Sufficient Conditions for a Symmetric Chain Order

1977 ◽  
Vol 32 (4) ◽  
pp. 807-809 ◽  
Author(s):  
Jerrold R. Griggs
Keyword(s):  
Sufficient Conditions ◽  
Chain Order ◽  
Symmetric Chain ◽  
Symmetric Chain Order

scholarly journals Some Quotients of Chain Products are Symmetric Chain Orders

10.37236/2430 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Dwight Duffus ◽  
Jeremy McKibben-Sanders ◽  
Kyle Thayer
Keyword(s):  
Automorphism Group ◽  
Symmetric Group ◽  
Boolean Lattice ◽  
Disjoint Cycles ◽  
Chain Order ◽  
Symmetric Chain ◽  
Straightforward Proof ◽  
Symmetric Chain Order

Canfield and Mason have conjectured that for all subgroups $G$ of the automorphism group of the Boolean lattice $B_n$ (which can be regarded as the symmetric group $S_n$), the quotient order $B_n/G$ is a symmetric chain order.  We provide a straightforward proof of a generalization of a result of K. K. Jordan: namely, $B_n/G$ is an SCO whenever $G$ is generated by powers of disjoint cycles. In addition, the Boolean lattice $B_n$ can be replaced by any product of finite chains. The symmetric chain decompositions of Greene and Kleitman provide the basis for partitions of these quotients.

scholarly journals Symmetric Chain Decomposition of Necklace Posets

10.37236/1178 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Vivek Dhand
Keyword(s):  
Disjoint Union ◽  
Cyclic Permutation ◽  
Chain Decomposition ◽  
Chain Order ◽  
Symmetric Chain Decomposition ◽  
Symmetric Chain ◽  
Symmetric Chain Order

A finite ranked poset is called a symmetric chain order if it can be written as a disjoint union of rank-symmetric, saturated chains. If $\mathcal{P}$ is any symmetric chain order, we prove that $\mathcal{P}^n/\mathbb{Z}_n$ is also a symmetric chain order, where $\mathbb{Z}_n$ acts on $\mathcal{P}^n$ by cyclic permutation of the factors.

scholarly journals Symmetric Chain Decompositions of Quotients by Wreath Products

10.37236/5073 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Dwight Duffus ◽  
Kyle Thayer
Keyword(s):  
Symmetric Group ◽  
Elementary Proof ◽  
Boolean Lattice ◽  
Wreath Products ◽  
Cyclic Groups ◽  
Chain Order ◽  
A Chain ◽  
Symmetric Chain ◽  
Symmetric Chain Order

Subgroups of the symmetric group $S_n$ act on $C^n$ (the $n$-fold product $C \times \cdots \times C$ of a chain $C$) by permuting coordinates, and induce automorphisms of the power $C^n$. For certain families of subgroups of $S_n$, the quotients defined by these groups can be shown to have symmetric chain decompositions (SCDs). These SCDs allow us to enlarge the collection of subgroups $G$ of $S_n$ for which the quotient $\mathbf{2}^n/G$ on the Boolean lattice $\mathbf{2}^n$ is a symmetric chain order (SCO). The methods are also used to provide an elementary proof that quotients of powers of SCOs by cyclic groups are SCOs.

Some Contributions to the Theory of Near-Critical Bisexual Branching Processes

2007 ◽  
Vol 44 (02) ◽  
pp. 492-505
Author(s):  
M. Molina ◽  
M. Mota ◽  
A. Ramos
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Positive Probability ◽  
Limit Laws ◽  
Bisexual Branching Processes ◽  
Probabilistic Evolution

We investigate the probabilistic evolution of a near-critical bisexual branching process with mating depending on the number of couples in the population. We determine sufficient conditions which guarantee either the almost sure extinction of such a process or its survival with positive probability. We also establish some limiting results concerning the sequences of couples, females, and males, suitably normalized. In particular, gamma, normal, and degenerate distributions are proved to be limit laws. The results also hold for bisexual Bienaymé–Galton–Watson processes, and can be adapted to other classes of near-critical bisexual branching processes.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Normal approximations to discrete unimodal distributions

1986 ◽  
Vol 23 (04) ◽  
pp. 1013-1018
Author(s):  
B. G. Quinn ◽  
H. L. MacGillivray
Keyword(s):  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Hypergeometric Distribution ◽  
Death Process ◽  
Normal Approximations ◽  
Discrete Random Variables ◽  
Birth Death

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

Budgeting in Socio-economic Development Strategies (Raising a Problem)

2008 ◽  
pp. 134-151
Author(s):  
A. Shastitko ◽  
M. Ovchinnikov
Keyword(s):  
Economic Policy ◽  
Economic Development ◽  
Social Change ◽  
Eastern Europe ◽  
Statistical Data ◽  
Sufficient Conditions ◽  
Transitional Period ◽  
Socio Economic Development ◽  
Necessary And Sufficient ◽  
Economic Development Strategies

The article proposes an approach to the analysis of social change and contributes to the clarification of concepts of economic policy. It deals in particular with the notion of "change of system". The author considers positive and normative aspects of the analysis of capitalist and socialist systems. The necessary and sufficient conditions for the system to be changed are introduced, their fulfillment is discussed drawing upon the historical and statistical data. The article describes both economic and political peculiarities of the transitional period in different countries, especially in Eastern Europe.

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