Another Note on Dilworth's Decomposition Theorem

Journal of Discrete Mathematics ◽

10.1155/2013/692645 ◽

2013 ◽

Vol 2013 ◽

pp. 1-4 ◽

Cited By ~ 2

Author(s):

Wim Pijls ◽

Rob Potharst

Keyword(s):

Decomposition Theorem ◽

New Method ◽

Flow Networks ◽

Maximal Antichain ◽

Dilworth’S Theorem

This paper proposes a new proof of Dilworth's theorem. The proof is based upon the minflow/maxcut property in flow networks. In relation to this proof, a new method to find both a Dilworth decomposition and a maximal antichain is presented.

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Formalization of some central theorems in combinatorics of finite sets

10.29007/r7fg ◽

2018 ◽

Author(s):

Abhishek Kr Singh

Keyword(s):

Ordered Set ◽

Decomposition Theorem ◽

Proof Assistant ◽

Combinatorial Objects ◽

Finite Poset ◽

The Coq Proof Assistant ◽

Coq Proof Assistant ◽

Finite Partially Ordered Set ◽

Dilworth’S Theorem ◽

Finite Posets

We present fully formalized proofs of some central theorems from combinatorics. These are Dilworth's decomposition theorem, Mirsky's theorem, Hall's marriage theorem and the Erdős-Szekeres theorem. Dilworth's decomposition theorem is the key result among these. It states that in any finite partially ordered set (poset), the size of a smallest chain cover and a largest antichain are the same. Mirsky's theorem is a dual of Dilworth's decomposition theorem, which states that in any finite poset, the size of a smallest antichain cover and a largest chain are the same. We use Dilworth's theorem in the proofs of Hall's Marriage theorem and the Erdős-Szekeres theorem. The combinatorial objects involved in these theorems are sets and sequences. All the proofs are formalized in the Coq proof assistant. We develop a library of definitions and facts that can be used as a framework for formalizing other theorems on finite posets.

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Partial observer canonical form for multi-output nonlinear forced system: a new method

Journal of Intelligent Manufacturing and Special Equipment ◽

10.1108/jimse-05-2020-0001 ◽

2020 ◽

Vol 1 (1) ◽

pp. 121-134

Author(s):

Haotian Xu ◽

Jingcheng Wang ◽

Hongyuan Wang ◽

Ibrahim Brahmia ◽

Shangwei Zhao

Keyword(s):

Nonlinear System ◽

Canonical Form ◽

Design Method ◽

Sufficient Conditions ◽

Decomposition Theorem ◽

Original System ◽

New Method ◽

Content Type ◽

Wide Range ◽

Necessary And Sufficient

PurposeThe purpose of this paper is to investigate the design method of partial observer canonical form (POCF), which is one of the important research tools for industrial plants.Design/methodology/approachMotivated by the two-steps method proposed in Xu et al. (2020), this paper extends this method to the case of Multi-Input Multi-Output (MIMO) nonlinear system. It decomposes the original system into two subsystems by observable decomposition theorem first and then transforms the observable subsystem into OCF. Furthermore, the necessary and sufficient conditions for the existing of POCF are proved.FindingsThe proposed method has a wide range of applications including completely observable nonlinear system, noncompletely observable nonlinear system, autonomous nonlinear system and forced nonlinear system. Besides, comparing to the existing results (Saadi et al., 2016), the method requires less verified conditions.Originality/valueThe new method concerning design POCF has better plants compatibility and less validation conditions.

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Remarks on Dilworth's theorem in relation to transversal theory

Glasgow Mathematical Journal ◽

10.1017/s0017089500003931 ◽

1980 ◽

Vol 21 (1) ◽

pp. 19-22 ◽

Cited By ~ 1

Author(s):

Hazel Perfect

Keyword(s):

Decomposition Theorem ◽

Transversal Theory ◽

Dilworth’S Theorem

In his book Transversal Theory [3], L. Mirsky has remarked that “At present, the relation between Dilworth's decomposition theorem…and transversal theory is rather tenuous; but further study may reveal unexpected connections”. Some of these connections can perhaps now be seen a little more clearly; and our purpose in this note is to make one or two observations in this regard. Throughout, all sets considered are finite.

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Selective Sampling and Orientation of Non-Homogeneous Tissues

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100100238 ◽

1968 ◽

Vol 26 ◽

pp. 98-99

Author(s):

C. C. Clawson ◽

L. W. Anderson ◽

R. A. Good

Keyword(s):

Electron Microscopy ◽

Electron Microscope ◽

Light Microscopy ◽

Random Sampling ◽

New Method ◽

Microscope Examination ◽

Small Areas ◽

Selective Sampling ◽

Large Numbers ◽

Specific Orientation

Investigations which require electron microscope examination of a few specific areas of non-homogeneous tissues make random sampling of small blocks an inefficient and unrewarding procedure. Therefore, several investigators have devised methods which allow obtaining sample blocks for electron microscopy from region of tissue previously identified by light microscopy of present here techniques which make possible: 1) sampling tissue for electron microscopy from selected areas previously identified by light microscopy of relatively large pieces of tissue; 2) dehydration and embedding large numbers of individually identified blocks while keeping each one separate; 3) a new method of maintaining specific orientation of blocks during embedding; 4) special light microscopic staining or fluorescent procedures and electron microscopy on immediately adjacent small areas of tissue.

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