a user-friendly sketching-based suggestive interface for untangling mathematical knots with complicated structures. Rather than treating mathematical knots as if they were 3D ropes, our interface is designed to assist the user to interact with knots with the right sequence of mathematically legal moves. Our knot interface allows one to sketch and untangle knots by proposing the Reidemeister moves, and can guide the user to untangle mathematical knots to the fewest possible number of crossings by suggesting the moves needed. The system highlights parts of the knot where the Reidemeister moves are applicable, suggests the possible moves, and constrains the user's drawing to legal moves only. This ongoing suggestion is based on a Reidemeister move analyzer, that reads the evolving knot in its Gauss code and predicts the needed Reidemeister moves towards the fewest possible number of crossings. For our principal test case of mathematical knot diagrams, this for the first time permits us to visualize, analyze, and deform them in a mathematical visual interface. In addition, understanding of a fairly long mathematical deformation sequence in our interface can be aided by visual analysis and comparison over the identified "key moments" where only critical changes occur in the sequence. Our knot interface allows users to track and trace mathematical knot deformation with a significantly reduced number of visual frames containing only the Reidemeister moves being applied. All these combine to allow a much cleaner exploratory interface for us to analyze and study mathematical knots and their dynamics in topological space.

Encoding and topological computation on textile structures

A textile structure is a periodic arrangement of threads in the thickened plane. A topological classification of textile structures is harder than for classical knots and links that are non-periodic and restricted to a bounded region. The first important problem is to encode all textile structures in a simple combinatorial way. This paper extends the notion of the Gauss code in classical knot theory, providing a tool for topological computation on these structures. As a first application, we present a linear time algorithm for determining whether a code represents a textile in the physical sense. This algorithm, along with invariants of textile structures, allowed us for the first time to classify all oriented textile structures woven from a single component up to complexity five.

Z-Knotted Triangulations of Surfaces

A zigzag in a map (a 2-cell embedding of a connected graph in a connected closed 2-dimensional surface) is a cyclic sequence of edges satisfying the following conditions: (1) any two consecutive edges lie on the same face and have a common vertex, (2) for any three consecutive edges the first and the third edges are disjoint and the face containing the first and the second edges is distinct from the face which contains the second and the third one. A map is z-knotted if it contains a single zigzag. Such maps are closely related to the Gauss code problem and have nice homological properties. We show that every triangulation of a connected closed 2-dimensional surface admits a z-knotted shredding.

Modelling Time-Varying Parameters in Panel Data State-Space Frameworks: An Application to the Feldstein–Horioka Puzzle

In this paper, we develop a very flexible and comprehensive state-space framework for modeling time series data. Our research extends the simple canonical model usually employed in the literature, into a panel-data time-varying parameters framework, combining fixed (both common and country-specific) and varying components. Under some specific circumstances, this setting can be understood as a mean-reverting panel time-series model, where the mean fixed parameter can, at the same time, include a deterministic trend. Regarding the transition equation, our structure allows for the estimation of different autoregressive alternatives, and include control instruments, whose coefficients can be set-up either common or idiosyncratic. This is particularly useful to detect asymmetries among individuals (countries) to common shocks. We develop a GAUSS code that allows for the introduction of restrictions regarding the variances of both the transition and measurement equations. Finally, we use this empirical framework to test for the Feldstein–Horioka puzzle in a 17-country panel. The results show its usefulness for solving complexities in macroeconomic empirical research.

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.

COLLAPSE RISK PREDICTION OF BUILDINGS ON FIRE USING KEY ELEMENT INDEX

It is difficult to estimate the redundant strength remaining in a building on fire, since the fire conditions such as their ranges and locations may be different in each case. A large-scale fire occurred on the New York World Trade Center 7 (WTC-7) after the 9.11 terrorist attacks in 2001. The building continued to burn for 7 hours, and ended in a total collapse. The official report on the collapse investigation of WTC-7, released by the National Institute of Standards and Technology (NIST) in 2005, had suggested that the total collapse was triggered by a severe damage of an important column of the building called the key element. On the other hand, there is also an example of a high-rise building on fire that avoided a total collapse, regardless of its main structure continuously heated by fire for a long time; the Windsor building of Madrid in 2005. From the viewpoint of preventing the fire-induced collapse of buildings, it is necessary to clarify the relationship between various structural parameters of buildings, fire locations, their ranges, and the scale of collapse. The purpose of this study is to predict collapse risks of buildings on fire using a key element index (KI). The KI is defined as the ratio of the ultimate yield strengths of the structure with one column eliminated and the initial, undamaged structure. It indicates the contribution of a structural column to the vertical capacity of the structure. We investigated the relationship between the sum of KI values in various cases of fire range and the sum of the height of remains after the collapse. We applied an adaptively shifted integration (ASI) - Gauss code utilizing linear Timoshenko beam elements to investigate the collapse behaviors of a ten-story steel framed building model with various fire patterns. Fracture contact, contact release and re-contact algorithms were implemented in the code. The reduction curves of elastic modulus and yield strength of steel related to elevated temperature, shown by NIST, were adopted to consider the heat effect of fire. Thermal expansion of materials was also considered. From the numerical results of the fire-induced collapse analyses, it is found that the scale of collapse highly depends on fire ranges and locations. The risk of total collapse tends to increase when a large range of fire occurs in the lower layer. In addition, the risk increases much higher in those cases when fire occurs at the peripheral area of the building, compared to those cases at the inner area. The sum of heights of remains seems to depend on the sum of KI values, and there is a specific threshold of the sum of KI values that relates to the initiation of a large scale collapse. Therefore, the sum of KI values may be used to predict and take measures to avoid the risk of collapse under various fire conditions.

Numerical distribution functions for seasonal unit root tests with OLS and GLS detrending

This paper implements the approach introduced by MacKinnon (J Bus Econ Stat 12:167–176, 1994, J Appl Econom 11:601–618, 1996) to estimate the response surface of the test statistics of seasonal unit root tests with OLS and GLS detrending for quarterly and monthly time series. The Gauss code that is available in the supplementary material of the paper produces p values for five test statistics depending on the sample size, deterministic terms and frequency of the data. A comparison with previous studies is undertaken, and an empirical example using airport passenger arrivals to a tourist destination is carried out. Quantile function coefficients are reported for simple computation of critical values for tests at 1, 5 and 10 % significance levels.

Operator Formalism for Topology-Conserving Crossing Dynamics in Planar Knot Diagrams

We address here the topological equivalence of knots through the so-called Reidemeister moves. These topology-conserving manipulations are recast into dynamical rules on the crossings of knot diagrams. This is presented in terms of a simple graphical representation related to the Gauss code of knots. Drawing on techniques for reaction-diffusion systems, we then develop didactically an operator formalism wherein these rules for crossing dynamics are encoded. {The aim is to develop new tools for studying dynamical behaviour and regimes in the presence of topology conservation}. This necessitates the introduction of composite paulionic operators. The formalism is applied to calculate some differential equations for {the time evolution} of densities and correlators of crossings, subject to topology-conserving stochastic dynamics. {We consider here the simplified situation of two-dimensional knot projections. However, we hope that this is a first valuable step towards} addressing a number of important questions regarding the role of topological constraints {and specifically of topology conservation} in dynamics through a variety of solution and approximation schemes. Further applicability arises in the context of the simulated annealing of knots. The methods presented here depart significantly from the invariant-based path integral descriptions often applied in polymer systems, {and, in our view, offer a fresh perspective on} the conservation of topological states and topological equivalence in knots.

Calculating colored homflypt invariants with image processing and a new form of Gauss code

Calculating colored homflypt invariants with image processing and a new form of Gauss code

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].

A homological solution for the Gauss code problem in arbitrary surfaces

Let P ¯ be a sequence of length 2 n in which each element of { 1 , 2 , … , n } occurs twice. Let P ′ be a closed curve in a closed surface S having n points of simple self-intersections, inducing a 4-regular graph embedded in S which is 2-face colorable. If the sequence of auto-intersections along P ′ is given by P ¯ , we say that P ′ is a 2-face colorable solution for the Gauss code P ¯ on surface S or a lacet for P ¯ on S. In this paper we show (by using surface homology theory mod 2), that the set of lacets for P ¯ on S are in 1–1 correspondence with the tight solutions of a system of quadratic equations over the Galois field GF ( 2 ) . If S is the 2-sphere, the projective plane or the Klein bottle, the corresponding quadratic systems are equivalent to linear ones. In consequence, algorithmic characterizations for the existence of solutions on these surfaces are available. For the two first surfaces this produces simple proofs of known results. The algorithmic characterization for the existence of solutions on the Klein bottle is new. We provide a polynomial algorithm to resolve the issue.

Symbolic computation with finite biquandles

A method of computing a basis for the second Yang–Baxter cohomology of a finite biquandle with coefficients in Q and Z p from a matrix presentation of the finite biquandle is described. We also describe a method for computing the Yang–Baxter cocycle invariants of an oriented knot or link represented as a signed Gauss code. We provide a URL for our Maple implementations of these algorithms.

On the self-intersections of an immersed sphere

A closed curve f: S1 → ℝ2 in general position gives rise to a word whose letters are the self-intersection points, each of them appearing exactly twice. Such a word is called a Gauss code. The problem of determining whether a given Gauss code is realisable or not was first proposed by Gauss and it has been settled a long time ago. The analogous question for immersions f: S2 → ℝ3 in general position is settled only in a special case when the immersion has no triple points. We give a necessary condition for a system of curves to be realisable by a general immersion f: S2 → ℝ3.

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