@techreport{SultanowKochCox2019,
  author    = {Sultanow, Eldar and Koch, Christian and Cox, Sean},
  title     = {Collatz Sequences in the Light of Graph Theory},
  edition   = {2nd version},
  doi       = {10.25932/publishup-43741},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-437416},
  pages     = {21},
  year      = {2019},
  abstract  = {The Collatz conjecture is a number theoretical problem, which has puzzled countless researchers using myriad approaches. Presently, there are scarcely any methodologies to describe and treat the problem from the perspective of the Algebraic Theory of Automata. Such an approach is promising with respect to facilitating the comprehension of the Collatz sequence's "mechanics". The systematic technique of a state machine is both simpler and can fully be described by the use of algebraic means. The current gap in research forms the motivation behind the present contribution. The present authors are convinced that exploring the Collatz conjecture in an algebraic manner, relying on findings and fundamentals of Graph Theory and Automata Theory, will simplify the problem as a whole.},
  language  = {en}
}
@techreport{SultanowKochCox2019,
  author    = {Sultanow, Eldar and Koch, Christian and Cox, Sean},
  title     = {Collatz Sequences in the Light of Graph Theory},
  doi       = {10.25932/publishup-43008},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-430089},
  pages     = {15},
  year      = {2019},
  abstract  = {The Collatz conjecture is a number theoretical problem, which has puzzled countless researchers using myriad approaches. Presently, there are scarcely any methodologies to describe and treat the problem from the perspective of the Algebraic Theory of Automata. Such an approach is promising with respect to facilitating the comprehension of the Collatz sequences "mechanics". The systematic technique of a state machine is both simpler and can fully be described by the use of algebraic means. The current gap in research forms the motivation behind the present contribution. The present authors are convinced that exploring the Collatz conjecture in an algebraic manner, relying on findings and fundamentals of Graph Theory and Automata Theory, will simplify the problem as a whole.},
  language  = {en}
}
@techreport{SultanowKochCox2020,
  author    = {Sultanow, Eldar and Koch, Christian and Cox, Sean},
  title     = {Collatz Sequences in the Light of Graph Theory},
  edition   = {3rd version},
  doi       = {10.25932/publishup-44185},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-441859},
  pages     = {29},
  year      = {2020},
  abstract  = {It is well known that the inverted Collatz sequence can be represented as a graph or a tree. Similarly, it is acknowledged that in order to prove the Collatz conjecture, one must demonstrate that this tree covers all (odd) natural numbers. A structured reachability analysis is hitherto not available. This paper investigates the problem from a graph theory perspective. We define a tree that consists of nodes labeled with Collatz sequence numbers. This tree will be transformed into a sub-tree that only contains odd labeled nodes. The analysis of this tree will provide new insights into the structure of Collatz sequences. The findings are of special interest to possible cycles within a sequence. Next, we describe the conditions which must be fulfilled by a cycle. Finally, we demonstrate how these conditions could be used to prove that the only possible cycle within a Collatz sequence is the trivial cycle, starting with the number 1, as conjectured by Lothar Collatz.},
  language  = {en}
}
@techreport{SultanowKochCox2020,
  author    = {Sultanow, Eldar and Koch, Christian and Cox, Sean},
  title     = {Collatz Sequences in the Light of Graph Theory},
  edition   = {4th version},
  doi       = {10.25932/publishup-44325},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-443254},
  pages     = {31},
  year      = {2020},
  abstract  = {It is well known that the inverted Collatz sequence can be represented as a graph or a tree. Similarly, it is acknowledged that in order to prove the Collatz conjecture, one must demonstrate that this tree covers all odd natural numbers. A structured reachability analysis is hitherto not available. This paper investigates the problem from a graph theory perspective. We define a tree that consists of nodes labeled with Collatz sequence numbers. This tree will be transformed into a sub-tree that only contains odd labeled nodes. The analysis of this tree will provide new insights into the structure of Collatz sequences. The findings are of special interest to possible cycles within a sequence. Next, we describe the conditions which must be fulfilled by a cycle. Finally, we demonstrate how these conditions could be used to prove that the only possible cycle within a Collatz sequence is the trivial cycle, starting with the number one, as conjectured by Lothar Collatz.},
  language  = {en}
}
@techreport{SultanowVolkovCox2017,
  author    = {Sultanow, Eldar and Volkov, Denis and Cox, Sean},
  title     = {Introducing a Finite State Machine for processing Collatz Sequences},
  edition   = {1st version},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-399223},
  year      = {2017},
  abstract  = {The present work will introduce a Finite State Machine (FSM) that processes any Collatz Sequence; further, we will endeavor to investigate its behavior in relationship to transformations of a special infinite input. Moreover, we will prove that the machine's word transformation is equivalent to the standard Collatz number transformation and subsequently discuss the possibilities for use of this approach at solving similar problems. The benefit of this approach is that the investigation of the word transformation performed by the Finite State Machine is less complicated than the traditional number-theoretical transformation.},
  language  = {en}
}
@techreport{SultanowKochCox2020,
  author    = {Sultanow, Eldar and Koch, Christian and Cox, Sean},
  title     = {Collatz Sequences in the Light of Graph Theory},
  edition   = {Fifth version},
  doi       = {10.25932/publishup-48214},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-482140},
  pages     = {47},
  year      = {2020},
  abstract  = {It is well known that the inverted Collatz sequence can be represented as a graph or a tree. Similarly, it is acknowledged that in order to prove the Collatz conjecture, one must demonstrate that this tree covers all odd natural numbers. A structured reachability analysis is hitherto unavailable. This paper investigates the problem from a graph theory perspective. We define a tree that consists of nodes labeled with Collatz sequence numbers. This tree will be transformed into a sub-tree that only contains odd labeled nodes. Furthermore, we derive and prove several formulas that can be used to traverse the graph. The analysis covers the Collatz problem both in it's original form 3x + 1 as well as in the generalized variant kx + 1. Finally, we transform the Collatz graph into a binary tree, following the approach of Kleinnijenhuis, which could form the basis for a comprehensive proof of the conjecture.},
  language  = {en}
}
@phdthesis{LopezValencia2023,
  author    = {Lopez Valencia, Diego Andres},
  title     = {The Milnor-Moore and Poincar{\´e}-Birkhoff-Witt theorems in the locality set up and the polar structure of Shintani zeta functions},
  doi       = {10.25932/publishup-59421},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-594213},
  school      = {Universit{\"a}t Potsdam},
  pages     = {147},
  year      = {2023},
  abstract  = {This thesis bridges two areas of mathematics, algebra on the one hand with the Milnor-Moore theorem (also called Cartier-Quillen-Milnor-Moore theorem) as well as the Poincar{\´e}-Birkhoff-Witt theorem, and analysis on the other hand with Shintani zeta functions which generalise multiple zeta functions. The first part is devoted to an algebraic formulation of the locality principle in physics and generalisations of classification theorems such as Milnor-Moore and Poincar{\´e}-Birkhoff-Witt theorems to the locality framework. The locality principle roughly says that events that take place far apart in spacetime do not infuence each other. The algebraic formulation of this principle discussed here is useful when analysing singularities which arise from events located far apart in space, in order to renormalise them while keeping a memory of the fact that they do not influence each other. We start by endowing a vector space with a symmetric relation, named the locality relation, which keeps track of elements that are "locally independent". The pair of a vector space together with such relation is called a pre-locality vector space. This concept is extended to tensor products allowing only tensors made of locally independent elements. We extend this concept to the locality tensor algebra, and locality symmetric algebra of a pre-locality vector space and prove the universal properties of each of such structures. We also introduce the pre-locality Lie algebras, together with their associated locality universal enveloping algebras and prove their universal property. We later upgrade all such structures and results from the pre-locality to the locality context, requiring the locality relation to be compatible with the linear structure of the vector space. This allows us to define locality coalgebras, locality bialgebras, and locality Hopf algebras. Finally, all the previous results are used to prove the locality version of the Milnor-Moore and the Poincar{\´e}-Birkhoff-Witt theorems. It is worth noticing that the proofs presented, not only generalise the results in the usual (non-locality) setup, but also often use less tools than their counterparts in their non-locality counterparts. The second part is devoted to study the polar structure of the Shintani zeta functions. Such functions, which generalise the Riemman zeta function, multiple zeta functions, Mordell-Tornheim zeta functions, among others, are parametrised by matrices with real non-negative arguments. It is known that Shintani zeta functions extend to meromorphic functions with poles on afine hyperplanes. We refine this result in showing that the poles lie on hyperplanes parallel to the facets of certain convex polyhedra associated to the defining matrix for the Shintani zeta function. Explicitly, the latter are the Newton polytopes of the polynomials induced by the columns of the underlying matrix. We then prove that the coeficients of the equation which describes the hyperplanes in the canonical basis are either zero or one, similar to the poles arising when renormalising generic Feynman amplitudes. For that purpose, we introduce an algorithm to distribute weight over a graph such that the weight at each vertex satisfies a given lower bound.},
  language  = {en}
}
@techreport{SultanowVolkovCox2017,
  author    = {Sultanow, Eldar and Volkov, Denis and Cox, Sean},
  title     = {Introducing a Finite State Machine for processing Collatz Sequences},
  edition   = {2nd version},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-404738},
  pages     = {17},
  year      = {2017},
  abstract  = {The present work will introduce a Finite State Machine (FSM) that processes any Collatz Sequence; further, we will endeavor to investigate its behavior in relationship to transformations of a special infinite input. Moreover, we will prove that the machine's word transformation is equivalent to the standard Collatz number transformation and subsequently discuss the possibilities for use of this approach at solving similar problems. The benefit of this approach is that the investigation of the word transformation performed by the Finite State Machine is less complicated than the traditional number-theoretical transformation.},
  language  = {en}
}