A totally ordered set of discrete abstractions for a given hybrid or continuous system

1997 ◽  
pp. 342-360 ◽  
Author(s):  
Jörg Raisch ◽  
Siu O'Young
Keyword(s):  
Ordered Set ◽  
Continuous System ◽  
Totally Ordered Set

ON RIBBON PRESENTATIONS OF RIBBON KNOTS

1994 ◽  
Vol 03 (02) ◽  
pp. 223-231
Author(s):  
TOMOYUKI YASUDA
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered ◽  
Totally Ordered Set

A ribbon n-knot Kn is constructed by attaching m bands to m + 1n-spheres in the Euclidean (n + 2)-space. There are many way of attaching them; as a result, Kn has many presentations which are called ribbon presentations. In this note, we will induce a notion to classify ribbon presentations for ribbon n-knots of m-fusions (m ≥ 1, n ≥ 2), and show that such classes form a totally ordered set in the case of m = 2 and a partially ordered set in the case of m ≥ 1.

scholarly journals The number of convex topologies on a finite totally ordered set

2015 ◽  
Vol 8 (1) ◽  
pp. 25-32 ◽  
Author(s):  
Tyler Clark ◽  
Tom Richmond
Keyword(s):  
Ordered Set ◽  
Totally Ordered Set

scholarly journals On the structure of the totally ordered set of unimodal cycles

2001 ◽  
Vol 25 (5) ◽  
pp. 323-329 ◽  
Author(s):  
Irene Mulvey
Keyword(s):  
Ordered Set ◽  
Totally Ordered Set

We continue the study of a class of unimodal cycles where each cycle in the class is forced by every unimodal cycle not in the class. For every order, we identify the cycle in the class of that order, which is maximal with respect to the forcing relation.

The π-Full Tight Riesz Orders on A(Ω)

1981 ◽  
Vol 24 (2) ◽  
pp. 137-151
Author(s):  
Gary Davis ◽  
Stephen H. McCleary
Keyword(s):  
Permutation Group ◽  
Ordered Set ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Totally Ordered Set

Let G be a lattice-ordered group (l-group), and let t, u∈ G+. We write tπu if t ∧ g = 1 is equivalent to u ∧ g = 1, and say that a tight Riesz order T on G is π-full if t ∈ T and t π U imply u∈T. We study the set of π-full tight Riesz orders on an l-permutation group (G, Ω), Ω a totally ordered set.

A Representation Theorem for Distributive l-Monoids

1984 ◽  
Vol 27 (2) ◽  
pp. 238-240 ◽  
Author(s):  
Marlow Anderson ◽  
C. C. Edwards
Keyword(s):  
Group Theory ◽  
Representation Theorem ◽  
Ordered Set ◽  
Totally Ordered Set

AbstractIn this note the Holland representation theorem for l-groups is extended to l-monoids by the following theorem: an l-monoid is distributive if and only if it may be embedded into the l-monoid of order-preserving functions on some totally ordered set. A corollary of this representation theorem is that a monoid is right orderable if and only if it is a subsemigroup of a distributive l-monoid; this result is analogous to the group theory case.

scholarly journals Boundaries in digital planes

1990 ◽  
Vol 3 (1) ◽  
pp. 27-55 ◽  
Author(s):  
Efim Khalimsky ◽  
Ralph Kopperman ◽  
Paul R. Meyer
Keyword(s):  
Jordan Curve ◽  
Ordered Set ◽  
Continuous Functions ◽  
Jordan Curve Theorem ◽  
Connected Sets ◽  
Digital Plane ◽  
Parameter Interval ◽  
Totally Ordered Set ◽  
Definition Of ◽  
Natural Way

The importance of topological connectedness properties in processing digital pictures is well known. A natural way to begin a theory for this is to give a definition of connectedness for subsets of a digital plane which allows one to prove a Jordan curve theorem. The generally accepted approach to this has been a non-topological Jordan curve theorem which requires two different definitions, 4-connectedness, and 8-connectedness, one for the curve and the other for its complement.In [KKM] we introduced a purely topological context for a digital plane and proved a Jordan curve theorem. The present paper gives a topological proof of the non-topological Jordan curve theorem mentioned above and extends our previous work by considering some questions associated with image processing:How do more complicated curves separate the digital plane into connected sets? Conversely given a partition of the digital plane into connected sets, what are the boundaries like and how can we recover them? Our construction gives a unified answer to these questions.The crucial step in making our approach topological is to utilize a natural connected topology on a finite, totally ordered set; the topologies on the digital spaces are then just the associated product topologies. Furthermore, this permits us to define path, arc, and curve as certain continuous functions on such a parameter interval.

scholarly journals Rigidity in order-types

1977 ◽  
Vol 24 (2) ◽  
pp. 203-215 ◽  
Author(s):  
J. L. Hickman
Keyword(s):  
Ordered Set ◽  
Characterization Theorem ◽  
Decomposition Theorem ◽  
Order Type ◽  
Partial Ordering ◽  
Limit Laws ◽  
Order Types ◽  
Totally Ordered Set ◽  
The Axiom Of Choice ◽  
Increasing Sequences

AbstractA totally ordered set (and corresponding order-type) is said to be rigid if it is not similar to any proper initial segment of itself. The class of rigid ordertypes is closed under addition and multiplication, satisfies both cancellation laws from the left, and admits a partial ordering that is an extension of the ordering of the ordinals. Under this ordering, limits of increasing sequences of rigid order-types are well defined, rigid and satisfy the usual limit laws concerning addition and multiplication. A decomposition theorem is obtained, and is used to prove a characterization theorem on rigid order-types that are additively prime. Wherever possible, use of the Axiom of Choice is eschewed, and theorems whose proofs depend upon Choice are marked.

Compatible Tight Riesz Orders II

1979 ◽  
Vol 31 (2) ◽  
pp. 304-307 ◽  
Author(s):  
A. M. W. Glass
Keyword(s):  
Ordered Set ◽  
Group A ◽  
Galley Proof ◽  
Totally Ordered Set

R. N. Ball (unpublished) and G. E. Davis and C. D. Fox [1] established that if Ω is a doubly homogeneous totally ordered set, the l-group A (Ω) of all orderpreserving permutations of Ω endures a compatible tight Riesz order. Specifically T = {g ∈ A(Ω)+ : supp (g) is dense in Ω} is a compatible tight Riesz order for A(Ω). Using this fact, I inserted Theorem 3.7 into [2; MR 53 (1977), #13070] at the galley proof stage. (It was also included in MR 54 (1977), #7350 and [3; p. 472].) Theorem 3.7 stated: Let Ω be homogeneous. Then A(Ω) endures a compatible tight Reisz order if and only if Ω is dense. I stated that it was obvious that if Ω were homogeneous and discrete, A(Ω) could not endure a compatible tight Riesz order. This “obvious” is neither obvious nor true.

scholarly journals Captive diffusions and their applications to order-preserving dynamics

2020 ◽  
Vol 476 (2241) ◽  
pp. 20200294
Author(s):  
Levent Ali Mengütürk ◽  
Murat Cahit Mengütürk
Keyword(s):  
Matrix Theory ◽  
Interacting Particle Systems ◽  
Ordered Set ◽  
Random Processes ◽  
Particle Systems ◽  
Full Generality ◽  
Interacting Particle ◽  
Bounded Functions ◽  
Totally Ordered Set ◽  
And Diffusion

We propose a class of stochastic processes that we call captive diffusions, which evolve within measurable pairs of càdlàg bounded functions that admit bounded right-derivatives at points where they are continuous. In full generality, such processes allow reflection and absorption dynamics at their boundaries—possibly in a hybrid manner over non-overlapping time periods—and if they are martingales, continuous boundaries are necessarily monotonic. We employ multi-dimensional captive diffusions equipped with a totally ordered set of boundaries to model random processes that preserve an initially determined rank. We run numerical simulations on several examples governed by different drift and diffusion coefficients. Applications include interacting particle systems, random matrix theory, epidemic modelling and stochastic control.

Sign in / Sign up

Export Citation Format

Share Document