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A treatise on quantum Clifford Algebras
"... on bilinear forms of arbitrary symmetry, are treated in a broad sense. Five alternative constructions of QCAs are exhibited. Grade free Hopf gebraic product formulas are derived for meet and join of GraßmannCayley algebras including comeet and cojoin for GraßmannCayley cogebras which are very e ..."
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on bilinear forms of arbitrary symmetry, are treated in a broad sense. Five alternative constructions of QCAs are exhibited. Grade free Hopf gebraic product formulas are derived for meet and join of GraßmannCayley algebras including comeet and cojoin for GraßmannCayley cogebras which are very efficient and may be used in Robotics, left and right contractions, left and right cocontractions, Clifford and coClifford products, etc. The Chevalley deformation, using a Clifford map, arises as a special case. We discuss Hopf algebra versus Hopf gebra, the latter emerging naturally from a biconvolution. Antipode and crossing are consequences of the product and coproduct structure tensors and not subjectable to a choice. A frequently used Kuperberg lemma is revisited necessitating the definition of nonlocal products and interacting Hopf gebras which are generically nonperturbative. A ‘spinorial ’ generalization of the antipode is given. The nonexistence of nontrivial integrals in lowdimensional Clifford cogebras is shown. Generalized cliffordization is discussed which is based on nonexponentially generated bilinear forms in general resulting in non unital, nonassociative products. Reasonable assumptions lead to bilinear forms based on 2cocycles. Cliffordization is used to derive time and normalordered generating functionals for the SchwingerDyson hierarchies of nonlinear spinor field theory and spinor electrodynamics. The relation between the vacuum structure, the operator ordering, and the Hopf gebraic counit is discussed. QCAs are proposed as the natural language for (fermionic) quantum field theory. MSC2000: 16W30 Coalgebras, bialgebras, Hopf algebras; 1502 Research exposition (monographs, survey articles);
SylvesterGallai theorem and metric betweenness
, 2002
"... Sylvester conjectured in 1893 and Gallai proved some forty years later that every finite set S of points in the plane includes two points such that the line passing through them includes either no other point of S or all other points of S. There are several ways of extending the notion of lines fro ..."
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Cited by 5 (0 self)
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Sylvester conjectured in 1893 and Gallai proved some forty years later that every finite set S of points in the plane includes two points such that the line passing through them includes either no other point of S or all other points of S. There are several ways of extending the notion of lines from Euclidean spaces to arbitrary metric spaces. We present one of them and conjecture that, with lines in metric spaces defined in this way, the SylvesterGallai theorem generalizes as follows: in every finite metric space, there is a line consisting of either two points or all the points of the space. Then we present slight evidence in support of this rash conjecture and finally we discuss the underlying ternary relation of metric betweenness. 1 The SylvesterGallai theorem Sylvester (1893) proposed the following problem: Prove that it is not possible to arrange any finite number of real points so that a right line through every two of them shall pass through a third, unless they all line in the same right line.
The growth of mathematical knowledge: an open world view
 The growth of mathematical knowledge, Kluwer, Dordrecht 2000
"... mathematical knowledge: “The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new ones, but to the continuous evolution of zoological types which develop ceaselessly and end by becoming unrecognizable to the common sight, but ..."
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Cited by 5 (5 self)
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mathematical knowledge: “The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new ones, but to the continuous evolution of zoological types which develop ceaselessly and end by becoming unrecognizable to the common sight, but where an expert eye finds always traces of the prior work of the centuries past ” (Poincaré 1958, p. 14). The view criticized by Poincaré corresponds to Frege’s idea that the development of mathematics can be described as an activity of system building, where each system is supposed to provide a complete representation for a certain mathematical field and must be pitilessly torn down whenever it fails to achieve such an aim. All facts concerning any mathematical field must be fully organized in a given system because “in mathematics we must always strive after a system that is complete in itself ” (Frege 1979, p. 279). Frege is aware that systems introduce rigidity and are in conflict with the actual development of mathematics because “in history we have development; a system is static”, but he sticks
Temporal Patterns and Modal Structure
, 1999
"... Temporal logic arose at the border of philosophy and linguistics. From the seventies onward, it became a major tool also in computer science and articial intelligence, which have turned into the most powerful source of new logical developments since. We discuss some recent themes demonstrating new c ..."
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Temporal logic arose at the border of philosophy and linguistics. From the seventies onward, it became a major tool also in computer science and articial intelligence, which have turned into the most powerful source of new logical developments since. We discuss some recent themes demonstrating new connections with modal logic. In the course of this, we also point out some new types of open research questions.
Art of Graph Drawing and Art
, 2001
"... this article. But this article is for Graph Drawing and also there is more here than meets the eye. The prime problem of Graph Drawing is to visualize properly and as accurately as possible the information which is given to us in a confusing, sometimes pictorial and sometimes mathematical, or \tech ..."
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this article. But this article is for Graph Drawing and also there is more here than meets the eye. The prime problem of Graph Drawing is to visualize properly and as accurately as possible the information which is given to us in a confusing, sometimes pictorial and sometimes mathematical, or \technical", way. This intended visualization should help us to understand, should attract and should explain. It is interesting to note that on such abstract gnoseological level these are the same aspects and problems which are facing an artist when he/she is trying to convey his/her message. Visualization, representation, the reality and its model are the code words. Clearly this is complemented by further aspects which make the whole picture more complicated and stresses the dierences of these areas but still the abstract core is similar if not the same. (At this place it is only proper to stress the following: one has to be careful with analogies and parallels. We try to stress only those points which one can see and prove beyond doubts. Otherwise we can easily slip to the level of an essay and unjusti ed speculations.) This text is based on the article Art of Drawing [28] which in turn is based on an invited talk delivered by the author at GD'99. That lecture was conceived as a multimedia show with slides, transparencies and CD projection (which operated uently thanks to Hubert de Fraysseix). These three parts of the lecture, projected on three 2 dierent screens, were called Samples, Stories and Souveniers. Since then some progress has been made and this lecture has been given in another context. Most notably in March 2001 in the Institute for Art History of the Academy of Sciences of the Czech Republic (thanks to an invitation of Mahulena Neslehova) and at E.H.E.S.S....
The History and Concept of Mathematical Proof
, 2007
"... A mathematician is a master of critical thinking, of analysis, and of deductive reasoning. These skills travel well, and can be applied in a large variety of situations—and in many different disciplines. Today, mathematical skills are being put to good use in medicine, physics, law, commerce, Intern ..."
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A mathematician is a master of critical thinking, of analysis, and of deductive reasoning. These skills travel well, and can be applied in a large variety of situations—and in many different disciplines. Today, mathematical skills are being put to good use in medicine, physics, law, commerce, Internet design, engineering, chemistry, biological science, social science, anthropology, genetics, warfare, cryptography, plastic surgery, security analysis, data manipulation, computer science, and in many other disciplines and endeavors as well. The unique feature that sets mathematics apart from other sciences, from philosophy, and indeed from all other forms of intellectual discourse, is the use of rigorous proof. It is the proof concept that makes the subject cohere, that gives it its timelessness, and that enables it to travel well. The purpose of this discussion is to describe proof, to put it in context, to give its history, and to explain its significance. There is no other scientific or analytical discipline that uses proof as readily and routinely as does mathematics. This is the device that makes theoretical mathematics special: the tightly knit chain of reasoning, following strict logical rules, that leads inexorably to a particular conclusion. It is proof that is our device for establishing the absolute and irrevocable truth of statements in our subject. This is the reason that we can depend on mathematics that was done by Euclid 2300 years ago as readily as we believe in the mathematics that is done today. No other discipline can make such an assertion.
Consistency  What's Logic Got to Do with It?
, 1996
"... this paper, I want to explore the origin of the modern conception of the idea of consistency in logic in the work of German mathematician David Hilbert. My interest in the development of the modern idea of consistency arises from my belief that an overriding concern with a strict requirement of cons ..."
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this paper, I want to explore the origin of the modern conception of the idea of consistency in logic in the work of German mathematician David Hilbert. My interest in the development of the modern idea of consistency arises from my belief that an overriding concern with a strict requirement of consistency, borrowed primarily from the rigors of modern developments in logic, has prevented latter day twentieth century philosophers from producing philosophical systems of the type produced in earlier times.
Reasoning about Cardinal Directions Using Grids as Qualitative Geographic Coordinates
, 1999
"... In this article we propose a calculus of qualitative geographic coordinates which allows reasoning about cardinal directions on gridbased reference systems in maps. Grids in maps can be considered as absolute reference systems. The analysis reveals that the basic information coded in these refe ..."
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In this article we propose a calculus of qualitative geographic coordinates which allows reasoning about cardinal directions on gridbased reference systems in maps. Grids in maps can be considered as absolute reference systems. The analysis reveals that the basic information coded in these reference systems is ordering information. Therefore, no metric information is required. We show that it is unnecessary to assume a coordinate system based on numbers in order to extract information like a point P is further north than a point Q. We investigate several grids in maps resulting from different types of projections. In addition, a detailed examination of the north arrow is given since it supplies a grid with ordering information. On this basis, we provide a general account on grids, their formalization and the inferences about cardinal directions drawn using qualitative geographic coordinates.
PARTNERS: FUNCTIONAL ANALYSIS AND TOPOLOGY
, 2005
"... Functional analysis and topology were born in the first two decades of the twentieth century and each has greatly influenced the other. Identifying the dual space—the space of continuous linear functionals—of a normed space played an especially important role in the formative years of functional ..."
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Functional analysis and topology were born in the first two decades of the twentieth century and each has greatly influenced the other. Identifying the dual space—the space of continuous linear functionals—of a normed space played an especially important role in the formative years of functional