tag:blogger.com,1999:blog-76949518489837439692016-09-19T07:15:49.834-06:00Einstein + Heisenberg = Petri & WeylCarl Adam Petri established 4 decades ago the pillars for a different way of thinking “Informatics” and “Physics”, deeply influenced by the ideas of Einstein, Heisenberg and Weyl. As my small contribution I'm working on a mathematical model, Q-Orders, that formally combines the discrete but combinatorial world of Petri and the underlying combinatorial yet continuous structure of General Relativity, a step may be towards Combinatorial General Relativity Theory.Cornelio.Hopmann at Gmail.comhttp://www.blogger.com/profile/08432273532834843773noreply@blogger.comBlogger31125tag:blogger.com,1999:blog-7694951848983743969.post-74201301011130250512016-01-22T15:41:00.001-06:002016-01-22T17:24:23.324-06:00Name it - Tame itAfter almost 3 years of silence -while I was like the mouse in the cage, wheeling around without getting anywhere, here is something new announcing that finally I found something that looks like the combinatorial equivalent of a Pseudo-Riemann Differential Manifolds. It appears as if all boiled down to find the most appropriate universe of discourse to name the abstract objects I was struggling Cornelio.Hopmann at Gmail.comhttp://www.blogger.com/profile/08432273532834843773noreply@blogger.com0tag:blogger.com,1999:blog-7694951848983743969.post-72373244875675631082012-10-11T10:27:00.001-06:002012-10-11T10:27:05.656-06:00More –this time mathematical- heuristicsLet me try to explain what I’m trying to do. [Especially not trying to re-invent the wheel when there are so many proven, useful models at hand]. I’m aiming at Combinatorial General Relativity, as Einstein mentioned at the end of some of his writings as an unsolved puzzle. Yet “my way” is not the usual top-down –i.e. trying to find quantization-rules on the very top- but “bottom-up” that is Cornelio.Hopmann at Gmail.comhttp://www.blogger.com/profile/08432273532834843773noreply@blogger.com0tag:blogger.com,1999:blog-7694951848983743969.post-75318018607659126212012-09-11T17:26:00.001-06:002012-09-11T17:39:00.653-06:00Topology and Geometry: HeuristicsThis is yet another attempt to define and deduce the two very basic ingredients of the type of structures we would like to use as common underpinning for General Relativity Theory (normally assumed to be continuous) and General Net Theory (for which the so called Petri-Nets are just one example). This time we will concentrate on Topology and Geometry. Be aware: this is work in progress! Cornelio.Hopmann at Gmail.comhttp://www.blogger.com/profile/08432273532834843773noreply@blogger.com0tag:blogger.com,1999:blog-7694951848983743969.post-69038064373848575232011-12-05T18:24:00.001-06:002011-12-05T18:31:15.070-06:00Linear Q-Space - Third IntentWe had again to modify the definition in two aspects: (1) Taking into account,as we did before when defining the topology that not all elements are places, and (2) introduce some means to connect the locally defined linear structures. While the Light-cones are the underlying combinatorial structure from General Relativity to model the Causal (or Conformal) Structure, it seems that totally Cornelio.Hopmann at Gmail.comhttp://www.blogger.com/profile/08432273532834843773noreply@blogger.com0tag:blogger.com,1999:blog-7694951848983743969.post-27274221425031724042011-11-28T13:13:00.001-06:002011-11-28T14:38:06.156-06:00Linear Q-Spaces – Second IntentWhile the Light-cones are the underlying combinatorial structure from General Relativity to model the Causal (or Conformal) Structure, it seems that Normal Neighborhoods and Normal Coordinate-Systems (alas the Inertial-Frames or the heuristic base for the Einstein Equivalence Principle) are the proper candidates upon which to model the combinatorial equivalent for the Projective Structure. In myCornelio.Hopmann at Gmail.comhttp://www.blogger.com/profile/08432273532834843773noreply@blogger.com0tag:blogger.com,1999:blog-7694951848983743969.post-88937537595741694062011-11-19T18:11:00.001-06:002011-11-20T03:44:06.473-06:00Combinatorial Pre-Weyl-SpacesIntroduction We are looking for a combinatorial framework that, in an essential way, includes the structure of Space-Time as a continuous model on one side and the structure of of Petri-Nets as a finite (countable) model on the other. Essential means that physically different Space-Times and logically different Petri-Nets shall have different models and that different models produce different Cornelio.Hopmann at Gmail.comhttp://www.blogger.com/profile/08432273532834843773noreply@blogger.com0tag:blogger.com,1999:blog-7694951848983743969.post-80703730227050144462011-11-19T09:51:00.001-06:002011-11-19T09:51:15.405-06:00Related approaches & their problemsFor Space-Times a seminal contribution of S. W. Hawking[1] introduced a unique combinatorial structure –a partial-order– attached to Lorentzian Manifolds with some additional restrictions, that up to conformal mappings defines the manifold (Alfonso García-Parrado and José M. M. Senovilla review[2] on Causal Space-times). David Malament[3] showed how this combinatorial structure alone, under Cornelio.Hopmann at Gmail.comhttp://www.blogger.com/profile/08432273532834843773noreply@blogger.com0tag:blogger.com,1999:blog-7694951848983743969.post-56661589920757192272010-12-14T12:41:00.001-06:002010-12-16T19:36:56.843-06:00Einstein SpacesIntroduction We are looking for a combinatorial framework that, in an essential way, includes the structure of Space-Time as a continuous model on one side and the structure of of Petri-Nets as a finite (countable) model on the other. Essential means that physically different Space-Times and logically different Petri-Nets shall have different models and that different models produce different Cornelio.Hopmann at Gmail.comhttp://www.blogger.com/profile/08432273532834843773noreply@blogger.com0tag:blogger.com,1999:blog-7694951848983743969.post-88019587344627687312010-09-08T11:31:00.001-06:002010-09-08T11:31:23.567-06:00Q-Spaces - ExamplesThe following examples are our Target i.e. when elaborating/working with the Axioms and the question arises whether to accept or reject a new formulation, I check whether all examples are still covered or not (sometimes by slight modifications of the example without changing its essence). Example 1 Real Line as Q-Space Example 2 Circle-Group as Q-Space Example 3 Genesis as Q-Space The Cornelio.Hopmann at Gmail.comhttp://www.blogger.com/profile/08432273532834843773noreply@blogger.com0tag:blogger.com,1999:blog-7694951848983743969.post-27664818844635514192010-09-07T17:28:00.001-06:002010-09-15T22:23:49.865-06:00A improved edition: Order & GeometryThe Axiom-Sets for Q-Spaces There has been another round of silence in this BLOG, partially to attend some bread-and-butter business, partially to solidify the inclusion of a second basic relation besides order, a relation that reflects locally geometry: while the relation Q clearly models the conformal invariant structure –the Topology- of Space-Time, it misses the projective invariant part, Cornelio.Hopmann at Gmail.comhttp://www.blogger.com/profile/08432273532834843773noreply@blogger.com0tag:blogger.com,1999:blog-7694951848983743969.post-90448922027161802142010-07-16T11:06:00.001-06:002010-07-16T11:20:37.803-06:00Carl Adam Petri dies – his ideas live on Carl Adam Petri Geboren/born am 12. Juli 1926 Gestorben/died am 02. Juli 2010 Mathematiker und Informatiker Reference to his homepage Cornelio.Hopmann at Gmail.comhttp://www.blogger.com/profile/08432273532834843773noreply@blogger.com0tag:blogger.com,1999:blog-7694951848983743969.post-34029225114579033842010-01-16T07:32:00.001-06:002010-01-16T07:32:20.263-06:00Q-Topology – IThe structure (Q;Q) as defined by Axioms I-VII enables the definition of a Topology for Q. Theorem 1 Q-Topology We will present this theorem, its definitions and proofs step by step. T 1.1-3 Intervals and Sub-Cone Neighborhoods Intervals J are complete pieces of lines, i.e. contain all elements of a line that are between some border-elements. [Closed] Points are elements that appearCornelio.Hopmann at Gmail.comhttp://www.blogger.com/profile/08432273532834843773noreply@blogger.com0tag:blogger.com,1999:blog-7694951848983743969.post-71897814222280079972009-12-22T22:20:00.001-06:002010-01-12T14:50:19.392-06:00Lines, Cuts and DedekindReformulated without changing its essential content. Axiom 7 finally ties together classically continuous and classically countable models. As shown in a later post, it defines a new type of Topology valid for both. Axiom 7 Continuous We will walk through the construction step by step. Step 1: Lines Here the second definition introduces sets, where all of its elements are q-related. Cornelio.Hopmann at Gmail.comhttp://www.blogger.com/profile/08432273532834843773noreply@blogger.com0tag:blogger.com,1999:blog-7694951848983743969.post-19557742188532614272009-12-22T15:10:00.001-06:002009-12-22T15:36:52.094-06:00Revised: Going in Circles – Part IIIThere is an old saying When the only tool at hand is a hammer, the world appears to be a bunch of nails. Our hammer are the sets {{a,b},{c,d}}, the nails are the points, Axiom 5 and 6 define then how the world appears to us. Axiom 6 Coherent Axiom 6 is still quite easy to understand: it claims that the relation Q is connected by q-sets. Whatever it’s cut into two pieces, there will be alwaysCornelio.Hopmann at Gmail.comhttp://www.blogger.com/profile/08432273532834843773noreply@blogger.com0tag:blogger.com,1999:blog-7694951848983743969.post-7150887389261164082009-12-22T14:28:00.001-06:002009-12-22T14:34:53.305-06:00Revised: Going in Circles – Part IIIn Going in Circles – Part I we presented the basic reasoning underlying Q-Orders by analyzing the configurations of elements on 1 Circle and the combination of 2 circles. Already in the discussion of Axiom 4 we said informally that the combination of circles should create new ones and specified which would be permitted based on pictures. Yet it would be a fruitless and hence useless task, to Cornelio.Hopmann at Gmail.comhttp://www.blogger.com/profile/08432273532834843773noreply@blogger.com0tag:blogger.com,1999:blog-7694951848983743969.post-68227949190533741142009-12-22T13:47:00.001-06:002009-12-22T14:02:19.421-06:00Revised: Going in Circles – Part IStrange as is may sound, to me Axioms are not just formal statements –obviously if we like to do Mathematics later on, they have to be also formal statements in some formal language-, but rather the intent to express as precisely as possible a concept found in reality, Plato would have said an idea. In this post we’re going to show for 3 of the first 5 Axioms how they have been found and what Cornelio.Hopmann at Gmail.comhttp://www.blogger.com/profile/08432273532834843773noreply@blogger.com0tag:blogger.com,1999:blog-7694951848983743969.post-42245351363662263262009-12-22T12:36:00.001-06:002010-01-12T14:57:40.100-06:00Revised Axioms for Q-Spaces: Q-OrdersThere have been month of silence in this BLOG, as I was running into serious troubles beyond the elementary Axioms for Q-Orders, i.e. those that equip Q-Spaces with a suitable topology. More over it turned out that –at least for the moment- an additional axiomatic relation may be needed: while the relation Q clearly models the conformal invariant structure –the Topology- of Space-Time, it misses Cornelio.Hopmann at Gmail.comhttp://www.blogger.com/profile/08432273532834843773noreply@blogger.com0tag:blogger.com,1999:blog-7694951848983743969.post-47544674798621679992009-04-16T14:44:00.001-06:002009-04-19T08:38:02.341-06:00The Great Simplification – Part IIAfter slashing the original definition of Axiom 6, here comes a similar reduction of Axiom 7, originally introduced in The trapped Arrow of Time – Part III. The new Axiom 7 Q-Loops Axiom 7.1 are technical definitions: completely Q-ordered sets and the closed hull of a set. Axiom 7.2-4 introduce a substitute for Jordan-Curves. 7.2 defines the property of being connect for a set inCornelio.Hopmann at Gmail.comhttp://www.blogger.com/profile/08432273532834843773noreply@blogger.com0tag:blogger.com,1999:blog-7694951848983743969.post-3272686874408147332009-04-14T15:33:00.001-06:002009-04-21T16:13:24.642-06:00The Great Simplification - Part I (corrections)As stated clearly in the presentation, this BLOG documents Work in Progress, not at all final results. Some decades ago, when I started to study seriously Mathematics, I always wondered: how the hell were those powerful initial axioms and definitions found, which then gave origin to such powerful theories? Most of my teachers (and most Text-Books) presented only the final results in the sequence Cornelio.Hopmann at Gmail.comhttp://www.blogger.com/profile/08432273532834843773noreply@blogger.com0tag:blogger.com,1999:blog-7694951848983743969.post-49961841155250523172009-04-04T15:31:00.001-06:002009-04-21T16:30:40.357-06:00A Glimpse of the Big PictureI’ve been asked whether there is a single text, that comprises the most essential of Q-Orders. Well – there is not, or not yet. To get an idea of what is and what not yet, may be the below picture helps. The Big Picture The left side shows in a very simplified manner the tower of mathematics beneath contemporary, classical General Relativity Theory. The right side, as far as I’m aware Cornelio.Hopmann at Gmail.comhttp://www.blogger.com/profile/08432273532834843773noreply@blogger.com0tag:blogger.com,1999:blog-7694951848983743969.post-69184284404015961882009-03-31T12:37:00.001-06:002009-04-15T00:31:33.624-06:00Going Backward, Going Forward – Part IIPartial obsolete – see below This post will be dedicated to 1-dimensional Q-Orders, studying the effects Axiom 6 Q-Topology has on them and preparing the ground for the n+1-dimensional case. Axiom 6 We will study 4 cases: the Rational Numbers, the Real Numbers, the Cyclic Group Zn and the Circle Group, using for the latter geometric representations on the Unit-Circle when Cornelio.Hopmann at Gmail.comhttp://www.blogger.com/profile/08432273532834843773noreply@blogger.com0tag:blogger.com,1999:blog-7694951848983743969.post-78172261484643692552009-03-29T10:46:00.001-06:002009-03-29T13:57:12.379-06:00Going Backward, Going Forward – Part IIn the following 3 posts we will go backward and forward through the seven axiom-sets, on one side to get a better feeling for Q-Orders, on the other to relate Q-orders with classically known concepts. The final post will show that the Hawking-Topology is a Q-Order. Let’s start in this post with some considerations about the Axioms 1 to 5 and their relation to partial orders. Axiom 1 Cornelio.Hopmann at Gmail.comhttp://www.blogger.com/profile/08432273532834843773noreply@blogger.com0tag:blogger.com,1999:blog-7694951848983743969.post-47225186518098243612009-03-17T21:13:00.006-06:002009-04-16T14:49:36.887-06:00The trapped Arrow of Time – Part IIIObsolete by the Great Simplification … yet still useful for heuristics The central result of the cited articles from Stephen Hawking(1) and David Malament(2) is the proof that the path-topology, and only the path-topology, of space-time defines the time-like curves and viceversa, i.e. the time-like curves define uniquely the topology, where in turn the metric Tensor g may be reconstructed up toCornelio.Hopmann at Gmail.comhttp://www.blogger.com/profile/08432273532834843773noreply@blogger.com0tag:blogger.com,1999:blog-7694951848983743969.post-71239556574573959742009-03-17T12:48:00.003-06:002009-04-14T15:38:16.524-06:00The trapped Arrow of Time – Part IIObsolete by the Great Simplification … yet still useful for heuristics After the long introduction of Part I with so many caveats based on painful experiences, here the Axiom 6 Topology As final goal we will construct Q-Path-Topologies, that is the category of topological spaces that correspond precisely to Q-Orders. The construction is done in two steps: first in this part we Cornelio.Hopmann at Gmail.comhttp://www.blogger.com/profile/08432273532834843773noreply@blogger.com0tag:blogger.com,1999:blog-7694951848983743969.post-38985183382987664582009-03-14T13:07:00.004-06:002009-04-14T15:37:01.511-06:00The trapped Arrow of Time – Part IObsolete by the Great Simplification … yet still useful for heuristics The Axioms 1-5 allow to speak about q-sets –our hammer- and points –our nails-, yet we don’t have a space where to put eventually constructed buildings –physical processes-. The wrong way to get a space would be simply assume it, as Einstein showed convincingly a century ago refuting thereby Immanuel Kant, who another Cornelio.Hopmann at Gmail.comhttp://www.blogger.com/profile/08432273532834843773noreply@blogger.com0