[Fis] _ Re: concluding by beginning
Louis H Kauffman
loukau at gmail.com
Thu Mar 31 09:49:20 CEST 2016
Dear Folks,
I will close with some comments about the relationship between recursive distinctioning and replication in biology.
This will be another example of the sort of modeling excursion that one can make by looking at patterns and analogies.
See
homepages.math.uic.edu/~kauffman/RD.html <http://homepages.math.uic.edu/~kauffman/RD.html>
RECURSIVE DISTINCTIONING
This folder contains links to papers related to Recursive Distinctioning. Recursive Distinctioning means just what it says. A pattern of distinctions is given in a space based on a graphical structure (such as a line of print or a planar lattice or given graph). Each node of the graph is occupied by a letter from some arbitrary alphabet. A specialized alphabet is given that can indicate distinctions about neighbors of a given node. The neighbors of a node are all nodes that are connected to the given node by edges in the graph. The letters in the specialized alphabet (call it SA) are used to describe the states of the letters in the given graph and at each stage in the recursion, letters in SA are written at all nodes in the graph, describing its previous state. The recursive structure that results from the iteration of descriptions is called Recursive Distinctioning. Here is an example. We use a line graph and represent it just as a finite row of letters. The Special Alphabet is SA = { =, [, ], O} where "=" means that the letters to the left and to the right are equal to the letter in the middle. Thus if we had AAA in the line then the middle A would be replaced by =. The symbol "[" means that the letter to the LEFT is different. Thus in ABB the middle letter would be replaced by [. The symbol "]" means that the letter to the right is different. And finally the symbol "O" means that the letters both to the left and to the right are different. SA is a tiny language of elementary letter-distinctions. Here is an example of this RD in operation where we use the proverbial three dots to indicate a long string of letters in the same pattern. For example,
... AAAAAAAAAABAAAAAAAAAA ... is replaced by
... =========]O[========= ... is replaced by
... ========]OOO[======== ... is replaced by
... =======]O[=]O[======= ... .
Note that the element ]O[ appears and it has replicated itself in a kind of mitosis. To see this in more detail, here is a link to a page from a mathematica program written by LK that uses a 'blank' or 'unmarked state' instead of the '=" sign. Program and Output <https://dl.dropbox.com/u/11067256/RDL.pdf>. Elementary RD patterns are fundamental and will be found in many structures at all levels. To see an cellular automaton example of this phenomenon, look at the next link. Here we see a replicator in 'HighLife' a modification of John Horton Conway's automaton 'Life'. The Highlife Replicator follows the same pattern as our RD Replicator! We can begin to understand how the RD Replicator works. This gives a foundation for understanding how the more complex HighLife Replicator behaves in its context. HighLife Replicator. <https://en.m.wikipedia.org/wiki/Highlife_(cellular_automaton)> Finally, here is an excerpt from a paper by LK about replication in biology and the role of RD. Excerpt. <https://dl.dropbox.com/u/11067256/KauffmanExcerpt.pdf>
See RDLetter. <http://homepages.math.uic.edu/~kauffman/RDLetter.pdf> This is the Isaacson-Kauffman report on RD, summarized in a letter-to-the-editor of JSP, Vol. 4, No. 1, Spring 2015, directly accessed on this server.See Patent. <https://dl.dropbox.com/u/11067256/JoelIsaacsonPatentDocument.pdf>This is Joel Isaacson's patent document for RD.
See Biological Replication. <https://dl.dropbox.com/u/11067256/KauffmanJPBM1033.pdf> This is a related paper by Kauffman.
You see above a very simple distinction making/using automaton that produces a ‘cell’ ]O[ from an elementary distinction (of B from the background of equal A’s),
and that this cell then undergoes mitosis. Then as an observer you must look again and note that the nothing that happens in this automaton is local. The cell happens
because of the global structure of the one-dimensional automata space. The apparent splitting from the inside of the cell is actually a consequence of the global
condition of the cell in the whole space. The entire evolution of the process is a repeated articulation of the distinctions that are present in the process. This is
a new holistic modeling paradigm and we are exploring with simple examples the extent to which it will apply to more complex phenomena.
A more extended paper by myself and Joel Isaacson will be available soon.
Best,
Lou Kauffman
> On Mar 30, 2016, at 7:18 AM, Pedro C. Marijuan <pcmarijuan.iacs at aragon.es> wrote:
>
> Sorry but the dancing time is over... maybe tomorrow or on Friday Lou could send some concluding comment, and next Monday Soeren would start the new part. The present Q. discussion can surface again during the coming session...
> best--Pedro
>
>
> El 30/03/2016 a las 1:06, Dr. Plamen L. Simeonov escribió:
>> I think you are right, Lou, with respect to Deutsch who actually met Everett III with the multiple universe hypothesis. The sole name “constructor theory” invoked associations beyond the quantum frame in me, but he did not went that far. As for Josephson, I am not quite sure about his notion. Brian remains firmly on the quantum level in the papers I referred earlier, but he often returns to Ilexa Yarley”s “circular theory” which offers a much broader interpretation in my opinion. I expected your mentioning of (the vibrations of) “thought forms”, which are supposed to invoke the emergence of word and action. I welcome your understanding for the necessity of a deeper QM to make the links between actuality and the bounded potentiality more comprehensive.
>>
>> Best,
>>
>> Plamen
>>
>>
>>
>>
>> On Wed, Mar 30, 2016 at 12:33 AM, Louis H Kauffman < <mailto:kauffman at uic.edu>kauffman at uic.edu <mailto:kauffman at uic.edu>> wrote:
>> Josephson and Deutsh are not ‘deeper than QM’. Deutsch for example is a very literal interpretation of QM that says that all the trajectories in the Feynman path sum are real, and they occur in parallel universes. This is a nice mathematical way to think, but it is not deeper than present QM!
>> Energy is conserved, but ‘particles’ and indeed universes can be created from vacuum. If we want to go to discussion of ‘holy spirit’ then one should look at the structure of thought itself. For it is at the level of thought that every concept has a life behind it. Every idea is real and alive. Platonism asserts this directly in the belief in the existence of form and this form is a living form that we interact with and we are. How these notions are related to QM probably does await the emergence of a deeper QM.
>>
>>> On Mar 29, 2016, at 4:43 PM, Dr. Plamen L. Simeonov <plamen.l.simeonov at gmail.com <mailto:plamen.l.simeonov at gmail.com>> wrote:
>>>
>>>
>>>
>>> Thank you for your responses, Lou and Stan. I am aware about the details of the autopoietic model. What I was actually addressing by the transition from abiotic to biotic structures and the later emergence of RNA and DNA was this elusive aspect of “mass action” which Stan mentioned, that in my opinion must have emerged out of the field of “triggered (by resonance) potentialities which deeper theories than QM are trying to develop (cf. Josephson and Deutsch mentioned earlier). This enigmatic emergence of action out of nothing (vacuum or pure potentiality) naturally allows the (co-)existence of such heretic ideas as the immaterial “Holy Spirit” or Hans Driesch”s vitalism, Jean Sharon’s eternal electron, or “The Matrix of Matter and Life”at the sub-Planckian scale. How about this possible link to Platonism, theology, logic and algebra?
>>>
>>> All the best,
>>>
>>> Plamen
>>>
>>> PS. I do not know why my notes appear twice on this list.
>>>
>>>
>>>
>>> On Tue, Mar 29, 2016 at 10:55 PM, Louis H Kauffman < <mailto:kauffman at uic.edu>kauffman at uic.edu <mailto:kauffman at uic.edu>> wrote:
>>> This is a reply to Plamen’s comment about autopoeisis. In their paper Maturana,Uribe and Varela give a working model (computer model) for autopoeisis.
>>> It is very simple, consisting of a subtrate of nodal elements that tend to bond when in proximity, and a collection of catalytic nodal elements that promote bonding in their vicinity. The result of this dynamics is that carapaces of linked nodal elements form around the catalytic elements and these photo-cells tend to keep surviving the perturbations built into the system. This model shows that cells can arise from a very simple dynmamic geometric/topological substrate long before anything as sophisticated as DNA has happened.
>>>
>>>> On Mar 29, 2016, at 2:54 PM, Stanley N Salthe < <mailto:ssalthe at binghamton.edu>ssalthe at binghamton.edu <mailto:ssalthe at binghamton.edu>> wrote:
>>>>
>>>> Plamen wrote:
>>>>
>>>> I begin to believe that the transition from abiotic to biotic structures, incl. Maturana-Varela.-Uribe’s autopoiesis may, really have some underlying matrix/”skeleton”/”programme” which has nothing in common with the nature of DNA, and that DNA and RNA as we know them today may have emerged as secondary or even tertiary “memory” of something underlying deeper below the microbiological surface. It is at least worth thinking in this direction. I do not mean necessarily the role of the number concept and Platonic origin of the universe, but something probably much more “physical”
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> S: An interesting recently published effort along these lines is:
>>>>
>>>> Alvaro Moreno and Matteo Mossio: Biological Autonomy: A Philosophical and Theoretical Enquiry (History, Philosophy and Theory of the Life Sciences 12) Springer
>>>>
>>>> They seek a materialist understanding of biology as a system, attempting to refer to the genetic system as little as possible.
>>>>
>>>> I have until very recently attempted to evade/avoid mechanistic thinking in regard to biology, but, on considering the origin of life generally while keeping Howard Pattee's thinking in mind, I have been struck by the notion that the origin of life (that is: WITH the genetic system) was the origin of mechanism in the universe. Before that coding system, everything was mass action. I think we still do not understand how this mechanism evolved.
>>>>
>>>> STAN
>>>>
>>>>
>>>> On Tue, Mar 29, 2016 at 7:40 AM, Dr. Plamen L. Simeonov < <mailto:plamen.l.simeonov at gmail.com>plamen.l.simeonov at gmail.com <mailto:plamen.l.simeonov at gmail.com>> wrote:
>>>>
>>>> Dear Lou, Pedro and All,
>>>>
>>>>
>>>> I am going to present a few opportunistic ideas related to what was said before in this session. Coming back to Pivar’s speculative mechano-topological model of life excluding genetics I wish to turn your attention to another author with a similar idea but on a sound mathematical base, Davide Ambrosi with his resume at https://www.uni-muenster.de/imperia/md/content/cim/events/cim-mathmod-workshop-2015_abstracts.pdf <https://www.uni-muenster.de/imperia/md/content/cim/events/cim-mathmod-workshop-2015_abstracts.pdf>:
>>>>
>>>> “Davide Ambrosi:
>>>>
>>>> A role for mechanics in the growth, remodelling and morphogenesis of living systems In the XX Century the interactions between mechanics in biology were much biased by a bioengineering attitude: people were mainly interested in evaluating the state of stress that bones and tissues undergo in order to properly design prosthesis and devices. However in the last decades a new vision is emerging. "Mechano-biology" is changing the point of view, with respect to "Bio-mechanics", emphasizing the biological feedback. Cells, tissues and organs do not only deform when loaded: they reorganize, they duplicate, they actively produce dynamic patterns that apparently have multiple biological aims.
>>>> In this talk I will concentrate on two paradigmatic systems where the interplay between mechanics and biology is, in my opinion, particularly challenging: the homeostatic stress as a driver for remodeling of soft tissue and the tension as a mechanism to transmit information about the size of organs during morphogenesis. In both cases it seems that mechanics plays a role which at least accompanies and enforces the biochemical signaling.”
>>>>
>>>>
>>>> Some more details about this approach can be found here:
>>>> http://rsta.royalsocietypublishing.org/content/367/1902/3335 <http://rsta.royalsocietypublishing.org/content/367/1902/3335>
>>>> http://biomechanics.stanford.edu/paper/MFOreport.pdf <http://biomechanics.stanford.edu/paper/MFOreport.pdf>
>>>> In other words, for the core information theorists in FIS, the question is: is there really only (epi)genetic evolution communication in living organisms. Stan Salthe and Lou Kauffman already provided some answers. I begin to believe that the transition from abiotic to biotic structures, incl. Maturana-Varela.-Uribe’s autopoiesis may, really have some underlying matrix/”skeleton”/”programme” which has nothing in common with the nature of DNA, and that DNA and RNA as we know them today
>>>>
>>>> <http://www.sciencedirect.com/science/article/pii/S0022519314006778>http://www.sciencedirect.com/science/article/pii/S0022519314006778 <http://www.sciencedirect.com/science/article/pii/S0022519314006778>
>>>> <http://www.sciencedirect.com/science/article/pii/S0022519316001260>http://www.sciencedirect.com/science/article/pii/S0022519316001260 <http://www.sciencedirect.com/science/article/pii/S0022519316001260>
>>>> <https://www.sciencedaily.com/releases/2015/01/150107101405.htm>https://www.sciencedaily.com/releases/2015/01/150107101405.htm <https://www.sciencedaily.com/releases/2015/01/150107101405.htm>
>>>> may have emerged as secondary or even tertiary “memory” of something underlying deeper below the microbiological surface. It is at least worth thinking in this direction. I do not mean necessarily the role of the number concept and Platonic origin of the universe, but something probably much more “physical” or at least staying at the edge between physical/material and immaterial such as David Deutsch’s constructor theory ( <http://constructortheory.org/>http://constructortheory.org/ <http://constructortheory.org/>) and Brian Josephson’s “structural/circular theory” ( <http://arxiv.org/ftp/arxiv/papers/1502/1502.02429.pdf>http://arxiv.org/ftp/arxiv/papers/1502/1502.02429.pdf <http://arxiv.org/ftp/arxiv/papers/1502/1502.02429.pdf>; <http://arxiv.org/ftp/arxiv/papers/1506/1506.06774.pdf>http://arxiv.org/ftp/arxiv/papers/1506/1506.06774.pdf <http://arxiv.org/ftp/arxiv/papers/1506/1506.06774.pdf>; <http://arxiv.org/pdf/1108.4860.pdf>http://arxiv.org/pdf/1108.4860.pdf <http://arxiv.org/pdf/1108.4860.pdf>) searching for the theories underpinning the foundations of the physical laws (and following Wheeler’s definition for a “Law without Law”.
>>>>
>>>> Some of you may say that QT and Gravitation Theory are responsible for such kind of strange effects, but I would rather leave the brackets open, because the recent discussion about potentialities and actualities in QM brings up the idea that there are still different ways of looking at those concepts (although they are strictly defined in their core domains). This was actually also the lesson from the last special issue on integral biomathics (2015) dedicated to phenomenology, with the different opinions of scientists and philosophers on obviously clear matters in their domains. This is why also the question of what we define as science needs to be probably revised in future to include also such issues that are “felt” rather than “reasoned”, even if we do not have the “proofs” yet, because the proofs also emerge as subjective (or perhaps “suggested”! – ask the psychologists for that aspect) thoughts in the minds of the mathematicians. I am really glad that we began such a phenomenological discussion on this aspect such as Hipolito’s paper ( <http://www.sciencedirect.com/science/article/pii/S0079610715000899>http://www.sciencedirect.com/science/article/pii/S0079610715000899 <http://www.sciencedirect.com/science/article/pii/S0079610715000899>) that was widely commented in the reviewer’s circle. In many cases when we have a “fuzzy” intuition about a certain relationship or analogy we miss the correct definitions and concepts, and so in a creative act to hold down the flying thought we move to using examples, metaphors, pictures. Pedro correctly addressed the explanatory problem of science which presupposes a certain causative and predicative “workflow” to derive a conclusion from the facts, and this is the way in which also proofs are (selectively) made. As a young scholar I often wondered how artificially people like Gauss, Cauchy and Weierstrass design their proofs, but then I got used to that style. I am thankful to Lou for his response on my question about using adequate “resonant” methods to model developmental biology, because this is also an important aspect of the biology (and physics as well) including the phenomenological/first-person view of an “observer-participant” (to use Vrobel’s term) which is crucial for understanding the process of self-reflection/recursion/cycle in science, which is usually led by what?: the intuition, also well recognized by such giants like Poincare and Einstein. Isn’t not “resonance” in the core of detecting such vibration between the observer and the observed? Because logic, back trace, prove come later.
>>>>
>>>> And finally, when looking at the clear simple mathematical abstractions of numbers, vectors, directions, sets, algebras, geometries, etc. used by many without scrutinizing when developing system (biological) models of yet another kind of mechanics/automation/machinery of the physical reality, I am asking myself which are the premises for using such tools to describe a model: the parameters, or the idea behind? It is probably not a commonly known fact (even for those who are engaged with such exciting disciplines as algebraic geometry and geometrical algebra, now considered to be very close to what we wish to express in biology) that William Hamilton, the inventor of the quaternions did not simply use the already known concept of “vector” in his method. Instead he used “step” with “direction” to express a duration of time (or “duree” as Husserl called it from the other side of the phenomenological divide) and action (to move from A to B): two very biology-related concepts at that time (although they may be considered as physical or computational today). He actually stated that if there is geometry as a pure science of space, then algebra must be the pure science of time [1]. What did we actually gain for biology from merging space and time in physics?
>>>>
>>>> Reference:
>>>>
>>>> [1] W. R. Hamilton, 1835. Theory of Conjugate Functions, or Algebraic Couples; with a Preliminary or Elementary Essay on Algebra as the Science of Pure Time. Trans. Royal Irish Acad., Vol. XVII, Part II. 292-422.
>>>>
>>>>
>>>> Best,
>>>>
>>>>
>>>> Plamen
>>>>
>>>>
>>>> I have a few provoking notes related to what was said before in this session. Coming back to Pivar’s speculative mechano-topological model of life excluding genetics I wish to turn your attention to another author with a similar idea but on a sound mathematical base, Davide Ambrosi with his resume at https://www.uni-muenster.de/imperia/md/content/cim/events/cim-mathmod-workshop-2015_abstracts.pdf <https://www.uni-muenster.de/imperia/md/content/cim/events/cim-mathmod-workshop-2015_abstracts.pdf>:
>>>>
>>>> “Davide Ambrosi:
>>>>
>>>> A role for mechanics in the growth, remodelling and morphogenesis of living systems In the XX Century the interactions between mechanics in biology were much biased by a bioengineering attitude: people were mainly interested in evaluating the state of stress that bones and tissues undergo in order to properly design prosthesis and devices. However in the last decades a new vision is emerging. "Mechano-biology" is changing the point of view, with respect to "Bio-mechanics", emphasizing the biological feedback. Cells, tissues and organs do not only deform when loaded: they reorganize, they duplicate, they actively produce dynamic patterns that apparently have multiple biological aims.
>>>> In this talk I will concentrate on two paradigmatic systems where the interplay between mechanics and biology is, in my opinion, particularly challenging: the homeostatic stress as a driver for remodeling of soft tissue and the tension as a mechanism to transmit information about the size of organs during morphogenesis. In both cases it seems that mechanics plays a role which at least accompanies and enforces the biochemical signaling.”
>>>>
>>>>
>>>> Some more details about this approach can be found here:
>>>> http://rsta.royalsocietypublishing.org/content/367/1902/3335 <http://rsta.royalsocietypublishing.org/content/367/1902/3335>
>>>> http://biomechanics.stanford.edu/paper/MFOreport.pdf <http://biomechanics.stanford.edu/paper/MFOreport.pdf>
>>>> In other words, for the core information theorists in FIS, the question is: is there really only (epi)genetic evolution communication in living organisms. Stan Salthe and Lou Kauffman already provided some answers. I begin to believe that the transition from abiotic to biotic structures, incl. Maturana-Varela.-Uribe’s autopoiesis may, really have some underlying matrix/”skeleton”/”programme” which has nothing in common with the nature of DNA, and that DNA and RNA as we know them today
>>>>
>>>> <http://www.sciencedirect.com/science/article/pii/S0022519314006778>http://www.sciencedirect.com/science/article/pii/S0022519314006778 <http://www.sciencedirect.com/science/article/pii/S0022519314006778>
>>>> <http://www.sciencedirect.com/science/article/pii/S0022519316001260>http://www.sciencedirect.com/science/article/pii/S0022519316001260 <http://www.sciencedirect.com/science/article/pii/S0022519316001260>
>>>> <https://www.sciencedaily.com/releases/2015/01/150107101405.htm>https://www.sciencedaily.com/releases/2015/01/150107101405.htm <https://www.sciencedaily.com/releases/2015/01/150107101405.htm>
>>>> may have emerged as secondary or even tertiary “memory” of something underlying deeper below the microbiological surface. It is at least worth thinking in this direction. I do not mean necessarily the role of the number concept and Platonic origin of the universe, but something probably much more “physical” or at least staying at the edge between physical/material and immaterial such as David Deutsch’s constructor theory ( <http://constructortheory.org/>http://constructortheory.org/ <http://constructortheory.org/>) and Brian Josephson’s “structural/circular theory” ( <http://arxiv.org/ftp/arxiv/papers/1502/1502.02429.pdf>http://arxiv.org/ftp/arxiv/papers/1502/1502.02429.pdf <http://arxiv.org/ftp/arxiv/papers/1502/1502.02429.pdf>; <http://arxiv.org/ftp/arxiv/papers/1506/1506.06774.pdf>http://arxiv.org/ftp/arxiv/papers/1506/1506.06774.pdf <http://arxiv.org/ftp/arxiv/papers/1506/1506.06774.pdf>; <http://arxiv.org/pdf/1108.4860.pdf>http://arxiv.org/pdf/1108.4860.pdf <http://arxiv.org/pdf/1108.4860.pdf>) searching for the theories underpinning the foundations of the physical laws (and following Wheeler’s definition for a “Law without Law”.
>>>>
>>>> Some of you may say that QT and Gravitation Theory are responsible for such kind of strange effects, but I would rather leave the brackets open, because the recent discussion about potentialities and actualities in QM brings up the idea that there are still different ways of looking at those concepts (although they are strictly defined in their core domains). This was actually also the lesson from the last special issue on integral biomathics (2015) dedicated to phenomenology, with the different opinions of scientists and philosophers on obviously clear matters in their domains. This is why also the question of what we define as science needs to be probably revised in future to include also such issues that are “felt” rather than “reasoned”, even if we do not have the “proofs” yet, because the proofs also emerge as subjective (or perhaps “suggested”! – ask the psychologists for that aspect) thoughts in the minds of the mathematicians. I am really glad that we began such a phenomenological discussion on this aspect such as Hipolito’s paper ( <http://www.sciencedirect.com/science/article/pii/S0079610715000899>http://www.sciencedirect.com/science/article/pii/S0079610715000899 <http://www.sciencedirect.com/science/article/pii/S0079610715000899>) that was widely commented in the reviewer’s circle. In many cases when we have a “fuzzy” intuition about a certain relationship or analogy we miss the correct definitions and concepts, and so in a creative act to hold down the flying thought we move to using examples, metaphors, pictures. Pedro correctly addressed the explanatory problem of science which presupposes a certain causative and predicative “workflow” to derive a conclusion from the facts, and this is the way in which also proofs are (selectively) made. As a young scholar I often wondered how artificially people like Gauss, Cauchy and Weierstrass design their proofs, but then I got used to that style. It was a question of overall convention. I am thankful to Lou for his response on my question about using adequate “resonant” methods to model developmental biology, because this is also an important aspect of the biology (and physics as well) including the phenomenological/first-person view of an “observer-participant” (to use Vrobel’s term) which is crucial for understanding the process of self-reflection/recursion/cycle in science, which is usually led by what?: the intuition, also well recognized by such giants like Poincare and Einstein. Isn’t not “resonance” in the core of detecting such vibration between the observer and the observed? Because logic, backtracing and proof come later.
>>>>
>>>>
>>>>
>>>> And finally, when looking at the clear simple mathematical abstractions of numbers, vectors, directions, sets, algebras, geometries, etc. used by many without scrutinizing when developing system (biological) models of yet another kind of mechanics/automation/machinery of the physical reality, I am asking myself which are the premises for using such tools to describe a model: the parameters, or the idea behind? It is probably not a commonly known fact (even for those who are engaged with such exciting disciplines as algebraic geometry and geometrical algebra, now considered to be very close to what we wish to express in biology) that William Hamilton, the inventor of the quaternions did not simply use the already known concept of “vector” in his method. Instead he used “step” with “direction” to express a duration of time (or “duree” as Husserl called it from the other side of the phenomenological divide) and action (to move from A to B): two very biology-related concepts at that time (although they may be considered as physical or computational today). He actually stated that if there is geometry as a pure science of space, then algebra must be the pure science of time [1]. What did we actually gain for biology from merging space and time in physics? And if we apply a specific mathematical-computational technique what is the key idea/intuition behind it?. Because, as a colleague pathologist told me this morning about the model correctness when predicting the development of tumors: the model can be assumed for being correct based on the interpretation of some (limited) set of data, but Ptolemy's system was also considered to be correct in its rather complex way of predicting the movement of the celestial bodies. Where is the difference? I am curious about your opinion.
>>>>
>>>>
>>>>
>>>> Reference:
>>>>
>>>> [1] W. R. Hamilton, 1835. Theory of Conjugate Functions, or Algebraic Couples; with a Preliminary or Elementary Essay on Algebra as the Science of Pure Time. Trans. Royal Irish Acad., Vol. XVII, Part II. 292-422.
>>>>
>>>>
>>>> Best,
>>>>
>>>>
>>>> Plamen
>>>>
>>>> ______________________
>>>>
>>>> 2015 JPBMB Special Issue on Integral Biomathics: Life Sciences, Mathematics and Phenomenological Philosophy <http://www.sciencedirect.com/science/journal/00796107/119/3>
>>>> (note: free access to all articles until July 19th, 2016)
>>>>
>>>>
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>>
>>
>
>
> --
> -------------------------------------------------
> Pedro C. Marijuán
> Grupo de Bioinformación / Bioinformation Group
> Instituto Aragonés de Ciencias de la Salud
> Centro de Investigación Biomédica de Aragón (CIBA)
> Avda. San Juan Bosco, 13, planta X
> 50009 Zaragoza, Spain
> Tfno. +34 976 71 3526 (& 6818)
> pcmarijuan.iacs at aragon.es <mailto:pcmarijuan.iacs at aragon.es>
> http://sites.google.com/site/pedrocmarijuan/ <http://sites.google.com/site/pedrocmarijuan/>
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