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The Dirichlet Hopf algebra of arithmetics
 JOURNAL OF KNOT THEORY AND ITS RAMIFICATIUONS
, 2006
"... Many constructs in mathematical physics entail notational complexities, deriving from the manipulation of various types of index sets which often can be reduced to labelling by various multisets of integers. In this work, we develop systematically the “Dirichlet Hopf algebra of arithmetics ” by dual ..."
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Many constructs in mathematical physics entail notational complexities, deriving from the manipulation of various types of index sets which often can be reduced to labelling by various multisets of integers. In this work, we develop systematically the “Dirichlet Hopf algebra of arithmetics ” by dualizing the addition and multiplication maps. Then we study the additive and multiplicative antipodal convolutions which fail to give rise to Hopf algebra structures, but form only a weaker Hopf gebra obeying a weakened homomorphism axiom. A careful identification of the algebraic structures involved is done featuring subtraction, division and derivations derived from coproducts and chochains using branching operators. The consequences of the weakened structure of a Hopf gebra on cohomology are explored, showing this has major impact on number theory. This features multiplicativity versus complete multiplicativity of number theoretic arithmetic functions. The deficiency of not being a Hopf algebra is then cured by introducing an ‘unrenormalized’ coproduct and an ‘unrenormalized ’ pairing. It is then argued that exactly the failure of the homomorphism property (complete multiplicativity) for noncoprime integers is a blueprint for the problems in quantum field theory (QFT) leading to the need for renormalization. Renormalization turns out to be the morphism from the algebraically sound Hopf algebra to the physical and number
Why sets?
 PILLARS OF COMPUTER SCIENCE: ESSAYS DEDICATED TO BORIS (BOAZ) TRAKHTENBROT ON THE OCCASION OF HIS 85TH BIRTHDAY, VOLUME 4800 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2008
"... Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a faraway planet. Would their mathematics be setbased? What are the alternatives to the settheoretic foundation of mathematics? Besi ..."
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Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a faraway planet. Would their mathematics be setbased? What are the alternatives to the settheoretic foundation of mathematics? Besides, set theory seems to play a significant role in computer science; is there a good justification for that? We discuss these and some related issues.
RENORMALIZATION: A NUMBER THEORETICAL MODEL
, 2006
"... ABSTRACT. We analyse the Dirichlet convolution ring of arithmetic number theoretic functions. It turns out to fail to be a Hopf algebra on the diagonal, due to the lack of complete multiplicativity of the product and coproduct. A related Hopf algebra can be established, which however overcounts the ..."
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ABSTRACT. We analyse the Dirichlet convolution ring of arithmetic number theoretic functions. It turns out to fail to be a Hopf algebra on the diagonal, due to the lack of complete multiplicativity of the product and coproduct. A related Hopf algebra can be established, which however overcounts the diagonal. We argue that the mechanism of renormalization in quantum field theory is modelled after the same principle. Singularities hence arise as a (now continuously indexed) overcounting on the diagonals. Renormalization is given by the map from the auxiliary Hopf algebra to the weaker multiplicative structure, called Hopf gebra, rescaling the diagonals.
The Information Flow Framework: New Architecture
, 2006
"... “Philosophy cannot become scientifically healthy without an immense technical vocabulary. We can hardly imagine our greatgrandsons turning over the leaves of this dictionary without amusement over the paucity of words with which their grandsires attempted to handle metaphysics and logic. Long before ..."
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“Philosophy cannot become scientifically healthy without an immense technical vocabulary. We can hardly imagine our greatgrandsons turning over the leaves of this dictionary without amusement over the paucity of words with which their grandsires attempted to handle metaphysics and logic. Long before that day, it will have become indispensably requisite, too, that each of these terms should be confined to a single meaning, which, however broad, must be free from all vagueness. This will involve a revolution in terminology; for in its present condition a philosophical thought of any precision can seldom be
Saunders MacLane Categories for the Working Mathematician [14]
"... The point of these observations is not the reduction of the familiar to the unfamiliar[...] but the extension of the familiar to cover many more cases. ..."
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The point of these observations is not the reduction of the familiar to the unfamiliar[...] but the extension of the familiar to cover many more cases.
CONTENTS
, 2006
"... ABSTRACT. We analyse the Dirichlet convolution ring of arithmetic number theoretic functions. It turns out to fail to be a Hopf algebra on the diagonal, due to the lack of complete multiplicativity of the product and coproduct. A related Hopf algebra can be established, which however overcounts the ..."
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ABSTRACT. We analyse the Dirichlet convolution ring of arithmetic number theoretic functions. It turns out to fail to be a Hopf algebra on the diagonal, due to the lack of complete multiplicativity of the product and coproduct. A related Hopf algebra can be established, which however overcounts the diagonal. We argue that the mechanism of renormalization in quantum field theory is modelled after the same principle. Singularities hence arise as a (now continuously indexed) overcounting on the diagonals. Renormalization is given by the map from the auxiliary Hopf algebra to the weaker multiplicative structure, called Hopf gebra, rescaling the diagonals.
On the Concrete Categories of Graphs
, 2014
"... In the standard Category of Graphs, the graphs allow only one edge to be incident to any two vertices, not necessarily distinct, and the graph morphisms must map edges to edges and vertices to vertices while preserving incidence. We refer to these graph morphisms as Strict Morphisms. We relax the co ..."
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In the standard Category of Graphs, the graphs allow only one edge to be incident to any two vertices, not necessarily distinct, and the graph morphisms must map edges to edges and vertices to vertices while preserving incidence. We refer to these graph morphisms as Strict Morphisms. We relax the condition on the graphs allowing any number of edges to be incident to any two vertices, as well as relaxing the condition on graph morphisms by allowing edges to be mapped to vertices, provided that incidence is still preserved. We call this broader graph category The Category of Conceptual Graphs, and define four other graph categories created by combinations of restrictions of the graph morphisms as well as restrictions on the allowed graphs. We investigate which Lawvere axioms for the category of Sets and Functions apply to each of these Categories of Graphs, as well as the other categorial constructions of free objects, projective objects, gen
Why sets are not collections PREPRINT
, 2014
"... Sets are often taken to be collections, or at least akin to them. This paper argues, against this, that although we cannot be sure what sets are (and the question, perhaps, does not even make sense), what we can be entirely sure of is that they are not collections of any kind. The central argument w ..."
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Sets are often taken to be collections, or at least akin to them. This paper argues, against this, that although we cannot be sure what sets are (and the question, perhaps, does not even make sense), what we can be entirely sure of is that they are not collections of any kind. The central argument will be that elementhood in a set and membership in a collection satisfy quite different axioms, and a brief logical investigation into how they are related is offered. The latter part of the paper concerns attempts to modify the ‘sets are collections ’ credo by use of idealization and abstraction, as well as the Fregean notion of sets as the extensions of concepts. These are all shown to be either unmotivated or unable to provide the desired support. We finish on a more positive note, with some ideas on what can be said of sets. The main thesis is that (i) sets are points in a set structure, (ii) a set structure is a model of a set theory, and (iii) set theory is a family of formal and informal theories, loosely defined by their axioms. 1