Derivation of Methods and Knowledge in Structures by Combinatorial Representations

2009 ◽  
Author(s):  
N. Ta'aseh ◽  
O. Shai
Keyword(s):  
Combinatorial Representations

scholarly journals Combinatorial representations

2013 ◽  
Vol 120 (3) ◽  
pp. 671-682 ◽  
Author(s):  
Peter J. Cameron ◽  
Maximilien Gadouleau ◽  
Søren Riis
Keyword(s):  
Combinatorial Representations

scholarly journals Approximation of bounds on mixed-level orthogonal arrays

2011 ◽  
Vol 43 (02) ◽  
pp. 399-421
Author(s):  
Ali Devin Sezer ◽  
Ferruh Özbudak
Keyword(s):  
Random Walk ◽  
Large Deviation ◽  
Recursive Algorithm ◽  
Markov Property ◽  
Orthogonal Arrays ◽  
Recursive Formula ◽  
Limit Problem ◽  
Simulation Algorithm ◽  
Bellman Equations ◽  
Combinatorial Representations

Mixed-level orthogonal arrays are basic structures in experimental design. We develop three algorithms that compute Rao- and Gilbert-Varshamov-type bounds for mixed-level orthogonal arrays. The computational complexity of the terms involved in the original combinatorial representations of these bounds can grow fast as the parameters of the arrays increase and this justifies the construction of these algorithms. The first is a recursive algorithm that computes the bounds exactly, the second is based on an asymptotic analysis, and the third is a simulation algorithm. They are all based on the representation of the combinatorial expressions that appear in the bounds as expectations involving a symmetric random walk. The Markov property of the underlying random walk gives the recursive formula to compute the expectations. A large deviation (LD) analysis of the expectations provides the asymptotic algorithm. The asymptotically optimal importance sampling (IS) of the same expectation provides the simulation algorithm. Both the LD analysis and the construction of the IS algorithm use a representation of these problems as a sequence of stochastic optimal control problems converging to a limit calculus of a variations problem. The construction of the IS algorithm uses a recently discovered method of using subsolutions to the Hamilton-Jacobi-Bellman equations associated with the limit problem.

A LOWER BOUND FOR THE NUMBER OF FORBIDDEN MOVES TO UNKNOT A LONG VIRTUAL KNOT

2013 ◽  
Vol 22 (06) ◽  
pp. 1350024 ◽  
Author(s):  
MYEONG-JU JEONG
Keyword(s):  
Lower Bound ◽  
Finite Type ◽  
Finite Sequence ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Finite Type Invariants ◽  
Combinatorial Representations ◽  
Congruent Modulo

Nelson and Kanenobu showed that forbidden moves unknot any virtual knot. Similarly a long virtual knot can be unknotted by a finite sequence of forbidden moves. Goussarov, Polyak and Viro introduced finite type invariants of virtual knots and long virtual knots and gave combinatorial representations of finite type invariants. We introduce Fn-moves which generalize the forbidden moves. Assume that two long virtual knots K and K′ are related by a finite sequence of Fn-moves. We show that the values of the finite type invariants of degree 2 of K and K′ are congruent modulo n and give a lower bound for the number of Fn-moves needed to transform K to K′.

scholarly journals Approximation of bounds on mixed-level orthogonal arrays

2011 ◽  
Vol 43 (2) ◽  
pp. 399-421 ◽  
Author(s):  
Ali Devin Sezer ◽  
Ferruh Özbudak
Keyword(s):  
Random Walk ◽  
Large Deviation ◽  
Recursive Algorithm ◽  
Markov Property ◽  
Orthogonal Arrays ◽  
Recursive Formula ◽  
Limit Problem ◽  
Simulation Algorithm ◽  
Bellman Equations ◽  
Combinatorial Representations

Mixed-level orthogonal arrays are basic structures in experimental design. We develop three algorithms that compute Rao- and Gilbert-Varshamov-type bounds for mixed-level orthogonal arrays. The computational complexity of the terms involved in the original combinatorial representations of these bounds can grow fast as the parameters of the arrays increase and this justifies the construction of these algorithms. The first is a recursive algorithm that computes the bounds exactly, the second is based on an asymptotic analysis, and the third is a simulation algorithm. They are all based on the representation of the combinatorial expressions that appear in the bounds as expectations involving a symmetric random walk. The Markov property of the underlying random walk gives the recursive formula to compute the expectations. A large deviation (LD) analysis of the expectations provides the asymptotic algorithm. The asymptotically optimal importance sampling (IS) of the same expectation provides the simulation algorithm. Both the LD analysis and the construction of the IS algorithm use a representation of these problems as a sequence of stochastic optimal control problems converging to a limit calculus of a variations problem. The construction of the IS algorithm uses a recently discovered method of using subsolutions to the Hamilton-Jacobi-Bellman equations associated with the limit problem.

On Offer Shai’s Contribution to Mechanical Engineering and Design

2017 ◽  
Author(s):  
Yoram Reich ◽  
Elad Hahn ◽  
Michael Slavutin
Keyword(s):  
Research Program ◽  
Mechanical Engineering ◽  
Discrete Systems ◽  
Building Blocks ◽  
New Methods ◽  
New Concepts ◽  
Combinatorial Representations

This paper presents the contribution of Offer Shai to mechanical engineering and design. Over a period of three decades Shai has created an impressive research program that is founded on solid mathematical grounds — combinatorial representations of systems. On this foundation he made contributions that ranged from inventing new concepts in mechanics (e.g., face force), new ways to characterize systems (e.g., singularity positions), new ways to create building blocks to model discrete systems (e.g., Assur graphs and their synthesis), and new methods in design (e.g., infused design). This paper summarizes some of these contributions in an attempt to describe the breadth and depth and attract researchers to continue develop his ideas.

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