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 Posted: Sun May 20, 2007 11:09 am Post subject: Homeomorphism and Abiogenesis |
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"In the mathematical field of topology a homeomorphism or topological
isomorphism (from the Greek words homoios = similar and morphe =
shape) is a special isomorphism between topological spaces which
respects topological properties. Two spaces with a homeomorphism
between them are called homeomorphic."
"In topology and related areas of mathematics a topological property
or topological invariant is a property of a topological space which is
invariant under homeomorphisms. That is, a property of spaces is a
topological property if whenever a space X possesses that property
every space homeomorphic to X possesses that property. Informally, a
topological property is a property of the space that can be expressed
using open sets. This article will show that properties such as
continuity of functions and compactness -- depending on convergence --
can be defined in terms of open sets.
A common problem in topology is to decide whether two topological
spaces are homeomorphic or not. To prove that two spaces are not
homeomorphic, it is sufficient to find a topological property which is
not shared by them."
Do single particles possess unique topological properties? Are these
topologies signficant when defining the origin of systems that derive
their form from the use of chemical energy? Is turbulence, viscosity,
shear patterns, low pressure, and high pressure as relevant in
abiogenesis (life from non-living matter) as it is in meteorology? Or
can the origin of life be reconstructed through purely from knowledge
of chemicals and stoichiometrics? |
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Guest
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| Posted: Sun May 20, 2007 11:10 am Post subject: Re: Homeomorphism and Abiogenesis |
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On May 20, 7:37 am, kmarina...@sailormoon.com wrote:
| Quote: |
"In the mathematical field of topology a homeomorphism or topological
isomorphism (from the Greek words homoios = similar and morphe =
shape) is a special isomorphism between topological spaces which
respects topological properties. Two spaces with a homeomorphism
between them are called homeomorphic."
"In topology and related areas of mathematics a topological property
or topological invariant is a property of a topological space which is
invariant under homeomorphisms. That is, a property of spaces is a
topological property if whenever a space X possesses that property
every space homeomorphic to X possesses that property. Informally, a
topological property is a property of the space that can be expressed
using open sets. This article will show that properties such as
continuity of functions and compactness -- depending on convergence --
can be defined in terms of open sets.
A common problem in topology is to decide whether two topological
spaces are homeomorphic or not. To prove that two spaces are not
homeomorphic, it is sufficient to find a topological property which is
not shared by them."
Do single particles possess unique topological properties?
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No, of course not. All single particles are topologically identical.
| Quote: |
Are these
topologies signficant when defining the origin of systems that derive
their form from the use of chemical energy?
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Not in the slightest.
| Quote: |
Is turbulence, viscosity,
shear patterns, low pressure, and high pressure as relevant in
abiogenesis (life from non-living matter) as it is in meteorology?
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Of course not.
| Quote: |
Or
can the origin of life be reconstructed through purely from knowledge
of chemicals and stoichiometrics?
|
No, it requires knowledge of many other factors as well.
Is this one of those posts which attempts by the use of complicated-
sounding terms whose meaning the author does not understand to "prove"
that abiogenesis is impossible?
RF |
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Perplexed in Peoria
Guest
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| Posted: Mon May 21, 2007 7:27 am Post subject: Re: Homeomorphism and Abiogenesis |
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<ediebur@rcn.com> wrote in message news:1179704557.747704.86460@r3g2000prh.googlegroups.com...
| Quote: |
On May 20, 5:39 pm, dkomo <dkomo...@comcast.net> wrote:
kmarina...@sailormoon.com wrote:
"In the mathematical field of topology a homeomorphism or topological
isomorphism (from the Greek words homoios = similar and morphe =
shape) is a special isomorphism between topological spaces which
respects topological properties. Two spaces with a homeomorphism
between them are called homeomorphic."
"In topology and related areas of mathematics a topological property
or topological invariant is a property of a topological space which is
invariant under homeomorphisms. That is, a property of spaces is a
topological property if whenever a space X possesses that property
every space homeomorphic to X possesses that property. Informally, a
topological property is a property of the space that can be expressed
using open sets. This article will show that properties such as
continuity of functions and compactness -- depending on convergence --
can be defined in terms of open sets.
A common problem in topology is to decide whether two topological
spaces are homeomorphic or not. To prove that two spaces are not
homeomorphic, it is sufficient to find a topological property which is
not shared by them."
Do single particles possess unique topological properties? Are these
topologies signficant when defining the origin of systems that derive
their form from the use of chemical energy? Is turbulence, viscosity,
shear patterns, low pressure, and high pressure as relevant in
abiogenesis (life from non-living matter) as it is in meteorology? Or
can the origin of life be reconstructed through purely from knowledge
of chemicals and stoichiometrics?
Abiogenesis and topology? You need to bring group theory into this.
Nature is all about symmetry, you know. So you should be looking into
topological groups. And then also nature is self-organizing and
emergent. So you should delve into chaos mathematics and
non-equilibrium thermodynamics. Let us know if discover anything.
--dk...@cris.com
Tensors and PDEs. The worked for relativity and quantum mechanics,
they will work for abiogenesis plus they sound great when you talk
about them.
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It is not so totally ridiculous as it sounds. The last chapter of
Sean Rice's book "Evolutionary Theory" is an attempt to apply tensors
to evo-devo. Well, tensor notation anyways - these aren't geometric
tensors that Elie Cartan would approve of.
But, in the context of abiogenesis, I have often wondered how to
best model the dynamics of membrane curvature. Gauss curvature doesn't
quite do it; I think I need to study up on curved 2-D Reimann curvature
with some other stuff thrown in to capture the Turing morphogenic
field idea.
The thing is, there are only two ways to deal with this kind of
complexity. The first is to say, "Everything affects everything
else. I mean, it is all, like, connected, you know?". And then,
when you get tired of saying that, you go off and study up on tensors. |
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r norman
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| Posted: Mon May 21, 2007 8:04 am Post subject: Re: Homeomorphism and Abiogenesis |
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On Mon, 21 May 2007 02:27:37 GMT, "Perplexed in Peoria"
<jimmenegay@sbcglobal.net> wrote:
| Quote: |
ediebur@rcn.com> wrote in message news:1179704557.747704.86460@r3g2000prh.googlegroups.com...
On May 20, 5:39 pm, dkomo <dkomo...@comcast.net> wrote:
kmarina...@sailormoon.com wrote:
"In the mathematical field of topology a homeomorphism or topological
isomorphism (from the Greek words homoios = similar and morphe =
shape) is a special isomorphism between topological spaces which
respects topological properties. Two spaces with a homeomorphism
between them are called homeomorphic."
"In topology and related areas of mathematics a topological property
or topological invariant is a property of a topological space which is
invariant under homeomorphisms. That is, a property of spaces is a
topological property if whenever a space X possesses that property
every space homeomorphic to X possesses that property. Informally, a
topological property is a property of the space that can be expressed
using open sets. This article will show that properties such as
continuity of functions and compactness -- depending on convergence --
can be defined in terms of open sets.
A common problem in topology is to decide whether two topological
spaces are homeomorphic or not. To prove that two spaces are not
homeomorphic, it is sufficient to find a topological property which is
not shared by them."
Do single particles possess unique topological properties? Are these
topologies signficant when defining the origin of systems that derive
their form from the use of chemical energy? Is turbulence, viscosity,
shear patterns, low pressure, and high pressure as relevant in
abiogenesis (life from non-living matter) as it is in meteorology? Or
can the origin of life be reconstructed through purely from knowledge
of chemicals and stoichiometrics?
Abiogenesis and topology? You need to bring group theory into this.
Nature is all about symmetry, you know. So you should be looking into
topological groups. And then also nature is self-organizing and
emergent. So you should delve into chaos mathematics and
non-equilibrium thermodynamics. Let us know if discover anything.
--dk...@cris.com
Tensors and PDEs. The worked for relativity and quantum mechanics,
they will work for abiogenesis plus they sound great when you talk
about them.
It is not so totally ridiculous as it sounds. The last chapter of
Sean Rice's book "Evolutionary Theory" is an attempt to apply tensors
to evo-devo. Well, tensor notation anyways - these aren't geometric
tensors that Elie Cartan would approve of.
But, in the context of abiogenesis, I have often wondered how to
best model the dynamics of membrane curvature. Gauss curvature doesn't
quite do it; I think I need to study up on curved 2-D Reimann curvature
with some other stuff thrown in to capture the Turing morphogenic
field idea.
The thing is, there are only two ways to deal with this kind of
complexity. The first is to say, "Everything affects everything
else. I mean, it is all, like, connected, you know?". And then,
when you get tired of saying that, you go off and study up on tensors.
|
Clearly there are many areas of biology where rather advanced
mathematics is useful and important. I recall studying the "backward
Kolmogorov equations" in population genetics. See, for example,
http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=1448842
And topological concepts are very useful in characterizing the
solution spaces to dynamic systems of which virtually any biological
system -- biochemical, cellular, physiological, developmental,
ecologic, or evolutionary -- is a good example.
That is all rather irrelevant when considering the nonsense proposed
in the original post where a rather abrupt and completely unexplained
and unjustified connection was made from topological homeomorphisms to
abiogenesis. Yes, meteorology involves complex differential equations
of flow and the notion of chaotic systems developed in large part from
Lorenz's meteorological studies. And the dynamic system we call "life"
certainly must be explained (in my opionion) in terms of a particular
basin of attraction in the system of interacting components. But
homeomorphism will explain abiogenesis about as well as it explains
hurricanes -- not at all. |
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