Checking Mobility and Decomposition of Linkages via Pebble Game Algorithm

2011 ◽  
Author(s):  
Adnan Sljoka ◽  
Offer Shai ◽  
Walter Whiteley
Keyword(s):  
Efficient Algorithm ◽  
Mechanical Engineering ◽  
Directed Graphs ◽  
Mathematical Concepts ◽  
Fast Analysis ◽  
3 Dimensional ◽  
New Methods ◽  
Analysis And Synthesis ◽  
Strongly Connected ◽  
Pebble Game

The decomposition of linkages into Assur graphs (Assur groups) was developed by Leonid Assur in 1914 - to decompose a linkage into fundamental minimal components for the analysis and synthesis of the linkages. In the paper, some new results and new methods are introduced for solving problems in mechanisms, bringing in methods from the rigidity theory community. Using these techniques, an investigation of Assur graphs and the decomposition of linkages has reworked and extended the decomposition using the well developed mathematical concepts from theory of rigidity and directed graphs. We recall some vocabulary and provide an efficient algorithm for decomposing 2-dimensional linkages into Assur components using strongly connected decompositions of graphs and a fast combinatorial Pebble Game Algorithm, which has been recently used in the study of rigidity and flexibility of structures and in fast analysis of large biomolecular structures such as proteins. Working on a one degree of freedom mechanism, we apply our algorithm to give the Assur decomposition. The Pebble Game Algorithm on such a mechanism is presented, along with an overview of the key properties and advantages of this elegant algorithm. We show how the pebble game algorithm can be used in the analysis and synthesis of linkages to mechanical engineering community. Core techniques and algorithms easily generalize to 3-dimensional structures, and can be further adapted to entire suite of other (body-bar) types of kinematic structures.

scholarly journals Convert a Strongly Connected Directed Graph to a Black-and-White 3-SAT Problem by the Balatonboglár Model

2020 ◽  
Author(s):  
Gábor Kusper ◽  
Csaba Biró
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Deep Insight ◽  
Communication Graph ◽  
Sat Problem ◽  
Weak Model ◽  
Know How ◽  
Black And White ◽  
Communication Graphs ◽  
Strongly Connected

In a previous paper we defined the Black-and-White SAT problem which has exactly two solutions, where each variable is either true or false. We showed that Black-and-White $2$-SAT problems represent strongly connected directed graphs. We presented also the strong model of communication graphs. In this work we introduce two new models, the weak model, and the Balatonbogl\'{a}r model of communication graphs. A communication graph is a directed graph, where no self loops are allowed. In this work we show that the weak model of a strongly connected communication graph is a Black-and-White SAT problem. We prove a powerful theorem, the so called Transitions Theorem. This theorem states that for any model which is between the strong and the weak model, we have that this model represents strongly connected communication graphs as Blask-and-White SAT problems. We show that the Balatonbogl\'{a}r model is between the strong and the weak model, and it generates $3$-SAT problems, so the Balatonbogl\'{a}r model represents strongly connected communication graphs as Black-and-White $3$-SAT problems. Our motivation to study these models is the following: The strong model generates a $2$-SAT problem from the input directed graph, so it does not give us a deep insight how to convert a general SAT problem into a directed graph. The weak model generates huge models, because it represents all cycles, even non-simple cycles, of the input directed graph. We need something between them to gain more experience. From the Balatonbogl\'{a}r model we learned that it is enough to have a subset of a clause, which represents a cycle in the weak model, to make the Balatonbogl\'{a}r model more compact. We still do not know how to represent a SAT problem as a directed graph, but this work gives a strong link between two prominent fields of formal methods: SAT problem and directed graphs.

scholarly journals TYPICAL CLASS METHODS FOR SOLVING EQUATIONS AND INEQUALITIES WITH DIFFERENT STRUCTURES

2021 ◽  
Vol 58 (3) ◽  
pp. 53-62
Author(s):  
A.K. Alpysov ◽  
◽  
A.K. Seytkhanova ◽  
I.Sh. Abishova ◽  
◽  
...  
Keyword(s):  
Problem Solving ◽  
Mathematical Thinking ◽  
Mathematical Concept ◽  
Thinking Skills ◽  
Practical Application ◽  
Mathematical Concepts ◽  
Theoretical Substantiation ◽  
New Methods ◽  
General Article ◽  
Solving Equations

The article discusses the ways of developing skills and abilities to effectively solve problems when describing methods for solving equations and inequalities, clarifying theoretical knowledge, the basics of forming skills for practical application. The formation of mathematical concepts through solving problems in teaching mathematics opens the way to the development of mathematical thinking, the application of knowledge in practice, and the development of search skills. To master a mathematical concept, along with its definition, it is necessary to know its features and properties. This can be achieved primarily through problem solving and exercise. Problem solving is based on the development of new methods, the creation of algorithms, ways of developing practical skills in the methods and techniques mastered with the help of tasks.In addition, transforming equations and inequalities through the development of thinking skills helps to identify common or special properties in order to draw correct conclusions. Solving various problems, it shows what operations should be used to determine the situation in which a solution was found, and what features of the solution allow choosing the most effective methods. Thanks to the theoretical substantiation of the general article, it is possible to master convenient methods for solving equations and inequalities of various structures.

A Mathematical Approach to Automatic Configuration of Machining Fixtures: Analysis and Synthesis

1989 ◽  
Vol 111 (4) ◽  
pp. 299-306 ◽  
Author(s):  
Y-C. Chou ◽  
V. Chandru ◽  
M. M. Barash
Keyword(s):  
Screw Theory ◽  
Theory Analysis ◽  
Workpiece Surface ◽  
3 Dimensional ◽  
Prismatic Parts ◽  
Analysis And Synthesis ◽  
Engineering Mechanics ◽  
Machining Fixtures ◽  
Machining Operations

Drawing on screw theory and engineering mechanics, a mathematical theory for automatic configuration of machining fixtures for prismatic parts is developed. There are two parts in the theory: analysis and synthesis. The following functions of fixtures are analyzed: deterministic workpiece location, clamping stability, and total restraint. The synthesis of fixtures includes the determination of locating and clamping points on workpiece surface and the determination of clamping forces. The theory deals with the general case of 3-dimensional parts, polygon support (as opposed to three-point support), and multiple machining operations.

Strongly Connected Spanning Subgraph for Almost Symmetric Networks

2017 ◽  
Vol 27 (03) ◽  
pp. 207-219
Author(s):  
A. Karim Abu-Affash ◽  
Paz Carmi ◽  
Anat Parush Tzur
Keyword(s):  
Directed Graph ◽  
Euclidean Distance ◽  
Directed Graphs ◽  
Minimum Weight ◽  
Shortest Paths ◽  
Strong Connectivity ◽  
Spanning Subgraph ◽  
Strongly Connected ◽  
Set Of Points ◽  
Symmetric Networks

In the strongly connected spanning subgraph ([Formula: see text]) problem, the goal is to find a minimum weight spanning subgraph of a strongly connected directed graph that maintains the strong connectivity. In this paper, we consider the [Formula: see text] problem for two families of geometric directed graphs; [Formula: see text]-spanners and symmetric disk graphs. Given a constant [Formula: see text], a directed graph [Formula: see text] is a [Formula: see text]-spanner of a set of points [Formula: see text] if, for every two points [Formula: see text] and [Formula: see text] in [Formula: see text], there exists a directed path from [Formula: see text] to [Formula: see text] in [Formula: see text] of length at most [Formula: see text], where [Formula: see text] is the Euclidean distance between [Formula: see text] and [Formula: see text]. Given a set [Formula: see text] of points in the plane such that each point [Formula: see text] has a radius [Formula: see text], the symmetric disk graph of [Formula: see text] is a directed graph [Formula: see text], such that [Formula: see text]. Thus, if there exists a directed edge [Formula: see text], then [Formula: see text] exists as well. We present [Formula: see text] and [Formula: see text] approximation algorithms for the [Formula: see text] problem for [Formula: see text]-spanners and for symmetric disk graphs, respectively. Actually, our approach achieves a [Formula: see text]-approximation algorithm for all directed graphs satisfying the property that, for every two nodes [Formula: see text] and [Formula: see text], the ratio between the shortest paths, from [Formula: see text] to [Formula: see text] and from [Formula: see text] to [Formula: see text] in the graph, is at most [Formula: see text].

scholarly journals Developing creative activity abilities of students in higher educational establishments

2021 ◽  
pp. 883-900
Author(s):  
Sergey Nikolaevich Dorofeev ◽  
Rustem Adamovich Shichiyakh ◽  
Leisan Nafisovna Khasimova
Keyword(s):  
Mathematics Education ◽  
Complex Problem ◽  
Professional Activity ◽  
Mathematical Education ◽  
State University ◽  
Creative Activity ◽  
Mathematical Concepts ◽  
Geometric Problems ◽  
Analysis And Synthesis ◽  
Theoretical Thinking

The article discusses methods for solving geometric problems with the active use of methods such as analysis and synthesis, analogy and generalization, based on theoretical thinking on the principle of ascent from simple to complex in order to develop students' ability to creative activity. The authors have developed systems of problems, focused on the formation of their ability to "make" independent discoveries both in the process of solving a problem and at the stage of researching the result of the solution. The developed system of problems is aimed at finding a way to solve a more complex problem, after a similar method has been used in relation to another simpler or particular problem. The participants in the experiment are future masters of pedagogical education (profile "Mathematical Education") at Togliatti State University. The article shows that the most effective methods of preparing future masters of mathematics education for creative professional activity can be such methods of scientific knowledge as analogy and generalization. It was revealed that in the process of learning to solve geometric problems included in the developed system, students demonstrate higher indicators of the level of formation of creative activity, as a result of the development of the ability of the future master of pedagogical education (profile "Mathematical Education") to analogy and its application in specific situations, his ability to use the established properties, skills and abilities formed, techniques and methods of action in relation to another object in new conditions and for new purposes, the use of mathematical concepts and theorems in more and more diverse specific problems.

A Study of the Impact of 3D Haptic-Augmented Learning Tools on Dynamics Course

2008 ◽  
Author(s):  
Weihang Zhu ◽  
Kendrick Aung ◽  
Bhavan Parikh ◽  
Jiang Zhou ◽  
Malur Srinivasan ◽  
...  
Keyword(s):  
Engineering Students ◽  
Force Feedback ◽  
Learning Tools ◽  
Mathematical Concepts ◽  
3 Dimensional ◽  
Undergraduate Study ◽  
Innovative Learning ◽  
Augmented Learning ◽  
Basic Course ◽  
The Impact

This paper presents our recent investigation on the impact of 3D haptic-augmented learning tools on Dynamics, which is a basic course in most of the engineering education program. Dynamics is considered to be one of the most difficult and non-intuitive courses that engineering students encounter during their undergraduate study because the course combines basic Newtonian physics and various mathematical concepts such as vector algebra, geometry, trigonometry, and calculus and these were applied to dynamical systems. Recent advances in Virtual Reality and robotics enable the human tactual system to be stimulated in a controlled manner through 3-dimensional (3D) force feedback devices, a.k.a. haptic interfaces. In this study, 3D haptic-augmented learning tools are created and used to complement the course materials in Dynamics course. Experiments are conducted with a group of Mechanical Engineering students in the Dynamics class. The assessment result shows that the innovative learning tools: 1) allow the students to interact with virtual objects with force feedback and better understand the abstract concepts by investigating the dynamics responses; 2) stimulate the students’ learning interests in understanding the fundamental physics theories.

Synchronizing Almost-Group Automata

2020 ◽  
pp. 1-22
Author(s):  
Mikhail V. Berlinkov ◽  
Cyril Nicaud
Keyword(s):  
Lower Bound ◽  
Efficient Algorithm ◽  
High Probability ◽  
Worst Case ◽  
Average Case ◽  
Model Of Computation ◽  
Letter Alphabet ◽  
Strongly Connected ◽  
Small Change ◽  
Random Automata

In this paper we address the question of synchronizing random automata in the critical settings of almost-group automata. Group automata are automata where all letters act as permutations on the set of states, and they are not synchronizing (unless they have one state). In almost-group automata, one of the letters acts as a permutation on [Formula: see text] states, and the others as permutations. We prove that this small change is enough for automata to become synchronizing with high probability. More precisely, we establish that the probability that a strongly-connected almost-group automaton is not synchronizing is [Formula: see text], for a [Formula: see text]-letter alphabet. We also present an efficient algorithm that decides whether a strongly-connected almost-group automaton is synchronizing. For a natural model of computation, we establish a [Formula: see text] worst-case lower bound for this problem ([Formula: see text] for the average case), which is almost matched by our algorithm.

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