423 research outputs found
Nonstandard Graphs, Revised
This is a revision of the paper archived previously on August 22, 2002. It
corrects a mistake in Sec. 8 concerning eccentricities of graphs.
From any given sequence of finite or infinite graphs, a nonstandard graph is
constructed. The procedure is similar to an ultrapower construction on an
internal set from a sequence of subsets of the real line, but now the
individual entities are the vertices of the graphs instead of real numbers. The
transfer principle is then invoked to extend several graph-theoretic results to
the nonstandard case. After incidences and adjacencies between nonstandard
vertices and edges are defined, several formulas regarding numbers of vertices
and edges, and nonstandard versions of Eulerian graphs, Hamiltonian graphs, and
a coloring theorem are established for these nonstandard graphs.
Key Words: Nonstandard graphs, transfer principle, ultrapower constructions
Nonstandard Transfinite Graphs and Networks of Higher Ranks
In Chapter 8 of the Book, ``Graphs and Networks: Transfinite and
Nonstandard'' (published by Birkhauser-Boston in 2004), nonstandard versions of
transfinite graphs and of electrical networks having such graphs were defined
and examined but only for the first two ranks, 0 and 1, of transfiniteness. In
the present work, these results are extended to higher ranks of transfinteness.
Such is done in detail for the natural-number ranks and also for the first
transfinite ordinal rank. Results for still higher ranks of transfiniteness can
be established in much the same way. Once the transfinite graphs of higher
ranks are established, theorems concerning the existence of hyperreal operating
points and the satisfaction of Kirchhoff's laws in nonstandard networks of
higher ranks can be proven just as they are for nonstandard networks of the
first rank.Comment: 8 pages, 0 figure
A Shorter Proof of the Transitivity of Transfinite Connectedness
A criterion is established for the transitivity of connectedness in a
transfinite graph. Its proof is much shorter than a prior argument published
previously for that criterion.Comment: 6 page
Ordinal Distances in Transfinite Graphs
An ordinal-valued metric taking its values in the set of all countable
ordinals can be assigned to a metrizable set of nodes in a transfinite graph.
Then, a variety of results concerning nodal eccentricities, radii, diameters,
centers, peripheries, and blocks can be extended to transfinite graphs
Hyperreal Waves on Transfinte, Terminated, Distortionless and Lossless, Transmission Lines
A prior work (see Chapter 8 of the book, ``Graphs and Networks: Transfinite
and Nonstandard,'' Birkhauser-Boston, Cambridge, Mass., USA, 2004) examined the
propagation of an electromagnetic wave on a transfinite transmission line,
transfinite in the sense that infinitely many one-way infinite transmission
lines are connected in cascade. That there are infinitely many such one-way
infinite lines results in the wave propagating without ever reflecting at some
discontinuity. The present work examines the cascade where the cascade
terminates after only finitely many one-way infinite transmission lines, with
the result that reflected waves are now produced at both the far end as well as
at the initial end of the transfinite transmission line. The question of
whether the reflected waves are infini tesimal or appreciable and whether they
sum to an infinitesimal or appreciable amount are resolved for both
distortionless and lossless lines. Finally, the generalizations to higher ranks
of transfiniteness is briefly summarized.Comment: 16 pages, 3 figure
Nonstandard Graphs
From any given sequence of finite or infinite graphs, a nonstandard graph is
constructed. The procedure is similar to an ultrapower construction of an
internal set from a sequence of subsets of the real line, but now the
individual entities are the vertices of the graphs instead of real numbers. The
transfer principle is then invoked to extend several graph-theoretic results to
the nonstandard case. After incidences and adjacencies between nonstandard
vertices are defined, several formulas regarding numbers of vertices and edges,
and nonstandard versions of Eulerian graphs, Hamiltonian graphs, and a coloring
theorem are established for these nonstandard graphs
The Galaxies of Nonstandard Enlargements of Infinite and Transfinite Graphs
The galaxies of nonstandard enlargements of conventionally infinite as well
as of transfinite graphs are defined, analyzed, and illustrated by some
examples. It is then shown that any such enlargement either has exactly one
galaxy, its principal one, or it has infinitely many galaxies. In the latter
case, the galaxies are partially ordered by their "closeness" to the principal
galaxy. If an enlargement has a galaxy different from its principal galaxy,
then it has a two-way infinite sequence of galaxies that are totally ordered
according to that "closeness" property. There may be many such totally ordered
sequences.Comment: 24 pages, 3 figure
Nonstandard Digraphs
Nonstandard graphs have been defined and examined in prior works. The present
work does the same for nonstandard digraphs. Since digraphs have more structure
than do graphs, the present discussion requires more complicated definitions
and yields a variety of results peculiar to nonstandard digraphs. A nonstandard
digraph can be obtained by means of an ultrapower construction based on a
sequence of digraphs or more elegantly by using the transfer principle. We use
either or both techniques in particular circumstances. As special cases, we
have the enlargement of a single infinite digraph and also hyperfinite digraphs
based on sequences of finite digraphs. Also examined are such ideas as
incidences and adjacencies for nonstandard arcs and vertices, connectedness,
components, and galaxies in nonstandard digraphs.Comment: 15 page
The Galaxies of Nonstandard Enlargements of Transfinite Graphs of Higher Rsnks
In a prior work, the galaxies of the nonstandard enlargements of
conventionally infinite graphs and also of transfinite graphs of the first rank
of transfiniteness were defined, examined, and illustrated by some examples. In
this work it is shown how the results of the prior work extend to graphs of
higher ranks.Comment: 12 page
A Circuit-Theoretic Anomaly Resolved by Nonstandard Analysis
An anomaly in electrical circuit theory is the disappearance of some of the
energy when two capacitors, one charged and the other uncharged, are connected
together through resistanceless wires. Nonstandard analysis shows that, when
the wires are taken to have infinitesimally small but nonzero resistance, the
energy dissipated in the wires equals that substantial amount of enregy that
had disappeared, and that all but an infinitesimal amount of this dissipation
occurs during an infinitesimal initial time period.Comment: 8 pages, 2 figure
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