scholarly journals Garside theory and subsurfaces: Some examples in braid groups

2019 ◽  
Vol 11 (2) ◽  
pp. 61-75
Author(s):  
Saul Schleimer ◽  
Bert Wiest
Keyword(s):  
Conjugacy Class ◽  
Geometric Property ◽  
Conjugacy Problem ◽  
Fixed Number ◽  
Braid Groups ◽  
Worst Case ◽  
Curve Graph ◽  
Precise Bound

Abstract Garside-theoretical solutions to the conjugacy problem in braid groups depend on the determination of a characteristic subset of the conjugacy class of any given braid, e.g. the sliding circuit set. It is conjectured that, among rigid braids with a fixed number of strands, the size of this set is bounded by a polynomial in the length of the braids. In this paper we suggest a more precise bound: for rigid braids with N strands and of Garside length L, the sliding circuit set should have at most {C\cdot L^{N-2}} elements, for some constant C. We construct a family of braids which realise this potential worst case. Our example braids suggest that having a large sliding circuit set is a geometric property of braids, as our examples have multiple subsurfaces with large subsurface projection; thus they are “almost reducible” in multiple ways, and act on the curve graph with small translation distance.

Technical Analysis of Ocean Towing Evolutions

2021 ◽  
Author(s):  
Bartley Eckhardt ◽  
Daniel Fridline ◽  
Richard Burke
Keyword(s):  
Failure Modes ◽  
Human Life ◽  
Single Point ◽  
Forensic Analysis ◽  
Sea State ◽  
Worst Case ◽  
Strain Monitoring ◽  
Load Limit ◽  
Offshore Oil

Ocean towing in general, and non-routine tows in particular, present unique technical challenges to towing vessel owners/operators, salvors, the offshore oil/gas and wind industries, and others. When such tows “go wrong”, the harm to human life, property and/or the environment can be significant. The authors have drawn from their work on the Towing Safety Advisory Committee’s investigation of the grounding of the MODU Kulluk to present methods and considerations in analyzing ocean towing evolutions, both “routine” and “non-routine”. (TASK 14-01) The methods and considerations presented should be employed in advance of a towing evolution, but can be used in accident reconstruction and forensic analysis when an evolution has failed. The methods presented are iterative, and consider 2 x 6 degree freedom of motion (of the towing vessel(s) and towed vessel respectively) and characteristics of the towline, and facilitate determination of: Worst Case Conditions. Extreme Towline Tension (ETT) as a function of sea state and speed. Limits of the Tow (Go-No Go Criteria). Recommended Catenary Length as a function of sea state and speed. Size and Selection of the Towing Vessel and Gear, including: Required Bollard Pull. Required Strength, Characteristics and Condition of the Towline. Limits and Set Points of the Towing Winch, Automatic or Manual. Required Strength and Characteristics of the Synthetic Emergency Towline and its methods of deployment and connection. Working Load Limit (WLL) of the Shackles, Delta Plate and Attachment Points. Required Strength and Characteristics of Bridles, Pendant and Surge Gear/Shock Lines. The authors further explore the implications of single point failure modes, redundancy in gear and towing vessel(s), high cycle fatigue, and strain monitoring.

Statistical Safe Braking Analysis

2011 ◽  
Author(s):  
David F. Thurston
Keyword(s):  
Control Systems ◽  
System Capacity ◽  
Worst Case ◽  
Stopping Distance ◽  
Train Control ◽  
Classic Theory ◽  
Braking Distance ◽  
Prime Determinant ◽  
Worst Case Approach

The main objective in optimizing train control is to eliminate the waist associated with classical design where train separation is determined through the use of “worst case” assumptions that are invariant to the system. In fact, the worst case approach has been in place since the beginning of train control systems. Worst case takes the most conservative approach to the determination of train stopping distance, which is the basis for design of virtually all train control. This leads to stopping distances that could be far more that actually required under the circumstances at the time the train is attempting to brake. Modern train control systems are designed to separate trains in order to provide safety of operation while increasing throughput. Calculations for the minimum distance that separates trains have traditionally been based on the sum of a series of worst case scenarios. The implication was that no train could ever exceed this distance in stopping. This distance is called Safe Braking Distance (SBD). SBD has always been calculated by static parameters that were assumed to be invariant. This is, however, not the case. Parameters such as adhesion, acceleration, weight, and reaction vary over time, location or velocity. Since the worst case is always used in the calculation, inefficiencies result in this methodology which causes degradation in capacity and throughput. This is also true when mixed traffic with different stopping characteristics are present at the same time. The classic theory in train control utilizes a SBD model to describe the characteristics of a stopping train. Since knowledge of these conditions is not known, poor conditions are assumed. A new concept in train control utilizes statistical analysis and estimation to provide knowledge of the conditions. Trains operating along the line utilize these techniques to understand inputs into their SBD calculation. This provides for a SBD calculation on board the train that is the shortest possible that maintains the required level of safety. The new SBD is a prime determinant in systems capacity. Therefore by optimizing SBD as describes, system capacity is also optimized. The system continuously adjusts to changing conditions.

Efficient Determination of Worst-Case Gust Loads Using System Norms

2017 ◽  
Vol 54 (3) ◽  
pp. 1205-1210 ◽  
Author(s):  
Andreas Knoblach ◽  
Gertjan Looye
Keyword(s):  
Worst Case ◽  
Gust Loads

scholarly journals Logic-based specification and verification of homogeneous dynamic multi-agent systems

2020 ◽  
Vol 34 (2) ◽  
Author(s):  
Riccardo De Masellis ◽  
Valentin Goranko
Keyword(s):  
Normal Form ◽  
Model Checking ◽  
Formal Specification ◽  
Formal Semantics ◽  
Fixed Number ◽  
State Transitions ◽  
Multi Agent Systems ◽  
Worst Case ◽  
Worst Case Complexity ◽  
Multi Agent

Abstract We develop a logic-based framework for formal specification and algorithmic verification of homogeneous and dynamic concurrent multi-agent transition systems. Homogeneity means that all agents have the same available actions at any given state and the actions have the same effects regardless of which agents perform them. The state transitions are therefore determined only by the vector of numbers of agents performing each action and are specified symbolically, by means of conditions on these numbers definable in Presburger arithmetic. The agents are divided into controllable (by the system supervisor/controller) and uncontrollable, representing the environment or adversary. Dynamicity means that the numbers of controllable and uncontrollable agents may vary throughout the system evolution, possibly at every transition. As a language for formal specification we use a suitably extended version of Alternating-time Temporal Logic, where one can specify properties of the type “a coalition of (at least) n controllable agents can ensure against (at most) m uncontrollable agents that any possible evolution of the system satisfies a given objective $$\gamma$$ γ ″, where $$\gamma$$ γ is specified again as a formula of that language and each of n and m is either a fixed number or a variable that can be quantified over. We provide formal semantics to our logic $${\mathcal {L}}_{\textsc {hdmas}}$$ L H D M A S and define normal form of its formulae. We then prove that every formula in $${\mathcal {L}}_{\textsc {hdmas}}$$ L H D M A S is equivalent in the finite to one in a normal form and develop an algorithm for global model checking of formulae in normal form in finite HDMAS models, which invokes model checking truth of Presburger formulae. We establish worst case complexity estimates for the model checking algorithm and illustrate it on a running example.

scholarly journals Holonomic and Legendrian parametrizations of knots

2000 ◽  
Vol 09 (03) ◽  
pp. 293-309 ◽  
Author(s):  
Joan S. Birman ◽  
Nancy C. Wrinkle
Keyword(s):  
Conjugacy Problem ◽  
Braid Groups ◽  
Algebraic Solution ◽  
Legendrian Knots

Holonomic parametrizations of knots were introduced in 1997 by Vassiliev, who proved that every knot type can be given a holonomic parametrization. Our main result is that any two holonomic knots which represent the same knot type are isotopic in the space of holonomic knots. A second result emerges through the techniques used to prove the main result: strong and unexpected connections between the topology of knots and the algebraic solution to the conjugacy problem in the braid groups, via the work of Garside. We also discuss related parametrizations of Legendrian knots, and uncover connections between the concepts of holonomic and Legendrian parametrizations of knots.

scholarly journals FAST ALGORITHMIC NIELSEN–THURSTON CLASSIFICATION OF FOUR-STRAND BRAIDS

2012 ◽  
Vol 21 (05) ◽  
pp. 1250043 ◽  
Author(s):  
MATTHIEU CALVEZ ◽  
BERT WIEST
Keyword(s):  
Conjugacy Class ◽  
Polynomial Time ◽  
Word Length ◽  
Conjugacy Problem ◽  
Time Solution

We give an algorithm which decides the Nielsen–Thurston type of a given four-strand braid. The complexity of our algorithm is quadratic with respect to word length. The proof of its validity is based on a result which states that for a reducible 4-braid which is as short as possible within its conjugacy class (short in the sense of Garside), reducing curves surrounding three punctures must be round or almost round. As an application, we give a polynomial time solution to the conjugacy problem for non-pseudo-Anosov four-strand braids.

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